June 8, 2001

Semiprime, prime and simple Jordan superpairs covered by grids

Dedicated to Kevin McCrimmon on the occasion of his sixtieth birthday

Esther García^{1}

Departamento de Matemáticas, Universidad de Oviedo C/ Calvo Sotelo s/n, 33007 Oviedo, Spain, egg@pinon.ccu.uniovi.es

Erhard Neher^{2}

Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada neher@uottawa.ca

Introduction

Summary. In this paper we continue the investigation of Jordan superpairs covered by a grid which have recently been described by the second author ([20]). For such a Jordan superpair we characterize semiprimeness, primeness and simplicity in terms of its associated supercoordinate system.

An important part of the present research activities in Jordan theory is devoted to simple Jordan superstructures, i.e., Jordan superalgebras, supertriples and superpairs. One now has a classification of finite dimensional simple Jordan superalgebras over algebraically closed fields of characteristic ≠ 2, due to the work of Kac [4], Martínez-Zelmanov [13] and Racine-Zelmanov [27], [26] (this latter paper also considers superalgebras over non-algebraically closed fields). The (infinite dimensional) graded simple Jordan superalgebras of growth one were recently described by Kac-Martínez-Zelmanov [5], and one has a classification of simple finite-dimensional Jordan superpairs over fields of characteristic 0 by Krutelevich [8].

While these works are devoted to classifying simple Jordan superalgebras and superpairs, the second author’s recent paper [20] developed a structure theory of Jordan superpairs over superrings covered by a grid. (Here and in the remainder of this introduction,

1 Research supported by an F.P.I. Grant (Ministerio de Ciencia y Tecnología) and also partially supported by the DGES, PB97-1069-C02-02 (Spain) 2 Research partially supported by a NSERC (Canada) research grant the term “grid” means a connected standard grid in the sense of [23].) Roughly speaking, this condition means that one has a “nice” family of idempotents whose simultaneous Peirce spaces make up the whole Jordan superpair. This type of Jordan superpairs is in general not simple, not even semiprime, nevertheless the presence of a grid allows one to give a precise classification. It is the goal of this paper to connect the theory of Jordan superpairs covered by grids to the works quoted above by answering the following questions: Among the Jordan superpairs covered by a grid, which are semiprime, prime or simple? Or, more generally, what are the ideals of Jordan superpairs covered by a grid?

Our answers to these questions will be given in terms of the supercoordinate system 𝒞 associated to any Jordan superpair V covered by a grid 𝒢. That this can be done is, at least in principle, not surprising since the results of [20] together with §1 show that V is determined by 𝒢 and 𝒞 (for Jordan pairs this goes back to [21]). It is the interplay between V and (𝒞, 𝒢) which lies at the heart of our paper.

The possible supercoordinate systems depend on the type of the covering grid 𝒢. A supercoordinate system always consists of a unital coordinate superalgebra, which is a Jordan or alternative superalgebra, plus possibly some additional structure, like a pair of orthogonal strongly connected idempotents in a Jordan superalgebra (if the grid is a triangle), or an involution and ample subspace of an alternative superalgebra (hermitian grids), or a quadratic form with base point (odd quadratic form grids). More intricate and large grids will force the coordinate superalgebra to be associative (rectangular grids, hermitian grids) or even supercommutative associative (quadratic form grids, alternating grids, Bi-Cayley grids or Albert grids). Our answer to the problem above is summarized in the following.

Theorem. Let V be a Jordan superpair covered by a grid and let 𝒞 be the associated supercoordinate system with coordinate superalgebra A. Then V is semiprime, prime or simple if and only if 𝒞 is respectively semiprime, prime or simple.

This theorem will be proven in §3 where we will determine precisely what it means for a supercoordinate system to be semiprime, prime or simple (3.4, 3.5, 3.7, 3.8, 3.10 and 3.13). For this introduction suffice it to say that in case 𝒞 just consists of a superalgebra A, i.e., the root system associated to the covering grid is simply-laced, 𝒞 is semiprime, prime or simple if and only if the superalgebra A has this property. The following example indicates that one can interpret our result by saying that semiprimeness, primeness and simplicity of Jordan superpairs covered by grids are Morita-invariant.

Example. A Jordan superpair V is covered by a rectangular grid of size J × K with |J| + |K| ≥ 4 if and only if V is isomorphic to a rectangular matrix superpair 𝕄_{JK}(A), consisting of J × K- respectively K × J-matrices over an associative superalgebra A with only finitely many non-zero entries. In this case, the supercoordinate system is 𝒞 = A. The ideals of 𝕄_{JK}(A) are 𝕄_{JK}(B) where B is an ideal of A, and 𝕄_{JK}(A) is semiprime, prime or simple if and only if the superalgebra A is so. In particular, using the structure of simple associative superalgebras ([30], see 2.5), 𝕄_{JK}(A) is a simple Jordan superpair if and only if either A is a simple algebra or A is the double of the simple algebra A¯0 (2.4), which means A = A¯0 ⊕ A¯1 with A¯1 = uA¯0 and u in the algebra centre of A satisfying u^{2} = 1. In the second case, 𝕄_{JK}(A) is the double (3.2) of the Jordan pair 𝕄_{JK}(A¯0): we have 𝕄_{JK}(A¯0 ⊕ uA¯0) = 𝕄_{JK}(A¯0) ⊕ u𝕄_{JK}(A¯0).

A review of the structure of Jordan superpairs covered by grids is given in §1. In this section we will also formally introduce supercoordinate algebras A and systems 𝒞. While their definition depends on some choices of idempotents from the covering grid, we will show that the coordinate superalgebra obtained from a different choice is either isomorphic to A or to its opposite algebra. The following section §2 contains a review and some new results on semiprime, prime and simple alternative or associative superalgebras, with or without involution. The theorem above is proven in §3 by first describing all the ideals of V, something which we feel is of independent interest. The results in §2 then allow us to work out the precise structure of the simple Jordan superpairs covered by a grid, at least in the case of finite-dimensional superpairs over algebraically closed fields (3.7, 3.9, 3.11 and 3.13).

In view of the close connection between Jordan superpairs V covered by grids and their supercoordinate systems 𝒞, one can expect that many properties of V can be expressed in terms of 𝒞 or the coordinate superalgebra A. For example, in a sequel to this paper we will show that the Gelfand-Kirillov dimensions of V and A coincide.

In another sequel we will describe the Tits-Kantor-Koecher superalgebras 𝔎(V) of Jordan superpairs V covered by a grid, and we will see how properties like (semi)primeness, simplicity or finite generation flow back and forth between V and 𝔎(V). We will also prove that, over fields, 𝔎(V) and V have the same Gelfand-Kirillov dimension. The Lie superalgebras 𝔎(V) are of interest because they provide examples of Lie superalgebras graded by 3-graded root systems. Indeed, generalizing results from [24], we will see that a Lie superalgebra is graded by a 3-graded root system R if and only if it is a perfect central extension of the TKK-superalgebra of a Jordan superpair covered by a grid with associated root system R.

1. Jordan superpairs covered by grids.

1.1. Setting. In this paper we will consider quadratic Jordan superpairs as introduced in [20]. All unexplained notions used here can be found there. Thus, k is an arbitrary base ring, k−salg denotes the category of superextensions of k, i.e., (super)commutative associative unital k-superalgebras. All Jordan superpairs are assumed to be defined over some S ∈ k−salg. More specifically, we will study Jordan superpairs V covered by a grid 𝒢 = {g_{α} : α ∈ R_{1}} where (R, R_{1}) is the associated 3-graded root system. We denote by V = ⨁_{α∈R_{1}} V_{α} the Peirce decomposition of V with respect to 𝒢. Recall g_{α} ∈ V_{α} = (V_{α}^{+}, V_{α}^{−}). Also, we denote by G the Grassmann algebra in a countable number of generators and by G(.) the Grassmann envelopes of algebraic structures.

Suppose (R, R_{1}) = ⨁_{i∈I}(R^{(i)}, R_{1}^{(i)}) is a direct sum of 3-graded root systems (R^{(i)}, R_{1}^{(i)}). Then, by [20, 4.5],

V = ⨁_{i∈I} V^{(i)}, V^{(i)} = ⨁_{α∈R_{1}^{(i)}} V_{α} ◁ V  (1)

is a direct sum of ideals V^{(i)} of V, each covered by the grids 𝒢^{(i)} = {g_{α} : α ∈ R_{1}^{(i)}}. By [22] or [10], every 3-graded root system is a direct sum of irreducible 3-graded root systems. The decompostion (1) then reduces the classification of Jordan pairs covered by a grid to the case of an irreducible root system R or, equivalently, of a connected grid 𝒢. It is no harm to assume that 𝒢 is a standard grid of which there are the following seven types:

(i) rectangular grid ℛ(J, K), 1 ≤ |J| ≤ |K|, (R, R_{1}) is the rectangular grading A_{I}^{J,K} where J ˙∪K = I ˙∪{0} for some element 0 ∉ I and R is a root system of type A and rank |I| (notation of [20]);

(ii) hermitian grid ℋ(I), 2 ≤ |I|, (R, R_{1}) is the hermitian grading of R = C_{I}; (iii) even quadratic form grid 𝒬_{e}(I), 3 ≤ |I|, (R, R_{1}) is the even quadratic form

grading of R = D_{I ˙∪{0}}; (iv) odd quadratic form grid 𝒬_{o}(I), 2 ≤ |I|, (R, R_{1}) is the odd quadratic form grading

of R = B_{I ˙∪{0}};

(v) alternating (= symplectic) grid 𝒜(I), 5 ≤ |I|, (R, R_{1}) is the alternating grading of R = D_{I};

(vi) Bi-Cayley grid ℬ, R = E_{6};

(vii) Albert grid 𝒜, R = E_{7}.

1.2. McCrimmon-Meyberg algebras, supercoordinate systems. Let V be a Jordan superpair over S covered by a connected standard grid 𝒢 with associated 3-graded root system (R, R_{1}). For the classification (= coordinatization) of these Jordan superpairs the concept of a McCrimmon-Meyberg algebra [20, 4.2] is important. This is an alternative superalgebra over S defined for every collinear pair (g_{α}, g_{β}) on V_{α}^{+} = V_{2}^{+}(g_{α}) ∩ V_{1}^{+}(g_{β}) by the product formula ab = {{a g_{α}^{−} g_{β}^{+}} g_{β}^{−}b}.

We will associate to V a supercoordinate system 𝒞. Its definition depends on the type of R. However, for a simply-laced R of rank R ≥ 2, equivalently 𝒢 is an ortho-collinear family of |𝒢| ≥ 2, we have the following uniform description

𝒞 = McCrimmon-Meyberg algebra of some collinear pair g_{α}, g_{β} ∈ 𝒢.  (1)

This algebra is associative for rank R ≥ 3 and even associative supercommutative, i.e. a superextension of S, for R of type D or E. For non-simply-laced root systems, 𝒞 will have more structure and will be defined in the review of the coordinatization theorems below.

Modulo isomorphisms and taking the opposite algebra, the McCrimmon-Meyberg algebra does not depend on the chosen collinear pair g_{α}, g_{β}, see 1.4. To establish this, the following lemma will be needed.

1.3. Lemma. Let 𝒢 ⊂ V be a connected covering standard grid, and let e, f ∈ 𝒢 be two collinear idempotents. Then the exchange automorphism t_{e,f}, [20, 4.2], preserves 𝒢 up to sign:

t_{e,f}(𝒢) ⊂ {±g; g ∈ 𝒢}.  (1)

Proof. Let g ∈ 𝒢. Since t := t_{e,f} exchanges e and f we can assume that g is distinct from e and f. The action of t depends on ij for g ∈ V_{i}(e) ∩ V_{j}(f). We therefore consider the various possibilities for ij:

ij = 21: This case cannot occur since e ≠ g ∈ V_{2}(e) implies g ⊣ e, and then g ⊣ e⊤f yields g ⊣ f or g ⊥ f, contradicting g ∈ V_{1}(f). By symmetry, ij = 12 does not occur either.

ij = 11: In this case t(g) = {ef{feg}}−g, and hence (1) holds if {feg} = 0. If not, then {feg} lies in a Peirce space V_{γ} with respect to 𝒢 and also lies in V_{2}(f)∩V_{0}(e) by the Peirce multiplication rules. The idempotent h ∈ 𝒢 ∩ V_{γ} therefore has the property f ⊢ h ⊥ e. It then follows that e⊤g⊤f, hence h ∈ V_{1}(g) and therefore h ⊣ g. Thus (h; f, e, g) is a diamond of idempotents, hence {feg} = 2h and so {ef{feg}} = {e f 2h} = 2g.

ij = 01: In this case e ⊥ g and f ⊣ g or f⊤g. Since f ⊣ g ⊥ e implies f ⊥ e we must have f⊤g. Because 𝒢 is a grid, there exists ε ∈ {±} and h ∈ 𝒢 such that (e, f, g, εh) is a quadrangle of idempotents. But then t(g) = −{efg} = −εh. The case ij = 10 follows by symmetry.

ij = 02: Here t(g) = Q¯0(e)Q¯0(f)g and f ⊣ g ⊥ e. Because 𝒢 is a grid, f, g imbed into a triangle (f; g, g̃) ⊂ 𝒢 of idempotents and therefore Q¯0(f)g = g̃. It follows that g̃ ∈ V_{2}(e). Hence e ⊢ g̃, and by the same argument there exists h ∈ 𝒢 such that (e; g̃, h) is a triangle of idempotents. It then follows that t(g) = h. The case ij = 20 follows again by symmetry.

ij = 00 or ij = 22: In these cases t = Id on V_{i}(e) ∩ V_{j}(f), finishing the proof of the lemma.

