The Structure of Quadratic Jordan Systems of Clifford Type

Alain D’Amour  Kevin McCrimmon †
Department of Mathematics & CS Department of Mathematics
Denison University  University of Virginia
Granville, OH 43023 USA  Charlottesville, VA 22903 USA

0. Introduction

Zel’manov’s classification of prime nondegenerate linear Jordan triple systems and pairs parallels in many ways the corresponding structure theory for Jordan algebras [18] in the sense that both theories follow the same basic structural flowchart. In [19], Zel’manov showed that prime nondegenerate quadratic Jordan systems (like their algebra counter-parts) are either i-special, or exceptional finite-dimensional over their centroid, and further established that whenever char ≠ 2,3, the i-special triples (resp. pairs) are forms of either one of 5 simple triples (resp. 2 pairs) of hermitian type or one of 4 simple triples (resp. 2 pairs) of Clifford type. (As it turns out, he inadvertently omitted two pairs—hence four triples— of the latter type.) The part of the argument that pertains to systems of hermitian type has been extended to quadratic Jordan systems over an arbitrary ring of scalars [5]; in the present work, we finalize the general classification by handling the case of quadratic systems of Clifford type.

By system of Clifford type (triple or pair), we mean a system all the homotopes of which strictly satisfy some Jordan Clifford identity, a Jordan polynomial that does not vanish on the split 3 × 3 hermitian matrices H(M_{3}(Φ), t). [ The dichotomy in the structure of simple i-special systems depends on the existence of Jordan PI’s which have both a Clifford and “hermitian” heritage, the so-called Zel’manov polynomials ; the construction of one such creature is carried out in [4] for quadratic systems.] Our basic philosophy consists in reducing the general setting of simple systems to systems of finite capacity.

† This author wishes to thank the Schweizerische Nationalfonds for their financial support, and his host Prof. Gottfried Ruettimann for his warm hospitality, during a productive month in Bern in the summer of 1994.

In section 2, we first narrow down Loos’ list of classical Jordan central simple pairs of finite capacity to the three kinds of i-special pairs capable of satisfying a Clifford identity: the rectangular pair V(M, R, f) (see (1.11.2)), the Clifford pair J(q) of a quadratic form q (see (1.9–10)) and the alternating pair A_{5}(Φ) (see (1.7–8)). This involves a scrutiny of small division pairs (section 2) and rectangular pairs (section 3).

In section 4, we reveal what perhaps ought to be considered the true nature of a “degree 2” Jordan system: the local algebras of each of our Cliffordian candidates are Jordan algebras of degree 2 or less. The usefulness of this information is best illustrated by the Primitive Form Theorem (section 5): it essentially asserts that if a central simple pair V is a scalar form of a pair Ṽ (over a field) all of whose local algebras are Clifford algebras, then V must be semiprimitive (it contains nonzero regular elements), hence primitive. We end the preparatory work in section 6 with the verification that some Clifford identity does indeed vanish on each member of our list, so all of them are bonafide pairs of Clifford type.

We devote section 7 to piecing together the structure of primitive pairs. Most of the ground work was laid out in [6]: a primitive system J with a (homotope) PI necessarily has nonzero socle, and if the PI is Clifford, then J = Soc(J) is actually simple of strict finite capacity ≤ 2. Section 8 establishes the pivotal fact that for a system with a polynomial identity over a large uncountable field, nondegeneracy and semiprimitivity are equivalent. This allows us to handle the delicate issue of scalar extensions in characteristic 2: given a simple pair V of Clifford type, we take a TIGHT scalar extension V_{Ω} over a suitably large field so that V_{Ω} is now semiprimitive (since V_{Ω} remains simple (by tightness) and inherits the Clifford identity); hence V_{Ω} is primitive classical of capacity ≤ 2 and by the Primitive Form Theorem, the same holds for V. The structure of simple triples then emerges from their categorical equivalence with Jordan pairs with involution (section 9).

Finally, section 10 of the paper explains why a prime nondegenerate system sits inside a simple one as a scalar form. Over a field, a locally finite-dimensional idempotent-finite Jordan system with no nonzero locallynilpotent ideals is a direct sum of finitely many simple ideals the nature of which is known from sections 8 and 9. We combine this with an ultrafilter argument that consists in embedding (as a scalar form) the strongly prime J in some ultraproduct J̃ that taps into the common algebraic features of its simple factors; by primeness, J must lie completely within one of J̃’s simple building blocks.

1. Review

We refer the reader to [9] for most of the basic material about quadratic Jordan systems found below. Unless specified otherwise, we take Φ to be an arbitrary ring of scalars. A Jordan triple system J over Φ is a Φmodule equipped with a product P_{x}y quadratic in x and linear in y (where the linearization P_{x,z} := P_{x+z} − P_{x} − P_{z} yields P_{x,z}(y) =: {xyz} =: L_{x,y}(z)) strictly satisfying the identities

(J1)  L_{x,y}P_{x} = P_{x}L_{y,x} = P_{P (x)y,x}
(J2)  L_{x,P (y)x} = L_{P (x)y,y}
(J3)  P_{P (x)y} = P_{x}P_{y}P_{x}

(hence (J1)-(J3) continue to hold in all scalar extensions of J).

In a unital Jordan algebra (a triple plus a choice of element 1 with P_{1} = Id), we write U_{x}y and V_{x,y}z for P_{x}y and L_{x,y}z respectively. By a Jordan algebra , we mean a subspace of some unital Jordan algebra closed under U_{x}y and the square x^{2} := U_{x}1. A Jordan pair V = (V^{+}, V^{−}) is a pair of Φ-modules acting on each other in a Jordan triple fashion; we traditionally denote its products by Q_{x^{ε}}y^{−ε} and D_{x^{ε},y^{−ε}}z^{ε}, ε = ±. We will rely heavily on the well-known categorical equivalences between triples and pairs: Jordan pairs are equivalent to polarized Jordan triples via the polarizing and de-polarizing functors

(1.1.1) V = (V^{+}, V^{−}) ⟹ P ol(V) = V^{+} ⊕ V^{−} :
P_{x^{+}⊕x^{−}}(y^{+} ⊕ y^{−}) = Q_{x^{+}}y^{−} ⊕ Q_{x^{−}}y^{+},

(1.1.2)  J = J^{+} ⊕ J^{−} ⟹ V(J, pol) = (J^{+}, J^{−}).

Here a polarized J is simple as triple iff it is simple as polarized triple (no polarized ideals) iff V is simple as pair. (Note that if I is a triple ideal, then \underline{I} ⊂ I ⊂ \overline{I} for polarized ideals \underline{I}^{ε} = I ∩ V^{ε}, \overline{I}^{ε} = π_{ε}(I). If \underline{I} = 0, \overline{I} = J, then P_{J}J = 0 since P_{I}J + P_{J}I + \{JJI\} ⊂ \underline{I}. The polarized ideals I^{+} ⊕ I^{−} of J are precisely the pair ideals (I^{+}, I^{−}) of V.)

Ordinary Jordan triples J are equivalent to Jordan pairs (V, η) with involution (i.e. a pair antiautomorphism η_{ε} : V^{ε} → V^{−ε} such that η_{−ε}∘η_{ε} = Id on V^{ε}, ε = ±) via

J a triple ⟹ (V(J), ex) is a pair with involution for (1.2.1) V(J) = (J, J) under Q_{x^{ε}}y^{−ε} = P_{x}y and exchange involution (x, y)^{ex} = (y, x), (the ex-invariant pair ideals of V(J) are precisely all V(I) = (I, I) for triple ideals I ◁ J),

(V, η) a pair with involution ⟹ J (V, η) = H(P ol(V), η) (1.2.2) is a triple isomorphic to the isotope V^{+(η_{+})} :

P_{x^{+}}y^{+} = Q_{x^{+}}η_{+}(y^{+}).

(1.2.3) These constructions are inverses: J (V(J), ex) ≃ J, (V(J (V), η), ex) ≃ (V, η).

(1.2.4) J polarized ⟺ V = W \overline{\underline{+}} W^{η} for W ◁ V

( then J = P ol(W)).

(If J = J^{+}⊕J^{−} then V = W ⊕W^{η} for W = (J^{+}, J^{−}), W^{η} = (J^{−}, J^{+}) where W is an ideal [hence orthogonal to the ideal W^{η}] by Q_{W} V ⊂ (P_{J^{+}}J, P_{J^{−}}J) ⊂ (J^{+}, J^{−}) and {W V V } ⊂ ({J^{+}JJ}, {J^{−}JJ}) ⊂ (J^{+}, J^{−}). Conversely, if V = W \overline{\underline{+}}W^{η} then J = J^{+}⊕J^{−} for J^{+} = W^{+}, J^{−} = (W^{η})^{+} so J^{+}+(J^{−})^{η} ⊂ W, J^{−} + (J^{+})^{η} ⊂ W^{η} so P_{J^{+}}J^{+} ⊂ Q_{W} W^{η} = 0, P_{J^{−}}J^{−} ⊂ Q_{W^{η}}W = 0, {J^{+}JJ^{−}} ⊂ {W V W^{η}} = 0 and J is polarized.)

Most of the important Jordan systems come from associative structures. Using the ambient product in an associative algebra (respectively triple or pair) A, we can Jordanify A in the usual way: either

(1.3.1)  A^{J} : P_{x}y = xyx,

or

(1.3.2)  A^{∗J} : P_{x}y = xy^{∗}x, ∗ an involution in A.

We say a Jordan system is special if it is a subsystem of some A^{J} (or A^{∗J}). For instance, examples of special triples include hermitian triples

(1.4)  H(A, ∗) = {x ∈ A | x^{∗} = x} ⊆ A^{∗J}, P_{x}y = xy^{∗}x = xyx,

and more generally ample hermitian triples

H_{∘}(A, ∗) ⊆ H(A, ∗), aH_{∘}a^{∗} ⊆ H_{∘} for all a ∈ A, and all traces

(1.5) a + a^{∗} lie in H_{∘}; if A is an algebra, then all the norms aa^{∗}

must also lie in H_{∘} (equivalently, for A unital, 1 ∈ H_{∘}).

(If 1/2 ∈ Φ, the only ample subspace is H_{∘} = H.) In particular, for A = M_{n}(Δ), − an involution of the coordinate algebra Δ and Δ_{∘} ⊆ H(Δ, −) an ample subspace, we can form the triple of hermitian matrices with diagonal entries in Δ_{∘}

(1.6)  H_{n}(Δ, Δ_{∘}) : P_{x}y = x\overline{y}^{t}x = xyx.

Similarly, we have the alternating triples consisting of all skewtraces

(1.7)  Alt(A, ∗) = {a − a^{∗} | a ∈ A} ⊆ A^{∗J}, P_{x}y = xy^{∗}x = −xyx;

e.g. if A = M_{n}(Ω), for Ω a field and ∗ the transpose involution, we recover the usual alternating matrices

(1.8.1)  A_{n}(Ω) : P_{x}y = xy^{t}x = −xyx.

This description is tied to a particular basis. We can think of M_{n}(Ω) as End(W) for W = Ω^{n} a finite-dimensional vector space over Ω with inner product σ(x, y) and transpose as the adjoint involution ∗ relative to σ. Then A_{n}(Ω) = Alt(W, σ) consists of the usual σ-alternating linear transformations on W,

Alt(W, σ) = {x ∈ End(W) | σ(x(w), w) = 0 ∀ w ∈ W }, (1.8.2)
P_{x}y = xy^{∗}x = −xyx

for any nondegenerate symmetric bilinear form σ. The Jordan pair V(Alt(W, σ)) has an even more intrinsic description without reference to any σ (just the natural pairing between W and W^{∗}): if we define a linear transformation x : W ⟶ W^{∗} (and dually for W^{∗} ⟶ W) to be alternating if the image x(w) of any w is a linear functional which kills its parent, [x(w)](w) = 0, then

V(Alt(W, σ)) ≃ (Alt(W, W^{∗}), Alt(W^{∗}, W)) under (1.8.3)
Q_{x}y = −xyx

via the isomorphism (x, y) → (s ∘ x, y ∘ s^{−1}) for s : W → W^{∗} the linear isomorphism s(w) = σ(w, ·) = σ(·, w). (It is easy to check that x ∈ Alt(W, σ) ⇔ s∘x ∈ Alt(W, W^{∗}) ⇔ (since σ is symmetric) x∘s^{−1} ∈ Alt(W^{∗}, W).) We could do the same construction for any nondegenerately paired spaces W^{+}, W^{−}.

Many of the arguments below will revolve around the quintessential degree 2 system, the Clifford triple J(q) built out of a quadratic form

(1.9)  J(q) : P_{x}y = q(x, y)x − q(x)y.

In characteristic 2, we may also take an algebra outer ideal I containing 1 in a Jordan Clifford algebra J(q, 1),

(1.10.1)  J(q, 1) : U_{x}y = q(x, \overline{y})x − q(x)\overline{y}, where q(1) = 1,
\overline{y} = t(y)1 − y (t(y) = q(1, y)).

In Jordan Clifford algebras the bar is always an algebra involution,

\overline{1} = 1, \overline{\overline{y}} = y, q(\overline{y}) = q(y), q(x, \overline{y}) = q(\overline{x}, y), (1.10.2)
\overline{U_{x}y} = U\overline{x}\overline{y}, U_{x}\overline{x} = q(x)x.

As a typical example of a subtriple system of an A^{∗J}, we have the triple of rectangular matrices M_{p,q}(Δ) ⊆ M_{p+q,p+q}(Δ)^{∗J} where x^{∗} = \overline{x}^{t}, − an involution on Δ. More generally in what follows we will need to consider the class of Jordan pairs of rectangular type , for which we adopt Loos’ notation [9, 6.4]: we let R =: R^{+} denote an associative Φ-algebra, R^{−} := R^{op}, a → \overline{a} the identity map from R^{ε} → R^{−ε}, M = (M^{+}, M^{−}) a pair of Φ-modules such that M^{+} is a left and M^{−} is a right R-module, f =: f^{+} : M^{+} × M^{−} → R an R-bilinear map and f^{−} : M^{−} × M^{+} → R defined by f^{−}(y, x) = \overline{f^{+}(x, y)}; then (M^{+}, M^{−}) becomes an associative pair

(1.11.1) A(M, R, f) : < xyz >= f^{ε}(x, y)z where for a ∈ R, ε = ±,

f^{ε}(ax, y) = af^{ε}(x, y), \overline{f^{ε}(x, y)} = f^{−ε}(y, x),

which naturally yields a Jordan pair

(1.11.2) V(M, R, f) := A(M, R, f)^{J} : Q_{x^{+}}y^{−} = f(x^{+}, y^{−})x^{+},
Q_{y^{−}}x^{+} = y^{−}f(x^{+}, y^{−}).

As is customary, we will denote direct products by ∏ J_{i}, subdirect products as

J ∼^{=} ∏ J_{i} (J_{i} = π_{i}(J) for homomorphisms π_{i} with ⋂ Ker π_{i} = 0)

and the Jordan system direct sum J_{1} × J_{2} as J_{1} \overline{\underline{+}} J_{2} (instead of the mere module direct sum J_{1} ⊕ J_{2}). A Jordan triple or pair comes with a slew of Jordan algebras, its homotopes

(1.12.1) b-homotope J^{(b)} :  U_{x}^{(b)}y = P_{x}P_{b}y, x^{(2,b)} = P_{x}b,
V_{x,y}^{(b)} = L_{x,P (b)y}, V_{x}^{(b)} = L_{x,b}.

To a certain extent, we can condense the bad behavior of the algebra J^{(b)} into the kernel of J at b

Ker_{J}(b) = {z ∈ J | P_{b}z = P_{b}P_{z}b = 0} ◁ J^{(b)},

Ker_{J}(b) = {z ∈ J | P_{b}z = 0} = Ker(P_{b}) if J (1.12.2)

is nondegenerate [5, 1.2.4], so that the local algebra of J at b, A_{b} := J^{(b)}/Ker_{J}(b) , inherits a good number of desirable properties from the original system J (see [6]).

Especially important in our Clifford situation are isotopes of Jordan Clifford algebras J(q, 1) as in (1.10.1):

if q(b) ≠ 0 then J(q, 1)^{(b)} = J(q^{(b)}, 1^{(b)}) for q^{(b)}(x) = q(b)q(x), 1^{(b)} = b^{−1}; here t^{(b)}(y) = q(y, b̄), ȳ^{(b)} = q(y, b̄)b^{−1} − y (1.12.3)

= q(b)^{−1}\overline{U_{b}y}.

Given an invertible symmetric structural transformation J⟶^{σ}J (a linear bijection such that P_{σx} = σP_{x}σ), we can form the σ-isotope

(1.13.1)  J^{(σ)} :  P_{x}^{(σ)} = P_{x}σ where P_{σx} = σP_{x}σ,

which is again a triple (but not necessarily an algebra). Such isotopes arise naturally when comparing pairs V(J) built out of triples (cf. [8]):

V(J_{1}) ≃ V(J_{2}) via (φ^{+}, φ^{−}) ⟺ J_{1} ≃ J_{2}^{(σ)} via φ^{+} (1.13.2)
(σ = φ^{−} ∘ (φ^{+})^{−1}).

Indeed, φ = (φ^{+}, φ^{−}) is a pair isomorphism V(J_{1}) ⟶ V(J_{2}) iff the φ^{ε} are linear bijections J_{1} ⟶ J_{2} with φ^{ε}(P_{x_{1}}y_{1}) = P_{φ^{ε}(x_{1})}φ^{−ε}(y_{1}), so φ^{+} : J_{1} ⟶ J_{2} is an isomorphism of J_{1} with the σ-isotope of J_{2} for σ = φ^{−} ∘ φ^{+}^{−1}. For, φ^{+}(P_{x_{1}}y_{1}) = P_{φ^{+}(x_{1})} φ^{−}(φ^{+}^{−1} (φ^{+}(y_{1}))) = P_{φ^{+}(x_{1})} σ(φ^{+}(y_{1})) = P_{φ^{+}(x_{1})}^{(σ)}

φ^{+}(y_{1}) where J_{1} ⟶^{σ} J_{2} satisfies P_{σ(x_{2})} = σ∘P_{x_{2}} ∘σ [write x_{2} = φ^{+}(x_{1}), y_{2} =

φ^{+}(y_{1}) so σ(P_{x_{2}}σ(y_{2})) = σ(P_{φ^{+}(x_{1})}φ^{−}(y_{1})) = σ(φ^{+}(P_{x_{1}}y_{1})) = φ^{−}(P_{x_{1}}y_{1}) = P_{φ^{−}(x_{1})}φ^{+}(y_{1}) = P_{σ(x_{2})}y_{2}]. Conversely, if J_{1} ≃ J_{2}^{σ} via φ then φ^{+} = φ, φ^{−} = σ ∘ φ is a pair isomorphism.

An inner ideal is a subspace K ⊆ J with P_{K}J ⊆ K, an outer ideal B ⊆ J has P_{J}B + L_{J,J}B ⊆ B, and an ideal I ◁ J is both inner and outer. A subspace C ⊆ J is a semi-ideal if P_{C}J + P_{J}P_{J}C + L_{J,J}C ⊆ C.

A Jordan system is semiprime if it has no trivial ideal P_{I}I = 0, is prime if it has no orthogonal ideals P_{I}K = 0, is nondegenerate if it has no trivial elements P_{z} = 0, is strongly prime if it is both prime and nondegenerate, is simple if P_{J}J ≠ 0 and J has no nonzero proper ideals, is a division system if its nonzero elements are invertible (x ≠ 0 ⇒ P_{x} is invertible), is locally-nilpotent if all its finitely generated subsystems are nilpotent, is pnil (properly nil) if all its homotopes are nil (for all z, y ∈ J there is n, depending on z and y, such that z^{(n,y)} = 0), is psnil (properly strictly nil) if all its homotopes J^{(y)} remain strictly nil, i.e. all scalar extensions are nil (for all z̃ in all J_{Ω} and all y ∈ J there is n such that z̃^{(n,y)} = 0), is pnil- or psnil-free if it has no nonzero pnil or psnil ideal, and is semiprimitive if it has no quasi-invertible ideal I consisting entirely of properly quasi-invertible elements z, where z is p.q.i. if B_{z,y} = Id − L_{z,y} + P_{z}P_{y} is invertible on J for all y ∈ J.

Most of these ideal-theoretic properties can be forced onto a system J by radical surgery: the Jacobson radical Rad(J) (respectively properlynil radical PN il(J), properly-strictly-nil radical PSN il(J), locally-nilpotent radical L(J), McCrimmon or lower radical Mc(J), Baer radical B(J)) is the smallest ideal I such that J/I is semiprimitive (respectively pnil-free, psnilfree, locally-nilpotent-free, nondegenerate, semiprime). Recall that

(1.14.1) Rad(J) ⊇ PN il(J) ⊇ PSN il(J) ⊇^{(∗)} L(J) ⊇^{(∗∗)} Mc(J) ⊇ B(J),

in general, and all coincide in the finite situation:

Rad(J) = PN il(J) = PSN il(J) = L(J) = Mc(J) = B(J) (1.14.2)

if J is finite-dimensional over a field.