1.4. Lemma. In the setting of 1.2 let (α′, β′) be another collinear pair in R_{1} and denote by A′ the McCrimmon-Meyberg algebra defined on V_{α′}^{+} by (g_{α′}, g_{β′}). Then A′ is isomorphic to A or to its opposite algebra A^{op}.

Proof. Our first goal is to see that, modulo an algebra isomorphism, we can assume α = α′. Because of [21, I Thm. 4.9] the roots α, β, α′, β′ all have the same length. Hence either α = α′ or α⊤α′ or α ⊥ α′. We claim that there is an automorphism φ of V and g_{γ} ∈ 𝒢 such that

φ(g_{α′}) = g_{α} and φ(g_{β′}) = ±g_{γ}.  (1)

This is clear for α = α′ and follows from 1.3 in case α⊤α′. If α ⊥ α′ we can take a product of two exchange automorphisms, thanks to:

if α ⊥ α′ there exists δ ∈ R_{1} such that α⊤δ⊤α′.  (2)

Indeed, since connectedness of 𝒢 is equivalent to R being irreducible, we know, e.g. from [10], that there exists δ ∈ R_{1} with α⊤δ⊤α′ or α ⊣ δ ⊢ α′. But in the latter case, β⊤α ⊣ δ implies β ⊣ δ ⊢ β′ and hence α⊤β⊤α′ by properties of 3-graded root systems, see e.g. [21, I Thm. 4.9] or [10].

We have now established (1). The definition of A and A′ shows that φ|A′ is an algebra isomorphism from A′ onto the McCrimmon-Meyberg algebra A′′ defined on V_{α}^{+} by the collinear pair g_{α}, g_{γ}. If γ = β the lemma is proven. Otherwise, γ ⊥ β or γ⊤β. By [20, 4.4.(8)] we have V_{α} = V_{2}(g_{α}) ∩ V_{1}(g_{β}) ∩ V_{1}(g_{γ}). Hence, by [17, 2.6], A′′ = A^{op} in case γ ⊥ β and A′′ = A in case γ⊤β and {V_{α}^{+} e_{β}^{−} e_{γ}^{+}} = 0. But if γ⊤β and {V_{α}^{+} e_{β}^{−} e_{γ}^{+}} ≠ 0 then 𝒢 has a non-zero Peirce space in V_{2}(g_{α}) ∩ V_{0}(g_{β}) ∩ V_{2}(g_{γ}) and therefore g_{α}, g_{β}, g_{γ} imbed in a diamond of idempotents (g_{δ}; g_{α}, g_{β}, g_{γ}). We will show that A′′ = A^{op} in this case.

By [21, I Thm. 2.11], the diamond of idempotents (g_{δ}; g_{α}, g_{β}, g_{γ}) generates a hermitian grid {h_{ij} ; 1 ≤ i ≤ j ≤ 3}, [20, 5.8]. We can arrange the notation so that g_{α} = h_{12}, g_{β} = h_{13} and g_{γ} = h_{23}. For a, b ∈ V_{α}^{+} let a · b be the McCrimmon-Meyberg algebra product with respect to h_{12}, h_{13} and a ∗ b the one with respect to h_{12}, h_{23}. Using h_{13}^{−} = {h_{12}^{−} h_{23}^{+} h_{33}^{−}} and the identity (JSP15) of [20, 3.2] together with the abbreviations a_{23} = {a h_{12}^{−} h_{13}^{+}} and b_{23} = {h_{23}^{+} h_{12}^{−} b} ∈ V_{γ}^{+} we obtain

a · b = {a_{23} h_{13}^{−} b} = {a_{23} {h_{12}^{−} h_{23}^{+} h_{33}^{−}} b}
= −{h_{23}^{+} h_{12}^{−} {a_{23} h_{33}^{−} b}} + {{h_{23}^{+} h_{12}^{−} a_{23}} h_{33}^{−} b} + {a_{23} h_{33}^{−}{h_{23}^{+} h_{12}^{−} b}}
= {a_{23} h_{33}^{−} {h_{23}^{+} h_{12}^{−} b}} (since {V_{α}^{+} h_{33}^{−} b} = 0 because h_{33} ⊥ h_{12})
= {{h_{13}^{+} h_{12}^{−} a}h_{33}^{−} b_{23}}
= {h_{13}^{+} h_{12}^{−} {a h_{33}^{−} b_{23}}} + {a {h_{12}^{−} h_{13}^{+} h_{33}^{−}} b_{23}} − {a h_{33}^{−}{h_{13}^{+} h_{12}^{−} b_{23}}}
= {a h_{23}^{−} b_{23}} (since {a h_{33}^{−} V_{α}^{+}} = 0 and {h_{12}^{−} h_{13}^{+} h_{33}^{−}} = h_{23}^{−})
= (−1)^{|a||b|} b ∗ a

by definition of ∗.

In order to establish the notation used in the following and to define the supercoordinate system in the other cases, we will now review the coordinatization theorems of Jordan superpairs covered by connected standard grids.

1.5. A_{1}-Coordinatization. A Jordan superpair V over S is covered by a single idempotent if and only if V is isomorphic to the superpair 𝕁 = (J, J) of a unital Jordan superalgebra J over S. In this case we put

𝒞 = J.  (1)

1.6. Rectangular grids and rectangular matrix superpairs. Let A be a unital alternative superalgebra over S and let J, K be arbitrary sets. We denote by Mat(J, K; A) the left A-module of all matrices over A of size J × K with only finitely many nonzero entries. By restriction of scalars, this becomes an S-supermodule with even part Mat(J, K; A¯0) and odd part Mat(J, K; A¯1). The rectangular matrix superpair of size J ×K and with supercoordinate algebra A is the Jordan superpair 𝕄_{JK}(A) = (Mat(J, K; A), Mat(K, J; A)) with product given by

Q+ ¯0(x¯0)y = x¯0(yx¯0); {x y z} = x(yz) + (−1)^{|x||y|+|x||z|+|y||z|}z(yx)  (1)

for homogeneous elements x, z ∈ V^{+}, y ∈ V^{−} and x¯0 ∈ V + ¯0 [20, 5.6]. One obtains Q− ; ¯0 and the other triple product {...} by shifting the brackets in the expressions above one position to the left. We require A to be associative if |J| + |K| ≥ 4. In this case, the brackets in the definition of the Jordan product are of course not necessary.

Let E_{jk} be the usual matrix units. Then e_{jk} = (E_{jk}, E_{kj}) ∈ 𝕄_{JK}(A) is an idempotent and ℛ(M, N) = {e_{jk} : j ∈ J, k ∈ J} is a rectangular grid of size J × K which covers 𝕄_{JK}(A). The joint Peirce spaces of ℛ(J, K) are (AE_{jk}, AE_{kj}). Moreover, A is the McCrimmon-Meyberg algebra of any collinear pair (e_{ij}, e_{ik}) for distinct j, k. The McCrimmon-Meyberg algebra of the collinear pairs (e_{ij}, e_{lj}) for distinct i, l is isomorphic to A^{op}.

By [20, 5.5, 5.7] a Jordan superpair V over S ∈ k−salg is covered by a rectangular grid ℛ(J, K), |J| + |K| ≥ 3, if and only if there exists a unital alternative superalgebra A over S, which is associative if |J| + |K| ≥ 4, such that V ≅ 𝕄_{JK}(A).

1.7. C_{2}-coordinatization. A grid with associated 3-graded root system (R, R_{1}) and R = B_{2} = C_{2} is a triangle of idempotents (g_{0}; g_{1}, g_{2}). By [20, 5.9] a Jordan superpair V over S ∈ k−salg is covered by a such a 𝒢 if and only if V ≅ 𝕁 = (J, J) where J is a Jordan superalgebra over S which contains two strongly connected supplementary orthogonal idempotents c_{1}, c_{2}. In fact, J is the (g_{1} + g_{2})^{−}-isotope of V and the two orthogonal idempotents c_{i} = g_{i}^{+} are connected by g_{0}^{+}. We have a Peirce decomposition 𝔓 of J with respect to (c_{1}, c_{2}) in the form 𝔓 : J = J_{11} ⊕ J_{12} ⊕ J_{22}. In this case, the supercoordinate system of V is

𝒞 = (J, 𝔓).  (1)

1.8. Hermitian grids ℋ(I), |I| ≥ 3 and hermitian matrix superpairs. Let V be a Jordan superpair covered by a hermitian grid ℋ(I) = {h_{ij} : i, j ∈ I}. The joint Peirce spaces of ℋ(I) coincide with the Peirce spaces V_{ij} of the orthogonal system {h_{ii} : i ∈ I}. For the purpose of coordinatization we need an additional assumption in this case: for all i, j ∈ I, i ≠ j the maps

D(h_{ij}^{σ}, h_{jj}^{−σ}) : V_{jj}^{σ} → V_{ij}^{σ} are injective  (1)

(e.g. V has vanishing extreme radical or V has no 2-torsion). Assuming (1), we associate to V the supercoordinate system

𝒞 = (A, A_{0}, π)  (2)

defined as follows:

(a) A is the McCrimmon-Meyberg algebra of a fixed collinear pair (h_{ij}, h_{ik}) which is necessarily associative for |I| ≥ 4;

(b) π is the involution of A given by a^{π} = Q+ ¯0(h_{ij}){h_{ii}^{−} a h_{jj}^{−}};

(c) A_{0} is the ample subspace A_{0} = D(h_{ij}^{+}, h_{jj}^{−})V_{jj}^{+} of (A, π). It is shown in [20, 5.12] that, assuming (1), V is isomorphic to a hermitian matrix superpair ℍ_{I}(A, A_{0}, π) = (H_{I}(A, A_{0}, π), H_{I}(A, A_{0}, π)), where

H_{I}(A, A_{0}, π) = {x = (x_{ij}) ∈ Mat(I, I; A) : x = x^{πT}, all x_{ii} ∈ A_{0}}.  (3)

Conversely, for data (A, A_{0}, π) as above, we can form the hermitian matrix superpair ℍ_{I}(A, A_{0}, π). The S-module H_{I}(A, A_{0}, π) is spanned by elements of type a[ij] = aE_{ij} + a^{π}E_{ji} (a ∈ A, i ≠ j) and a_{0}[ii] = a_{0}E_{ii} (a_{0} ∈ A_{0}). The elements h_{ii} = (1[ii], 1[ii]) and h_{ij} = (1[ij], 1[ij]) = h_{ji},i ≠ j, are idempotents of ℍ_{I}(A, A_{0}, π) such that ℋ(I) = {h_{ij}; i, j ∈ I} is a covering hermitian grid of ℍ_{I}(A, A_{0}, π) satisfying (1). The joint Peirce spaces are (A[ij], A[ij]) for i ≠ j and (A_{0}[ii], A_{0}[ii]). For a suitable choice, the supercoordinate system of ℍ_{I}(A, A_{0}, π) is (A, A_{0}, π).

From now on, we will assume that (1) holds whenever we speak of the supercoordinate system of (V, 𝒢) for a hermitian grid 𝒢.

1.9. Even quadratic form grids and even quadratic form superpairs. An even quadratic form superpair 𝔼ℚ_{I}(A), A ∈ k−salg, has the form 𝔼ℚ_{I}(A) = (H(I, A), H(I, A)) where H(I, A) is the free A-module with basis {h_{±i}; i ∈ I} and 𝔼ℚ_{I}(A) is the quadratic form superpair associated to the hyperbolic form q_{I} : H(I, A) → A. In 𝔼ℚ_{I}(A) the pairs g_{i} = (h_{+i}, h_{−i}) and g_{−i} = (h_{−i}, h_{+i}) are idempotents, and the family 𝒬_{e}(I) = {g_{±i}; i ∈ I} is an even quadratic form grid which covers 𝔼ℚ_{I}(A).