The inclusion (∗∗) emerges as a byproduct of Kostrikin and Zel’manov’s work on Lie sandwiches (as a consequence, a simple triple or pair is necessarily nondegenerate) and (∗) holds since locally-nilpotent ideals remain so in scalar extensions and L(J) is not only nil but pnil in J: given z ∈ L(J), w ∈ J, we have < z, P_{w}z > ⊆ L(J) ⇒ < z, P_{w}z > nilpotent of some index n, so all products of degree ≥ n vanish, 0 = z^{(n,P_{w}z)} = z^{(2n−1,w)}, so z is nilpotent in J^{(w)}. Let us also mention that if J̃ ⊇ J is a tight cover of J,

(1.15)  all ideals of J̃ hit J : 0 ≠ Ĩ◁ J̃ ⟹ Ĩ∩ J ≠ 0,

then J̃ behaves like J in many ways,

(1.16) if J is semiprime, prime, nondegenerate, pnil-free or psnil-free, so is any tight cover J̃.

We say a Jordan pair V = (V^{+}, V^{−}) is ε-core-free if

Q_{V^{ε}}(z^{−ε}) = Q_{V^{ε}}Q_{z^{−ε}}V^{ε} = 0 ⟹ z^{−ε} = 0

and ε-primitive at b^{−ε} if it is ε-core-free and it has a b^{−ε}-primitizer, that is, a proper inner ideal K^{ε} < V^{ε} modular in the homotope V^{ε(b^{−ε})} with some modulus b^{ε}, which complements nonzero pair ideals:

B_{b^{ε},b^{−ε}}V^{ε} ⊆ K^{ε} (1.17.1)
b^{ε} − Q_{b^{ε}}b^{−ε} ∈ K^{ε} (1.17.2)
D_{b^{ε},b^{−ε}}K^{ε} ⊆ K^{ε} (1.17.3)
{D_{x^{ε},b^{−ε}} − D_{b^{ε},Q(b^{−ε})x^{ε}}}K^{ε} ⊆ K^{ε} for all x^{ε} ∈ V^{ε} (1.17.4)
0 ≠ I ◁ V ⟹ I^{ε} + K^{ε} = V^{ε}. (1.17.5)

V is primitive if it is ε-primitive for some ε. (Note that the superscriptless version of (1.17.1-5) provides the definition of primitivity for triples.) Primitivity of V and P ol(V) are equivalent [6, 5.5]:

(1.18) V = (V^{+}, V^{−}) is a primitive pair

⟺ P ol(V) = V^{+} ⊕ V^{−} is a primitive triple system.

Finally, polynomial identities will play a crucial role in our classification. A Jordan system (triple or pair) is i-special if it satisfies all the identities satisfied by all special Jordan systems (the so-called s-identities). An sidentity is just a Jordan polynomial f whose image f^{s} in the free special Jordan system is zero. To avoid speciality and torsion issues for scalars from a general ring Φ, one always restricts the term polynomial identity to polynomials which are s-monic, where f is s-monic if its image f^{s} is not merely nonzero, but is monic (has a leading coefficient 1, i.e. one of its associative monomials of maximal total x-degree appears with coefficient 1). We say that a Jordan system J satisfies a homotope PI if there is an s-monic Jordan polynomial f^{(y)}(x_{1}, …, x_{n}) in the y-homotope of the free Jordan system such that

(1.19) f^{(y)}(x_{1}, …, x_{n}) vanishes in all scalar extensions
J_{Ω} of J, i.e. f^{(b)}(a_{1}, …, a_{n}) = 0 for all a_{i}, b ∈ J_{Ω}

(hence all homotopes J^{(b)} satisfy an ordinary PI). Note that for us, “satisfies” always means “strictly satisfies”.

A Jordan system is of Clifford type if it satisfies a Clifford homotope PI, i.e. an s-monic polynomial that does not vanish on the split matrix algebra H_{3} = H(M_{3}(Φ), t) of hermitian 3×3 matrices over Φ; in particular, by definition (1.19), such a polynomial is not satisfied by any system J having a scalar extension ΩJ containing H_{3}.

In the presence of any homotope PI (not necessarily Clifford), a primitive system possesses a nonzero socle, a fact that we record for future reference (this is the main result of [6], Theorems 6.1 / 6.2 pp. 232 / 233; also see [10] for generalities on socles):

(1.20) if a primitive triple (or pair) J satisfies a homotope PI, then Soc(J) ≠ 0.

We will see that if the identity is actually Clifford, then J has finite capacity ≤ 2.

2. Classical pairs of Clifford type

To establish a list of candidates for our classification, we use Loos’ list [9, p.138] of classical central simple Jordan pairs as a starting point. Loos’ list was based on Jacobson’s list of simple Jordan algebras of finite capacity, which included Type (0), division algebras; from the later work of Zel’manov we know these are all subsumed under the remaining types. Loos’ list did not specify the centroid. Jordan centroids were examined in detail in [14, Theorem 5.9] specifically to culminate in:

Loos’ Classical List 2.1. If V is a central simple Jordan pair of finite capacity over the field Φ, then V is isomorphic to one of the following:

(I) Rectangular Pairs: V(M, R, f) for a central simple Artinian Φ-algebra R ≃ M_{n}(Δ) for some central division algebra Δ over Φ, f a nondegenerate R-bilinear pairing,

(II) Alternating Pairs: V(A_{n}(Φ)) for n ≥ 5, (III) Clifford Pairs: V(J) for a central-simple Clifford Jordan algebra J (a

tight scalar form of some simple J(\overline{q}, 1) for a nondegenerate quadratic form \overline{q} of dimension ≠ 2 over an algebraically closed extension field \overline{Φ} of Φ), (IV) Hermitian Pairs: V(H_{n}(Δ, Δ_{∘})) for n ≥ 2 and some central division

algebra Δ with involution over Ω and ample subspace Δ_{∘}, Φ = C(Δ, Δ_{∘}) = {α ∈ C(Δ) | αΔ_{∘} ⊂ Δ_{∘}},

(V) BiCayley Pairs: V(M_{1,2}(C)) for some 8-dimensional Cayley algebra C

over Φ, (VI) Albert Pairs: V(A) for some 27–dimensional exceptional simple Albert

algebra A over Φ.

Since we are investigating i-special systems, we immediately eliminate the BiCayley and Albert pairs; this leaves us with types (I)-(IV). Next, we remark that many small systems can be subsumed under the Clifford case III: H_{2}(Δ, Φ) is Clifford as is H_{1}(Δ, Δ_{∘}) if Δ is a composition algebra over Φ. Similarly, A_{n}(Φ) is Clifford for n ≤ 4 since A_{4}(Φ) ≃ H_{2}(Q(Φ), Φ), hermitian 2 × 2 matrices over a split quaternion algebra. [In fact, the even alternating systems A_{2n}(Φ) ≃ H_{n}(Q(Φ), Φ, s) can all be represented as hermitian matrices with symplectic—rather than transpose—involution, and thus if we enlarged Type (IV) to include H_{n}(D, D_{∘}) for D ∗-central simple with ample division subspace D_{∘}, then we would only need to keep odd alternating systems in Type (II).]

We now determine which of the remaining possibilities can satisfy a Clifford homotope PI. In this case no homotope of the pair V can contain a copy of H_{3}(Φ), and since any scalar extension V_{Ω} continues to satisfy the Clifford identity, V_{Ω} can’t contain H_{3}(Ω) either. This rules out all alternating A_{n}(Φ) for n ≥ 6 (they contain A_{6}(Φ) ≃ H_{3}(Q(Φ), Φ)) and in view of the above comment for n ≤ 4, only A_{5}(Φ) remains. Next, we claim the hermitian pairs can be eliminated: they are either “too big” or already Clifford. The following general theorem explains why and also provides a useful argument for division and rectangular pairs.

Theorem 2.2 (Hermitian-Clifford). If R is a ∗-simple Artinian algebra over its ∗-center Φ_{∘} and J = H_{∘}(R, ∗) an ample Φ_{∘}-subspace of symmetric elements which satisfies a Clifford homotope PI, then J is a Jordan Clifford algebra J(q, 1) over Φ_{∘} of dimension 1, 3, 4 or 6. In particular (applying this to R ⊕R^{op} under the exchange involution, Φ_{∘} = Φ), if R is a simple Artinian algebra over its center Φ then R^{+} satisfies a Clifford PI if and only if either R = Φ1 is 1–dimensional or R = Q is a 4–dimensional quaternion algebra over Φ.

PROOF. From associative theory we recall some facts about splitting fields. Any central simple Artinian algebra is R ≃ M_{n}(Δ) over a division algebra Δ with center Φ and degree r (finite or ∞), so some Δ_{Ω} contains M_{m}(Ω) for all finite m ≤ r. Then R has degree s = nr so that some scalar extension R_{Ω} = Ω ⊗_{Φ} R contains M_{m}(Ω) for all finite m ≤ s. We denote this situation by R ≈ M_{s}(Ω). If J ⊃ trace(R, ∗) satisfies a Jordan homotope PI then R satisfies an associative ∗-PI, hence by Amitsur an ordinary PI. Once R strictly satisfies some polynomial identity, then s < ∞ and R is finite-dimensional over its center (of dimension s^{2}). When s ≤ 2, this says R = Φ1 of dimension 1 (s = nr = 1 ⟹ n = r = 1) or R = Q of dimension 4 (s = 2 implies either n = 1, r = 2 and Q = Δ a quaternion division algebra, or n = 2, r = 1 and Q = M_{2}(Φ) split quaternions):

(1)  s = 1 ⟹ R = Φ1,  s = 2 ⟹ R = Q quaternions.

Next we bring in the involution. If (R, ∗) is ∗-simple over its ∗-center Φ_{∘}, any involution is of exchange, orthogonal or symplectic type.

Now ∗ has exchange type iff it is an “involution of second kind”, where its center Φ > Φ_{∘} is bigger than its ∗-center (i.e. ∗ doesn’t fix the center); here [Φ : Φ_{∘}] = 2 and already (R, ∗) ⊗_{Φ_{∘}} Φ ≃ (R \overline{\underline{+}} R^{op}, ex) splits into the form of the exchange involution, so for a splitting extension Ω of Φ we have (R, ∗)_{Ω} ≃ ((R, ∗)_{Φ})_{Ω} ≃ (R \overline{\underline{+}} R^{op}, ex)_{Ω} = (R_{Ω} \overline{\underline{+}} R_{Ω}^{op}, ex) ≃ (M_{s}(Ω) \overline{\underline{+}} M_{s}(Ω)^{op}, ex), hence

H(R, ∗)_{Ω} ≃ M_{s}(Ω)^{+}.

Here the only ample subspace under the exchange involution (where all symmetric elements are traces) is the full H(R, ∗)_{Ω}, and this satisfies a Clifford PI iff s ≤ 2: if s ≥ 3 then M_{s}(Ω) ⊃ H_{3}(Φ), thus s = 1 or 2 and by (1) R = Φ or Q, so

if J = H_{∘}(R, ∗) of exchange type satisfies a Clifford identity (2) then R = Φ or R = Q for a quaternion algebra Q, J = Φ_{∘}

or H_{∘}(Q, ∗), so J = J(q, 1) for q of dimension 1 or 4 over Φ_{∘}.

Now consider ∗ of orthogonal type (of “first kind”, so center = ∗- center and Φ = Φ_{∘}). Then (R, ∗)_{Ω} ≃ (M_{s}(Ω), t) i.e. ∗ becomes the transpose involution,

H(R, ∗)_{Ω} ≃ H_{s}(Ω, t),

whose only ample Ω-subspace is H_{5}(Ω, t) itself. Again if H_{∘}(R, ∗) (hence ΩJ = H_{5}(Ω, t)) satisfies a Clifford homotope PI then s ≤ 2 since if s ≥ 3 the ample H_{5}(Ω, t) would contain H_{3}(Φ, t). Once more by (1), R = Φ or Q and H(R, ∗) = Φ1 or H(Q, ∗), and the ample Φ-subspace J is all of H(R, ∗) of dimension 1 or 3,

if J = H_{∘}(R, ∗) of orthogonal type satisfies a Clifford identity

(3) then R = Φ or Q, J = Φ or J = H(Q, ∗) for a quaternion algebra Q, so J = J(q, 1) for q of dimension 1 or 3 over Φ.

Finally, consider ∗ of symplectic type; we have (R, ∗)_{Ω} ≃ (M_{s}(Ω),sp) = (M_{2r}(Ω), sp) where s = 2r must be even and the symplectic involution is the alternating isotope of the transpose:

X^{sp} = SX^{t}S^{−1} for S = diag \{( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , …, ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )\} .

For Jordan theory, a more useful description is M_{2r}(Ω) ≃ M_{r}(M_{2}(Ω)) = M_{r}(Q(Ω)) for the split quaternion algebra Q(Ω) = M_{2}(Ω); the standard quaternion involution \overline{x} = tr(x)1 − x has

( \begin{matrix}α & β \\ γ & δ \end{matrix} )^{−} = ( \begin{matrix}δ & −β \\ −γ & α \end{matrix} ) = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ( \begin{matrix}α & γ \\ β & δ \end{matrix} ) ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) = sx^{t}s^{−1}

in the split case, and the symplectic involution sp on M_{2r}(Ω) becomes the standard conjugate transpose X^{∗_{s}} = \overline{X}^{t} on M_{r}(Q(Ω)),

H(R, ∗)_{Ω} ≃ H_{r}(Q(Ω), ∗_{s}).

Here the Ω-ample subspace ΩJ ⊃ H_{r}(Ω) (any ample Ω-subspace has at least Ω down its diagonal), so again a Clifford PI forces r ≤ 2 (this time s ≤ 4). If r = 1, s = 2, then R = Q, H(R, ∗) = H(Q, ∗_{s}) has dimension 1 (in characteristic ≠ 2) or 3 (in characteristic 2), and the Φ-subspace J = H_{∘}(Q, ∗) has dimension 1 (in characteristic ≠ 2) or dimension 1 or 3 (in characteristic 2).

Now we come to the hard part, the case r = 2, s = 4. In characteristic ≠ 2 the only ample subspace of H(Q(Ω), −) = Ω1 is itself, and the only ample

subspace of H_{2}(Q(Ω), ∗_{s}) is H_{2}(Q(Ω), Ω, ∗_{s}) = \{X = ( \begin{matrix}α & a \\ \overline{a} & β \end{matrix} ) | α, β ∈ Ω\}

of dimension 1 + 1 + 4 = 6; this is J(det, 1) for the quadratic form det(X) = αβ − a\overline{a} ∈ Ω1. Then J = H_{2}(Q, Φ, ∗_{s}) is also J(q, 1) of dimension 6 for q the restriction of det.

In characteristic 2, in addition to the 1–dimensional ample subspace Ω1 of H(Q(Ω), −) as above, there is the entire 3–dimensional space H(Q(Ω), −)

= \{Y = ( \begin{matrix}α & β \\ γ & α \end{matrix} ) ∈ Q(Ω)\}; the corresponding ample subspace of H_{2}(Q(Ω),

∗_{s}) is H_{2}(Q(Ω), ∗_{s}) itself of dimension 3 + 3 + 4 = 10: all ( \begin{matrix}h & a \\ \overline{a} & k \end{matrix} ) for a ∈ Q

and h, k ∈ H(Q(Ω), −). Here there are only two orthogonal idempotents, yet the algebra is NOT an algebra J(q, 1)! But the homotope PI hypothesis comes to our aid: all homotopes (J_{Ω})^{(b)} must still satisfy the Clifford PI. When r = 2 we have s = 4 and somewhere there should be four idempotents

to rescue us. We claim that if we take the diagonal isotope by b = ( \begin{matrix}u & 0 \\ 0 & u \end{matrix} ) ∈

H_{2}(Q(Ω), ∗_{s}) (u = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ∈ Q(Ω)) we will find the missing idempotents.

We will show

(4)  (M_{2}(Q(Ω))^{(b)}, ∗_{s}) ≃ M_{2}(Q(Ω), t)

so H_{2}(Q(Ω), ∗_{s})^{(b)} = H(M_{2}(Q(Ω))^{(b)}, ∗_{s}) ≃ H(M_{2}(Q(Ω)), t) = H(M_{4}(Ω), t) ⊃ H_{4}(Φ, t) which shows this 10–dimensional J_{Ω} does not satisfy the Clifford homotope PI.

To establish (4), note that for a diagonal b = diag(u, …, u) we have M_{n}(R)^{(b)} = M_{n}(R^{(u)}), so M_{2}(Q(Ω))^{(b)} = M_{2}(Q(Ω)^{(u)}) where (Q(Ω)^{(u)}, −) ⟶^{L_{u}} (Q(Ω),t) is a ∗-algebra isomorphism: L_{u}(x ·_{u} y) = uxuy = (L_{u}x)(L_{u}y) and L_{u}(\overline{x}) = u\overline{x}uu^{−1} = u(\overline{ux})u^{−1} (since \overline{u} = u) = (\overline{L_{u}x})′ for \overline{a}′ = u\overline{a}u^{−1} = usa^{t}s^{−1}u^{−1} (as mentioned earlier for Q = Q(Ω) split) = a^{t} SINCE IN

CHARACTERISTIC 2, u = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) COINCIDES WITH s = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )

AND ITS INVERSE. Thus (Q(Ω)^{(u)}, −) ≃ (Q(Ω), t) and (M_{2}(Q(Ω))^{(b)}, ∗_{s}) = (M_{2}(Q(Ω)^{(u)}), ∗_{s}) ≃ M_{2}(Q(Ω), t) as in (4), and we are done.

Let J = H_{∘}(R, ∗) of symplectic type satisfy a Clifford identity. If s = 2 then R = Q, J = Φ of dimension 1, or H_{∘}(Q, ∗) of (5) dimension 3 (only in characteristic 2), so J = J(q, 1) for q of

dimension 1 or 3. If s = 4 then R = M_{2}(Q), J = H_{2}(Q, Φ, ∗_{s})

= J(q, 1) for q of dimension 6 over Φ.

Thus applying the above theorem to the case R = M_{n}(Δ), all the hermitian pairs (type IV) satisfying a Clifford homotope PI are Clifford pairs (type III). Note that any full J(q, 1) over a field Φ is a tight scalar form of a simple J(\overline{q}, \overline{1}) over \overline{Φ} by the Clifford Tightening Theorem [15, Thm. 3.4], and that over an algebraically closed field J(\overline{q}, \overline{1}) is simple iff \overline{q} is nondegenerate of dimension ≠ 2.

Remark 2.3. In getting rid of division pairs from Loos’ list, we could avoid the high-powered tool of the Zel’manov classification by using big field extensions. If J is a division algebra over its centroid Φ and we form a tight scalar extension J ⊂ J̃ = Φ̃J for Φ̃ ⊃ Φ a big algebraically closed field, then by tightness J̃ remains simple but is no longer a division algebra unless we have the trivial case J̃ = Φ̃1 (in which case J = Φ̃_{∘}1 is tiny Clifford): if x̃ ∉ Φ̃1 then all ω̃1 − x̃ are nonzero, Φ̃ ⊂ Resolvent(x̃) and by Amitsur’s Resolvent Theorem (see Theorem 18.5 below) and bigness, x̃ is algebraic over Φ̃, hence by algebraic closure p(x̃) = ∏(x̃ − α̃_{i}1) = 0 ⇒ some x̃ − α̃_{i}1 = 0 and x̃ = α̃_{i}1 ∈ Φ̃1 after all.

Thus, by imposing a Clifford homotope PI on the classical pairs of 2.1, we have shrunk our list of Cliffordian candidates to

List 2.4. If V is an i-special central simple Jordan pair of finite capacity over the field Φ satisfying a Clifford homotope PI, then V is isomorphic to one of the following:

(I) Rectangular Pairs: V(M, R, f) for a central simple Artinian Φ-algebra

R ≃ M_{n}(Δ) for some central division algebra Δ over Φ, (II) Alternating Pair: V(A_{5}(Φ)), (III) Clifford Pairs: V(J) for a central-simple Clifford Jordan algebra J (a

tight scalar form of some simple J(\overline{q}, 1) for a nondegenerate quadratic form \overline{q} of dimension ≠ 2 over an algebraically closed extension field \overline{Φ} of Φ).

3. Rectangular Pairs

Seeking a more explicit description of rectangular pairs, we now probe into the structure of V(M, R, f) when R is a simple Artinian associative algebra: it can be realized as a (potentially very large) pair of rectangular matrices M_{p,q}(Δ) := (M_{p,q}(Δ), M_{q,p}(Δ)) where we may always assume p ≤ q. Since a Clifford identity puts a limit on the number of pair idempotents, a rectangular pair of Clifford type must have p ≤ 2 (while q may actually be infinite). We begin with

Proposition 3.1. If V = V(M, R, f) for a simple Artinian algebra R ≃ M_{n}(Δ), then either

(i) V ≃ M_{n,q}(Δ) for some q,

or (ii) V ⊃ M_{n,q}(Δ) for all q.

When V ⊃ M_{n,q}(Δ), there exists b^{−} ∈ V^{−} with rank(b^{−}) = min(n, q).

PROOF. Abstractly we know that since M^{+} is a unital left R-module, it is a direct sum of irreducibles isomorphic to Re_{11} ≃ M_{n,1}(Δ) : M^{+} = ⊕_{i∈I} M_{i}^{+} ≃ ⊕_{i∈I} M_{n,1}(Δ) =: M_{n,I}(Δ), and dually M^{−} ≃ M_{J,n}(Δ). However, this module decomposition completely disregards f. We must use a straightening argument to construct a “dual” or “orthonormal” basis.

We do this by induction. We start with no basis vectors at all: q − 1 for q = 1. Suppose we have constructed dual basis vectors m_{k}^{±} for k = 1, 2, …, q − 1 such that

(1)  f(m_{i}^{+}, m_{j}^{−}) = δ_{ij}e_{11} (m_{i}^{+} = e_{11}m_{i}^{+}, m_{j}^{−} = m_{j}^{−}e_{11}).

We claim we can split off the R-span of these vectors:

M^{+} = Rm_{1}^{+} ⊕ … ⊕ Rm_{q−1}^{+} ⊕ M_{q}^{+}, where Rm_{i}^{+} ≃ M_{n,1}(Δ)
and M_{q}^{+} = {x^{+} | f(x^{+}, m_{j}^{−}) = 0 for j = 1, …, q − 1}, (2)
M^{−} = m_{1}^{−}R ⊕ … ⊕ m_{q−1}^{−}R ⊕ M_{q}^{−}, where m_{i}^{−}R ≃ M_{1,n}(Δ)

and M_{q}^{−} = {x^{−} | f(m_{i}^{+}, x^{−}) = 0 for i = 1, …, q − 1}.

Indeed, any m^{+} can be written as m^{+} = ∑_{i=1}^{q−1} a_{i}m_{i}^{+} +n^{+} (a_{i} := f(m^{+}, m_{i}^{−}) = f(m^{+}, m_{i}^{−}e_{11}) = a_{i}e_{11}) where

f(n^{+}, m_{j}^{−}) = f(m^{+} − ∑_{i=1}^{q−1} a_{i}m_{i}^{+}, m_{j}^{−})

= f(m^{+}, m_{j}^{−}) − ∑_{i=1}^{q−1} a_{i}f(m_{i}^{+}, m_{j}^{−})

= a_{j} − ∑_{i=1}^{q−1} a_{i}δ_{ij}e_{11} (by (1))

= a_{j} − a_{j}e_{11} = 0.

Thus n^{+} ∈ M_{q}^{+}, and M^{+} = ∑_{i=1}^{q−1} Rm_{i}^{+} + M_{q}^{+} is a sum of modules. The sum is direct since ∑ a_{i}m_{i}^{+} + n^{+} = 0 ⇒ 0 = f(∑ a_{i}m_{i}^{+} + n^{+}, m_{j}^{−}) = a_{j}e_{11} = a_{j} for j = 1, …, q − 1, hence n^{+} = 0 too. We have Rm_{i}^{+} ≃ Re_{11} since φ_{i}^{+}(rm_{i}^{+}) := f(rm_{i}^{+}, m_{i}^{−}) = re_{11} and ψ_{i}^{+}(re_{11}) := re_{11}m_{i}^{+} = rm_{i}^{+} are well-defined inverse R-linear maps. Dually for M^{−}.

If M_{q}^{+} = M_{q}^{−} = 0 then we are done: M^{+} ≃ ⊕_{i=1}^{q−1} M_{n,1}(Δ) = M_{n,q}(Δ), M^{−} ≃ M_{q,n}(Δ) where f(m^{+}, n^{−}) = f(∑_{i} a_{i}m_{i}^{+}, ∑_{j} m_{j}^{−}b_{j}) = ∑_{i,j} a_{i} f(m_{i}^{+}, m_{j}^{−}) b_{j} = ∑_{j} a_{j}e_{11}b_{j} corresponds to the matrix product φ^{+}(m^{+})φ^{−}(n^{−}) = [r_{ij}] [s_{kl}] for r_{ij} = e_{ii}φ_{j}^{+}(a_{j}m_{j}^{+}) = φ_{j}^{+}(e_{ii}a_{j}m_{j}^{+}) = e_{ii}a_{j}e_{11}, s_{jk} = φ_{j}^{−}(m_{j}^{−}b_{j})e_{kk} = e_{11}b_{j}e_{kk} [the ik-entry of the matrix product is

∑_{j} r_{ij}s_{jk} = ∑_{j} e_{ii}a_{j}e_{11} · e_{11}b_{j}e_{kk} = ∑_{j} e_{ii}a_{j}e_{11}b_{j}e_{kk} = e_{ii}(∑_{j} a_{j}e_{11}b_{j})e_{kk}

which equals e_{ii}(f(m^{+}, n^{−}))e_{kk}]. Thus V ≃ V((M_{n,q}(Δ), M_{q,n}(Δ)), R, π) under π(A, B) = AB.

If one of M_{q}^{±} is nonzero so is the other since they are nondegenerately paired: we have f(n^{+}, M_{q}^{−}) = 0 ⇒ f(n^{+}, M^{−}) = 0 (n^{+} ⊥ all m_{j}^{−}R) ⇒ n^{+} = 0 by left nondegeneracy of f, and dually on the other side. Thus f(M_{q}^{+}, M_{q}^{−}) is a nonzero ideal of the simple algebra R, so it must be all of R. In particular, there exist nonzero vectors n^{+} in M_{q}^{+} with e_{11}n^{+} = n^{+}, and for any such we can complete it to form a larger orthonormal basis: 0 ≠ f(n^{+}, M_{q}^{−}) = e_{11}f(n^{+}, M_{q}^{−}) ⊂ e_{11}R ⇒ f(n^{+}, M_{q}^{−}) = e_{11}R (by minimality of the right ideal e_{11}R) ⇒ f(n^{+}, n^{−}) = e_{11} for n^{+} = e_{11}n^{+}, n^{−} = n^{−}e_{11}; since these are in M_{q}^{±} orthogonal to the m_{i}^{∓} for i = 1, …, q − 1, taking n_{q}^{+} = n^{+}, n_{q}^{−} = n^{−} we obtain an orthonormal basis of length q, as in (1).

Thus either this process comes to a halt at some q, and we have V ≃ M_{n,q}(Δ), or else V ⊃ M_{n,q}(Δ) for all q.

Note that if r = min(n, q) then b^{−} = ∑_{i=1}^{r} E_{ii} lies in M_{n,q}(Δ) ⊂ V^{−}, c^{+} = ∑_{i=1}^{r} E_{ii} lies in M_{q,n}(Δ) ⊂ V^{+}, and f(c^{+}, b^{−}) = ∑_{i=1}^{r} E_{ii} has rank r = rank(b^{−}).

In the above situation, if q = ∞ (i.e. V ⊃ M_{n,q}(Δ) for all q) then clearly r = min(n, q) = n, but if q < n then the local algebras A_{b^{−}} ≃ M_{r}(Δ) for r ≤ q always comes out smaller than R = M_{n}(Δ)! This just means that we described V wrong: instead of V ≃ M_{n,q}(Δ) = V((M_{n,q}(Δ), M_{q,n}(Δ)), M_{n}(Δ), π), we should have described it as V ≃ M_{q,n}(Δ) = V((M_{q,n}(Δ), M_{n,q}(Δ)), M_{q}(Δ), π).

Renormalization Lemma 3.2. In the case of a rectangular pair of finite capacity V ≃ M_{p,q}(D) = (M_{p,q}(D), M_{q,p}(D)) for an associative algebra D, we may always assume p ≤ q: indeed, M_{p,q}(D) ≃ M_{q,p}(D^{op}) via x ⟶ x^{t}.

PROOF. The transpose sets up linear bijections φ^{+} : M_{p,q}(D) ⟶ M_{q,p}(D^{op}), φ^{−} : M_{q,p}(D) ⟶ M_{p,q}(D^{op}), and these yield a pair isomorphism since they preserve products: for x ∈ M_{p,q}(D), y ∈ M_{q,p}(D) we have φ^{+}(Q_{x}y) = (xyx)^{t} = x^{t}y^{t}x^{t} = φ^{+}(x)φ^{−}(y)φ^{+}(x) and φ^{−}(Q_{y}x) = (yxy)^{t} = y^{t}x^{t}y^{t} = φ^{−}(y)φ^{+}(x)φ^{−}(y), recalling (u ·_{D} v)^{t} = v^{t} ·_{D^{op}} u^{t} for u ∈ M_{r,s}(D), v ∈ M_{s,t}(D) [their ij-entries are (u ·_{D} v)_{ji} = ∑_{k=1}^{s} u_{jk}v_{ki} and (v^{t} ·_{D^{op}} u^{t})_{ij} = ∑_{k=1}^{s}(v^{t})_{ik} ·_{D^{op}} (u^{t})_{kj} = ∑_{k=1}^{s} u_{jk} ·_{D} v_{ki} respectively].

Thus we always choose to represent V(M, R, f) as M_{p,q}(Δ) for p ≤ q (allowing q to be “infinity”). We say V(M, R, f) is of type M_{p,q}(Δ) if it is ≃ M_{p,q}(Δ) for finite p ≤ q < ∞, and of type M_{p,∞}(Δ) if it contains copies of M_{p,q}(Δ) for all finite q (then abstractly M^{+} ≃ M_{p,I}(Δ), M^{−} ≃ M_{J,p}(Δ) where both index sets I, J are infinite). Combining our matrix description with a Clifford homotope PI, we obtain

Theorem 3.3 (Rectangular-Clifford). If V = V(M, R, f) for a central simple Artinian algebra R over Φ satisfies a Clifford homotope PI, then V has type M_{1,q}(Φ) for q ≥ 1, M_{1,q}(Q) for q ≥ 1 and Q a quaternion division algebra over Φ, or M_{2,q}(Φ) for q ≥ 2. Moreover, the pairs M_{1,1}(Φ), M_{1,1}(Q) and M_{2,2}(Φ) have invertible elements and are all isomorphic to a Clifford pair of type III for some nondegenerate quadratic form.

PROOF. The last statement is clear since M_{r,r}(D) = (M_{r}(D), M_{r}(D)) ≃ V(M_{r}(D)^{+}), and M_{1}(Φ)^{+} ≃ Φ, M_{1}(Q)^{+} ≃ Q^{+} = J(q, 1) (q(X) = det(X)), M_{2}(Φ)^{+} = J(q, 1) (again for q(X) = det(X)).

The first statement is again a consequence of Theorem 2.2. We know V is of type M_{n,q}(Δ) for some q ≥ n, in particular V ⊃ W ≃ M_{n,n}(Δ) = V(M_{n}(Δ)^{+}). We now apply 2.2 to R = M_{n}(Δ) and deduce that R = Φ1 or R = Q quaternions over Φ; consequently, either n = 1 and Δ = Φ or Δ = Q quaternions over Φ (in which case V ≃ M_{1,q}(Φ) or M_{1,q}(Q), 1 ≤ q ≤ ∞) or n = 2 and Δ = Φ (in which case V ≃ M_{2,q}(Φ), 2 ≤ q ≤ ∞).

In view of Theorem 3.3, we can now refine List 2.4:

Clifford List 3.4. If V is an i-special central simple Jordan pair of finite capacity over a field Φ satisfying a Clifford homotope PI, then V is isomorphic to one of the following:

(I) Rectangular Pairs: V(M, R, f) for a central simple Artinian Φ-algebra R of degree ≤ 2, which thus has one of the three forms (Ia) M_{1,q}(Φ), 1 ≤ q ≤ ∞, (Ib) M_{1,q}(Q), Q a quaternion division algebra over Φ, 1 ≤ q ≤ ∞, (Ic) M_{2,q}(Φ), 2 ≤ q ≤ ∞,

(II) Alternating Pair: V(A_{5}(Φ)), (III) Clifford Pairs: V(J) for a central-simple Clifford Jordan algebra J (a

tight scalar form of some simple J(\overline{q}, 1) for a nondegenerate quadratic form \overline{q} of dimension ≠ 2 over an algebraically closed extension field \overline{Φ} of Φ).

All these classical pairs are locally finite-dimensional: every finitely generated subpair is finite-dimensional. For our ultimate use in (10.12) of an ultraproduct, we need to know more precisely that there is a bound (depending on n) for these dimensions. We say a system J has locally d-bounded dimension over Φ for a fixed function d : N ⟶ N if every subsystem B ⊂ J generated over Φ by n elements is spanned over Φ by d(n) elements (when Φ is a field this means dim_{Φ}(B) ≤ d(n)).

Theorem 3.5. All the central-simple classical pairs over Φ on the Clifford List 3.4 have locally d-bounded dimension for

d(n) = (2n) 2^{n (2n+1)}.

PROOF. We need to show that any finitely generated subpair B = Φ[(x_{1}, y_{1}), …, (x_{n}, y_{n})] is contained in some subspace C = Φ(x_{1}, y_{1}) + … + Φ(x_{d}, y_{d}) (d = d(n)).

In rectangular pairs (I) M_{p,q}(Δ), p ≤ q the formulas Q_{x^{+}}(y^{−}) = f(x^{+}, y^{−})x^{+}, Q_{y^{−}}(x^{+}) = y^{−}f(x^{+}, y^{−}) show that B ⊂ C = (Rx_{1} + … + Rx_{n}, y_{1}R + … + y_{n}R) for R = M_{p,p}(Δ), so dim_{Φ}(C) = n dim_{Φ}(R); in (Ia) R = M_{1,1}(Φ) has dimension 1, in (Ib) R = M_{1,1}(Q) has dimension 4, and in (Ic) R = M_{2,2}(Φ) has dimension 4, so in all cases dim(C) ≤ 4n.

In alternating pairs (II) A_{5}(Φ) we trivially have B ⊂ C = V with dim(C) = 10.

In Clifford pairs (III) V(J) = (J, J) for a simple J = J(q, 1) in characteristic not 2, the formula U_{x}y = q(x, \overline{y})x − q(x)\overline{y} = (t(x)t(y) − q(x, y))x − q(x)t(y)1 + q(x)y for q(J) ⊂ Φ shows B ⊂ C = (K, K) for K = Φ1 + Φx_{1} + … + Φx_{n} + Φy_{1} + … + Φy_{n} of dimension 2n + 1, so dim(C) ≤ 2n + 1.

Thus in characteristic ≠ 2 the pairs have locally d-bounded dimensions for d(n) = max(4n, 10). The characteristic 2 case is messier, and the bound d(n) = (2n) 2^{n (2n+1)} becomes enormous. Note that for all n ≥ 1 we have (2n) 2^{n (2n+1)} ≥ (2n) 2^{3} = 16n = 6n + 10n > 4n + 10 > max(4n, 10), so d(n) continues to serve as bound for all non-Clifford pairs in the List.

In characteristic 2, by [18, 6.5 p.163] the central-simple Clifford form J is contained in a full Clifford algebra J(q̃, Φ̃) over a field Φ̃ with Φ ⊂ Φ̃ ⊂ \overline{Φ}, Φ̃^{2} ⊂ Φ. In particular, q̃(J) ⊂ Φ̃ implies q̃(J) + q̃(J, J) + q̃(J, \overline{J}) + q̃(\overline{J}) ⊂ Φ̃ (even though we don’t know \overline{J} ⊂ J, we do have q̃(\overline{J}) = q̃(J) and q̃(J, \overline{J}) = q̃(J, 1) q̃(J, 1) − q̃(J, J)), so

(1)  q̃(J)^{2}, q̃(\overline{J})^{2}, q̃(J, J)^{2}, q̃(\overline{J}, \overline{J})^{2}, q̃(J, \overline{J})^{2} ⊂ Φ.

Let

(2)  Ω := Φ[q̃(x_{i}), q̃(x_{i}, x_{j}), q̃(\overline{y_{i}}), q̃(\overline{y_{i}}, \overline{y_{j}}), q̃(x_{i}, \overline{y_{j}})]

be the Φ-subalgebra of Φ̃ generated by the ñ = n + n(n − 1)/2 + n + n(n − 1)/2 + n^{2} = 2n + n(n − 1) + n^{2} = 2n^{2} + n = n(2n + 1) different norms of x’s and \overline{y}’s. For uniformity we set

(3)  z_{i} := x_{i},  z_{i+n} := \overline{y_{i}}  (1 ≤ i ≤ n)

so (2) simplifies to

(4)  Ω = Φ[q̃(z_{i}), q̃(z_{i}, z_{j})]  (1 ≤ i, j ≤ 2n).

Introduce

K^{+} := ∑_{i=1}^{n} Ωx_{i} + ∑_{i=1}^{n} Ω\overline{y_{i}} = ∑_{i=1}^{2n} Ωz_{i} (5)

K^{−} := \overline{K^{+}} = ∑_{i=1}^{n} Ω\overline{x_{i}} + ∑_{i=1}^{n} Ωy_{i} = ∑_{i=1}^{2n} Ω\overline{z_{i}}

so by definition (4) the norms of the generators fall in Ω and

(6)  q̃(K^{+}) = q̃(K^{−}) ⊂ Ω.

This guarantees that C = (K^{+}, K^{−}) is an Ω-subpair: Q_{K^{+}}K^{−} ⊂ ∑ ΩQ_{z_{i}} \overline{z_{k}} + ∑ ΩQ_{z_{i},z_{j}} \overline{z_{k}} = ∑ Ω[q̃(z_{i}, z_{k})z_{i} − q̃(z_{i})z_{k}] + ∑ Ω[q̃(z_{i}, z_{k})z_{j} + q̃(z_{j}, z_{k})z_{i} −q̃(z_{i}, z_{j})z_{k}] ⊂ ∑ Ωz_{i} (by (6)) = K^{+}, applying bar gives Q_{K^{−}}K^{+} = \overline{Q_{K^{+}}K^{−}} ⊂ \overline{K^{+}} = K^{−}. Since C is a subpair containing (x_{i}, y_{i}) = (z_{i}, \overline{z_{i+n}}) it contains the subpair they generate:

B = Φ[(z_{1}, \overline{z_{1+n}}), (z_{2}, \overline{z_{2+n}}), …, (z_{n}, \overline{z_{2n}})] (7)

⊂ ( ∑_{i=1}^{2n} Ωz_{i}, ∑_{i=1}^{2n} Ω\overline{z_{i}}) = C.

Thus dim_{Φ}(B) ≤ dim_{Φ}(C) ≤ ∑_{i=1}^{2n} dim_{Φ}(Ωz_{i}) ≤ (2n) dim_{Φ}(Ω). Now Ω is generated by ñ elements w_{m} = q̃(z_{i}) or q̃(z_{i}, z_{j}) with w_{m}^{2} ∈ Φ by (1), hence (as in Clifford or exterior algebras) Ω is spanned over Φ by 2ñ monomials w_{m_{1}}w_{m_{2}} … w_{m_{r}} for m_{1} < m_{2} < … < m_{r}, 0 ≤ r ≤ ñ. Thus dim_{Φ}(B) ≤ (2n) 2ñ = (2n) 2^{n(2n+1)} ≤ d(n) as claimed in the theorem.

Thus there is an upper bound d(n) for the dimension of n-generated subpairs, which is usually 4n but explodes for Clifford forms in characteristic 2.

4. Local Algebras

The Primitive Form Theorem in §5 asserts that a simple scalar form V of a classical pair Ṽ of Clifford type must be primitive, a crucial fact for classification purposes. The proof hinges on the “local-Cliffordness” of those classical pairs (all their local algebras are Clifford algebras). We now proceed to show that this is the case if Ṽ is one of the pairs of List 3.4. We begin with the rectangular case; for those, it amounts to carrying out the argument of example 1.9 of [6] for pairs. Recall that by (1.12.2) Ker_{J}(b) = Ker(P_{b}) for nondegenerate (e.g. simple) systems.

Proposition 4.1. If V = V(M, R, f) for a simple Artinian R ≃ M_{n}(Δ), then for each b^{−} ∈ V^{−} there is an idempotent e_{b} ∈ R with f(M^{+}, b^{−}) = Re_{b}, and

A_{b} ≃ (e_{b}Re_{b})^{+} ≃ M_{r}(Δ) where r = rank(e_{b}).

PROOF. f(M^{+}, b^{−}) is a left ideal in R, hence has the desired form Re_{b}. There are lots of choices for e_{b}, but notice that they all preserve b^{−}: b^{−} = b^{−}e since b^{−}(1 − e) = 0 by nondegeneracy of f, f(M^{+}, b^{−}(1 − e)) = f(M^{+}, b^{−})(1 − e) = Re(1 − e) = 0. We claim

(1)  V^{+(b^{−})} ⟶^{φ} (e_{b}Re_{b})^{+} via φ(x^{+}) = e_{b}f(x^{+}, b^{−})

is an epimorphism with kernel precisely Ker_{J}(b^{−}) = Ker(Q_{b^{−}}), and so induces an isomorphism of A_{b} with (e_{b}Re_{b})^{+}.