It is shown in [20, 5.14] that a Jordan superpair V over S is covered by an even quadratic form grid 𝒬_{e}(I), |I| ≥ 3 if and only if V is isomorphic to the even quadratic form superpair 𝔼ℚ_{I}(A) where A is the McCrimmon-Meyberg algebra of (V, 𝒬_{e}(I)) which necessarily is a superextension of S.

1.10. Odd quadratic form grids 𝒬_{o}(I),|I| ≥ 2 and odd quadratic form superpairs. Let 0 ∉ I. An odd quadratic form grid can be written in the form 𝒬_{o}(I) = {g_{0}} ˙∪{g_{±i} : i ∈ I} with g_{0} ⊢ g_{±i}. The supercoordinate system associated to a Jordan superpair V covered by 𝒬_{o}(I) is

𝒞 = (A, X, q_{X})  (1)

defined as follows:

(a) A is the McCrimmon-Meyberg algebra of g_{1}, g_{2} ∈ 𝒬_{o}(I) where 1, 2 are two distinct elements of I (note that A = V_{2}^{+}(g_{1}) as S-module), A necessarily is a superextension of S;

(b) X is the A-supermodule X = ⋂_{i∈I,σ=±} V_{1}^{+}(g_{σi}) with the canonical induced ℤ_{2}- grading and the A-action a.x = {a g_{1}^{−} x} for a ∈ A, x ∈ X.

(c) q_{X} = (b_{X}, q_{X¯0}) is the A-quadratic form on X given by b_{X}(x, x′) = {x g_{−1}^{−} x′} and q_{X¯0}(x¯0) = Q¯0(x¯0)g_{−1}^{−}, where g_{−1} ∈ 𝒬_{0}(I) is the unique idempotent orthogonal to g_{1}. The element g_{0}^{+} is a base point of q_{X}, i.e., q_{X¯0}(g_{0}^{+}) = 1.

The supercoordinate system 𝒞 determines V: by [20, 5.16] V is isomorphic over S to the odd quadratic form superpair 𝕆ℚ_{I}(A, q_{X}). This Jordan superpair is the superpair of the A-quadratic form q_{I} ⊕ q_{X} where q_{I} is the hyperbolic form on the hyperbolic superspace over A of rank 2|I|, see 1.9. Conversely, 𝕆ℚ_{I}(A, q_{X}) is covered by a odd quadratic form grid.

1.11. S-extensions. Let U be a Jordan pair over k and let A be a superextension of S. By [20, 3.8] there exists a unique Jordan superpair structure on U_{A} = (A⊗_{k} U^{+}, A⊗_{k} U^{−}), called the A-extension, such that its Grassmann envelope G(U_{A}) is canonically isomorphic to the G(A)-extension U_{G(A)} of U.

The Jordan superpairs arising in the coordinatization theorems for the remaining three types of grids, alternating, Bi-Cayley and Albert grids, are all A-extensions for suitable split Jordan pairs over k [20, 4.9]. The supercoordinate system is given by 1.2.1.

1.12. Alternating grids and alternating matrix superpairs. By [20, 5.18], a Jordan superpair is covered by an alternating grid 𝒜(I), |I| ≥ 4, if and only if it is isomorphic to an alternating matrix superpair 𝔸_{I}(A) where A is the McCrimmon-Meyberg algebra of some collinear pair in 𝒜(I). Here 𝔸_{I}(A) is the A-extension of the split Jordan pair 𝔸_{I}(k) = (Alt(I, k), Alt(I, k)) of alternating matrices over k. Recall that in 𝔸_{I}(k) the family of all e_{ij} = (E_{ij} − E_{ji}, E_{ji} − E_{ij}), i < j, where < is some total order on I, is an alternating covering grid.

1.13. Bi-Cayley grids and Bi-Cayley superpairs. By [20, 5.20] a Jordan superpair over S is covered by a Bi-Cayley grid ℬ if and only if it is isomorphic to the Bi-Cayley superpair 𝔹(A) = A ⊗_{k} 𝕄_{12}(𝕆_{k}), the A-extension of the rectangular matrix superpair 𝔹(k) = 𝕄_{12}(𝕆_{k}) for 𝕆_{k} the split Cayley algebra over k. The corresponding McCrimmon-Meyberg algebra is A for a suitable collinear pair in ℬ.

1.14. Albert grids and Albert superpairs. By [20, 5.22] a Jordan superpair V over S is covered by an Albert grid 𝒜 if and only if there exists a superextension A of S such that V is isomorphic to the Albert superpair 𝔸𝔹(A) = A ⊗_{k} 𝔸𝔹(k), the A-extension of the split Jordan pair 𝔸𝔹(k) = ℍ_{3}(𝕆_{k}, k · 1, π) where 𝕆_{k} is the split Cayley algebra over k with canonical involution π. The corresponding McCrimmon-Meyberg algebra is A for a suitable collinear pair in 𝒜.

1.15. Standard examples. It will be useful to have a common notation for the Jordan superpairs arising in the coordinatization theorems. We denote by 𝕍(𝒢, 𝒞) and call it a standard example the following Jordan superpairs over S ∈ k−salg:

(i) a Jordan superpair 𝕁 = (J, J) of a unital Jordan superalgebra J over S, 1.5, or of a unital Jordan superalgebra with a Peirce decomposition 𝔓 as defined in 1.7;

(ii) a rectangular matrix superpair 𝕄_{JK}(A) for |J| + |K| ≥ 3, 1.6; (iii) a hermitian Jordan superpair ℍ_{I}(A, A_{0}, π), 1.8; (iv) an even or odd quadratic form superpair 𝔼ℚ_{I}(A), 1.9, or 𝕆ℚ_{I}(A, q_{X}), 1.10;

(v) an alternating matrix superpair 𝔸_{I}(A), 1.12, a Bi-Cayley superpair 𝔹(A) or an Albert superpair 𝔸𝔹(A);

The coordinatization theorems can then be summarized by saying that V is a Jordan superpair over S covered by a connected standard grid 𝒢 if and only if V ≅ 𝕍(𝒢, 𝒞) where 𝒞 is the supercoordinate system of V. (Recall that we assume 1.8.1 in the hermitian case.)

2. Semiprime, prime and simple superalgebras.

In this section we will consider semiprime, prime and simple alternative superalgebras and superalgebras with involutions, with emphasis on associative superalgebras. It is not the goal to present a complete theory, rather the results of this section serve as preparation for the following section §3 where we will describe semiprime, prime and simple Jordan superpairs covered by a grid.

Unless stated otherwise, all superalgebras are defined over some superextension S of a base ring k. All alternative superalgebras are assumed to be unital, unless stated otherwise.

2.1. Definitions. By definition, any ideal in a superalgebra is ℤ_{2}-graded. We recall that an alternative or Lie superalgebra A is called

(i) semiprime if II ≠ 0 for any non-zero ideal I of A, (ii) prime if IJ ≠ 0 for any two non-zero ideals I, J of A, or (iii) simple if every ideal I of A is trivial, I = 0 or I = A, and A^{2} ≠ 0.

For prime alternative superalgebras one has the fundamental result 2.2 below.

For an associative superalgebra A the usual proof in the non-supercase shows that semiprimeness and primeness have the following elementwise characterizations ([25, p. 595]):

(1) A is semiprime if and only if a_{μ}Aa_{μ} ≠ 0 for all 0 ≠ a_{μ} ∈ A_{μ}, μ ∈ {¯0, ¯1}.

(2) A is prime if and only if a_{μ}Ab_{ν} ≠ 0 for all 0 ≠ a_{μ} ∈ A_{μ}, 0 ≠ b_{ν} ∈ A_{ν} and μ, ν ∈ {¯0, ¯1}.

More results on prime associative superalgebras over rings containing \frac{1}{2} are given in [19, 1].

Recall that an involution π of a superalgebra A is an S-linear map satisfying the conditions A_{μ}^{π} ⊂ A_{μ} for μ = ¯0, ¯1, (ab)^{π} = (−1)^{|a||b|}b^{π}a^{π} and (a^{π})^{π} = a for homogeneous a, b ∈ A. One says A is π-semiprime or π-prime or π-simple if the conditions (i) or (ii) or (iii) above hold respectively for π-invariant ideals I = I^{π}.

2.2. Theorem. (Shestakov-Zelmanov [31, Thm. 2]) A prime alternative superalgebra A over a field of characteristic ≠ 2, 3 is either associative or A = A¯0 is a Cayley-Dickson ring.

This result is no longer true in characteristic 2, 3. For example, let 𝕆 = ℍ ⊕ vℍ be a Cayley-Dickson algebra over a field of characteristic 2, obtained by the Cayley-Dickson process from a quaternion subalgebra ℍ. Then 𝕆 with respect to the ℤ_{2}-grading 𝕆¯0 = ℍ, 𝕆¯1 = vℍ is a simple alternative superalgebra. Nonassociative alternative superalgebras over arbitrary fields which are simple or which are prime and satisfy an additional assumption are classified in [29].

We next describe the structure of simple associative superalgebras. We start with two important classes of examples.

2.3. Matrices over associative superalgebras. The concept of matrices over an associative superalgebra A, introduced in [11, Ch. 3, 1.7] for matrices of finite size, can easily be extended to finite matrices of arbitrary size. As in 1.6 we denote by Mat(J, K; A) the A-bimodule of matrices of size J × K, where J and K are arbitrary sets, with entries from A which are finite in the sense that only finitely many entries are non-zero. In 1.6 we have used the ℤ_{2}-grading of Mat(J, K; A) given by the grading of A, i.e., Mat(J, K; A)_{μ} = Mat(J, K; A_{μ}) for μ = ¯0, ¯1. This can be generalized to a ℤ_{2}-grading which, in addition to the grading of A, uses two partitions J = M ˙∪N, K = P ˙∪Q and which will therefore be denoted Mat(M|N, P |Q; A). For this new grading, a matrix x ∈ Mat(J, K; A), written as

\begin{matrix}x = & ( \begin{matrix}x_{1} & x_{2} \\ x_{3} & x_{4} \end{matrix} ) \begin{matrix}M \\ N \end{matrix} \\ P & Q \end{matrix}

with x_{1} ∈ Mat(M, P; A), x_{2} ∈ Mat(M, Q; A), x_{3} ∈ Mat(N, P; A) and x_{4} ∈ Mat(N, Q; A) , is even (respectively odd) if all entries in x_{1} and x_{4} are even (respectively odd) and if all entries in x_{2} and x_{3} are odd (respectively even). Thus, symbolically,

Mat(M|N, P |Q; A)_{μ} = ( \begin{matrix}Mat(M, P; A_{μ}) & Mat(M, Q; A_{μ+¯1}) \\ Mat(N, P; A_{μ+¯1}) & Mat(N, Q; A_{μ}) \end{matrix} )  (1)

for μ = ¯0, ¯1. We note that ([11, III.1.7])

Mat(M|N, P |Q; A) ≅ Hom_{A}(A^{(P |Q)}, A^{(M|N)})  (2)

where A^{(M|N)} = A^{(M)}⊕(ΠA)^{(N)} and ΠA denotes the regular module with changed parity. Thus, A_{μ}^{(M|N)} = (A_{μ})^{(M)} ⊕ (A_{μ+¯1})^{(N)} for μ = ¯0, ¯1 and A^{(M|N)} has a basis composed of an even part of size M and an odd part of size N. With the usual matrix product

Mat_{P |Q}(A) := Mat(P |Q, P |Q; A)  (3)

becomes an associative (not necessarily unital) superalgebra. As in 3.8 one can show that the ideals of Mat_{P |Q}(A) are given by Mat_{P |Q}(B) where B is an ideal of A, and hence

Mat_{P |Q}(A) is semiprime, prime or simple  (4)

⟺ A is semiprime, prime or simple.

Following standard practice, we will replace the sets M, N, P, Q in the notations above with their cardinality in case they are all finite. Thus, we will write Mat_{p|q}(A) = Mat_{P |Q}(A) if |P | = p < ∞ and q = |Q| < ∞. Note that in this case Mat_{p|q}(A) is unital.

2.4. Doubles of associative algebras. Let B be an associative algebra. The superalgebra Mat_{1|1}(B) has a canonical subalgebra

𝔻(B) : = \{( \begin{matrix}a & b \\ b & a \end{matrix} ) : a, b ∈ B\}  (1)
= B ⊕ Bu for u = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )  (2)

which we call the double of B. For B = Mat(n, n; k) this superalgebra already occurs in [4] where it is denoted Q_{n}(k) because of its relation to the simple Lie superalgebras Q_{n}, see [1]. Note that u is a central element of the associative algebra 𝔻(B), i.e., xu = ux for all x ∈ 𝔻(B), but does not supercommute with all elements, in fact [u, u] = 2. In terms of the representation (2), the superalgebra 𝔻(B) can equivalently be described as follows:

𝔻(B)¯0 = B , 𝔻(B)_{1} = Bu.  (3)

with associative product · given by (a, b ∈ B)

a · b = ab, a · bu = (ab)u = au · b, au · bu = ab.  (4)

These formulas show that this doubling process is closely related to the Kantor doubling process in which B is replaced by a superextension S of k and au · bu is given by a second product on S, see [6], [7], [28] or the recent preprint [12].

In the following we will discuss some easily established properties of doubles. Any ideal of 𝔻(B) has the form 𝔻(I) = I ⊕ Iu where I is an ideal of B. It then follows easily that 𝔻(B) is semiprime, prime or simple if and only if B is respectively semiprime, prime or simple. A double 𝔻(B) is a division superalgebra if and only if B is a division algebra.

Suppose π is an involution of 𝔻(B). Then there exists a unique element z ∈ Z(B), the centre of B, satisfying

z^{2} = −1, z^{π} = −z and (a ⊕ bu)^{π} = a^{π} ⊕ b^{π}zu  (5)

for a, b ∈ B. Conversely, given an associative algebra B with involution π and an element z ∈ Z(B) with z^{2} = −1 and z^{π} = −z the extension of π to 𝔻(B), defined as in (5) above, is an involution. In particular, if 2B = 0 then 𝔻(B) always has an involution given by z = 1, and this is the only one in case B is simple. If B is an algebra over a field of characteristic ≠ 2 then a necessary condition for the existence of an involution on 𝔻(B) is that B has an involution of second kind, i.e., π|Z(B) ≠ Id. In particular, if B is central in the sense that Z(B) = k then 𝔻(B) does not have an involution. The symmetric elements of 𝔻(B) with respect to the involution π are given by

H(𝔻(B), π) = H(B, π) ⊕ u \{ \begin{matrix}H(B, π) & if z = 1 \\ (1 + z)H(B, π) & if \frac{1}{2} ∈ S. \end{matrix}  (6)

Any 𝔻(B)-module X satisfies X¯1 = uX¯0, and hence we can put X = 𝔻(X¯0) with obvious meaning. Observe 𝔻(B)^{(P |Q)} ≅ 𝔻(B^{(P ˙∪Q)}) for free 𝔻(B)-modules.

For two 𝔻(B)-modules X = 𝔻(X¯0), Y = 𝔻(Y¯0) we have Hom_{𝔻(B)}(X, Y) = Hom_{B}(X¯0, Y¯0)⊕u Hom_{B}(X¯0, Y¯0) = 𝔻(Hom_{B}(X¯0, Y¯0)). For endomorphisms between free modules this leads to the following identification of matrix superalgebras:

Mat_{P |Q}(𝔻(B)) ≅ 𝔻(Mat(P ˙∪Q, P ˙∪Q; B))  (7)

with respect to the map Mat_{P |Q}(𝔻(B)) → 𝔻(Mat(P ˙∪Q, P ˙∪Q; B) given by

( \begin{matrix}x_{1}u^{s} & x_{2}u^{s+1} \\ x_{3}u^{s+1} & x_{4}u^{s} \end{matrix} ) ↦ u^{s} ( \begin{matrix}x_{1} & x_{2} \\ x_{3} & x_{4} \end{matrix} ) , s = 0, 1.

A double 𝔻(B) is supercommutative if and only if B is commutative and 2B = 0. Assume this to be the case. Then 𝔻(B) is commutative as an algebra and a 𝔻(B)-quadratic form q = (b, q¯0) on a 𝔻(B)-module X = 𝔻(X¯0) is uniquely determined by q¯0 and its polar b¯0 = b|X¯0 × X¯0: b is the 𝔻(B)-extension of b¯0.

2.5. Proposition. (Wall) An associative superalgebra A is simple as superalgebra if and only if either A is simple as algebra or A = 𝔻(A¯0) and A¯0 is simple.

In particular, an associative superalgebra A over an algebraically closed field k is finite dimensional and simple if and only if either A ≅ 𝔻(Mat(m, m; k)) or A ≅ Mat_{p|q}(k) for finite numbers m, p and q.

Proof. The description of a simple associative superalgebra is given in [30, Lemma 3]. The two cases are mutually exclusive since in the second case {a¯0 ⊕ a¯0u : a¯0 ∈ A¯0} is a proper ideal.

The finite-dimensional result is implicit in [30]. It can also be derived from 2.7 and 2.4.7 using the easily established fact that a finite-dimensional associative division superalgebra D over k is either isomorphic to k or to 𝔻(k).

2.6. Corollary. Let A be a superextension of S, and put Tor_{2}(A) = {a ∈ A : 2a = 0}.

(a) A is semiprime if and only if A does not have non-zero homogeneous nilpotent elements. In this case, A¯1 ≠ 0 ⟹ Tor_{2}(A) ≠ 0.

(b) A is prime if and only if A does not contain non-zero homogeneous zero divisors. In this case, the algebra A is commutative and A¯1 ≠ 0 ⟹ 2A = 0, i.e., Tor_{2}(A) = A.

(c) The following are equivalent:

(i) A is a simple superalgebra, (ii) every non-zero homogeneous element is invertible, (iii) either A = A¯0 is a field or A = 𝔻(A¯0) where A¯0 is a field of characteristic 2.

(d) In particular, if \frac{1}{2} ∈ A then A is prime or simple if and only if A = A¯0 is an integral domain or a field.

Proof. (a) By 2.1(1) we have A is semiprime if and only if a_{μ}^{2} ≠ 0 for all 0 ≠ a_{μ} ∈ A_{μ}, μ ∈ {¯0, ¯1}. This in turn is equivalent to the stated condition. We have a2 ¯1 = −a2 ¯1 whence a_{1}^{2} ∈ Tor_{2}(A) for all a¯1 ∈ A¯1.

(b) The characterization of prime superextensions A is immediate from 2.1(2). The algebra A is a commutative algebra if and only if A2 ¯1 ⊂ Tor_{2}(A), a condition trivially fulfilled if A = A¯0. Also, Tor_{2}(A) and 2A are ideals of A with Tor_{2}(A)(2A) = 0. Hence Tor_{2}(A) = 0 or 2A = 0 in any prime A. In the first case A = A¯0 by (a), while in the second case we have A = Tor_{2}(A) and again commutativity follows.

(c) Suppose that A is simple, and let 0 ≠ a_{μ} ∈ A_{μ}. Then A = a_{μ}A by simplicity, and hence there exists b_{μ} ∈ A_{μ} with a_{μ}b_{μ} = 1. If μ = ¯1 we have 2A = 0 by (b), and so also b_{μ}a_{μ} = 1. Hence in all cases a_{μ} is invertible. The converse is obvious. That (i) ⇔ (iii) follows from (b) and 2.5.

(d) is a special case of (b) and (c).

More can be said in the case of associative superalgebras which are Artinian, i.e., they satisfy the descending chain condition for right ideals ([25]). Let us also recall that a superalgebra is a division superalgebra if every non-zero homogeneous element is invertible. Clearly, division superalgebras are simple superalgebras. Associative division superalgebras are described in [25, p. 605].

2.7. Theorem. (Racine) An associative superalgebra A is Artinian and simple if and only if there exists an associative division superalgebra D such that one of the following two alternatives holds:

(i) D¯1 ≠ 0, and there exists a finite m such that A ≅ Mat(m, m; D) where the ℤ_{2}-grading is given by Mat(m, m; D)_{μ} = Mat(m, m; D_{μ}) for μ = ¯0, ¯1;

(ii) D¯1 = 0, and there exist finite p, q such that A ≅ Mat_{p|q}(D).

Proof. It is shown in [25, Thm. 3] that an Artinian simple associative superalgebra is isomorphic, as superalgebra, to End_{D} V where D is a division superalgebra and V is supermodule over D which is finite-dimensional as D¯0-module. As is mentioned in the discussion before [25, Thm. 3] this implies the two cases above. Indeed, if D¯1 ≠ 0 then D¯1 = xD¯0 for any non-zero x ∈ D¯1 and V¯1 = xV¯0 follows, implying that V ≅ D^{m} as supermodules and A ≅ Mat(m, m; D). The converse is clear.

We now turn to superalgebras with involutions. The following characterization of πsimple superalgebras (2.1) can be proven in the same way as the corresponding result for algebras ([25, Lemma 11] for A associative).

2.8. Lemma. A superalgebra A with involution π is π-simple if and only if either A is simple as superalgebra or A = B ⊕ B^{π} for a simple ideal B of A.

We note that in the second case we have A ≅ B ⊕ B^{op} with exchange involution π. More precise information can be obtained for alternative superalgebras containing an ample subspace.

2.9. Lemma. Let A be an alternative superalgebra which is π-simple with respect to an involution π of A, and let A_{0} be an ample subspace of (A, π). Then there are the following two possibilities:

(i) A ≅ B ⊕B^{op} with π the exchange involution, B a simple associative superalgebra and A_{0} = H(A, π), or

(ii) A is a simple alternative superalgebra (see 2.2).

Proof. We can assume that A ≅ B ⊕ B^{op} where B is a simple superalgebra and π is the exchange involution. Then H(A, π) = {b ⊕ b : b ∈ B} = {a + a^{π} : a ∈ A} ⊂ A_{0} ⊂ H(A, π), whence H(A, π) = A_{0}. Since A_{0} ⊂ N(A), the nucleus of A, and since N(A) = N(B) ⊕ N(B^{op}), we obtain A = N(A), i.e., A is associative.

2.10. Theorem. Let A be a π-simple associative superalgebra where π is an involution π of A. Then precisely one of the following cases holds:

(i) A is not a simple superalgebra: there exists a simple ideal B of A such that A = B ⊕ B^{π}.

(ii) A = 𝔻(A¯0) for a simple A¯0 and there exists z ∈ Z(A¯0) such that 2.4.5 holds. (iii) A is simple as algebra, A¯0 = B_{1} ⊕ B_{2} is a direct sum of non-zero simple ideals

B_{i} and A¯1 = C_{1} ⊕ C_{2} is a direct sum of irreducible A¯0-bimodules C_{i} = e_{i}Ae_{j} where e_{i} is the unit element of B_{i} and (i, j) = (1, 2) or (i, j) = (2, 1). Moreover,

either B_{i}^{π} = B_{j}, C_{i}^{π} = C_{i} or B_{i}^{π} = B_{i}, C_{i}^{π} = C_{j}.  (1)

(iv) A and A¯0 are simple as algebras.

Proof. By 2.8 we either have (i) or A is a simple superalgebra with involution. In the second case we can apply 2.5: either we have (ii) or A is simple as an algebra. If A is a simple algebra we either have (iv) or A¯0 is not simple. In the following we can therefore assume that A is a simple algebra but A¯0 is not. We will see that this leads to (iii).

Let then B be a non-trivial ideal of A¯0. By [30, Lemma 1], we have

A¯0 = A2 ¯1 = B + A¯1BA¯1 and A¯1 = A¯1B + BA¯1.  (2)

In particular, this implies that A¯1BA¯1 is a non-zero ideal. If the ideal I = B ∩ A¯1BA¯1 is non-zero, it is proper and hence (2) also holds for I leading to the contradiction A¯1IA¯1 ⊂ A2 ¯1BA2 ¯1 ⊂ B and A¯0 = B. Therefore A¯0 = B ⊕ A¯1BA¯1 is a direct sum of ideals. Since this holds for any proper ideal it is immediate that B_{1} := B and B_{2} := A¯1BA¯1 are simple. It follows that either B_{i}^{π} = B_{i} or B_{i}^{π} = B_{j} for i, j as in (iii). Any x ∈ A¯1B ∩ BA¯1 has the property xA¯1 ∈ BA2 ¯1 ∩ A¯1BA¯1 = B_{1} ∩ B_{2} = 0, whence A¯1 = BA¯1 ⊕ A¯1B. Also observe A¯1B_{1} = A¯1BA2 ¯1 = B_{2}A¯1 and then B_{1}A¯1 = A¯1B_{2} by symmetry. We can now show the missing part of (1), putting C_{1} = BA¯1 = B_{1}A¯1 and C_{2} = A¯1B = B_{2}A¯1: if B_{i} = B_{i}^{π} then C_{i}^{π} = (B_{i}A¯1)^{π} = A¯1B_{i} = C_{j} while for B_{i}^{π} = B_{j} we get C_{i}^{π} = A¯1B_{j} = C_{i}. Let e_{i} ∈ B_{i} such that 1 = e_{1} ⊕ e_{2} ∈ A_{0} = B_{1} ⊕ B_{2}. Then e_{1}, e_{2} are orthogonal idempotents. Their Peirce spaces A_{ij} = e_{i}Ae_{j} are A_{11} = B_{1}, A_{12} = C_{1}, A_{21} = C_{2} and A_{22} = B_{2}. Therefore we have the following multiplication rules

B_{i}C_{i} = C_{i} = C_{i}B_{j}, C_{i}C_{j} = B_{i}, 0 = B_{i}C_{j} = C_{i}B_{i} = C_{i}C_{i}  (3)

for (i, j) as in (iii). Using these, it is straightforward to show that any proper A¯0-sub-bimodule D of C_{i} gives rise to an ideal DC_{j} ⊕ C_{j}D ⊕ D ⊕ C_{j}DC_{j} of A, proving that C_{i} is irreducible.