First, φ is a Jordan homomorphism (using be = b): φ(P_{x^{+}}^{(b)}y^{+}) = φ(Q_{x}Q_{b}y) = φ(f(x, Q_{b}y)x) = φ(f(x, bf(y, b))x) = φ(f(x, b)f(y, b)x) = e f(x, b) f(y, b) f(x, b) = e f(x, be) f(y, be)f(x, b) = ef(x, b)ef(y, b)ef(x, b) = φ(x) φ(y) φ(x), and φ(x^{+}^{(2,b)}) = φ(Q_{x}b) = φ(f(x, b)x) = ef(x, be)f(x, b) = ef(x, b)ef(x, b) = φ(x)^{2}.

Second, φ is surjective: e ∈ Re = f(M^{+}, b^{−}) ⇒ e = f(c^{+}, b^{−}) for some c^{+} ∈ M^{+} ⇒ φ(rc^{+}) = ef(rc^{+}, b^{−}) = erf(c^{+}, b^{−}) = ere.

Finally, the kernel of φ is the kernel of Q_{b}: if ef(x, b) = 0 then Q_{b}x = bf(x, b) = (be)f(x, b) = 0, and conversely if Q_{b}x = 0 then 0 = f(c, Q_{b}x) = f(c, bf(x, b)) = f(c, b) f(x, b) = ef(x, b).

Corollary 4.2. If a central-simple pair V = V(M, R, f) of finite capacity satisfies a Clifford homotope identity, then its local algebras have dimension 1 or 4 as follows:

(Ia) V = M_{1,q}(Φ): if b^{−} ≠ 0, A_{b^{−}} ≃ Φ^{+},
(Ib) V = M_{1,q}(Q): if b^{−} ≠ 0, A_{b^{−}} ≃ Q^{+}, Q a quaternion

division algebra,

(Ic) V = M_{2,q}(Φ): if rank (b^{−}) = 1, A_{b^{−}} ≃ Φ^{+}, if rank (b^{−}) = 2, A_{b^{−}} ≃ M_{2}(Φ)^{+}.

PROOF. In (3.4) (Ia) and (Ib) the only nonzero e_{b} is e = 1, so A_{b^{−}} ≃ (e_{b}Re_{b})^{+} = R^{+}. In (Ic), if b^{−} has rank 1 then so does e_{b}, and A_{b^{−}} ≃ (e_{b}Re_{b})^{+} = Φe_{b}^{+} ≃ Φ^{+}; if b^{−} has rank 2 then e_{b} = 1, A_{b^{−}} ≃ R^{+}.

We now turn to the alternating pair V = V(A_{5}(Φ)). We think of V as the alternating linear maps between a 5–dimensional Φ-space W and its dual W^{∗} as in (1.8.3). We have

Proposition 4.3. If V = V(A_{5}(Φ)) = Alt(W, W^{∗}) then a local algebra of V is isomorphic to either Φ^{+} of dimension 1, or J(q, 1) of dimension 6.

PROOF. An element b^{−} ∈ Hom(W^{∗}, W) has even (matrix) rank 0, 2 or 4 (rank(b) = 0 ⇒ b = 0). If rank(b) = 2, then there exists some basis B of W such that the matrix B = Mat_{B,B^{∗}}(b) of b with respect to B and its dual B^{∗} is

B = diag \{( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ) , 0\}

and Ker_{J}(b) = Ker(P_{b}) = all alternating maps x^{+} ∈ Hom(W, W^{∗}) whose 5 × 5 matrix X = Mat_{B,B^{∗}}(x^{+}) has X_{11} = X_{12} = X_{21} = X_{22} = 0, so that A_{b} = Φc̃, where c ∈ Hom(W, W^{∗}) has the same matrix Mat_{B,B^{∗}}(c) = B; since Mat_{B,B^{∗}}(−cbc) = −BBB = B = Mat_{B,B^{∗}}(c) we have c̃^{2} = P̃_{c}b = −c̃bc = c̃ which shows that A_{b} ≃ Φ^{+}.

If rank(b) = 4, then some basis of W yields the matrix representation

Mat_{B,B^{∗}}(b) = B := diag \{( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , 0\}

in which case Ker(P_{b}) = all alternating maps x^{+} whose 5 × 5 matrix X = Mat_{B,B^{∗}}(x^{+}) has X_{ij} = X_{55} = 0, 1 ≤ i, j ≤ 4, and

A_{b} ≃ \{( \begin{matrix}αs & a \\ −a^{t} & βs \end{matrix} ) s = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , a ∈ M_{2}(Φ), α, β ∈ Φ\}

≃ A_{4}(Φ) ≃ H_{2}(Q(Φ), Φ, s)

≃ \{ all ( \begin{matrix}α & a \\ \overline{a} & β \end{matrix} )\} = J(q, 1)

for q (( \begin{matrix}α & a \\ \overline{a} & β \end{matrix} )) = det ( \begin{matrix}α & a \\ \overline{a} & β \end{matrix} ) = αβ −a\overline{a} ∈ Φ, \overline{a} = sa^{t}s^{−1} the quaternion

involution, and q is nondegenerate of dimension 1 + 1 + 4 = 6.

Finally, in the case of Clifford Jordan pairs, the local algebras are again Clifford forms by [15, 6.2(ii)]. We have

Proposition 4.4. If V = V(J) for a central-simple Clifford Jordan algebra (a tight scalar form of some simple J(\overline{q}, 1) for a nondegenerate \overline{q} of dimension ≠ 2 over an algebraically closed extension field \overline{Φ} of Φ), then all the local algebras of V are Clifford forms: they have tight scalar extensions \overline{A} = \overline{Φ}A ⊃ A which are either

(i) tiny \overline{Φ}^{+} of dimension 1, (ii) small H_{2}(\overline{Φ}) or H(Q(\overline{Φ}), −) of dimension 3 in characteristic 2 only, (iii) large J(\overline{q}, \overline{1}) of dimension ≥ 3 (actually ≥ 4 in characteristic 2).

Theorem 4.5 (Local Classical Algebras). If V is a classical centralsimple Jordan pair from the Clifford List 3.4, then all its local algebras A_{b} for b ≠ 0 are Clifford forms: they are full Clifford J(q, 1) of dimensions 1, 4 or 6, unless V is itself a simple Clifford form V(J). In all cases they have tight scalar extensions \overline{A} ≃ J(\overline{q}, \overline{1}) of dimension ≠ 2 over the algebraically closed field \overline{Φ} .

PROOF. Let 0 ≠ b ∈ V^{−} for a V from the list. Using Cor. 4.2, if V is M_{1,q}(Φ) of type (Ia) then all A_{b} are Φ^{+} of dimension 1; if V is M_{1,q}(Q) of type (Ib) (Q a quaternion division algebra over Φ) they are all Q^{+} ≃ J(q, 1) of dimension 4; if V is M_{2,q}(Φ) of type (Ic) then all A_{b} are Φ^{+} of dimension 1 or M_{2}(Φ)^{+} ≃ J(q, 1) of dimension 4. Using Prop. 4.3, if V is A_{5}(Φ) the A_{b} are J(q, 1)’s of dimension 1 or 6. Only the case V = V(J) for a simple Clifford form J is messy; if J were a full Clifford Jordan algebra J(q, 1), as in characteristic ≠ 2, we could give an easy direct proof, but in characteristic 2 the matter is more complex, and we have to settle for (perhaps non-full) Clifford forms instead of full Clifford algebras J(q, 1). For this we fall back on [15, Thm. 6.2]. In all cases the local A_{b}’s are at least Clifford forms, hence (by [18, 6.1 p.160], [15, 3.5]) are tight scalar forms of simple J(\overline{q}, \overline{1})’s.

5. Primitive Form Theorem

In this section we will examine simple pairs V that have a tight scalar extension Ṽ = V Φ̃ of classical type. In characteristic ≠ 2 one usually takes ordinary tensor products Ṽ = V ⊗_{Φ} Φ̃; when the centroid Φ contains 1/2 then V is outer-simple, and the usual argument for associative algebras (using the Jacobson Density Theorem for V as an irreducible M(V)-module, M the multiplication algebra of V) shows Ṽ is again (outer-) simple over Φ̃. The directness of the imbedding of V in V ⊗_{Φ} Φ̃ makes it fairly easy to show that if Ṽ is of a certain type over Φ̃, then V is of the same type over Φ. This argument breaks down in characteristic 2: V need not be outer-simple, and even worse, tensor products V ⊗_{Φ} Φ̃ over the centroid need not remain simple. So we replace free scalar extensions with tight ones (1.15) (any scalar extension, e.g. the free V ⊗_{Φ} Φ̃, can be tightened by dividing out by an ideal Ĩ maximal with respect to Ĩ∩ V = 0).

We begin by deriving a series of lemmas concerned with how scalars ω̃ ∈ Φ̃ in a pair Ṽ over Φ̃ interact with a central simple Φ-subpair V.

Lemma 5.1. If x (resp. x^{ε} ∈ V^{ε}) is any (resp. homogeneous) element of a Jordan triple system or algebra J (resp. Jordan pair V) then I_{x} = M(J)P_{x}J (resp. I_{x^{ε}} = M(V)Q_{x^{ε}}V^{−ε}) is an ideal in J (resp. V) while I_{x}^{∘} = M(J)x (resp. I_{x^{ε}}^{∘} = M(V)x^{ε}) is an outer ideal; if J (resp. V) is simple (hence nondegenerate) then I_{x} = J for all x ≠ 0 (resp. I_{x^{ε}} = V for all x^{ε} ≠ 0), and if J (resp. V) is outer-simple (= simple when 1/2 ∈ Φ) then I_{x}^{∘} = J for all x ≠ 0 (resp. I_{x^{ε}}^{∘} = V for all x^{ε} ≠ 0).

PROOF. Clearly I_{x}^{∘} (resp. I_{x^{ε}}^{∘}) is a (resp. polarized) outer ideal since this just means invariant under M; if the system is outer-simple and x ≠ 0 this outer ideal must be the whole system. The harder part concerns the quadratic expression for I_{x} (note that we aren’t trying to find an expression for the ideal generated by x; that would be MP_{x}J + Mx [in the algebra case we would have to add Mx^{2} as well]). Since nonzero elements in a simple system can’t be absolute zero-divisors, the (resp. polarized) space I_{x} is nonzero for x ≠ 0; once it is an ideal it will be all of J (resp. V) in the simple case.

I_{x} is by construction an outer ideal spanned by elements MP_{x}y for structural M’s (products of Id, P_{z}, B_{z,w} which in the pair case can be taken as Q_{z^{ε}}, B_{z^{ε},w^{−ε}}, and in the algebra case B_{z,1} as well). Once P_{c}J ⊂ I_{x} for a spanning set of c’s for the outer ideal I_{x}, I_{x} is an inner ideal as well, so we only need to prove innerness P_{c}a ∈ I_{x} for a ∈ J (resp. a ∈ Ĵ in the algebra case), c = MP_{x}y ∈ I_{x} (resp. c ∈ I_{x^{ε}}^{η}, a ∈ V^{−η} in the pair case). But by structurality of M we have P_{c}a = MP_{x}P_{y}P_{x}M^{∗}a ∈ MP_{x}J (even if a ∈ Ĵ in the algebra case).

This will allow us to show that if ω̃x lies in a simple system J (for ω̃ a scalar acting on some larger system J̃ ⊃ J over Φ̃ ⊃ Φ) then ω̃^{2} lies in the centroid Φ: P_{x} absorbs ω̃^{2} (but not ω̃) to produce a multiplication in J, ω̃^{2}P_{x} = P_{ω̃x} ∈ M(J). It is clear that if J ⊂ J̃ and T̃ ∈ Γ(J̃) leaves J invariant, then the restriction T = T̃ |_{J} belongs to Γ(J).

Lemma 5.2. Let J be a simple Jordan triple system (resp. algebra, pair) over its centroid Φ, and assume J ⊂ J̃ for a system over a field Φ̃ ⊃ Φ.

(i) If some ω̃ ∈ Φ̃ multiplies a single (resp. homogeneous) element of J back into J, then ω̃^{2} ∈ Φ (or: ω̃ ∈ Φ if J is outer-simple):

0 ≠ ω̃x ∈ J for x ∈ J (resp. x^{ε} ∈ V^{ε}) ⟹ ω̃^{2} ∈ Φ (or: ω̃ ∈ Φ).

(ii) If some ω̃ ∈ Φ̃ multiplies a local element of A_{y} back into A_{y} for y ∈ J (resp. y ∈ V^{−}) then ω̃^{2} ∈ Φ (or: ω̃ ∈ Φ if J is outer-simple):

0 ≠ ω̃a = b ∈ A_{y} for a ∈ A_{y} ⟹ ω̃^{2} ∈ Φ (or: ω̃ ∈ Φ).

(iii) Scalar-regularity breeds regularity: if some x ∈ J (resp. x^{ε} ∈ V^{ε}) is scalar-regular via z ∈ J (resp. z^{−} ∈ V^{−}), or some a in a local algebra A_{y} for y ∈ J (resp. y^{−} ∈ V^{−}) is scalar-regular via b ∈ A_{y}, then that scalar multiple is regular in J or A_{y},

P_{x}z = ω̃x ≠ 0 (resp. z^{−} ∈ V^{−}) ⟹ ω̃x regular in J, or
U_{a}b = ω̃a ≠ 0 ⟹ ω̃a regular in A_{y}.

(iv) If x ∈ J has 0 ≠ P_{x}^{n} ∈ Γ(J̃) for some n ≥ 1, then x is invertible (hence regular) in J.

(v) If J is radical then neither J nor any of its local algebras A_{y} can have scalar-regular elements.

PROOF. For (i), it suffices to prove ω̃^{2}J ⊂ J (or: ω̃J ⊂ J). But by (5.1 (1)) J = MP_{x}J so ω̃^{2}J = MP_{ω̃x}J ⊂ J (or: J = Mx, so ω̃J = M(ω̃x) ⊂ J).

For (ii), let us first recall Ker_{J}(y) = Ker(P_{y}) by (1.12.2). Now, if a = \overline{x}, b = \overline{z} for x, z ∈ J (resp. x, z ∈ V^{+}) and ω̃a = b then ω̃x = z + k̃ (z ∈ J, k̃ ∈ Ker(P̃_{y}) in J̃). Therefore ω̃P̃_{y}x = P_{y}z + 0, so ω̃(P_{y}x) = P_{y}z ≠ 0 is in J (resp. V^{−}) [if P_{y}z = 0 then z ∈ Ker(P_{y}), b = \overline{z} = \overline{0} in A_{y}, whereas by hypothesis b ≠ 0]. Thus by (i), ω̃^{2} ∈ Φ (or ω̃ ∈ Φ in the outer-simple case).

For (iii), from P_{x}z = ω̃x ≠ 0 we get 0 ≠ ω̃^{2} = α ∈ Φ by (i), where α is invertible in Φ, since Φ is a field by simplicity of J. Thus P_{ω̃x}J = ω̃^{2}P_{x}J = P_{x}J (αJ = J) ∋ P_{x}z = ω̃x and ω̃x is regular. Similarly 0 ≠ ω̃a = U_{a}b ∈ A = A_{y} ⇒ ω̃^{2} = α ∈ Φ (by (ii)) ⇒ U_{ω̃a}A = ω̃^{2}U_{a}A = U_{a}(αA) = U_{a}(A) ∋ U_{a}b = ω̃a so ω̃a is regular in A_{y}.

For (iv) 0 ≠ P_{x}^{n} ∈ Γ(J̃) ∩ End(J) ⊂ Γ(J) = Φ a field implies P_{x}^{n} (so also P_{x} and x) invertible on J.

Finally, (v) follows from (iii): if J is radical, so is any homotope J^{(b)} and any local algebra A_{b}, and radical systems can’t contain nonzero regular elements.

Note in (iv) that if J is a polarized triple and x = (x^{+}, x^{−}), then odd

powers of P_{x} = ( \begin{matrix}0 & P_{x^{+}} \\ P_{x^{−}} & 0 \end{matrix} ) are never central, but powers of P_{x}P_{x} =

( \begin{matrix}P_{x^{+}}P_{x^{−}} & 0 \\ 0 & P_{x^{−}}P_{x^{+}} \end{matrix} ) will be central if P_{x^{ε}}P_{x^{−ε}} = γ̃ Id_{V^{ε}} for some γ̃ ∈ Γ̃.

It will be important in (5.5) that we allow P_{x}^{n} to be in Γ(J̃) (which may not be a field), not just in the field Φ̃.

Our ultimate goal is to show that central simple forms of classical pairs of Clifford type are actually primitive. The next three lemmas aim at showing that such forms (along with their local algebras) contain regular elements and therefore cannot be radical (in the sense of Jacobson). By (5.2 (iii)), it will be enough to establish the existence of scalar-regular elements.

Lemma 5.3. If A is a nondegenerate Jordan Φ-subalgebra of a Clifford Jordan algebra Ã = J(q̃, ˜1) of a quadratic form q̃ over a field Φ̃ ⊃ Φ, an element a ≠ 0 is scalar-regular in A (i.e. some 0 ≠ ω̃a ∈ U_{a}A) if and only if either (i) q̃(a) = 0, or (ii) q̃(a) ≠ 0 and \overline{a} = μ̃^{−1}b for some μ̃ ∈ Φ̃, b ∈ A (in which case 0 ≠

U_{a}b^{2} ∈ A ∩ Φ̃1). In particular, (iii) a ≠ 0 is scalar-regular if it is symmetric or skew (\overline{a} = ±a, e.g. if

t̃(a) = 0). (iv) A will have scalar-regular elements unless q̃, t̃ are anisotropic on it, it

is asymmetric, and all its inner derivations vanish, D_{A,A} = 0.

PROOF. (i) If q̃(a) = 0 then U_{a}A = q̃(a, \overline{A})a ⊂ Φ̃a; by nondegeneracy U_{a}A ≠ 0 (unless a = 0, which is trivially regular) so q̃(a, \overline{b}) = ω̃ ≠ 0 for some b ∈ A and some ω̃ ∈ Φ̃, and U_{a}b = ω̃a ≠ 0.

(ii) If q̃(a) ≠ 0 then U_{a}b = q̃(a, \overline{b})a − q̃(a)\overline{b} is of the form ω̃a iff (q̃(a, \overline{b}) − ω̃)a = q̃(a)\overline{b} iff \overline{b} = μ̃a ∈ \overline{A} for some μ̃ (since q̃(a) is invertible in Φ̃) iff b = μ̃\overline{a} ∈ A. Here ω̃a ≠ 0 iff μ̃ ≠ 0: μ̃ = 0 ⟹ b = 0 ⟹ ω̃a = U_{a}b = 0 and ω̃a = 0 ⟹ 0 = U_{a}b = μ̃U_{a}\overline{a} = μ̃q̃(a) ⟹ μ̃ = 0. Thus U_{a}b = ω̃a ≠ 0 iff \overline{a} = μ̃^{−1}b for some μ̃ ≠ 0 in Φ̃. When b = μ̃\overline{a} we have U_{a}b^{2} = μ̃^{2}U_{a}\overline{a}^{2} = μ̃^{2}q̃(a)^{2}1 ∈ Φ̃1 nonzero by μ̃, q̃(a) ≠ 0.

(iii) If a is symmetric or skew then a is scalar-regular automatically by (i) if q̃(a) = 0, and by (ii) if \overline{a} = ±a where q̃(a) ≠ 0.

Finally, (iv) follows from (i), (iii) since inner derivations produce skew elements.

So far A was an arbitrary nondegenerate subalgebra. In the case we are interested in, A is a tight scalar form of a Jordan algebra of a nondegenerate quadratic form (in which case A is automatically nondegenerate).

Lemma 5.4. Let A be a tight scalar form (AΦ̃ = Ã) of a Clifford Jordan algebra Ã = J(q̃, ˜1) for a nondegenerate quadratic form q̃ over a field Φ̃ ⊃ Φ. Then A contains scalar-regular elements unless Ã is 2-dimensional over Φ̃ with q̃, t̃ anisotropic on A.

PROOF. Assume A contains no scalar-regular elements. Since A is nondegenerate, we can apply (5.3 (iv)) to conclude q̃, t̃ are anisotropic on A, and A is asymmetric without inner derivations. Then D_{A,A}A = 0 ⟹ D_{Ã,Ã}Ã = 0 too since Ã is a scalar extension of A.

Now t̃anisotropic certainly implies t̃≠ 0; but for a nondegenerate traceful form this forces dimΦ̃(Ã) ≤ 2. Indeed, if dimΦ̃(Ã) were ≥ 3 then for any x̃ ∈ Ã, Ã ⊄ Φ̃1 + Φ̃x̃, yet by [15, (2.2.1)] D_{x̃,Ã}x̃ = 0 ⟹ [4q̃(x̃) − t̃(x̃)^{2}]Ã ⊂ Φ̃1 + Φ̃x̃, so for all x̃ we would have

4q̃(x̃) − t̃(x̃)^{2} = 0.