Remarks. (a) This theorem generalizes [25, Thm. 12] which describes π-simple associative superalgebras A for which A¯0 is not π-simple, i.e., the cases (i) and (iii) with the second alternative in (1). Involutions of associative division superalgebras are studied in Prop. 9 and 10 of [25]. These are special cases of (ii) and (iv) above. We point out that case (iv) can only occur if A is infinite-dimensional over the even part of the centre of the associative algebra A ([25, Lemma 8 and Remark p. 607]). Involutions of simple associative superalgebras are also studied in [3, 3] and [2, 2].

(b) It is convenient to view A in case (iii) as a Morita context,

A = ( \begin{matrix}B_{1} & C_{1} \\ C_{2} & B_{2} \end{matrix} )  (4)

with involutions given by

( \begin{matrix}b_{1} & c_{1} \\ c_{2} & b_{2} \end{matrix} ) ↦ ( \begin{matrix}b_{2}^{π} & c_{1}^{π} \\ c_{2}^{π} & b_{1}^{π} \end{matrix} ) or ( \begin{matrix}b_{1} & c_{1} \\ c_{2} & b_{2} \end{matrix} ) ↦ ( \begin{matrix}b_{1}^{π} & c_{2}^{π} \\ c_{1}^{π} & b_{2}^{π} \end{matrix} )  (5)

For A = Mat_{p|q}(D) with D = D¯0 a division algebra and p, q > 0 more precise information on these two types of involutions is given in Prop. 13 and Prop. 14 of [25]. In particular, both types do occur, see also [25, Example p. 602] for the first type. Note that by 2.4 and 2.5 any finite-dimensional simple associative superalgebra over k of characteristic ≠ 2 which has an involution is isomorphic to some Mat_{p|q}(k).

(c) The theorem above is similar to the description of (π, φ)-simple associative algebras with commuting involution π and involutorical automorphism φ ([18, Prop. 2.8]).

3. Semiprime, prime and simple Jordan superpairs covered by grids.

In this section we will describe semiprime, prime and simple Jordan superpairs covered by a grid. Our analysis will first reduce the problem to the case of a connected grid, whence one of the standard examples introduced in §1. We will then describe all ideals in the standard examples in terms of the associated supercoordinate system. Their knowledge will easily allow us to determine the conditions for semiprimeness, primeness and simplicity.

Unless stated otherwise, all Jordan superpairs are defined over some superextension S of a base ring k.

3.1. Semiprime and prime Jordan superpairs. The definition of an ideal in a Jordan superpair is given in [20, 3.3]. In particular, any ideal in a Jordan superpair is ℤ_{2}-graded. For ideals I, J of a Jordan superpair V we define their Jordan product I ⋄ J = ((I ⋄ J)^{+}, (I ⋄ J)^{−})) by

(I ⋄ J)^{σ} = Q¯0(Iσ ; ¯0)J^{−σ} + {I^{σ} , J^{−σ} , I^{σ}}.

We note that I ⋄ J is in general not an ideal. We will say V is (i) semiprime if the Jordan cube I ⋄ I ≠ 0 for any non-zero ideal I of V, (ii) prime if I ⋄ J ≠ 0 for any two non-zero ideals I and J of V.

Primenness can also be defined in terms of annihilators, where for an ideal I in a Jordan superpair V the annihilator

Ann_{V} (I) = (Ann(I)+ ¯0 ⊕ Ann(I)+ ¯1, Ann(I)− ; ¯0 ⊕ Ann(I)− ; ¯1)

is given by z ∈ Ann(I)_{μ}^{σ} for σ = ± and μ ∈ {¯0, ¯1} if and only if

0 = D(z^{σ}, V^{−σ}) = D(V^{−σ}, z^{σ}) = Q(z^{σ}, V^{σ})
= Q¯0(I−σ ; ¯0)z = Q¯0(Iσ ; ¯0)Q¯0(V −σ ; ¯0)z , and in addition for μ = ¯0 :
0 = Q¯0(z)I^{−σ} = Q¯0(I−σ ; ¯0)Q¯0(z) .

The Grassmann envelope of Ann(I) coincides with the annihilator of the ideal G(I) in G(V). Since the annihilator of an ideal in a Jordan pair is again an ideal, [16, §4], Ann_{V} (I) is an ideal of the Jordan superpair V. As in [16, Prop. 1.6] one can show that the following conditions are equivalent for a Jordan superpair V:

(i) V is prime, (ii) V is semiprime and any two non-zero ideals of V have non-zero intersection, and (iii) V is semiprime and the annihilator of any non-zero ideal of V vanishes.

3.2. Doubles of Jordan pairs. The double 𝔻(B) of an associative algebra B as defined in 2.4 can be viewed as the tensor product algebra 𝔻(B) = 𝔻(k) ⊗ B. This motivates the following definition. For a Jordan pair U over k, the double of U is defined as the superextension

𝔻(U) := U_{𝔻(k)} = 𝔻(k) ⊗_{k} U  (1)

of U in the sense of [20, 3.8]. We recall that 𝔻(U) is a Jordan superpair over 𝔻(k) with 𝔻(U)¯0 = U, 𝔻(U)¯1 = Uu ≅ U and whose Jordan triple product satisfies {x_{1} ⊗ v_{1}, x_{2} ⊗ v_{2}, x_{3} ⊗ v_{3}} = (x_{1}x_{2}x_{3}) ⊗ {v_{1}, v_{2}, v_{3}}.

The relation between ideals of U and 𝔻(U) is not as simple as in the case of superalgebras (2.4). A submodule I_{0} ⊕ I_{1}u is an ideal of 𝔻(U) if and only if I_{0} is an ideal and I_{1} is an outer ideal of U, related to each other by {U U I_{1}} + {U I_{1} U} ⊂ I_{0} and Q(I_{0})U + {U U I_{0}} + {U I_{0} U} ⊂ I_{1}. In particular, if I is an ideal of U then 𝔻(I) = I ⊕ Iu is an ideal of 𝔻(U). We have 𝔻(I) ⋄ 𝔻(J) = 𝔻(I ⋄ J) for two ideals I, J of U. It is then easy to see that U is semiprime, prime or simple if 𝔻(U) is so, and the converse is true if \frac{1}{2} ∈ k. We note that for any field F of characteristic 2, the Jordan pair U = (F, F) is simple but Uu is a nontrivial ideal of 𝔻(U) with Uu ⋄ Uu = 0, showing that 𝔻(U) is not even semiprime.

Let B be an extension of k, i.e., a unital associative commutative k-algebra. Then B ⊗_{k} U is a Jordan pair over B and we have a canonical isomorphism of Jordan superpairs over 𝔻(B)

𝔻(B ⊗_{k} U) = 𝔻(B) ⊗_{B} (B ⊗_{k} U) ≅ 𝔻(B) ⊗_{k} U  (2)

where 𝔻(B) ⊗_{k} U is the 𝔻(B)-extension of U. Similarly, taking doubles commutes with taking rectangular matrices, in the following sense. Let 𝕄_{JK}(B) be a rectangular matrix pair with an associative coordinate algebra B. Then

𝔻(𝕄_{JK}(B)) ≅ 𝕄_{JK}(𝔻(B)).  (3)

Recall 2.4 that a supercommutative superalgebra 𝔻(B) necessarily has 2B = 0 and hence is commutative. In this case, any 𝔻(B)-quadratic form q on X = 𝔻(X¯0) is uniquely given by the quadratic form q¯0 on X¯0. This easily implies that

𝕆ℚ_{I}(𝔻(B), q_{X}) = 𝔻(𝕆ℚ_{I}(B, q¯0)).  (4)

After these preparations we now consider Jordan superpairs covered by a grid. Our first result will reduce our study to the case of connected grids.

3.3. Proposition. Let V be a Jordan superpair covered by a grid 𝒢 = {g_{α} : α ∈ R_{1}} with associated 3-graded root system (R, R_{1}) and joint Peirce spaces V_{α}, whence V = ⨁_{α∈R_{1}} V_{α}. Then any ideal I of V splits in the sense that

I = ⨁_{α∈R_{1}} (I ∩ V_{α}), I ∩ V_{α} = (I^{+} ∩ V_{α}^{+}, I^{−} ∩ V_{α}^{−}).  (1)

(a) Suppose (R, R_{1}) = ⨁_{c∈C}(R^{(c)}, R_{1}^{(c)}) is a direct sum of 3-graded root systems (R^{(c)}, R_{1}^{(c)}) so that

V = ⨁_{c∈C} V^{(c)}, V^{(c)} = ⨁_{α∈R_{1}^{(c)}} V_{α} ◁ V

is a direct sum of ideals V^{(c)} of V, covered by the grids 𝒢^{(c)} = {g_{α} : α ∈ R_{1}^{(c)}}, [20, 4.5]. Then I = ⨁_{c∈C} (I ∩ V^{(c)}) and each I ∩ V^{(c)} is an ideal of the Jordan superpair V^{(c)}. In particular,

V semiprime ⟺ every V^{(c)} is semiprime, and  (2)
V prime ⟹ R is irreducible.  (3)

(b) Let (α, β) be a collinear pair and let A_{α} be the associated McCrimmon-Meyberg algebra on V_{α}^{σ}. Then I ∩ V_{α} is an ideal of A_{α}.

Proof. Any x ∈ I^{σ} is a finite sum x = ∑_{α∈R_{1}} x_{α}, x_{α} ∈ V_{α}^{σ}. Suppose there exists x ∈ I for which x_{α} ∉ I^{σ} for some α ∈ R_{1}. Among all such x we choose one which has a minimal number of non-zero components x_{α}, whence all non-zero x_{α} ∉ I^{σ}. Let E_{i}^{σ}(α) be the projection operator onto the Peirce space V_{i}^{σ}(g_{α}), i = 0, 1, 2, [9, 5.4]. Since all E_{i}^{σ}(α) are multiplication operators we have E_{i}^{σ}(α)I^{σ} ⊂ I^{σ}. In particular, if V_{2}(g_{α}) = V_{α}, i.e., α is a long root, x_{α} = E_{2}^{σ}(α)x_{α} = E_{2}^{σ}(α)x ∈ I^{σ}. Thus, for all non-zero components x_{α} we have V_{α}^{σ} ⫋ V_{2}^{σ}(g_{α}). The connected component of α is therefore either a hermitian grading or an odd quadratic form grading. In the first case, α imbeds in a triangle (α; β, γ) such that V_{2}(g_{α}) = V_{α} ⊕ V_{β} ⊕ V_{γ}, and V_{α}^{σ} = E_{1}^{σ}(β)E_{1}^{σ}(γ)V^{σ}. As before this implies x_{α} ∈ I^{σ}, contradicting our choice of x. Hence, we now know that for all non-zero x_{α} the connected component C_{α} of the root α is an odd quadratic form grading, α is the unique short root in C_{α} ∩ R_{1} and for every other non-zero x_{α′} the root α′ lies in a different connected component. Since then E_{2}^{σ}(α)x_{α′} = 0 it follows again that x_{α} ∈ I^{σ}, leading to the final contradiction and hence to the proof of (1). The proof of (a) is then immediate from the definitions, using that every 3-graded root system is a direct sum of irreducible 3-graded root systems, see [22] or [10]. (b) follows from the definition of the product in A_{α}, [20, (4.2.2)].

In view of this result we will assume for the remainder of this section that V is a Jordan superpair covered by a connected grid 𝒢, equivalently, the associated 3-graded root system of 𝒢 is irreducible. Hence, V is one of the standard examples of §1. We start with the case |𝒢| = 1, i.e., R = A_{1}.

3.4. Example: V = 𝕁 = (J, J) for a unital Jordan superalgebra J. The ideals of 𝕁 are easily described:

(I^{+}, I^{−}) is an ideal of (J, J) if and only if I^{+} = I^{−} is an ideal of J.

Indeed, let (I^{+}, I^{−}) be an ideal of V = (J, J). We then have I^{σ} = U¯0(1)I^{σ} ∈ Q¯0(V −σ ; ¯0)I^{σ} ⊂ I^{−σ}, i.e., I^{+} = I^{−}. Moreover, I = I^{+} = I^{−} is an ideal of J since U¯0(I¯0)J = Q¯0(I+ ¯0)V^{−} ⊂

I^{+} = I, U¯0(J¯0)I = Q¯0(V − ; ¯0)I^{+} ⊂ I^{−} = I, {I, J, J} = {I^{+}, V^{−}, V^{+}} ⊂ I^{+} = I and

{J, I, J} = {V^{+}, I^{−}, V^{+}} ⊂ I^{−} = I.

Semiprimeness and primeness of a unital Jordan superalgebra J are defined in the same way as for Jordan superpairs. It therefore follows from the above that

𝕁 is semiprime, prime or simple

⟺ J is respectively semiprime, prime or simple.  (1)

We next consider the case R = B_{2}.

3.5. Jordan superpairs covered by a triangle. By 1.7 these are the Jordan superpairs V ≅ 𝕁 = (J, J) for a unital Jordan superalgebra J over S which contains two strongly connected supplementary orthogonal idempotents c_{1} and c_{2} and hence has Peirce decomposition 𝔓, which for convenient notation we write in the form J = J_{2} ⊕ J_{1} ⊕ J_{0} where J_{i} = J_{i}(c_{1}) denotes the Peirce spaces of c_{1}.

We have seen in 3.4 that the ideals of superpairs 𝕁 are of the form 𝕀 = (I, I), where I is an ideal of J. By 3.3.1, I is a direct sum of its Peirce components

I = I_{2} ⊕ I_{1} ⊕ I_{0} where I_{i} = I ∩ J_{i}.  (1)

The criterion 3.4.1 also holds in this case.

Since I ⊂ J is an ideal if and only if the Grassmann envelope G(I) is an ideal of G(J) the Ideal Criterion in [15, 1.5] provides a description of the ideals of J.

We will next consider a class of Jordan superpairs which are superextensions of Jordan pairs in the sense of 1.11. This class covers 4 types of standard examples, see 3.7 below.

3.6. Theorem. Suppose A is a superextension of S and V = A⊗_{k}U is the A-extension of a Jordan pair U over k which is split of type 𝒢, i.e., U^{σ} = ⨁_{α∈R_{1}} k·g_{α}^{σ} for σ = ±, where 𝒢 = {g_{α} : α ∈ R_{1}} is a connected ortho-collinear standard grid and A is the McCrimmon-Meyberg algebra of some collinear pair g_{α}, g_{β} ∈ 𝒢.

(a) The ideals of V are precisely the spaces B ⊗_{k} U where B is an ideal of the superextension A of S.

(b) V is semiprime, prime or simple if and only if A is respectively semiprime, prime and simple. In particular:

(i) If \frac{1}{2} ∈ S then V is semiprime or prime if and only if V¯1 = 0 and V = V¯0 = A¯0 ⊗_{k} U where A¯0 is, respectively, a semiprime or prime extension.

(ii) V is simple if and only if either A = A¯0 is a field (and hence V = V¯0 is a simple Jordan pair) or A¯0 is a field of characteristic 2 and V = 𝔻(A¯0 ⊗_{k} U).

Proof. (a) Let ⊗ = ⊗_{k} throughout. It is easily checked that B ⊗ U is an ideal of V for any ideal B of A. Conversely, by 3.3, any ideal I of V has the form I^{σ} = ⨁_{α∈R_{1}} I_{α}^{σ} where each I_{α}^{σ} is an S-submodule of V_{α}^{σ} = A ⊗ kg_{α}^{σ} and can therefore be written in the form I_{α}^{σ} = B_{α}^{σ} ⊗ kg_{α}^{σ} where (B_{α}^{σ} : σ = ±, α ∈ R_{1}) is a family of S-submodules of A. Since Q¯0(1 ⊗ g_{α}^{σ})Q¯0(1 ⊗ g_{α}^{−σ})|V_{2}^{σ}(g_{α}) = Id we have Q¯0(1 ⊗ g_{α}^{σ})I_{α}^{−σ} = I_{α}^{σ} and hence B^{−σ} ⊗ kg_{α}^{σ} = Q¯0(1 ⊗ g_{α}^{σ})(B_{α}^{−σ} ⊗ kg_{α}^{−σ}) = Q¯0(1 ⊗ g_{α}^{σ})I_{α}^{−σ} = I^{σ} ∩ V_{α}^{σ} = B_{α}^{σ} ⊗ kg_{α}^{σ}. We can therefore assume B_{α}^{+} = B_{α}^{−} =: B_{α} for all α ∈ R_{1}.

Suppose α⊤β. Since the exchange automorphism t_{αβ} = t_{g_{α},g_{β}} acts on V_{α}^{+} as the endomorphism D(g_{β}^{+}, g_{α}^{−})|V_{α}^{+} it maps I_{α}^{+} onto I_{β}^{+} which implies B_{α} ⊗ kg_{β}^{+} = t_{αβ}(B_{α} ⊗ kg_{α}^{+}) = B_{β} ⊗ kg_{β}^{+}. Because 𝒢 is connected, i.e., (R, R_{1}) is irreducible, any two distinct roots α, β ∈ R_{1} are connected by a chain of collinear roots, in fact either α⊤β or α ⊥ β in which case there exists γ ∈ R_{1} such that α⊤γ⊤β. Therefore B_{β} ⊗ kg_{β}^{+} = B_{α} ⊗ kg_{β}^{+} and we can assume that B_{α} =: B is independent of α and so I = B ⊗ U. Since for some collinear pair α, β, A is the McCrimmon-Meyberg algebra it follows from 3.3(b) that B is an ideal of A.

(b) The simplicity criterion is immediate from (a). The proof for semiprimeness and primeness can be done at the same time. Let B and C be ideals of A with B = C in the case of semiprimeness. First suppose V is semiprime or prime and that BC = 0. It then follows from the definition of the product in V that the ideals B ⊗U and C ⊗U of V satisfy (B ⊗U)⋄(C ⊗U) = 0. By (semi)primeness, B ⊗U = 0 or C ⊗U = 0 and consequently B or C is zero. Conversely, suppose that A is prime and that (B ⊗U)⋄(C ⊗U) = 0. Let g_{α} and g_{β} in 𝒢 be two collinear idempotents defining A. For all b, b̃ ∈ B and all c ∈ C, we have that b⊗g_{α}^{+}, b̃⊗g_{β}^{+} ∈ B ⊗U^{+} and c⊗g_{α}^{−} ∈ C ⊗U^{−}, so 0 = {b⊗g_{α}^{+} , c⊗g_{α}^{−} , b̃⊗g_{β}^{+}} = (bcb̃)⊗g_{β}^{+}, and bcb̃ = 0 follows. Thus BCB = 0 and since A is prime, B or C are zero. The statements (i) and (ii) follow from 2.6 and 3.2.2.

3.7. Example: Jordan superpairs covered by an even quadratic form grid, an alternating grid, a Bi-Cayley grid or an Albert grid. In all these cases the assumptions of Theorem 3.6 are satisfied. Hence we know:

(1) The ideals of an alternating matrix superpair 𝔸_{I}(A), |I| ≥ 4, are exactly the subpairs 𝔸_{I}(B) where B is an ideal of A.

(2) The ideals of an even quadratic form superpair 𝔼ℚ_{I}(A), |I| ≥ 3, are exactly the subpairs of the form 𝔼ℚ_{I}(B) where B is an ideal of A.

(3) The ideals of a Bi-Cayley superpair 𝔹(A) are exactly the subpairs of the form 𝔹(B), where B is an ideal of A.

(4) The ideals of an Albert superpair 𝔸𝔹(A) are exactly the subpairs 𝔸𝔹(B) where B is an ideal of A.

Moreover, in these cases semiprimeness, primeness and simplicity is determined by 3.6(b) and 2.6. In particular, if \frac{1}{2} ∈ A a semiprime Jordan superpair V covered by an even quadratic form grid, an alternating grid, a Bi-Cayley grid or an Albert grid necessarily has V¯1 = 0 and hence is a Jordan pair. For prime Jordan superpairs this can be considered a (weak) analogue of 2.2.

In the following we can restrict our attention to the case of rectangular, hermitian and odd quadratic form grids.

3.8. Proposition. The ideals of a rectangular matrix superpair 𝕄_{JK}(A), |J| + |K| ≥ 3, are exactly the spaces 𝕄_{JK}(B), where B is an ideal of A. Moreover, 𝕄_{JK}(A) is semiprime, prime or simple if and only if A is respectively semiprime, prime and simple. In particular:

(i) A prime rectangular matrix superpair over a field of characteristic ≠ 2, 3 either has a prime associative coordinate superalgebra or is of type 𝕄_{12}(A) for A = A¯0 a prime Cayley-Dickson algebra and hence has 𝕄_{12}(A)¯1 = 0.

(ii) 𝕄_{JK}(A) is simple if and only if either A is simple as algebra or 𝕄_{JK}(A) = 𝔻(𝕄_{JK}(A¯0)) and A¯0 is simple.

Proof. This proof is parallel to the proof of 3.6. Nevertheless, for the convenience of a reader uninitiated to the techniques used there, we include the details. Let I = (I^{+}, I^{−}) be an ideal of V = 𝕄_{IJ}(A). By 3.3 we then have I^{σ} = ⊕_{jk} I_{jk}^{σ} where I_{jk}^{σ} = I^{σ} ∩V_{2}^{σ}(e_{jk}) = B_{jk}^{σ}e_{jk}^{σ} for graded subspaces B_{ij}^{σ} of A and ℛ = {e_{jk} : j ∈ J, k ∈ K} the usual rectangular grid. If be_{jk}^{σ} ∈ I^{σ}, then:

{e_{lk}^{σ} , e_{jk}^{−σ} , be_{jk}^{σ}} = be_{lk}^{σ} ∈ I^{σ}, for l ≠ j,
{e_{jm}^{σ} , e_{jk}^{−σ} , be_{jk}^{σ}} = be_{jm}^{σ} ∈ I^{+}, for m ≠ k, and
Q¯0(e_{jk}^{σ})(be_{jk}^{−σ}) = be_{jk}^{σ} ∈ I^{−σ}.

These formulas imply that B_{jk}^{+} = B_{jk}^{−} is independent of i, j and that

I = 𝕄_{JK}(B) = (Mat(J, K; B), Mat(K, J; B))

for some submodule B of A. By 3.3(b), B is an ideal of A. Conversely, if B is an ideal of A, it is immediate to see that 𝕄_{JK}(B) is an ideal of 𝕄_{JK}(A). The remaining part is proven in the same way as 3.6(b). In the last part, one obtains B(CB) = 0 and one can then use [32, 5.5 Prop.1] to conclude that B = 0 or C = 0. The statement (i) follows from 2.2, (ii) is a consequence of 2.5 and 3.2.3.

3.9. More examples of Jordan superpairs. The description of the rectangular matrix superpairs 𝕄_{JK}(A) for a simple Artinian A is best done in the following more general set-up.

Let E be an associative superalgebra over S. A 3-grading of E is a ℤ-grading E = E_{1} ⊕ E_{0} ⊕ E_{−1} in the sense of [20, 2.4]. Thus, E_{i}E_{j} ⊂ E_{i+j} with the understanding that E_{i+j} = 0 if i + j ∉ {±1, 0}. Note that the E_{i} are assumed to be S-submodules, hence respecting the ℤ_{2}-grading of E. If E = E_{1} ⊕ E_{0} ⊕ E_{−1} is a 3-grading one easily sees that (E_{1}, E_{−1}) is a subpair of the special Jordan superpair (E, E) and hence in particular a Jordan superpair.

A natural example of a 3-graded associative superalgebra E = End_{A}(ℳ ⊕ 𝒫) where A is an associative superalgebra and ℳ and 𝒫 are two ℤ_{2}-graded A-bimodules. Here E has a 3-grading given by E_{1} = Hom_{A}(𝒫, ℳ), E_{0} = End_{A}(ℳ, ℳ) ⊕ End_{A}(𝒫, 𝒫) and E_{−1} = Hom_{A}(ℳ, 𝒫). In particular, if ℳ and 𝒫 are free the E_{i} can be identified with matrices over A, 2.3.2, and we obtain the Jordan superpair

𝕄_{M|N,P |Q}(A) = (Mat(M|N, P |Q; A), Mat(P |Q, M|N; A))  (1)

for which the homogenous parts of 𝕄_{M|N,P |Q}(A)^{+} are given by 2.3.1. For A = A_{0} = k a field of characteristic 0 this type of Jordan superpairs is one of the examples in [8].