In characteristic 2 any x ≠ 0 in A has 4q̃(x)−t̃(x)^{2} = t̃(x)^{2} ≠ 0 by anisotropy of the trace on A. In characteristic ≠ 2 (assuming the dimension is at least 3) by nondegeneracy of q̃ we can find x̃ ∈ Ã with t̃(x̃) = 0 ≠ q̃(x̃), again leading to a contradiction 4q̃(x) − t̃(x)^{2} = 4q̃(x) ≠ 0. Thus the assumption that the dimension is 3 or more is untenable.

On the other hand, if dimΦ̃(Ã) = 1 then A ⊂ Ã = Φ̃1 is entirely symmetric, contrary to asymmetry. Thus Ã can only be 2-dimensional over Φ̃.

It is a bedevilingly subtle point that while we have scalar-regularity in A, we cannot expect regularity just at the level A ⊂ Ã (we will give two examples to emphasize this point): it requires a background level J ⊂ J̃ with simplicity of J. Only in this situation do we know by (5.2) that scalar-regularity over Φ̃ breeds regularity in A. If Ã has dimension 2 then it is a tiny commutative algebra Ω̃, and its tight algebraic closure splits into \overline{A} = \overline{Φ}e⊕\overline{Φ}f. If we knew that A = A_{b} was simple or central over Φ then we would have A ↪ \overline{Φ}e ≃ \overline{Φ}^{+} where everything is scalar-regular over \overline{Φ}. But this scalar-regularity cannot be carried back to Ã over Φ̃.

First we give an example of A ⊂ Ã ≃ Ω̃^{+} a 2-dimensional field extension of Φ̃ where A has no Φ̃-regular elements at all.

Example 5.4.1. Let Ω̃ be a quadratic extension field of Φ̃ and a ∈ Ω̃ with t = t̃(a), q = q̃(a) algebraically independent over the field Φ. [For example, take Ω̃, Φ̃, Φ to be the complexes, reals and rationals, a = τ + iσ with real τ, σ algebraically independent over the rationals, so t = 2τ, q = τ^{2} + σ^{2} are also algebraically independent over the rationals.] Then A = Φ[a]_{0} = ∑_{i=1}^{∞} Φa^{i} is an infinite-dimensional domain over Φ which is a scalar form of Φ̃A = Ã = Ω̃ a field, yet A contains no nonzero Φ̃-scalar-regular elements or quasi-invertible elements at all.

PROOF SKETCH. Note that a is not a scalar ρ in Φ̃1, since then t = 2ρ, q = ρ^{2} would be algebraically dependent. An easy induction from a^{2} = ta−q1 shows a^{n} = t_{n−1}a − q_{n}1 where the {t_{n}}_{n≥0} are congruent to t^{n} mod q, so are linearly independent over Φ, hence the {a^{n}}_{n≥1} are linearly independent over Φ modulo Φ̃1; by usual results on polynomials with zero constant terms, A has no scalar-invertible or quasi-invertible elements (b, c ≠ 0 with bc = α̃1 or bc = b + c).

The above example is semiprimitive. We can construct a totally radical example inside a split Ã.

Example 5.4.2. Let Ã = Φ̃e_{1} ⊕ Φ̃e_{2} be a split J(q̃, ˜1) of dimension 2 over a field Φ̃, and let Φ_{1} ⊂ Φ̃ be a radical Jordan Φ-subalgebra with nontrivial Φlinear Jordan monomorphism Φ_{1} ⟶^{σ} Φ̃. Then A = {a(α) := αe_{1} ⊕ σ(α)e_{2} | α ∈ Φ_{1}} is a form of Ã = AΦ̃ which is a radical domain A = Rad(A) (hence without regular elements) isomorphic to Φ_{1}^{+} via a(α) ⟶ α, and q̃ is anisotropic on A. If in addition σ has no nonzero fixed points (so Φ_{1} cannot be unital), then A has no Φ̃-regular elements either (since the centroid of A need not be a field, A can have Φ̃-regular elements which are not regular). [For example, let Φ̃ = Φ((t, s)) be the Laurent series in 2 independent variables, Φ_{1} = Φ[[t]]_{0} ⊂ Φ̃ the power series with zero constant term in the single variable t, and σ(f(t)) = f(s) induced from the automorphism of Φ̃ which switches s and t.]

PROOF SKETCH. Nontriviality of σ guarantees e_{2} ∈ AΦ̃ (hence AΦ̃ = Ã) since σ(α) − α ≠ 0 ⟹ 0 ≠ (σ(α) − α)σ(α)e_{2} = a(α^{2}) − αa(α) ∈ AΦ̃. Absence of fixed points of σ guarantees A ∩ Φ̃1 = 0, and hence by (5.3) (ii) A has no nonzero Φ̃-regular elements.

In fact the dimension 2 situation is not a problem at the level of V.

Remark 5.5 (Dimension 2). Let V be a simple Jordan pair over its centroid Φ with tight scalar extension Ṽ = V Φ̃ ≃ V(J(q̃, ˜1)) for q̃ a nondegenerate quadratic form of dimension 2 over Φ̃, such that q̃ and t̃ are anisotropic on V. Then V has regular (even invertible) elements.

PROOF. A 2-dimensional J(q̃, ˜1) with trivial involution is traceless of characteristic 2, but our J̃ is not traceless because by hypothesis t̃ is anisotropic on V. Thus our 2-dimensional J̃ = J(q̃, ˜1) has nontrivial involution, so is a form of a commutative split J(\overline{q}, \overline{1}) = \overline{Φ}e_{1} ⊕ \overline{Φ}e_{2} ≃ \overline{Φ}^{+} ⊕ \overline{Φ}^{+} so all Ux̃ lie in the centroid. Then for all x^{+}, x^{−} ≠ 0 in V the element x = (x^{+}, x^{−}) ∈ T (V) has U_{x^{ε}} invertible on J̃ (by hypothesis q̃ is anisotropic on V) and hence Q_{x}Q_{x} = (U_{x^{+}}U_{x^{−}}, U_{x^{−}}U_{x^{+}}) ≠ 0 in Γ(Ṽ) so by (5.2)(iv) x is regular (even invertible) in V.

We are now ready to put the pieces together; our main result is

Primitive Form Theorem 5.6. If V is a simple Jordan pair over its centroid Φ which is a scalar form of some classical pair Ṽ of Clifford type over its centroid, a field Φ̃ ⊃ Φ, then V is primitive.

PROOF. Assume Ṽ = Φ̃V is classical (in particular, central-simple) over Φ̃ of Types (Ia-c), (II), (III) in (3.4). It suffices to prove that Rad(V) ≠ V [so that by simplicity, Rad(V) = 0 and V (being semiprimitive and simple) is in fact primitive]. Assume to the contrary that Rad(V) = V is entirely radical. Then by (5.2) (v) no local algebra A_{b} = V_{b^{−}}^{+} of V can have scalarregular elements in any extension \overline{Φ}. On the other hand, the local algebras A = A_{b} are tight scalar forms of the local algebras Ã = (Ã)_{b} of a classical Ṽ and by the Local Classical Algebra Theorem 4.5 we know these Ã are tight scalar forms of \overline{A} = J(\overline{q}, \overline{1}) simple of dimension ≠ 2 over an algebraically closed field \overline{Φ}, so A too is a tight scalar form of J(\overline{q}, \overline{1}). Then by 5.4 A has scalar-regular elements in the extension \overline{Φ}, a contradiction.

6. Zel’manov polynomial on classical pairs

We now address the question of whether each classical pair of List 3.4 is necessarily of Clifford type by showing that any such pair satisfies the Zel’manov polynomial constructed in [4]. We will avoid dealing with superscripts ± as much as possible and carry out most calculations in a triple system setting.

We denote by ST = ST (X) the free special Jordan triple system on an infinite set X of indeterminates, and we fix b ∈ X. Let [x, y]^{(b)} := xby − ybx denote a commutator in the b -homotope A(X)^{(b)} of the free associative algebra A(X). We first use an inner derivation

(6.1)  D_{x,y}(z) := (V_{x,y}^{(b)} − V_{y,x}^{(b)})z = [[x, y]^{(b)}, z]^{(b)}

of ST^{(b)} to construct a tetrad-eater a in the algebra ST^{(b)} [18] which is then combined to a composite element c into yet another b-commutator (note that although commutators are not Jordan elements, their P -operators are Jordan):

a_{31} = a := [[D_{x,y}^{2}(z_{1})^{(2,b)}, D_{x,y}(z_{2})]^{(b)}, D_{x,y}(z_{3})]^{(b)},
c_{69} = c := [L_{a,z_{4}}^{(b)}, L_{a,z_{5}}^{(b)}]z_{6}, (6.2)
d_{101} = d := [a, c]^{(b)},
m_{613} = m := P_{P (d)P (b)d}^{(b)}z_{7},
z_{1841} = z := m^{(3,b)}

where x, y, z_{i} are 9 distinct elements of X. Then z ≠ 0 is a Zel’manov polynomial, i.e. a hearty pentad-eater which is also a Clifford homotope PI [4].

We can give a general argument why the pairs on the Clifford List satisfy the Zel’manov polynomial.

Theorem 6.3. (i) All Classical Jordan Pairs on the Clifford List 3.4 (as well as the BiCayley Pair) satisfy c ≡ 0 (hence z ≡ 0).

(ii) Indeed, whenever all local algebras of a Jordan system J are forms of quadratic form algebras, then J satisfies m ≡ 0 (hence the Zel’manov polynomial z ≡ 0 too).

(iii) More generally, if a_{0}(X) is any identity in the local algebra at b,

\overline{a_{0}^{(b)}(X)} ≡ \overline{0} in A_{b}, then

for any c_{0} ∈ U_{a_{0}}^{(b)}(J) (e.g. c_{0} = V_{a_{0},z_{4}}^{(b)} V_{a_{0},z_{5}}^{(b)}z_{6} or (6.3.1)

[V_{a_{0},z_{4}}^{(b)}, V_{a_{0},z_{5}}^{(b)}]z_{6}, since V_{a_{0},J}^{(b)}V_{a_{0},J}^{(b)}J ⊂ U_{a_{0}}^{(b)}J) we have
m_{0} := P_{d_{0}}t = 0, z_{0} := m_{0}^{(3,b)} = 0 for any s, t (d_{0} = [s, c_{0}]^{(b)});

if P_{a_{0}}P_{b}J = 0 (e.g. if b = e^{−} for a maximal idempotent e (6.3.2)

in a pair J = V, V_{0}(e) = 0) then already c_{0} = 0 for any
c_{0} ∈ V_{a_{0},J}^{(b)}V_{a_{0},J}^{(b)}J.

PROOF. (ii) implies z = 0 in (i) since all local algebras A_{b} are Clifford forms by the Local Classical Algebra Theorem 4.5. Since squares of commutators lie in the center in Clifford forms, the Clifford identity

a_{0} := a_{15}(x, y, z_{1}, z_{2}, z_{3}; b) = [[D_{x,y}(z_{1})^{(2,b)}, z_{2}]^{(b)}, z_{3}]^{(b)}

(hence also a_{31}(x, y, z_{1}, z_{2}, z_{3}; b) = a_{15}(x, y, D_{x,y}(z_{1}), D_{x,y}(z_{2}), D_{x,y}(z_{3}); b), or indeed any Clifford identity a_{0}^{(b)}) vanishes in each A_{b}. Thus (iii) in each A_{b} implies (ii), so we concentrate on (iii).

For (iii), we start from the vanishing of a_{0} in A_{b} = J^{(b)}/Ker_{J}(b):

a_{0} ∈ Ker_{J}(b) : P_{b}a_{0} = P_{b}P_{a_{0}}b = 0.

These orthogonalities imply others:

P_{b}a = 0 ⟹ V_{J,a}^{(b)} = V_{J,U_{a}^{(b)}J}^{(b)} = 0, (6.3.3)
P_{b}P_{a}b = 0 ⟹ V_{J,a^{(2,b)}}^{(b)} = 0, (6.3.4)
P_{b}a = P_{b}P_{a}b = 0 ⟹ V_{U_{a}^{(b)}J}^{(b)} = 0, (6.3.5)
P_{b}a = P_{b}P_{a}b = 0 ⟹ V_{U_{a}^{(b)}J,J}^{(b)} = 0. (6.3.6)

For (3), note V_{x,a}^{(b)} = L_{x,P_{b}a} = 0 and V_{x,U_{a}^{(b)}y}^{(b)} = L_{x,P_{b}(P_{a}P_{b}y)} = L_{x,P_{P_{b}a}y} = 0.

For (4), V_{x,a^{(2,b)}}^{(b)} = L_{x,P_{b}(P_{a}b)} = 0. For (5), linearizing x ⟶ x, 1 in V_{x,U_{a}x} = V_{U_{x}a,a} in the unital hull of the Jordan algebra J^{(b)} gives V_{U_{a}x} + V_{x,a^{2}} = V_{a∘x,a}; then V_{U_{a}J} ⊂ V_{J,a} + V_{J,a^{2}} = 0 by (3), (4). It is easy to see why (6) holds in special systems: V_{U^{(b)}(a)x,y}^{(b)}(z) = (abxba)(byb) z + z(byb)(abxba) = abx(bab)ybz + zby(bab)xba = 0 using only the condition bab = 0. To

get (6) for general Jordan systems when \frac{1}{2} ∉ Φ we need both parts of the definition of Ker_{J}(b): V_{U_{a}x,y} = −V_{y,U_{a}x} + V_{y∘U_{a}x} where V_{J,U_{a}J} = 0 by (3) and U_{a}x ∘ J = 0 by (5).

Going back to (1), we now can see why V_{a_{0},J}^{(b)}V_{a_{0},J}^{(b)} ⊂ U_{a_{0}}^{(b)}J: (6) shows that in the unital hull of J^{(b)} we have V_{a,x}V_{a,y} = V_{U(a)x,y} + U_{a}U_{x,y} = 0 + U_{a}U_{x,y} ∈ U_{a}U_{J}. Now

c ∈ U_{a}^{(b)}J = P_{a}P_{b}J (P_{b}a = P_{b}P_{a}b = 0) (6.3.7)
⟹ L_{c,b} = P_{c}P_{b} = P_{b}P_{c} = 0

since L_{c,b} ∈ L_{P (a)P (b)J,b} = V_{U^{(b)}(a)J}^{(b)} = 0 by (5), and P_{c} ∈ P_{a}P_{b}P_{J}P_{b}P_{a} is killed on either side by P_{b} since P_{b}P_{a}P_{b} = P_{P_{b}a} = 0.

In the special case, c ∈ abJba implies bc = cb = 0, hence d_{0} = [s, c]^{(b)} = sbc − cbs = 0 and so P_{d_{0}} = 0. Again this holds in the general case: P_{d_{0}} = P_{[s,c]^{(b)}} := P_{{s,b,c}} − 2(P_{s}P_{b}P_{c} + P_{c}P_{b}P_{s}) = 0 by (7). Therefore any m_{0} = P_{d_{0}}t = 0, z_{0} = m_{0}^{(3,b)} = 0 and (1) is established.

For (2), note that if U_{a_{0}}^{(b)} = P_{a_{0}}P_{b} = 0 then any c_{0} ∈ V_{a_{0},J}^{(b)}V_{a_{0},J}^{(b)}J ⊂

V_{U_{a_{0}}^{(b)}J,J}^{(b)} + U_{a_{0}}^{(b)}U_{J,J}^{(b)} = 0, and the Zel’manov polynomial already vanishes at

the second stage. If b = e^{−} for an idempotent e = (e^{+}, e^{−}) in V, then Ker_{J}(b) = V_{1}^{+}(e) + V_{0}^{+}(e), and when V_{0}(e) = 0 this reduces to Ker_{J}(b) = V_{1}^{+}; then a ∈ Ker_{J}(b) ⟹ a = a_{1}^{+} so that U_{a}^{(b)}(V^{+}) = Q_{a_{1}^{+}}Q_{e^{−}}(V_{2}^{+}) ⊂

Q_{V_{1}}Q_{V_{2}}V_{2} ⊂ V_{0} = 0 and U_{a}^{(b)} = 0.

This shows z ≡ 0 in (i). But (i) claims that the Zel’manov polynomial already vanishes at the second stage c_{0} = 0 for all the systems on the Clifford list. By (6.3.2) it suffices to show that Q_{a_{0}}Q_{b} = 0 in each of these for a dense set of b.

I. Rectangular Pairs: c_{0} ≡ 0 since U_{a_{0}}^{(b)} = 0. Let V be the pair M_{1,q}(Φ) (resp. M_{1,q}(Q), M_{2,q}(Φ)), let R denote the associative algebra Φ (resp. Q, M_{2}(Φ)) and set J := V^{+(b)} for some b ∈ V^{−} with the operations

(6.4)  x^{(2,b)} = xbx, U_{x}^{(b)}y = xbybx, V_{x,y}^{(b)}z = xbybz + zbybx.

J is naturally a left R-module (scalar multiplication is the usual matrix product) and J ⟶^{T} R defined by T (x) = xb is clearly an R-module homomorphism. In those terms, (6.4) becomes

x^{(2,b)} = T (x)x, U_{x}^{(b)}y = T (x)T (y)x, (6.5)
V_{x,y}^{(b)}z = T (x)T (y)z + T (z)T (y)x.

We obtain an associative algebra (the “b-homotope” of the associative pair V) by setting A := J with product xy = T (x)y = xby, in which case J = A^{J} (as in (1.3.1)) and A^{J} ⟶^{T} R^{J} is a Jordan homomorphism: T (U_{x}^{(b)}y) = T (T (x)T (y)x) = T (x)T (y)T (x) = U_{T (x)}T (y) and T (x^{(2,b)}) = T (T (x)x) = T (x)^{2}. Thus applying T to an inner derivation D_{x,y}(z), we obtain T (D_{x,y}(z)) = D_{T (x),T (y)}T (z) = [[T (x), T (y)], T (z)], and so by (6.5)

(6.6)  T (D_{x,y}(z)^{(2,b)}) = [[T (x), T (y)], T (z)]^{2} ∈ Φ1

which in turn implies (recall (6.2))

(6.7)  T (a_{0}) = [[T (D_{x,y}(z_{1})^{(2,b)}), T (z_{2})], T (z_{3})] = 0,

so U_{a_{0}}^{(b)}y = T (a_{0})T (y)a_{0} (by (6.5)) = 0 (by (6.7)), and we have U_{a_{0}}^{(b)} ≡ 0.

Remark 6.8. In the case V = M_{1,q}(Φ), the commutativity of R = Φ makes it easy to see that a_{0} already vanishes: here, D_{x,y}(z) = T (z)(T (y)x − T (x)y) so T (D_{x,y}(z)) = T (z)[T (y), T (x)] = 0 by commutativity and D_{x,y}^{2}(z) = T (D_{x,y}(z))(T (y)x − T (x)y) = 0 so D_{J,J}^{2} = 0 and a_{0} = 0.

II. Alternating Pair: c_{0} ≡ 0 since U_{a_{0}}^{(b)} = 0. Let V = V(A_{5}(Φ)) with product Q_{x}y = −xyx; let b ∈ A_{5}(Φ) and set J = A_{5}(Φ)^{(b)} with product U_{x}y = Q_{x}Q_{b}y. We will denote Z := Ker_{J}(b).

The element b^{−} ∈ A_{5}(Φ) has (matrix) rank 0, 2 or 4. By a Zariski density argument (rank is a polynomial function on A_{5}(Φ)) it is enough to show that U_{a_{0}}^{(b)} vanishes in J for b of maximal rank 4. Once again, as in (1.8.3) we identify V with alternating maps between dual 5-dimensional Φ-spaces W, W^{∗}. We recall from the rank 4 case of Proposition 4.3 that we may assume b = E_{12}−E_{21}+E_{34}−E_{43} and Z = {X ∈ A_{5}(Φ) | X_{ij} = X_{55} = 0, 1 ≤ i, j ≤ 4} from which we easily see that

(6.9)  U_{a_{0}}^{(b)}J ⊂ J ∩ ΦE_{55} = 0 for all a_{0} ∈ Z.

Remark 6.10. If rank(b) = 2, we actually get a_{0} ≡ 0 on J. As in Prop. 4.3, A_{b} := J/Z ≃ Φ so we have D_{x,y}(z) = 0 in A_{b}, hence d := D_{x,y}(z) ∈ Z. But then in the special system V, [(D_{x,y}(z))^{(2,b)}, u]^{(b)} = (dbd)bu − ub(dbd) = 0 since bdb ∈ P_{b}Z = 0; thus a_{0} ≡ 0.