It is straightforward, using the techniques of the proof of 3.8, to show that the ideals of 𝕄_{M|N,P |Q}(A) are given by 𝕄_{M|N,P |Q}(B) where B is an ideal of A. Hence, 𝕄_{M|N,P |Q}(A) is semiprime, prime or simple if and only if A is so.

Note that the rectangular matrix superpairs with associative coordinate algebras A are special cases of this construction: 𝕄_{JK}(A) = 𝕄_{J|∅,K|∅}(A). But these are not the only examples of 𝕄_{M|N,P |Q}(A) covered by a rectangular grid. Indeed, for the associative superalgebra Mat_{P |Q}(A) of 2.3.3 as coordinate algebra we obtain

𝕄_{JK}(Mat_{P |Q}(A)) ≅ 𝕄_{J×P |J×Q,K×P |K×Q}(A).  (2)

Note that by 3.8 and 2.7 this in particular describes the structure of the simple rectangular matrix superpairs 𝕄_{JK}(A) with a simple Artinian associative A. For example it follows from 2.5 and 3.2.3 that the finite-dimensional simple rectangular matrix superpairs over an algebraically closed field k are isomorphic to

(i) 𝕄_{JK}(𝔻(k)) = 𝔻(𝕄_{JK}(k)) as in 3.2.3 for finite sets J, K, or to (ii) 𝕄_{JK}(Mat_{p|q}(k)) as in (2).

3.10. Proposition. Let V = ℍ_{I}(A, A_{0}, π) be a hermitian matrix superpair with |I| ≥ 3.

(a) The ideals of V are exactly the ℍ_{I}(B, B_{0}, π) = (H_{I}(B, B_{0}, π), H_{I}(B, B_{0}, π)), defined as in 1.8.3, where B is a π-invariant ideal of A and B_{0} ⊂ A_{0} ∩ B satisfies the conditions

(1) b¯0a_{0}bπ ; ¯0 ∈ B_{0} if b¯0 ∈ B¯0 and a_{0} ∈ A_{0},

(2) a¯0b_{0}aπ ; ¯0 ∈ B_{0} if b_{0} ∈ B_{0} and a¯0 ∈ A¯0, and

(3) b + b^{π} ∈ B_{0} for b ∈ B.

In particular, if \frac{1}{2} ∈ S then B_{0} = A_{0} ∩ B.

(b) V is semiprime or prime if and only if A is respectively π-semiprime or π-prime.

(c) If V is simple, then A is a π-simple superalgebra. Conversely, if A is π-simple and A_{0} is the span of all traces and norms, i.e., A_{0} = A_{0,min} as defined in [20, 5.10], then V is simple.

In particular, if \frac{1}{2} ∈ A then V is simple if and only if A is π-simple, and if S = k is a field of characteristic ≠ 2, 3 then there are exactly the following possibilities for a simple V:

(i) V ≅ 𝕄_{II}(B) for a simple associative superalgebra B, (ii) A is a simple associative superalgebra, or (iii) A = A¯0 is a simple Cayley-Dickson algebra and hence V = V¯0.

Proof. (a) can be proven by taking Grassmann envelopes and applying [21, IV Lemma 1.6] to the polarized Jordan triples system associated to G(ℍ_{I}(A, A_{0}, π)). For the convenience of the reader we include a direct proof.

Let L = (L^{+}, L^{−}) an ideal of V = ℍ_{I}(A, A_{0}, π). By 3.3 we have L^{σ} = ⊕_{ij}L_{ij}^{σ} where L_{ij}^{σ} = L^{σ} ∩ A[ij] for i ≠ j, L_{ii}^{σ} = L^{σ} ∩ A_{0}[ii] and, as usual, σ = ±. We will first show that the off-diagonal L_{ij}^{σ} are independent of i, j. This follows from the following two formulas where bh_{ij}^{σ} ∈ L^{σ}, i ≠ j:

{bh_{ij}^{σ} , h_{ji}^{−σ} , h_{ik}^{σ}} = bh_{ik}^{σ} ∈ L^{σ} for all k ≠ i,  (1)
{bh_{ij}^{σ} , h_{ij}^{−σ} , h_{jk}^{σ}} = b^{π}h_{jk}^{σ} = bh_{kj}^{σ} ∈ L^{σ} for all k ≠ j.  (2)

Next, we claim L_{ij}^{+} = L_{ij}^{−}. Indeed, if bh_{ij}^{σ} ∈ L^{σ} then Q¯0(h_{ij}^{−σ})bh_{ij}^{σ} = b^{π}h_{ij}^{−σ} ∈ L^{−σ}. For k ≠ i, j we then obtain bh_{ki}^{−σ} = b^{π}h_{ik}^{−σ} ∈ L^{−σ} by (1). We then apply again (1) to get that bh_{kj}^{−σ} ∈ L^{−σ} and finally bh_{ij}^{−σ} ∈ L^{−σ} by (2).

Similarly, we have L_{ii}^{σ} = L_{jj}^{τ} for all i, j ∈ I and σ, τ = ±. Indeed, if b_{0}h_{ii}^{σ} ∈ L^{σ}, then Q¯0(h_{ji}^{−σ})b_{0}h_{ii}^{σ} = b_{0}[jj] = b_{0}h_{jj}^{−σ} ∈ L^{−σ} for all j ∈ I, and applying Q¯0(h_{jj}^{σ}) again, we also have that b_{0}h_{jj}^{σ} ∈ L^{σ} for all j ∈ I.

We have now shown that L = ℍ_{I}(B, B_{0}, π) for certain subspaces B ⊂ A and B_{0} ⊂ A_{0}. In fact, since {b_{0}[ii] , [ij] , [jj]} = b_{0}[ij] for i ≠ j we also have B_{0} ⊂ B. Since A is the McCrimmon-Meyberg algebra for some collinear pair (h_{ij}, h_{ik}) it follows from 3.3(b) that B is a π-invariant ideal of A. The conditions (1) and (2) then follow from Q¯0(a¯0[ij])b_{0}[jj] = a¯0b_{0}aπ ; ¯0[ii] while (3) is a consequence of {a[ij], b[jk], c[ki]} = (a(bc) + (−1)^{|a||b|+|a||c|+|b||c|}(c^{π}b^{π})a^{π})[ii].

Conversely, it is easy to check that ℍ_{I}(B, B_{0}, π) is an ideal of ℍ_{I}(A, A_{0}, π) whenever B and B_{0} satisfy the conditions above. If \frac{1}{2} ∈ A then A_{0} = H(A, π)∩N(A) and H(B, π) ⊂ B_{0} by (3), proving B_{0} = A_{0} ∩ B.

(b) For any π-invariant ideal B of A, B_{0} = B ∩ A_{0} satisfies the conditions (1)–(3) of (a), hence gives rise to an ideal of V. We can then argue as in the proof of 3.6(b) above to establish (b).

(c) It is immediate that V simple implies A π-simple. For the converse, suppose we have a non-zero ideal L of V given by B and B_{0} as above. Then 0 ≠ B and hence B = A by π-simplicity. Moreover, B_{0} must contain at least all traces and norms of A. Thus A_{0} = B_{0} and L = V follows. If \frac{1}{2} ∈ A then A_{0} is indeed the span of all traces and norms, [20, 5.10]. The last statement follows from 2.2 and 2.9. Indeed, in case (i) of 2.9 we have case (i) above.

Remark. Our simplicity result provides a partial generalization of results in [2, §6]. It easily follows from the results proven there that H_{I}(A, A_{0}, π) is a simple Jordan algebra and hence ℍ_{I}(A, A_{0}, π) is a simple Jordan pair whenever |I| ≥ 2 and A is π-simple and associative over a ring k containing \frac{1}{2}.

3.11. Examples of simple hermitian matrix superpairs. We will describe simple superpairs ℍ_{I}(A, A_{0}, π) for a simple associative superalgebra A. Since ℍ_{I}(A, A_{0}, π) = (H_{I}(A, A_{0}, π), H_{I}(A, A_{0}, π)) is the Jordan superpair associated to the simple (by 3.4.1) Jordan superalgebra H_{I}(A, A_{0}, π) it suffices to describe the latter. Recall from 2.10 that there are three possibilities for a simple A with involution, namely cases (ii), (iii) and (iv). We will consider (ii) and (iii) only.

(a) Suppose A = 𝔻(B) for a simple associative algebra B. Since Z(B) is a field, there are two cases here.

(a.1) Z(B) has characteristic ≠ 2: Then A_{0} = H(A, π) = B_{0} ⊕ (1 + z)uB_{0} for B_{0} = H(B, π) by 2.4.6. Using 2.4.7 we obtain

H_{I}(𝔻(B), 𝔻(B)_{0}, π) = H_{I}(B, B_{0}, π) ⊕ u(1 + z)H_{I}(B, B_{0}, π).

Thus, the corresponding matrix pair ℍ_{I}(A, A_{0}, π) is a (slightly generalized) double of the hermitian matrix pair ℍ_{I}(B, B_{0}, π). Note that this case does not occur if S = k is an algebraically closed field.

(a.2) Z(B) has characteristic 2. Then (b_{1} ⊕ b_{2}u)^{π} = b_{1}^{π} + ub_{2}^{π} for b_{i} ∈ B by 2.4, hence H(A, π) = H(B, π) ⊕ uH(B, π). Also the ample subspace A_{0} = B_{0} ⊕ uC_{0} where B_{0} is an ample subspace of B, C_{0} contains all traces b + b^{π} for b ∈ B and ac_{0}b^{π} + bc_{0}a^{π} ∈ B_{0} for a, b ∈ B and c_{0} ∈ C_{0}. We have

H_{I}(𝔻(B), 𝔻(B)_{0}, π) = H_{I}(B, B_{0}, π) ⊕ uH_{I}(B, C_{0}, π),

and so again ℍ_{I}(A, A_{0}, π) is a (slightly generalized) double.

(b) In case (iii) of 2.10 we can write A = (B_{1} ⊕ B_{2}) ⊕ (C_{1} ⊕ C_{2}) as a Morita context

A = ( \begin{matrix}B_{1} & C_{1} \\ C_{2} & B_{2} \end{matrix} ) ⊂ Mat_{1|1}(A).

We identify Mat(I, I; Mat_{1|1}(A)) = Mat_{1|1}(Mat(I, I; A)). Also, we extend the involution of A to Mat(I, I; A) by (a_{ij})^{π} = (a_{ji}^{π}). There are again two cases here, depending on whether π exchanges the B_{i} or the C_{i}. For simplicity, we will abbreviate H_{I}(A, π) = H_{I}(A, H(A, π), π).

(b.1) Suppose B_{1}^{π} = B_{2} and C_{i}^{π} = C_{i} for i = 1, 2. We let C_{i,min} = {c_{i} + c_{i}^{π} : c_{i} ∈ C_{i}} denote the set of all traces. Recall ([20, 5.10]) that the ample subspace A_{0} of (A, π) contains A_{0,min}, the span of all traces a + a^{π} and norms aa^{π} for a ∈ A, and is itself contained in A_{0,max} = H(A, π). Hence in our situation

A_{0,min} = \{( \begin{matrix}b_{1} & c_{1} \\ c_{2} & b_{1}^{π} \end{matrix} ) : b_{1} ∈ B_{1}, c_{i} ∈ C_{i,min}\} ⊂ A_{0}
A_{0} ⊂ A_{0,max} = \{( \begin{matrix}b_{1} & c_{1} \\ c_{2} & b_{1}^{π} \end{matrix} ) : b_{1} ∈ B_{1}, c_{i} ∈ H(C_{i}, π|C_{i})\} .