III. Clifford Pairs: a_{0} ≡ 0. First suppose V = V(J(q)) is full Clifford with product Q_{x}y = q(x, y)x − q(x)y. Set J := J(q)^{(b)} and t(x) := q(x, b) (so t(b) = 2q(b)). We first compute expressions for x^{2}, U_{x}y, and {xyz} [15, 1.5-1.6]: they are respectively

x^{(2,b)} = Q_{x}b = q(x, b)x − q(x)b = t(x)x − q(x)b,
U_{x}y = Q_{x}Q_{b}y
= q(x, [q(b, y)b − q(b)y])x − q(x)[q(b, y)b − q(b)y]
= [t(x)t(y) − q(b)q(x, y)]x − t(y)q(x)b + q(x)q(b)y,

and so

\begin{matrix}\{xyz\} = & [t(z)t(y) − q(b)q(z, y)]x + [t(x)t(y) − q(b)q(x, y)]z \\ − t(y)q(x, z)b + q(x, z)q(b)y. \end{matrix}

From those, we easily compute,

D_{x,y}(z) = {xyz} − [ [t(x)t(z) − q(b)q(z, x)]y
+ [t(x)t(y) − q(b)q(x, y)]z − t(x)q(y, z)b + q(b)q(y, z)x ]
= [t(y)t(z) − 2q(b)q(y, z)]x − [t(x)t(z) − 2q(b)q(x, z)]y
+ [t(x)q(y, z) − t(y)q(x, z)]b.

With this last expression, we remark that inner derivations are traceless:

(6.11)  t(D_{x,y}(z)) = q(D_{x,y}(z), b) = 0

since t(D_{x,y}(z)) = [t(x)t(y)t(z) − 2q(b)q(y, z)t(x)] − [t(x)t(y)t(z) − 2 q(b) q(x, z) t(y)] + [2q(b)t(x)q(y, z) − 2q(b)t(y)q(x, z)] = 0 (using t(b) = 2q(b)); from (6.11), we then obtain

(6.12)  D_{x,y}(z)^{(2,b)} = t(D_{x,y}(z))D_{x,y}(z) − q(D_{x,y}(z))b
= −q(D_{x,y}(z))b ∈ Φb.

Next, we observe that

(6.13)  D_{b,w}(u) = 0 :

indeed D_{b,w}(u) = [t(w)t(u) − 2q(b)q(w, u)]b − [t(b)t(u) − 2q(b)q(b, u)]w + [t(b)q(w, u) − t(w)q(b, u)]b = 0. Combining (6.12) and (6.13), we get

D_{(D_{x,y}(z))^{(2,b)},w}u = D_{−q(D_{x,y}(z))b,w}u ∈ ΦD_{b,w}u = 0;

in particular, a_{0} vanishes. Since any Clifford V(J) of 3.4 (III) is a tight scalar form of V(J(\overline{q})), we see a_{0} vanishes on V(J(\overline{q})) and hence on the subpair V(J).

6.14 BiCayley Pairs: c_{0} ≡ 0 since U_{a_{0}}^{(b)} = 0. Our Zel’manov polynomial cannot possibly vanish on an Albert pair (Type (VI)) [since an Albert algebra becomes H_{3}(C, Ω) in some scalar extension]. Although we have included ispeciality in our definition of pair of Clifford type, the BiCayley pairs (Type (V)) also satisfy c_{0} ≡ 0 and hence z ≡ 0†.

Here V = V(J) for the Jordan triple J = M_{1,2}(K) = [K, K], for a Cayley algebra K over Φ, under P_{[a,b]}[c, d] = [a(\overline{c}a) + b(\overline{d}a), a(\overline{c}b) + b(\overline{d}b)]. It suffices to prove c_{0} ≡ 0 for the Zariski-dense set of b ∈ V^{−} of maximal rank dim_{Φ}(Q_{b}V^{+}) = 8 and by Petersson’s Conjugacy Theorem any such b is conjugate to b = [1, 0]. When b = [1, 0] we have Ker(b) = [0, K], A_{b} ≃ K^{+} = J(q, 1) for an 8-dimensional quadratic form, and U_{a_{0}}^{(b)}x = P_{a_{0}}P_{b}x ∈ P_{[0,K]}P_{[1,0]}[K, K] = P_{[0,K]}[\overline{K}, 0] = [0, 0].

7. Primitive pairs

In this section and the next, we will show that for Jordan pairs of Clifford type, primitivity and simplicity are equivalent. Our first goal is to prove that a primitive pair V with a Clifford homotope PI must be simple (recall it was established in [2] that primitive quadratic Jordan PI algebras are simple and unital). The argument revolves around Loos’ theory of the socle of Jordan pairs [10]. We begin with a general result about the socle of a strongly prime system (algebra, triple or pair). In what follows, the capacity of a Jordan triple system J is defined as the capacity of the corresponding Jordan pair V(J) = (J, J).

Proposition 7.1. (1) If J is a nondegenerate Jordan system (algebra, triple or pair) and I an ideal with a covering tripotent (I ⊂ J_{2}(e)+J_{1}(e) for e ∈ I) then J = I \overline{\underline{+}} J_{0}(e); if in addition J is prime and I ≠ 0, then J = I.

(2) If J is strongly prime and Soc(J) ≠ 0 is a nonzero system of finite capacity, then J = Soc(J) is simple.

PROOF. In (1), it suffices to prove the first assertion. By hypothesis I = I_{2} ⊕ I_{1} ⊂ J_{2} ⊕ J_{1} ⊕ J_{0} = J, and e ∈ I ◁ J implies I ⊃ J_{2} + J_{1}, so I = J_{2} ⊕ J_{1} and J = I ⊕ J_{0} is a module direct sum. We must prove that it is a direct sum of systems

† We would like to thank Robert Korty, an REU at the University of Virginia, for using Maple to reassure us computationally about our arguments.

P_{I}J_{0} = P_{J_{0}}I = {IIJ_{0}} = {IJ_{0}J_{0}} = 0 (7.1.1)

( for algebras we also need I ∘ J_{0} = 0).

In the prime nondegenerate case (which is what we need in this paper) we easily get J_{0} = 0: I ≠ 0 ⇒ 0 = Ann_{J}(I) (by primeness) = {z ∈ J | P_{z}I = 0} (by nondegeneracy, [16, 1.7(i)]), and already P_{J_{0}}I = 0 by the Peirce relations.

The result J = I \overline{\underline{+}} J_{0} still holds in the general nondegenerate case, but we have to look more carefully at the Peirce relations: from P_{J_{1}}J_{2} + {J_{1}J_{1}J_{0}} ⊂ I ∩ J_{0} = 0 we get

(7.1.2)  P_{J_{1}}J_{2} = {J_{1}J_{1}J_{0}} = 0,

and by Peirce orthogonality of J_{2} and J_{0} the conditions (7.1.1) reduce to

P_{J_{1}}J_{0} = 0, (7.1.3) {J_{2}J_{1}J_{0}} = 0, (7.1.4) {J_{1}J_{0}J_{0}} = 0, (7.1.5) (7.1.6) J_{1} ∘ J_{0} = 0 (in the algebra case).

(7.1.3) vanishes by nondegeneracy: z_{2} = P_{x_{1}}x_{0} has P_{z_{2}}J = P_{x_{1}}P_{x_{0}}P_{x_{1}}J_{2} = 0 by (7.1.2). From this, (7.1.4) also vanishes by nondegeneracy: z_{1} = {x_{2}x_{1}x_{0}} has P_{z_{1}}J = (P_{x_{2}}[P_{x_{1}}P_{x_{0}}]+P_{x_{0}}[P_{x_{1}}P_{x_{2}}]+L_{x_{0},x_{1}}P_{x_{2}}L_{x_{1},x_{0}} −P_{P (x_{0})P (x_{1})x_{2},x_{2}})J (see [4, 0.7]) ⊂ 0 + 0 + L_{x_{0},x_{1}}P_{x_{2}}L_{x_{1},x_{0}}J − 0 (by (7.1.3); (7.1.2); (7.1.2)) = 0 [since L_{x_{1},x_{0}}J_{2} = 0 by Peirce orthogonality, L_{x_{1},x_{0}}J_{1} = 0 by linearized (7.1.3), and P_{x_{2}}L_{x_{1},x_{0}}J_{0} ⊂ P_{J_{2}}J_{1} = 0]. This has (7.1.5) as consequence: {x_{1}x_{0}y_{0}} = {y_{0}x_{0}{x_{1}ee}} = −{{x_{1}ey_{0}}x_{0}e} + {x_{1}e{y_{0}x_{0}e}} + {y_{0}{ex_{1}x_{0}}e} = −0 + 0 + {y_{0}{ex_{1}x_{0}}e} where {ex_{1}x_{0}} ∈ {J_{2}J_{1}J_{0}} = 0 by (7.1.4). In an algebra, (7.1.6) is always a consequence of (7.1.4) since x_{1} ∘ x_{0} = {ex_{1}x_{0}}.

For (2), we apply (1) to I = Soc(J); by finite capacity there is a maximal idempotent e which covers I, I ∩ J_{0}(e) = 0, so J = I \overline{\underline{+}} J_{0}, and then by primeness and I ≠ 0 we have J_{0} = 0, J = I = Soc(J). Now the socle is always a direct sum of simple ideals [10, theorem 2], so by primeness it must reduce to a single simple ideal.

To get a primitive pair V with Clifford homotope PI to be simple, we first observe that Soc(V) is simple of finite capacity,

then we show that we actually have Soc(V) = V.

Proposition 7.2. If V is strongly prime with a Clifford homotope PI and Soc(V) ≠ 0, then Soc(V) is simple and has finite capacity ≤ 2.

PROOF. If V is strongly prime, so is Soc(V) (see [16]) and hence Soc(V) is simple and inherits the Clifford homotope PI. By Litoff’s theorem for Jordan pairs [10, theorem 3], Soc(V) must have finite capacity ≤ 2.

Our main result is

Theorem 7.3. If V is a primitive pair with Clifford homotope PI, then V = Soc(V) is simple classical of capacity ≤ 2.

PROOF. By (1.20) V has nonzero socle, and by (7.2) (primitivity implies strong primeness) Soc(V) is simple of capacity ≤ 2; hence, by (7.1, (2)) V = Soc(V).

8. Simple pairs

In the previous section, we established that primitive pairs of Clifford type are simple (and classical); we now wish to show that simple pairs are primitive (hence themselves classical) . This is done in two stages: we first argue that simple pairs are forms of classical ones, then we invoke the Primitive Form Theorem. The first part of the argument requires a scalar extension that inherits semiprimitivity, a condition difficult to confront head-on. But nondegeneracy is automatically inherited by tight extensions; therefore we choose our field of scalars so as to shrink the Jacobson radical into the lower radical. The following radical preliminaries explain how this is done.

Theorem 8.1. (1) If A is a Jordan algebra satisfying a polynomial identity, then

Mc(A) = L(A) = SN il(A)

so A is nondegenerate iff it is free of strictly-nil ideals.

(2) If J is a Jordan triple system or pair satisfying a homotope polynomial identity, then

Mc(J) = L(J) = PSN il(J)

so J is nondegenerate iff it is free of psnil ideals.

PROOF. (1) is simply [18, 17.1] (in view of the chain (1.14.1)), and (2) follows by applying (1) to all the PI algebras J^{(y)}: by radical surgery and Mc(J) ⊂ PSN il(J) it suffices to prove J nondegenerate ⇒ J is free of psnil ideals, so assume I ◁ J is psnil, i.e. I remains strictly-nil in all J^{(y)}. By (1) we have I ⊂ Mc(J^{(y)}), hence I ⊂ ∩_{y∈J}Mc(J^{(y)}) = Mc(J) (see [7]) = 0 (by nondegeneracy).

Over an uncountable field, nilness and strict-nilness are the same.

Theorem 8.2. Over an uncountable field Φ,

(1) SN il(A) = N il(A) for any Jordan algebra A,

(2) PSN il(J) = PN il(J) for any Jordan triple or pair J.

PROOF. Again the only thing to prove is the inclusion ⊃ since ⊂ always holds. Also note that (1) ⇒ (2) by applying it to all homotopes A = J^{(y)} (I pnil in J ⇒ I nil in all A = J^{(y)} ⇒ I snil in all A ⇒ I psnil in J). The argument for Jordan algebras is the same as that for associative algebras. Assume A is nil over an uncountable field; to show strict nilness, it suffices to prove each t_{1}a_{1} + … + t_{n}a_{n} ∈ A[t_{1}, …, t_{n}] is nilpotent,

(8.3)  A nil over an uncountable field ⟹ A[t_{1}, …, t_{n}] nil ,

and by induction it suffices to show nilness is preserved under adjunction of one indeterminate,

(8.4)  A nil ⟹ A[t] nil .

So consider any x(t) = a_{0} + a_{1}t + … + a_{n}t^{n} ∈ A[t]. For any λ ∈ Φ, the element x(λ) = a_{0} + a_{1}λ + … + a_{n}λ^{n} ∈ A is nilpotent of some index d(λ), so d : Φ → N is defined and Φ = ∪_{i=1}^{∞}Φ_{i} where Φ_{i} = {λ ∈ Φ | d(λ) = i}. Since Φ is uncountable, one of the countably many Φ_{i} must be infinite (even uncountable).

Assume Φ_{d} is such. Then x(λ)^{d} = 0 for infinitely many values of λ, in particular for more than dn values; x(t)^{d} = ∑_{k=0}^{nd} p_{k,d}(a_{0}, …, a_{n})t^{k} is a polynomial of degree ≤ nd in t, so if it has more than nd roots in Φ it must be identically zero as a polynomial, and x(t) is nilpotent in A[t] of index ≤ d. [If y(t) = ∑_{k=0}^{m} b_{k}t^{k} has y(λ) = 0 for more than m values λ from a field Φ, then y(t) = 0 in A[t]: for any linear functional φ ∈ A^{∗}, φ(g(t)) = ∑_{k=0}^{m} β_{k}t^{k} is an ordinary polynomial in Φ[t] (β_{k} = φ(b_{k})) with more than m roots, so φ(g(t)) ≡ 0 in Φ[t] and all β_{k} = 0. Thus φ(b_{k}) = 0 for all φ ∈ A^{∗} and all k, so b_{k} = 0 for all k, and y(t) = 0.]

The shrinkage of the radical into the nil radical is standard using

Theorem 8.5 (Amitsur’s Resolvent). If A is a Jordan algebra over a field Φ and a ∈ A has large resolvent ρ(a) = {λ | λˆ1 − a invertible in Â} ⊃ {λ ≠ 0 | ˆ1 − λ^{−1}a invertible in Â}

| ρ(a) | > 1 + dim_{Φ}(A),

then a is algebraic over Φ.

Next, we shrink the Jacobson radical into the properly-nil radical. Recall that given a Jordan system J over a field Φ, we say that Φ is large if |Φ| > 2 + dim_{Φ}(J) [8, 4.5.8].

Theorem 8.6. For any Jordan system J (algebra, triple or pair) over a large field,

Rad(J) = PN il(J),

so J is semiprimitive iff it is free of properly-nil ideals.

PROOF. [13, p.3542] We need only prove Rad(J) ⊂ PN il(J): if z ∈ Rad(J) then all λ^{−1}z ∈ Rad(J) are quasi- invertible in all J^{(y)}, so ˆ1 − λ^{−1}z is invertible in Ĵ^{(y)}, |ρ(z)| ⊃ |Φ\{0}| > (2 + dim_{Φ}(J)) − 1 (by largeness) = 1 + dim_{Φ}(J^{(y)}), so by Amitsur z is algebraic in Ĵ^{(y)}; but it also remains in the radical of Ĵ^{(y)}, and the only algebraic elements of the radical are nilpotents. Thus z is nilpotent in each J^{(y)}, i.e. it is properly-nilpotent.

Putting these together we have our main radical shrinkage theorem.

Theorem 8.7. If J is a Jordan system (algebra, triple or pair) satisfying a homotope polynomial identity over a large uncountable field, then

Mc(J) = Rad(J)

so J is nondegenerate iff it is semiprimitive.

Corollary 8.8. Any simple Jordan system satisfying a homotope PI over a large uncountable field is primitive.

PROOF. J is simple hence nondegenerate; by (8.7) above, J is therefore semiprimitive, hence primitive.

Remark 8.9. We advocate using “really large” fields, i.e. fields which are large and (in case J has finite dimension) also uncountable. Even better would be rich fields, those Φ which contain a large uncountable algebraically closed subfield Ω (they need not be algebraically closed themselves). These fields not only shrink radicals, but they also remove division algebras:

Proposition 8.10. The only Jordan division system over a rich field Φ is Φ itself.

PROOF. If J is a division system then for c ≠ 0, A = J^{(c)} [resp. V^{+(c^{−})} in the case of pairs] is a Jordan division algebra with unit 1^{c} = c^{−1}, and it suffices to prove A = Φ1. But if a ∉ Φ1 then all λ1−a for λ ∈ Ω are nonzero, hence invertible, so by (8.5) and largeness of Ω the element a is algebraic over Ω: p(a) = 0 for p(t) ≠ 0 in Ω[t]. But since Ω is algebraically closed, p(t) = ∏(t − λ_{i}) factors into linear factors λ_{i} ∈ Ω and p(a) = ∏(a − λ_{i}1) is invertible, a contradiction. Hence some λ1 − a = 0 and a ∈ Φ1.

We also note in passing that

Proposition 8.11. If J is a prime nondegenerate Jordan system then it is a tight cover of all its local algebras A_{b} for b ≠ 0:

0 ≠ I ◁ J ⟹ 0 ≠ \overline{I} ◁ A_{b} (b ≠ 0).

PROOF. If I ◁ J, then I ◁ J^{(b)} ⇒ \overline{I} ◁ A_{b}, hence \overline{I} = 0 ⇒ I ⊂ Ker(P_{b}) ⇒ P_{b}I = 0 ⇒ b ∈ Ann(I) (by nondegeneracy [16, 1.7]) = 0 (by primeness and I ≠ 0). This contradicts the assumption b ≠ 0.

This is false in general for a non-prime J: if J = J_{1} \overline{\underline{+}} J_{2} and b = b_{1} ⊕ 0, then Ker_{J}(b) = Ker(b_{1}) ⊕ J_{2}, A_{b} = (J_{1})_{b_{1}} and the nonzero ideal J_{2} ◁ J has \overline{J_{2}} = 0 in A_{b}.

From this we can avoid “proper” nilness (nilness at each b) and get by with nilness at a single b, provided that J is prime.

Proposition 8.12. Let J be a prime nondegenerate Jordan system. If an ideal I is strictly-nil in a single PI local algebra A_{b} for b ≠ 0, then I = 0.

PROOF. If I ≠ 0 then 0 ≠ \overline{I} ◁ A_{b} by primeness and (8.11); if \overline{I} is strictly-nil then \overline{I} ⊂ SN il(A_{b}) = Mc(A_{b}) (by (8.1) since A_{b} is PI) = 0 (J nondegenerate ⇒ A_{b} nondegenerate), a contradiction.

Assume now that V is a simple pair over its centroid Φ. The obstacle to primitivity is the obstacle to semiprimitivity: for all we know we may have a simple radical pair (such exist for associative algebras by Sasiada’s examples, but of course these are not PI). From the above results, we can get free of the radical by having enough scalars.

Theorem 8.13 (Tight Scalar Extension). Any simple pair V satisfying a Clifford homotope PI over its centroid Φ is a scalar form of a classical pair: any tight scalar extension Ṽ = V Φ̃ by a rich field Φ̃ ⊃ Φ is again simple but also primitive and retains the Clifford homotope PI, so Ṽ is classical Clifford over its centroid Ω̃ ⊃ Φ̃ (and Ṽ = V Ω̃ is also a tight scalar extension).

PROOF. By tightness Ṽ inherits simplicity as well as nondegeneracy, by scalarness it inherits the homotope PI (remember all PI’s are assumed to hold strictly); since it is nondegenerate over a rich field it is semiprimitive by (8.7), hence primitive with Clifford homotope PI, so by Theorem 7.3 the pair Ṽ is classical of Clifford type.

Combining Theorem 8.13 with the Primitive Form Theorem 5.6, we obtain

Theorem 8.14 (Simple Pairs). Any simple Jordan pair V of Clifford type is primitive hence classical over its centroid Φ. It is therefore isomorphic to one of the pairs of the Clifford List 3.4:

(I) Rectangular Pairs: V(M, R, f) for a central simple Artinian Φ-algebra R of degree ≤ 2, which thus has one of the three forms (Ia) M_{1,q}(Φ), 1 ≤ q ≤ ∞, (Ib) M_{1,q}(Q), Q quaternion division algebra over Φ, 1 ≤ q ≤ ∞, (Ic) M_{2,q}(Φ), 2 ≤ q ≤ ∞,

(II) Alternating Pair: V(A_{5}(Φ)), (III) Clifford Pairs: V(J) for a simple Clifford Jordan algebra J (a tight

scalar form of some simple J(\overline{q}, 1) for a nondegenerate quadratic form \overline{q} of dimension ≠ 2 over an algebraically closed extension field \overline{Φ} of Φ).

In particular, the simple pairs of Clifford type are not merely i-special but are in fact special.

9. Simple triple systems

Instead of building the structure of triples from scratch, we will rely on our knowledge of pairs and look at V(J). We would first like to point out that, as in the case of pairs, primitive triples of Clifford type are simple.

Theorem 9.1. If J is a primitive Jordan triple system of Clifford type, then J is simple of finite capacity.

PROOF. Given a primitive Jordan triple J with a Clifford homotope PI, form V = V(J) = (J, J). Then Soc(J) ≠ 0 by (1.20), so Soc(V) = (Soc(J), Soc(J)) ≠ 0 and by (7.1) we need only prove the Clifford homotope PI forces Soc(V) to have finite capacity. By [5, Theorem 5.8 p.228] there are two possibilities: either V itself is primitive with Clifford homotope PI, or V ∼^{=} W ⊕ W^{op} (W = V/I for I ◁ V maximal with respect to I ∩ I^{∗} = 0) for W primitive with Clifford homotope PI. In the first case Soc(V) has finite capacity by (7.3). In the second case, Soc(W) = W is again of finite capacity by (7.3). But then Soc(V) also has finite capacity: if {e_{i}} is an infinite family of orthogonal division idempotents in V, then all but a finite number of {e_{i} + I} are zero in W (by finite capacity there), so e_{i} ∈ I, i > n, for some integer n, and dually, e_{j} ∈ I^{∗} j > m, some integer m, and thus e_{k} ∈ I ∩ I^{∗} = 0 for k > max(m, n).

Remark 9.2. In fact, any primitive homotope-PI Jordan triple system which isn’t of hermitian type is simple (and equals its socle by (1.20)). Indeed from Zel’manov’s first classification paper [19], we know that a strongly prime Jordan triple J is either i-special or its central closure Γ^{−1}J is simple exceptional finite-dimensional over the field Γ^{−1}Γ. Since the i-special systems are either of hermitian type or Clifford type (by [5]), it suffices by (9.1) to handle the case of an exceptional triple J.

We first observe that for a map T in the centroid Γ of J,

(1)  J is nondegenerate ⟹ Ker(T) ◁ J

[Ker(T) is clearly outer, and if x ∈ J, z ∈ Ker(T), then P_{z}x ∈ Ker(T) since P_{T (P (z)x)} = T^{2}P_{z}P_{x}P_{z} = P_{T (z)}P_{x}P_{z} = 0 forcing T (P_{z}x) = 0 by nondegeneracy], and

(2)  J is strongly prime ⟹ the centroid acts faithfully on J

[since T ≠ 0 ⇒ 0 ≠ T (J) ◁ J and P_{T (J)}Ker(T) = T^{2}P_{J}Ker(T) = 0 ⇒ Ker(T) = 0 by (1) and primeness].

Assume now that J is primitive with a homotope PI and Γ^{−1}J is simple finite-dimensional over Γ^{−1}Γ. Some local algebra A_{b} is primitive with a polynomial identity, hence A_{b} is a simple unital Jordan algebra [2] and by [3, lemma 3.5] the heart H of J is nonzero. Now always

(3) J semiprime with nonzero heart H ⟹ H = P_{H}H + P_{J}P_{H}H

since I = P_{H}H + P_{J}P_{H}H is an ideal of J contained in H, and is nonzero by semiprimeness of J, so H ⊂ I by definition of heart, and H = I. Then

(4) J strongly prime with heart H ≠ 0 ⟹ T (H) = H for all T ≠ 0 in Γ

since 0 ≠ T (H) (by (2) and T ≠ 0) is an ideal of J, so again H ⊂ T (H) by definition of heart, yet by (3) T (H) ⊂ P_{H}T (H) + P_{J}P_{H}T (H) ⊂ H, hence T (H) = H ( ⇔ H = T^{−1}H).

Thus 0 ≠ Γ^{−1}H ◁ Γ^{−1}J implies Γ^{−1}H = Γ^{−1}J by simplicity. In particular J ⊆ Γ^{−1}H, that is, for any x ∈ J we have x = T^{−1}h for some T ∈ Γ, h ∈ H, so x ∈ T^{−1}(H) = H; in other words, the minimal ideal is H = J, and J is simple.

From here on, we assume that J is a simple Jordan triple system of Clifford type over its centroid Φ. By (1.2.1) the Jordan pair V = V(J) = (J, J) with exchange involution (x, y)^{∗} = (y, x) is ∗-simple, and has J ≃ J (V, ∗) by (1.2.3). Either V is simple in its own right, or it is not simple but merely ∗-simple, in which case it is the direct sum V = W \overline{\underline{+}} W^{op} of a simple W and its opposite under the exchange involution [if 0 ≠ W is a proper ideal of V, then W + W^{∗}, W ∩ W^{∗} are ∗-ideals, hence W + W^{∗} = V and W ∩ W^{∗} = 0 and V = W \overline{\underline{+}} W^{∗} ≃ W \overline{\underline{+}} W^{op}, ∗ ≃ ex].

The case V = W \overline{\underline{+}} W^{op} is easy: by (1.2.4) this just means J = P ol(W) is the polarized triple associated to the simple pair W.

We now examine the case where V = V(J) is already simple. By our classification (8.14), V is isomorphic to either V(M, R, f), V(A_{5}(Φ)) or V(I) for a simple Clifford Jordan algebra I (a tight scalar form of some simple J(\overline{q}, 1) for a nondegenerate form \overline{q} of dimension ≠ 2 over an algebraically closed extension field \overline{Φ} of Φ), and J ≃ J (V, ∗) for some involution ∗. If V ≃ V(M, R, f) is rectangular of type M_{1,q}(Δ) for q > 1, we have a rather precise description for the involutions.

Proposition 9.3. If V = (M^{+}, M^{−}) is of type M_{1,q}(Δ) for Δ = Φ or Q (quaternion division algebra) and q > 1, then any involution ∗ on V has the form (x^{+}, y^{−})^{∗} = (T^{−1}(y^{−}), T (x^{+})) for a Φ-linear bijection T : M^{+} ⟶ M^{−} which satisfies T (ax^{+}) = T (x^{+})\overline{a} for an involution − on Δ, and the Jordan triple J = J (V, ∗) ≃ (V^{+})^{(∗)} under P_{x^{+}}y^{+} = Q_{x^{+}}(y^{+})^{∗} has the form

J = T (M, Δ, h) : P_{x}y = h(x, y)x

for a nondegenerate hermitian form h on the left Δ-space M = M^{+}: h(ax, yb) = ah(x, y)\overline{b}, h(x, y) = \overline{h(y, x)}.

PROOF. V^{+} = M is a left vector space over Δ, and we have a bijection V^{+} = M ⟶^{∗} V^{−} which can be identified with a subspace of the dual of M via the given bilinear form f : V^{+} × V^{−} ⟶ Δ. We claim

(1)  (ax)^{∗} = x^{∗}σ_{x}(a) (a, σ_{x}(a) ∈ Δ, x ∈ M^{+}).

This is trivial if x = 0 (take σ_{0}(a) = a). As soon as x ≠ 0, it has f(x, M^{−}) = Δ, so as in the proof of Prop. 3.1 (with e_{11} = 1) we can find M^{+} = Δm_{1}^{+} ⊕ M_{1}^{+}, M^{−} = m_{1}^{−}Δ ⊕ M_{1}^{−} for m_{1}^{+} = x, f(m_{1}^{+}, m_{1}^{−}) = 1, f(M_{1}^{+}, m_{1}^{−}) = f(m_{1}^{+}, M_{1}^{−}) = 0. Then (ax)^{∗} = [f(m_{1}^{+}, m_{1}^{−}a)m_{1}^{+}]^{∗} = [Q_{m_{1}^{+}}(m_{1}^{−}a)]^{∗} = Q_{m_{1}^{+}}^{∗}(m_{1}^{−}a)^{∗} = (m_{1}^{+})^{∗} f((m_{1}^{−}a)^{∗}, m_{1}^{+}^{∗}) = x^{∗}σ_{x}(a) where σ_{x}(a) = f((m_{1}^{−}a)^{∗}, x^{∗}). Note if x ≠ 0 then σ_{x}(a) is uniquely determined by (1) [since x ≠ 0 ⇒ x^{∗} ≠ 0], and always σ_{x}(1) = 1, σ_{x}(0) = 0. Clearly σ_{x}(a) = f((m_{1}^{−}a)^{∗}, x^{∗}) is Φ-linear in a, and by uniqueness x^{∗}σ_{x}(ab) = {(ab)x}^{∗} = {a(bx)}^{∗} = (bx)^{∗}σ_{bx}(a) = x^{∗}σ_{x}(b)σ_{bx}(a) from (1) shows

(2) if x ≠ 0 : σ_{x}(ab) = σ_{x}(b)σ_{bx}(a), σ_{x}(a + b) = σ_{x}(a) + σ_{x}(b),
σ_{x}(αa) = ασ_{x}(a),

and [f(x, m^{−})x]^{∗} = x^{∗}σ_{x}(f(x, m^{−})) equals [Q_{x}m^{−}]^{∗} = Q_{x^{∗}}(m^{−})^{∗} = x^{∗} f(m^{−}^{∗}, x^{∗}) which shows

(3)  f(m^{−}^{∗}, x^{∗}) = σ_{x}(f(x, m^{−}))

(at least for x ≠ 0, but it is trivially true if x = 0 too). Moreover

(4)  x^{∗}, y^{∗} independent in M^{−} ⟹ σ_{x} = σ_{y}

since then in particular x, y, x+y ≠ 0 and x^{∗}σ_{x}(a)+y^{∗}σ_{y}(a) = (ax)^{∗} +(ay)^{∗} = [a(x+y)]^{∗} = (x+y)^{∗}σ_{x+y}(a) = x^{∗}σ_{x+y}(a)+y^{∗}σ_{x+y}(a) forces the coefficient of x^{∗}, y^{∗} on both sides to coincide, σ_{x}(a) = σ_{x+y}(a), σ_{y}(a) = σ_{x+y}(a), hence σ_{x}(a) = σ_{y}(a).

Thus by our hypothesis that q > 1 we have dim(M^{−}) > 1, and

(5)  dim(M^{−}) > 1 ⟹ σ_{x} = σ all have a common value

since if x^{∗}, y^{∗} are dependent there is z^{∗} independent of them both, hence by (4) σ_{x} = σ_{z} = σ_{y}.

If all σ_{x} for x ≠ 0 have a common value σ, then − := σ

(6) is an involution on Δ and h(x, y) := f(x, y^{∗}) is a nondegenerate hermitian form on M^{+}.

Indeed, h(x, M^{+}) = f(x, M^{+}^{∗}) = f(x, M^{−}) ≠ 0 if x ≠ 0 so h is left nondegenerate. It is hermitian: h(y, x) = f(y, x^{∗}) = f(m^{−}^{∗}, x^{∗}) (for some m^{−} ∈ M^{−} since ∗ maps V^{−} onto V^{+} as well) = \overline{f(x, m^{−})} (by (3)) = \overline{f(x, y^{∗})} (since m^{−} = (m^{−})^{∗∗} = y^{∗}) = \overline{h(x, y)}. [In particular, this shows h is nondegenerate in the second variable as well.] Certainly h is Δ-linear in the first variable, and it is conjugate-linear in the second since h(x, ay) = f(x, (ay)^{∗}) = f(x, y^{∗}\overline{a}) (by (1), (5)) = f(x, y^{∗})\overline{a} = h(x, y)\overline{a}. (This also follows directly from hermitianity once we note that − is an anti-homomorphism.)

(2) shows − is a Φ-linear anti-homomorphism (note σ(ab) = σ(a) σ(b) is trivial if b = 0), and it is involutory (\overline{\overline{a}} = a): ah(x, y) = h(ax, y) = \overline{h(y, ax)} = \overline{h(y, x)\overline{a}} = \overline{\overline{a}}\overline{h(y, x)} = \overline{\overline{a}}h(x, y) so choosing h(x, y) ≠ 0 we get \overline{\overline{a}} = a.

Remarks 9.4. The result fails for q = 1: V = M_{1,1}(Δ) = (Δm_{1}^{+}, m_{1}^{−}Δ) has invertible elements and is obtained by doubling the Jordan algebra Δ^{+}. The involutions in this case are all (T, T^{−1}) for T in the structure group of Δ^{+}. Here T = L_{a}A for a = T (1) and A an automorphism of Δ^{+}, hence (by Hua’s theorem) either an isomorphism or anti-isomorphism of Δ. In the case where Δ has no involution at all (e.g. Δ ≄ Δ^{op}), none of the involutions on M_{1,1}(Δ) come from involutions on Δ.

A complete description of involutions on rectangular pairs V(R, M, f) for simple artinian R is given in [17].

In (1.13.2) we saw that if we know V(J) up to isomorphism, we know J up to isotopy:

(9.5)  V(J_{1}) ≃ V(J_{2}) ⟹ J_{1} ≃ J_{2} isotopic.

For example, if V(J) ≃ M_{1,1}(Φ) or M_{1,1}(Q), then V(J) is a Clifford pair and J is isotopic to a Clifford triple system built out of a quadratic form. This covers the case left out by (9.3).

More generally, if V(J) ≃ V(I) for I a tight scalar form of a Clifford Jordan algebra J(\overline{q}, 1), then J ≃ I^{(σ)}. Comparing (1.9), (1.10.1) shows J(\overline{q}, 1) = J(\overline{q})^{(−)} so J(\overline{q}, 1)^{(σ)} = J(\overline{q})^{(τ)} for τ = − ∘ σ structural but not necessarily symmetric: τ^{∗} = σ ∘ −.

If V(J) ≃ V(A_{5}(Φ)), then J ≃ A_{5}(Φ)^{(σ)} under P_{x}σ(y) = −xσ(y)x = x τ(y)x where τ = −σ is just as symmetric and structural as σ.

Gathering the above remarks, we are now in a position to state

Theorem 9.6 (Simple triple systems). J is a simple Jordan triple system of Clifford type over its centroid Φ iff J satisfies one of the following:

(I) J ≃ J (V(J), ∗) where V(J) is isomorphic to either M_{1,q}(Δ) (Δ = Φ or

Q) or M_{2,q}(Φ). In the first case, J has the form

J = T (M, Δ, h), P_{x}y = h(x, y)x for a nondegenerate hermitian form h on the left Δ-space M.

(II) J ≃ I^{(τ)} is isotopic to a tight scalar form I of a Clifford Jordan algebra J(\overline{q}, 1) of a nondegenerate quadratic form \overline{q} over an algebraically closed field \overline{Φ} ⊃ Φ where P_{x}y = q(x, τ(y))x − q(x)τ(y) (τ^{∗} = τ ∈ Str(J(q))).

(III) J ≃ A_{5}(Φ)^{(τ)} is isotopic to A_{5}(Φ), where P_{x}y = xτ(y)x [τ^{∗} = τ ∈

Str(A_{5}(Φ)] (IV) J ≃ V^{+} ⊕ V^{−} = P ol(V) (c.f. (1.1.1)) where V is one of the simple pairs

of Clifford type found in (8.14).

10. Strongly prime systems

We refer the reader to [9] for generalities concerning Peirce decompositions. We recall that in a Jordan pair V, two idempotents e and f are associates (denoted e ≈ f) if e and f have the same Peirce spaces, e is indecomposable if it can’t be written as the sum of two orthogonal idempotents, primitive if all its associates are indecomposable, and local if V_{2}(e) is a local Jordan pair.

Certainly, any finite-dimensional nondegenerate pair V contains some idempotent, since Rad(V) = Mc(V) = 0 by (1.14.2) and by [9]:

(10.1) a non-radical pair V ≠ Rad(V) with dcc on principal inner ideals must contain a local idempotent.

Our next result will involve the lifting of idempotents and it is important that this be done orthogonally ([11], Proposition 1):

if I ◁ V is a pnil ideal, then countable sets of

(10.2) orthogonal idempotents lift orthogonally from V/I to V.

A pair V (resp. triple system J) is said to be idempotent-finite if V (resp. V(J)) contains no infinite family of pairwise orthogonal idempotents. Note that for triple systems this is stronger than tripotent-finiteness (tripotents of J are just the symmetric idempotents of (J, J)): it amounts to finiteness of “orthogonal” regular elements x_{i}, where orthogonality brings in the pairing elements y_{i} such that (x_{i}, y_{i}) are pair idempotents. For idempotentfinite systems we have

(10.3) if an idempotent-finite Jordan triple or pair contains an idempotent, then it contains a primitive idempotent

(otherwise given e, some associate of e decomposes orthogonally into nonzero idempotents e_{1} + ẽ_{1}, then some associate of ẽ_{1} decomposes, and we have e_{1} + e_{2} + ẽ_{2}, and so on, leading to an infinite orthogonal family).

Our goal in this section is to imbed a strongly prime system J into some ultraproduct the factors of which behave well in the sense of the following theorem (see [19, §3, lemma 18]).

Theorem 10.4. Let V be a nondegenerate idempotent-finite locally finitedimensional Jordan pair of Clifford type over a field Φ. Then V = I_{1} ⊕ I_{2} ⊕ … ⊕ I_{m} is a direct sum of finitely many simple ideals (where the I_{i} are precisely all the minimal ideals of V). Any maximal orthogonal family of primitive idempotents must have at least m members.

PROOF. By (10.1) and (10.3), V contains at least one primitive idempotent; let ξ = {e_{i}}_{i=1}^{n} be any maximal orthogonal family of primitive idempotents in V (such exist by idempotent-finiteness). Note that ξ is actually maximal among all orthogonal families of nonzero idempotents (not just primitive ones): if e ≠ 0 were orthogonal to ξ, idempotent-finiteness of V_{2}(e) would guarantee by (10.3) a primitive e_{0} ∈ V_{2}(e), which would automatically still be orthogonal to ξ ⊂ V_{0}(e), and ξ′ = {e_{0}, e_{1}, …, e_{n}} would be an orthogonal family of primitive idempotents, contrary to maximality of ξ.

Define a partial ordering i ≺ j and an equivalence relation i ∼ j on the indices {1, …, n} based on the ideals Id_{V} (e_{i}) generated by the e_{i}’s in V:

(10.5) i ≺ j iff Id_{V} (e_{i}) ⊂ Id_{V} (e_{j}) ( equivalently e_{i} ∈ Id_{V} (e_{j})),
i ∼ j iff Id_{V} (e_{i}) = Id_{V} (e_{j}).

[Note that we haven’t yet established that the Id_{V} (e_{i})’s are simple, so we don’t know containment Id_{V} (e_{i}) ⊂ Id_{V} (e_{j}) implies equality, so we must distinguish the two relations.] Partition the indices {1, …, n} = N_{1} ∪ N_{2} ∪ … ∪ N_{m} into equivalence classes with respect to ∼. Note that automatically n ≥ m. We pick i_{k} ∈ N_{k}, 1 ≤ k ≤ m, and we single out e_{i_{1}}, …, e_{i_{m}}; renumbering if necessary, we may assume these to be e_{1}, …, e_{m}. We claim first that the corresponding ideals span V:

(10.6)  V = I_{1} + I_{2} + … + I_{m}, I_{k} := Id_{V} (e_{k}), 1 ≤ k ≤ m.

Set I := I_{1} + … + I_{m}. Note that all e_{j} ∈ Id_{V} (e_{j}) ⊂ I_{k} ⊂ I, so e = (e^{+}, e^{−}), e^{ε} := ∑_{k=1}^{n} e_{k}^{ε} lies in I, and the Peirce spaces V_{2}^{ε}(e) + V_{1}^{ε}(e) = Q_{e^{ε}}V + {e^{ε}e^{−ε}V^{ε}} ⊂ I so V = I +V_{0}(e) and V_{0} := V_{0}(e) has no primitive idempotent (by maximality of the e_{i}’s) hence no idempotents at all (by (10.3)); thus by (10.1) and local-finiteness, finitely-generated subsystems of V_{0} are all radical and V_{0} is locally-nilpotent, i.e. V_{0} = L(V_{0}) = Mc(V_{0}) = 0 (by (8.1 (2)), since V_{0} has a Clifford homotope PI, and by Peirce inheritance of nondegeneracy) and V = I.

We wish to improve on (10.6) and prove that the sum is actually direct (and that the I_{k}’s are the minimal ideals of V). We remark that if the indices i ≺ j as in (10.5), then e_{i} ∈ Id_{V} (e_{j}) = M(V)e_{j} + M(V)Q_{e_{j}}V (where M(V) stands for the multiplication algebra of the system V), that is e_{i} = M_{ij}(e_{j}) is a “multiple” of e_{j}. (Here a multiple of x is a sum of terms W (x) and W Q_{x}w for multiplication operators W and elements w.) We let

(10.7) B_{0} :=< e_{1}^{±}, …, e_{n}^{±}, and all elements involved
in the multiples M_{ij} when i ≺ j > ⊂ V.

Thus e_{i} can be “ connected” to e_{j} within B_{0}. The following lemma shows how it suffices to concentrate our efforts on certain finitely-generated subpairs.