Therefore A_{0} can be written as

A_{0} = \{( \begin{matrix}b_{1} & c_{1} \\ c_{2} & b_{1}^{π} \end{matrix} ) : b_{1} ∈ B_{1}, c_{i} ∈ C_{i0}\}

for submodules C_{i0} with C_{i,min} ⊂ C_{i0} ⊂ H(C_{i}, π|C_{i}). It follows that

( \begin{matrix}b_{1} & c_{1} \\ c_{2} & b_{2} \end{matrix} ) ∈ H_{I}(A, A_{0}, π) ⟺ \{ \begin{matrix}b_{i} = b_{j}^{π} ∈ Mat(I, I; B_{i}) & for \{i, j\} = \{1, 2\} \\ c_{i} ∈ H_{I}(C_{i}, C_{i0}, π|C_{i}) & for i = 1, 2. \end{matrix}

In particular, for A = Mat_{Q|Q}(B) and B an associative algebra, we have the supertranspose involution

A ∋ ( \begin{matrix}w & x \\ y & z \end{matrix} ) ⟼^{τ} ( \begin{matrix}z^{T} & −x^{T} \\ y^{T} & w^{T} \end{matrix} ) ∈ A,  (1)

where z^{T} is the usual transpose of z ([25, Prop.13]). As fixed point set of an involution,

H(Mat_{Q|Q}(B), τ) = \{( \begin{matrix}a & b \\ c & a^{T} \end{matrix} ) ∈ Mat_{Q|Q}(B) : b skew, c symmetric\}
=: P_{Q}(B)  (2)

is a Jordan superalgebra. We have a canonical isomorphism

H_{I}(Mat_{Q|Q}(B), τ) ≅ P_{I×Q}(B).  (3)

By 3.10(c) above and 3.4.1 we know that H_{I}(Mat_{Q|Q}(B) is a simple Jordan superalgebra if |I| ≥ 3 and B is simple and defined over k containing \frac{1}{2}. Because of (3), this can also be deduced from [2, §6]. For finite Q and B = k an algebraically closed field of characteristic ≠ 2 the simple Jordan superalgebra P_{Q}(k) appears (of course) also in [26]. We note that if one replaces the involution z ↦ z^{T} above by a symplectic involution, assuming that Q has finite even order, one gets a Jordan superalgebra isomorphic to P_{n}(k), [26]. Indeed, if s is the block diagonal matrix inducing the symplectic involution d ↦ sd^{T}s^{T}, i.e.,

s = diag (( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , ⋯, ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , ⋯) ,  (4)

and we change τ to the conjugate involution

A ∋ ( \begin{matrix}w & x \\ y & z \end{matrix} ) ⟼^{τ̂} ( \begin{matrix}sz^{T}s^{T} & sx^{T}s^{T} \\ −sy^{T}s^{T} & sw^{T}s^{T} \end{matrix} ) ∈ A,

the fixed point set with respect to τ̂ is

H(Mat_{Q|Q}(k), τ̂)) = \{( \begin{matrix}a & b \\ c & sa^{T}s^{T} \end{matrix} ) ∈ Mat_{Q|Q}(k) : b = sb^{T}s^{T}, c = −sc^{T}s^{T}\} ,

and an isomorphism between H(Mat_{Q|Q}(k), τ̂) and P_{2Q}(k) is given by

H(Mat_{Q|Q}(k), τ̂) →  P_{2Q}(k)
( \begin{matrix}a & b \\ c & sa^{T}s^{T} \end{matrix} ) ↦ ( \begin{matrix}a & bs \\ s^{T}c & a^{T} \end{matrix} ) .

(b.2) Suppose B_{i}^{π} = B_{i} and C_{1}^{π} = C_{2}. Then

A_{0} = \{( \begin{matrix}b_{1} & c_{1} \\ c_{1}^{π} & b_{2} \end{matrix} ) : b_{i} ∈ B_{i0}, c_{1} ∈ C_{1}\}

where B_{i0} are ample subspaces of (B_{i}, π|A_{i}) and

( \begin{matrix}b_{1} & c_{1} \\ c_{2} & b_{2} \end{matrix} ) ∈ H_{I}(A, A_{0}, π) ⟺ \{ \begin{matrix}b_{i} ∈ H_{I}(B_{i}, B_{i0}, π|B_{i}) & for i = 1, 2, \\ c_{i} = c_{j}^{π} ∈ Mat(I, I; C_{i}) & for \{i, j\} = \{1, 2\} . \end{matrix}

In particular, for A = Mat_{P |Q}(B) and B an associative algebra, an involution of this type is the so-called orthosymplectic involution

( \begin{matrix}a & b \\ c & d \end{matrix} ) ⟼^{σ} ( \begin{matrix}a^{T} & c^{T}s^{T} \\ sb^{T} & sd^{T}s^{T} \end{matrix} ) ,  (5)

where s is the matrix (4) and Q is either of finite even order or infinite. The fixed point superalgebra

H(Mat_{P |Q}(B), σ) = \{( \begin{matrix}a & b \\ sb^{T} & d \end{matrix} ) ∈ Mat_{P |Q}(B) : a = a^{T}, d = sd^{T}s^{T}\}
=: OSP_{P |Q}(B)  (6)

is called the orthosymplectic (Jordan) superalgebra and denoted BC in [4]. We have

H_{I}(Mat_{P |Q}(B)) ≅ OSP_{I×P |I×Q}(B)  (7)

which is a simple Jordan superalgebra if |I| ≥ 3 and B is simple over a ring k containing \frac{1}{2}.

For finite non-empty P, Q and B a division algebra, the involutions of A = Mat_{P |Q}(B) leaving the two ideals of A¯0 invariant are described in [25, Prop. 14]. In particular, if B = k is an algebraically closed field then all such involutions are isomorphic to the orthosymplectic involution σ.

It remains to consider the standard example of an odd quadratic form superpair. To do so, we will need the following definition.

3.12. Quadratic form superpairs. Let A be a superextension of S. An A-quadratic form q = (q¯0, b) : X → A, as defined in [20, 2.8], is called nondegenerate if for x_{μ} ∈ X_{μ}, μ = ¯0, ¯1 we have

q¯0(A_{μ}x_{μ}) = 0 = b(x_{μ}, X) ⟹ x_{μ} = 0.

Of course, q¯0(A¯0x¯0) = 0 ⟺ q¯0(x¯0) = 0. We note that the orthogonal sum of nondegenerate forms is again nondegenerate and that any hyperbolic form [20, 5.13] is nondegenerate.

Let V = (X, X) be the corresponding quadratic form superpair, [20, 3.12]. Then

V semiprime ⟹ q nondegenerate.  (1)

Indeed, if x_{μ} ∈ X_{μ} for μ = ¯0, ¯1 is such that q¯0(A_{μ}x_{μ}) = 0 = b(x_{μ}, X) then (Ax_{μ}, Ax_{μ}) is an ideal of V whose Jordan cube is zero whence x_{μ} = 0.

3.13. Proposition. Let V = 𝕆ℚ_{I}(A, q_{X}) be an odd quadratic form superpair with |I| ≥ 2.

(a) The ideals of V are exactly the subpairs of the form Y ⊕ 𝔼ℚ_{I}(B), where B is an ideal of A and Y is an A-submodule of X satisfying

(i) q_{X}(Y¯0) ⊂ B¯0, (ii) b_{X}(Y, X) ⊂ B, where b( , ) is the polar of q_{X}, and (iii) BX ⊂ Y.

(b) If V is semiprime then q_{X} is nondegenerate, equivalently, q_{I} ⊕q_{X} is nondegenerate.

(c) Suppose q_{X} is nondegenerate. If A is semiprime, prime or simple then V is respectively semiprime, prime or simple.

(d) Suppose \frac{1}{2} ∈ A or A = A¯0. Then V is semiprime, prime or simple if and only if q_{X} is nondegenerate and A is respectively semiprime, prime or a field.

Proof. This follows by considering the Grassmann envelope and applying [21, IV Lemma 1.7]. A direct proof goes as follows. Let L = (L^{+}, L^{−}) be an ideal of 𝕆ℚ_{I}(A, q_{X}). By 3.3 we have that L^{σ} = (X ∩ L^{σ}) ⊕ (⊕_{±i} L_{±i}^{σ}) where L_{±i}^{σ} = L^{σ} ∩ V_{2}^{σ}(g_{±i}) for σ = ±. In particular, ⨁_{±i} L_{±i} is an ideal of the subpair ⨁_{±i} V_{2}(g_{±i}) = 𝔼ℚ_{I}(A) which, by 3.7, is therefore of the form 𝔼ℚ_{I}(B) for an ideal B of A. Moreover, if x ∈ X ∩ L^{σ} then {g_{+i}^{−σ}, x, g_{−i}^{−σ}} = x ∈ L^{−σ}, i.e., X ∩ L^{+} = X ∩ L^{−}. Therefore L = (Y, Y) ⊕ 𝔼ℚ_{I}(B) for a certain S-subspace Y = X ∩ L^{σ}. Now, we just need to check the properties satisfied by B and Y:

– Y is an A-submodule of X: For x ∈ Y we know that x ∈ L^{+}, so for a ∈ A we get {ag_{+i}^{+} , g_{+i}^{−} , x} = ax ∈ L^{+}, i.e., ax ∈ Y.

– q_{X}(Y¯0) ⊆ B: If x ∈ Y¯0, then x ∈ L^{+}, so Q¯0(x)g_{+i}^{−} = q_{X}(x)g_{−i}^{+} ∈ L^{+}, i.e., q_{X}(x) ∈ B¯0.

– b_{X}(Y, X) ⊆ B: Take x ∈ Y and y ∈ X. Then x ∈ L^{+} and {x, y, g_{+i}^{+}} = b_{X}(x, y)h_{+i} ∈ L^{+}, so b_{X}(x, y) ∈ B.

– BX ⊆ Y: Take b ∈ B (so bg_{+i}^{+} ∈ L^{+}) and x ∈ X. Then {bg_{+i}^{+} g_{+i}^{−} x} = bx ∈ L^{+}, so bx ∈ Y. Conversely, it is straightforward to check that any pair B, Y satisfying the conditions

above gives rise to an ideal of 𝕆ℚ_{I}(A, q_{X}).

(b) is immediate from 3.12.

For the proof of (c) let us first observe that any non-zero ideal of V, given by Y, B as in (a) above has B ≠ 0. Indeed, if B = 0 nondegeneracy of q_{X} implies Y = 0. Now suppose we have two ideals of V, given by Y, B and Z, C respectively, such that their product vanishes. Then BCB = 0 follows. So one of B or C must be zero and the corresponding ideal of V is zero. This proves the statements concerning semiprimeness and primeness. For simplicity, consider a non-zero ideal U of V, given by Y, B. As we just observed, B is then a non-zero ideal of A whence B = A and consequently X = AX ⊂ Y, proving U = V.

(d) Because of (b) and (c) we only have to prove that if V is semiprime, prime or simple then so is A. In general, for B an ideal of A, the ideal of V generated by 𝔼ℚ_{I}(B) ⊂ V is given by the data

\begin{aligned}B′ & = B + q(B¯1x¯1) + A¯1q_{X}(B¯1X¯1) ⊂ A and \\ Y & = BX + q_{X}(B¯1X¯1)X. \end{aligned}

Under our assumptions B′ = B and Y = BX. It is then straightforward to see that two ideals B, C of A generate ideals of V whose Jordan product is zero, whence B or C vanishes, and A is semiprime or prime if V is so. Also, if V is simple then so is A, and it follows from 2.6(c) that A is a field.

Examples. (a) If A = A¯0 we do not necessarily have V = V¯0. Rather, V is a quadratic form superpair of a quadratic form q defined on a superspace M = M¯0 ⊕ M¯1 which is an orthogonal sum of a quadratic form on M¯0 = H_{I}(A) ⊕ X¯0 and an alternating form on M¯1 = X¯1. In particular, by 2.6(a) we have A = A¯0 as soon as \frac{1}{2} ∈ A and V is semiprime.

(b) Suppose A is simple. We then either have A = A¯0 or A = 𝔻(A¯0) where A¯0 is a field of characteristic 2. In the latter case 𝕆ℚ(A, q_{X}) = 𝔻(𝕆ℚ_{I}(A¯0, q¯0)) is the double of an odd quadratic form pair by 3.2.4.

3.14. Concluding remarks. Let V be a standard example V = 𝕍(𝒢, 𝒞) with supercoordinate system 𝒞, as introduced in section §1. The results obtained in this section show that within a fixed type, ideals are determined in a uniform way by data depending on 𝒞 and not on 𝒢.

Suppose S = k is a field of characteristic ≠ 2, 3 and V is simple. It is remarkable that then either V = V¯0 or V = 𝕁 = (J, J) for a simple Jordan superalgebra or V is a subpair of a rectangular matrix superpair, and hence obviously special, or V is a quadratic form superpair and hence also special by McCrimmon’s recent result [14]: a Jordan superpair is special if and only if its Grassmann envelope is a special Jordan pair over G¯0.

Acknowledgements

The major part of this paper was written while the first author visited the University of Ottawa in the fall of 2000, with a stipend from an F.P.I. Grant (Ministerio de Ciencia y Tecnología, España). She would like to thank her co-author for his hospitality during her visit and would also like to thank Prof. J. A. Anquela for his valuable comments during the final preparation of this paper.

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