Lemma 10.8. In order to prove that the sum V = I_{1} + … + I_{m} is direct and the I_{k}’s are the minimal ideals of V, it suffices to show that for every finitely-generated subpair B ⊃ B_{0}, we have

\overline{B} := B/L(B) = \overline{B ∩ I_{1}} ⊕ … ⊕ \overline{B ∩ I_{m}}, (10.9)

where \overline{B ∩ I_{k}} are the minimal ideals of \overline{B}.

PROOF. To show the directness of the sum, suppose x_{1} + … + x_{m} = 0 in V for x_{i} ∈ I_{i}; for any finitely-generated B ⊃ B_{00} :=< B_{0}, x_{1}, …, x_{m} > we have \overline{x_{1}} + … + \overline{x_{m}} = \overline{0} in \overline{B} with \overline{x_{i}} ∈ \overline{B ∩ I_{i}}, so \overline{x_{i}} = 0 and x_{i} ∈ L(B) by directness in (10.9), 1 ≤ i ≤ m. To show each x_{i} vanishes it suffices to show each x_{i} lies in L(V) = Mc(V) = 0 by (8.1 (2) and nondegeneracy). But this will follow from

(10.10) x ∈ L(V) ⟺ x ∈ L(B) for all finitely generated subpairs B of V containing a fixed B_{00} ∋ x.

The implication ⟹ is easy; for the converse, if x lies in all L(B) then Id_{V} (x) is locally-nilpotent since any < y_{1}, …, y_{r} >⊂ Id_{V} (x) is nilpotent: if we write y_{j} = M_{j}(x) as a multiple of x and let

B =< B_{00} and all elements in the multipliers M_{j} > ,

then all y_{j} ∈ Id_{B}(x) (BY CONSTRUCTION OF B) ⊂ L(B) (since x ∈ L(B) BY HYPOTHESIS) imply < y_{1}, …, y_{r} >⊂ L(B) is nilpotent. This establishes (10.10) hence directness.

Always in a semiprime direct sum, the pieces are simple iff they are minimal, in which case they are all the minimal ideals. Indeed, a simple ideal is always minimal, a minimal and nontrivial direct summand is always simple, and if I is a minimal ideal which involves the simple direct summand I_{k} (π_{k}(I) ≠ 0) then π_{k}(I) = I_{k} by simplicity, so I_{k} = Q_{I_{k}}I_{k} (simplicity) = Q_{I_{k}}π_{k}(I) = Q_{I_{k}}I (directness) ⊂ I implies I equals I_{k} by minimality.

For minimality of each I := I_{k} = Id_{V} (e_{k}) := Id(e), let 0 ≠ L ◁ V such that L ⊂ I; by nondegeneracy, L(V) = 0, so L cannot be locally nilpotent, and there exists a non-nilpotent < x_{1}, …, x_{n} >⊂ L. Then < x_{1}, …, x_{n} >⊄ L(B) for B =< B_{0}, x_{1}, …, x_{n} >⊃ B_{0}, 0 ≠ \overline{< x_{1}, …, x_{n} >} ⊂ \overline{L ∩ B} forces \overline{L ∩ B} = \overline{I ∩ B} (by minimality in (10.9)) ⇒ \overline{e} ∈ \overline{L ∩ B} ⇒ e = x + z for some x ∈ L ∩ B and z ∈ L(B). But then e ∈ L ∩ B (hence I = Id(e) ⊂ L so I = L is minimal) by the

Lemma 10.11. If e is an idempotent in a Jordan pair V of the form e = x+z where

(1) z is nilpotent in V^{(e)} (e.g. z is properly-nil in B ∋ e, or z ∈ L(B)),

(2) x lies in some subalgebra C ⊂ V^{(e)} with Q_{e}Q_{e}C ⊂ C,

then e ∈ C.

PROOF. We have e = Q_{e}Q_{e}e = y + w for y = Q_{e}Q_{e}x in C ∩ V_{2}(e) and w = Q_{e}Q_{e}z ∈ V_{2}(e) still nilpotent in V^{(e)} [by the Shifting Principle for T = Q_{e}Q_{e} we see w^{(n,e)} = T (z)^{(n,e)} = T (z^{(n,T^{∗}(e))}) = T (z^{(n,e)}) eventually vanishes when z is nilpotent in V^{(e)}]. This moves us into the unital algebra A = V_{2}^{(e)}: y, w ∈ A with e = 1 so 0 = w^{n} = (1 − y)^{n} =

1 + ∑_{k=1}^{n} ( \begin{matrix}n \\ k \end{matrix} ) (−1)^{k}y^{k} and e = 1 = − ∑_{k=1}^{n} ( \begin{matrix}n \\ k \end{matrix} ) (−1)^{k}y^{k} ∈ C.

We now proceed to establish (10.9). Let B be an arbitrary finitely generated (hence finite-dimensional since V is assumed locally-finite-dimensional in (10.4)) subsystem containing B_{0} and form \overline{B} = B/L(B) (L(B) = Rad(B) by (1.14.2)). Then \overline{B} is a direct sum of a finite number of simple ideals \overline{B_{1}} ⊕ … ⊕ \overline{B_{s}} and \overline{e_{i}} is at least an indecomposable idempotent: indeed, \overline{B}_{2}(\overline{e_{i}}) = \overline{B_{2}(e_{i})} is nondegenerate, finite-dimensional (hence finite capacity), so if \overline{e_{i}} = \overline{f} + \overline{f}′ decomposes for nonzero orthogonal idempotents in \overline{B_{2}}(\overline{e_{i}}) we can by (10.2) lift to orthogonal f, f′ ∈ B_{2}(e_{i}) so g := f + f′ has \overline{g} = \overline{e_{i}} hence g is invertible in B_{2}(e_{i}) (by [11, Lemma 1]) and g ≈ e_{i}, which would contradict primitivity of e_{i} [11, Prop. 2]. In particular, each \overline{e_{i}} must lie entirely in one of the \overline{B_{k}}’s, 1 ≤ k ≤ s, in which case \overline{B}_{k} = Id\overline{B}(\overline{e}_{i}) by simplicity of \overline{B}_{k}. [If \overline{e_{i}} = \overline{f_{1}} ⊕ … ⊕ \overline{f_{s}} then by directness each \overline{f_{k}} is an idempotent, so by indecomposability only one \overline{f_{k}} appears.]

Each \overline{B_{j}} must contain at least one \overline{e_{i}} [if some \overline{B_{j}} didn’t, directness of the sum would provide an idempotent in \overline{B_{j}} orthogonal to the maximal family \overline{ξ}, maximality which is ensured by that of ξ among all orthogonal families and the fact (10.2) that idempotents lift orthogonally]. Finally, we can show that i ≺ j implies i ∼ j:

i ∼ j ⟺ i ≺ j
(10.12) ⟺ \overline{e_{i}} and \overline{e_{j}} lie in the same \overline{B_{k}}, 1 ≤ k ≤ s
⟺ Id\overline{B}(\overline{e_{i}}) = Id\overline{B}(\overline{e_{j}}) = \overline{B}_{k}.

Clearly i ∼ j ⇒ i ≺ j by (10.5). To see i ≺ j ⇒ \overline{e_{i}}, \overline{e_{j}} lie in the same \overline{B_{k}}, recall e_{i} = M_{ij}(e_{j}) where M_{ij} is a B_{0} multiplication by definition (10.7), hence \overline{e_{i}} = \overline{M_{ij}}(\overline{e_{j}}) where \overline{M_{ij}} is a \overline{B} multiplication, so if \overline{e_{j}} lies in \overline{B_{k}} ◁ \overline{B} so does \overline{e_{i}}. Clearly \overline{e_{i}}, \overline{e_{j}} ∈ \overline{B_{k}} ⇔ Id\overline{B}(\overline{e_{i}}) = Id\overline{B}(\overline{e_{j}}) = \overline{B}_{k} by simplicity of \overline{B}_{k}. To show Id\overline{B}(\overline{e_{i}}) = Id\overline{B}(\overline{e_{j}}) ⇒ i ∼ j, it suffices to show Id\overline{B}(\overline{e_{i}}) ⊂ Id\overline{B}(\overline{e_{j}}) ⇒ i ≺ j (since by (10.5) i ∼ j ⇔ i ≺ j and j ≺ i); but when \overline{e_{i}} ∈ Id\overline{B}(\overline{e_{j}}) then \overline{e_{i}} = \overline{N}_{ij}(\overline{e_{j}}) is a \overline{B}-multiple of \overline{e_{j}} and z_{i} := e_{i} − N_{ij}(e_{j}) ∈ L(B). Then e_{i} = x_{i} + z_{i} for z_{i} ∈ L(B) and x_{i} := N_{ij}(e_{j}) ∈ Id_{V} (e_{j}), so by Lemma (10.11) we see e_{i} ∈ Id_{V} (e_{j}) and so i ≺ j.

Thus the equivalence classes N_{k} are in 1-1 correspondence with the \overline{B}_{k}, and we may assume the numbering is chosen so that

(10.13)  i ∈ N_{k} ⟺ \overline{e_{i}} ∈ \overline{B}_{k} (k = 1, 2, … m = s).

To complete the proof of 10.9 we must show the simple minimal ideals \overline{B}_{k} are precisely the \overline{B ∩ I_{k}}. But \overline{B ∩ I_{k}} = \overline{B ∩ Id_{V} (e_{k})} ⊃ \overline{Id_{B}(e_{k})} = Id\overline{B}(\overline{e_{k}}) = \overline{B}_{k}. To show conversely that \overline{B ∩ I_{k}} ⊂ \overline{B}_{k} we need to show that b = M_{V} (e_{k}) ∈ B ∩ I_{k} = B ∩ Id_{V} (e_{k}) has \overline{b} entirely contained in \overline{B}_{k}. The difficulty is that M_{V} is a multiplication from V, not necessarily from B, so we don’t immediately know \overline{b} = \overline{M}_{V} (\overline{e_{k}}) ∈ \overline{B}_{k}, only that \overline{b} = \overline{b}_{1} ⊕ … ⊕ \overline{b}_{m} for \overline{b}_{i} ∈ \overline{B}_{i}. Instead we argue as follows. By simplicity of \overline{B}_{i}, we have \overline{b}_{i} ≠ 0 ⟹ Id\overline{B}(\overline{b}_{i}) = \overline{B}_{i} ∋ \overline{e_{i}} (recall our renumbering), so \overline{e_{i}} = M\overline{B}(\overline{b}_{i}) for some \overline{B}-multiplication, \overline{e_{i}} = Q\overline{e_{i}}Q\overline{e_{i}}M\overline{B}(\overline{b}_{i}) = Q\overline{e_{i}}Q\overline{e_{i}}M\overline{B}(\overline{b}) = \overline{Q_{e_{i}}Q_{e_{i}}M_{B}(b)} (all other Q\overline{e_{i}}Q\overline{e_{i}}M\overline{B}(\overline{b}_{j}) for j ≠ i fall in Q\overline{e_{i}}Q\overline{e_{i}} \overline{B}_{j} = 0 by orthogonality Q_{\overline{B}_{i}} \overline{B}_{j} = 0). Thus e_{i} = Q_{e_{i}}Q_{e_{i}}e_{i} = x + z for z ∈ L(B) nilpotent in V^{(e_{i})}, x = Q_{e_{i}}Q_{e_{i}}M_{B}(b) ∈ C := Id_{B}(b). Applying (10.11) shows e_{i} ∈ C = Id_{B}(b) = Id_{B}(M_{V} (e_{k})) ⊂ Id_{V} (e_{k}), hence by our original definition (10.5) i ≺ k, so i ∼ k by (10.12), and since e_{i}, e_{k} are equivalence class representatives we must have i = k. Thus \overline{b}_{i} = 0 for i ≠ k and \overline{b} reduces to \overline{b} = \overline{b}_{k} ∈ \overline{B}_{k}. This ends the proof of (10.9) and with it the proof of Theorem 10.4.

We are now ready to obtain the structure of strongly prime Jordan triples and pairs. We formulate the triple case in terms of ∗-pairs (pairs with involution), where primeness or simplicity for a ∗-pair means ∗-primeness (no orthogonal ∗-ideals) or ∗-simplicity (nontriviality and no proper ∗-ideals).

Theorem 10.14. A prime nondegenerate Jordan pair V (resp. ∗-pair, triple) of Clifford type over Φ is a Φ-form of a simple Jordan pair (resp. ∗-pair, triple) of Clifford type.

PROOF. The first step is to pass from Φ to a large uncountable field F ⊃ Φ, i.e. |F | > N where V is spanned over Φ by N elements. We form the tensor extension V_{F} = F ⊗_{Φ} V and then tighten it to get a tight scalar extension \overline{V} (V ⊂ \overline{V} with F V = \overline{V}). By tightness \overline{V} is a prime nondegenerate pair (resp. ∗-pair), by scalar extension \overline{V} remains of Clifford type (it strictly satisfies the same Clifford identity), and F remains just as large for \overline{V} (if V is spanned over Φ by N elements, they also span \overline{V} over F). If we can prove that \overline{V} is a scalar form of a simple pair (resp. ∗-pair) \overline{\overline{V}}, then so is V (\overline{\overline{V}} = \overline{\overline{F}} \overline{V} = \overline{\overline{F}} \overline{F} V = \overline{\overline{F}} V). Thus we may replace Φ, V by F, \overline{V} from the start and assume that Φ = F is a large uncountable field. Note that by (8.7) the ∗-invariant radicals Mc, L and Rad all coincide on such V, so V is nondegenerate ⇔ L = 0 ⇔ V is semiprime.

The second step is to imbed V in a “finite environment” (having localfinite-dimensionality and idempotent-finiteness as required for (10.4)). We noted V is semiprimitive, so V ∼^{=} ∏_{α∈S} V_{α} for V_{α} := V/I_{α} primitive over its centroid F_{α} ⊇ F for ideals I_{α} with ∩_{α∈S}I_{α} = 0. [For ∗-pairs the I_{α} need not be ∗-ideals, but the L_{α} = I_{α} ∩ I_{α}^{∗} are ∗-ideals and W_{α} := V/L_{α} ∼^{=} V_{α} \overline{\underline{+}} V_{α}^{∗} is a ∗-pair imbedded in the direct sum of V_{α} and a copy of its opposite under the exchange involution, so W_{α} is ∗-primitive.] By (7.3) each V_{α} (hence its dual V_{α}^{∗}) is simple classical of Clifford type, in particular by (3.5) has bound 2 on families of orthogonal idempotents and is locally d-bounded-dimensional over F_{α}. [Being imbedded in the direct sum of 2 such pairs, W_{α} remains of Clifford type but now has bound 4 on idempotents and is locally 2d-bounded-dimensional over F_{α} (where a centroidal γ ∈ F_{α} acts via (γ, γ^{∗}) on W_{α} ⊂ V_{α} \overline{\underline{+}} V_{α}^{∗} for γ^{∗} = ∗ ∘ γ ∘ ∗).]

Thus each V_{α} [resp. W_{α}] is “finite”, but because the index set S is usually infinite the product ∏_{α∈S} V_{α} does not have idempotent-finiteness, and has locally d-bounded dimension only over ∏_{α∈S} F_{α}, which is not a field. We must shrink the direct product using an ultrafilter. By subdirectness, the support set F_{a} = {α ∈ S | π_{α}(a) ≠ 0} for π_{α} : V → V_{α} [resp. π_{α} : V → W_{α}] is nonempty for a ≠ 0 in V, and by primeness is directed downwards. [For the case of ∗-pairs, ∗-primeness guarantees a, b ≠ 0 generate nonzero ∗-ideals I, K with Q_{I}K ≠ 0, so any c ≠ 0 in Q_{I}K has π_{α}(c) ≠ 0 ⇒ π_{α}(Q_{I}K) ≠ 0 ⇒ π_{α}(I), π_{α}(K) ≠ 0 ⇒ π_{α}(a), π_{α}(b) ≠ 0 in W_{α}, therefore F_{c} ⊂ F_{a} ∩ F_{b}.] Thus {F_{a} | a ≠ 0 in V } can be enlarged to an ultrafilter F, with V imbedded in the ultraproduct pair Ṽ = (∏_{α} V_{α})/F [resp. ∗-pair (∏_{α} W_{α})/F] over the field F̃ = (∏_{α} F_{α})/F containing F. The ultraproduct Ṽ is nondegenerate since each V_{α}, W_{α} is, but we don’t know Ṽ remains prime [resp. ∗-prime]. [Once the structure theory is complete - which is of no help to us in this paper- we know by [1, Theorem 1.10, p.28] that there is an elemental condition for strong primeness so the ultraproducts do inherit it.] Since ultraproducts inherit all elementary properties common to all factors, we at least know Ṽ is of Clifford type with bound 2 [resp. 4] on orthogonal idempotents and is locally d-[resp. 2d]- dimensional over the field F̃. [Note having fixed bounds d, 2d is crucial here: ultraproducts need not inherit mere idempotent-finiteness or local-finite-dimensionality from their factors, witness the subsystem generated by a single z = ∏_{n=1}^{∞} z_{n} for z_{n} nilpotent of index n. Ṽ inherits the fixed Clifford polynomial identity from V, or by (6.3) we know it inherits the Zel’manov polynomial identity from all the primitive V_{α}.] We may as well discard Ṽ and keep only the subpair F̃V. This inherits Clifford type, bound 2 [resp. 4] on idempotents, and local d-[resp. 2d]-bounded-dimensionality, and F̃ ⊃ F remains large (since F̃V is again spanned by N elements).

So far we have sacrificed everything (primeness and nondegeneracy) for finiteness of F̃V. We can restore nondegeneracy, at least by radical surgery: the quotient \overline{V} = F̃V/L(F̃V) is a nondegenerate pair [resp. ∗-pair, since L is always ∗-invariant] (since L(\overline{V}) = 0 by radical surgery, hence Mc(\overline{V}) = 0 too using largeness of F̃ and (8.7)). It is still a scalar extension of V (note V remains imbedded V ↪^{φ} \overline{V} since V ∩ L(F̃V) ⊂ L(V) = Mc(V) = 0 by (8.7) again). Any quotient automatically inherits the Clifford identity and local d-[resp. 2d]-bounded-dimensionality, and because L(F̃V) is pnil we know the quotient inherits the bound 2 [resp. 4] on idempotents by (10.2). Thus we have all the ingredients necessary for (10.4), and \overline{V} = I_{1} \overline{\underline{+}} … \overline{\underline{+}} I_{m} is a finite direct sum of its (simple) minimal ideals.

[In the ∗-prime case, the induced involution on \overline{V} must send minimal ideals to minimal ideals, so either I_{k}^{∗} = I_{k} (and S_{k} := I_{k} is itself a simple ∗-pair), or I_{k}^{∗} ≠ I_{k} and S_{k} := I_{k} \overline{\underline{+}} I_{k}^{∗} is a ∗- simple but not simple ∗-pair). Thus \overline{V} = S_{1} \overline{\underline{+}} … \overline{\underline{+}} S_{r} is a direct sum of ∗-simple pairs S_{k}.]

Finally, we restore primeness by showing that V imbeds into one of the simple pieces I_{k} or S_{k} as a form via V ↪^{π_{k}∘φ} I_{k} (π_{k} the projection onto I_{k} or S_{k}). Here, K_{k} := Ker(π_{k} ∘ φ) are ideals [resp. ∗-ideals] of V with ∩_{k}K_{k} = 0 since φ is an imbedding. But by primeness of V its nonzero ideals [resp. ∗- ideals] satisfy the finite intersection property, so some K_{k} is already 0, and π_{k} ∘ φ is injective. Thus V ↪^{π_{k}∘φ} I_{k} [resp. S_{k}] with F̃π_{k}(V) = π_{k}(F̃V) = π_{k}(\overline{V}) = I_{k} [resp. S_{k}]. We have shown V is a scalar form of the simple pair I_{k} [resp. ∗-pair S_{k}], and the theorem is established for pairs and ∗-pairs.

Since the category of triples is equivalent to that of ∗-pairs, the theorem for triples follows immediately. In more detail, if J is a strongly prime triple then V = V(J) = (J, J) is a strongly ∗-prime ∗-pair, hence by the above is a scalar form of a ∗-simple ∗-pair \overline{V} ≃ (\overline{J}, \overline{J}) = V(\overline{J}) for a triple \overline{J} which is a scalar extension \overline{J} = F̃J of J. If \overline{V} is actually simple, then J is simple (with “tight cover” \overline{V}); if \overline{V} is merely ∗-simple, then \overline{V} = \overline{W} \overline{\underline{+}} \overline{W}^{∗} is a direct sum of a simple \overline{W} and its opposite under an exchange involution, so by (1.2.4) \overline{J} = P ol(\overline{W}) is the polarized triple of the simple pair \overline{W} (with “loose cover” \overline{V}), hence by (1.1.2) \overline{J} is simple. Thus J is a form of a simple triple \overline{J} of Clifford type. This completes the proof of the theorem.

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