THE ROOT SYSTEM AND THE CORE OF

AN EXTENDED AFFINE LIE ALGEBRA

Bruce N. Allison^{1}and Yun Gao^{2}

An extended affine Lie algebra (EALA) is a complex Lie algebra L which has an invariant nondegenerate form (·, ·) and a finite dimensional abelian ad-diagonalizable subalgebra H which satisfies certain natural axioms (see Section 1 for the details). These Lie algebras were first introduced by Høegh-Krohn and Torresani in [HT] where they were called irreducible elliptic quasi-simple Lie algebras. As examples, finite dimensional simple Lie algebras, affine Kac-Moody Lie algebras and toroidal Lie algebras are EALA’s. There are many other examples which have been described in [HT], [BGK], [BGKN] and [AABGP].

Given an EALA L, one can associate to L two basic objects: the root system R of L and the Lie algebra L_{c}/Z(L_{c}), where L_{c} is the core of L and Z(L_{c}) is the centre of L_{c}. It has been seen in the papers [BGK] and [BGKN] on EALA’s of simply laced type that a description of the structure of R and L_{c}/Z(L_{c}) is the first main stage in the determination of the structure of the EALA L. In this paper, we give a complete description of R and L_{c}/Z(L_{c}) for EALA’s of nonsimply laced reduced type.

In order to be more precise about our results, we begin by briefly recalling some basic definitions and earlier results. First, the root system R of an EALA L is by definition the set of weights of the adjoint action of H on L. R is in general infinite. However, if we factor out the radical V^{0} of the real space V spanned by R, the image R̄ in the quotient space V̄ = V/V^{0} is a finite (possibly not reduced) root system in the usual sense (except that 0 ∈ R̄ since 0 ∈ R). The type of R̄ is called the type of L. If we let Λ be the additive subgroup of V generated by the set of isotropic roots in R, then Λ is a lattice in V^{0} with ν := dim_{ℝ} V^{0} = dim_{ℤ} Λ. Moreover, we can lift R̄ to an isomorphic copy Ṙ in V so that R is a subset of Ṙ + Λ.

Next the core of L is defined to be the subalgebra L_{c} of L generated by the nonisotropic root spaces of L. L_{c} is an ideal of L. Throughout this paper, we will denote the quotient algebra L_{c}/Z(L_{c}) by K. As a familiar example, if L is a nontwisted affine Kac-Moody Lie algebra, then L_{c} is the derived algebra of L, Z(L_{c}) is 1-dimensional and K is the loop algebra.

The first structural results on EALA’s were proved in [BGK] and [BGKN] (see also [G] where certain degenerate EALA’s are considered). The authors studied EALA’s L of simply laced type and rank ≠ 1. Their analysis proceeded in two main stages. The first stage was the determination of the structure of R and K. They observed that for the types they were considering the structure of R is quite simple, namely R = Ṙ + Λ (see [AABGP] for details). They also completely determined the possibilities for the Lie algebra K. In the second stage they showed how to construct all possible tame EALA’s L from a given K by forming a central extension of K to obtain L_{c} and then adding a subalgebra of derivations to L_{c} to obtain L (as one obtains an affine Kac-Moody Lie algebra from a loop algebra K by forming a 1-dimensional central extension K̂ of K and then adding the degree derivation to K̂). The hypothesis that L is tame, which means that the centralizer of L_{c} in L is contained in L_{c}, is necessary in the second stage to give the core some control over the structure of the full EALA L. Thus, the

Typeset by AMS-TEX

1The first author is grateful for the support of NSERC Grant #A8465 and the hospitality of the University of Ottawa during the preparation of this paper.

2The second author acknowledges support from a research fellowship from the Alexander von Humboldt Foundation and thanks Professor Peter Slodowy for his hospitality.

structure of tame EALA’s of simply laced type and rank ≠ 1 was completely described in [BGK] and [BGKN].

In this paper, we carry out the first stage of the above program for EALA’s of non-simply laced reduced type, i.e. type B_{l}(l ≥ 3), C_{l}(l ≥ 2), F_{4} and G_{2}. As in [BGK] and [BGKN], our study of K uses the fact that K is graded by the finite root system Ṙ. Lie algebras graded by finite root systems of nonsimply laced reduced type have been described by Benkart and Zelmanov [BZ] in terms of associative, Jordan and alternative coordinate algebras (see also [N] for types ≠ F_{4} and G_{2}). Moreover the Lie algebra K is also graded by the lattice Λ, and so it follows that the coordinate algebra of K is graded by Λ. In fact the Λ-grading on the coordinate algebra has rather strong properties which reflect properties of the root space decomposition of L. Our main task then is to develop these properties and use them to classify the Λ-graded coordinate algebras which can occur for each type.

For type C_{l}(l ≥ 2), it follows from [BZ] or [N] that K can be constructed using the Tits-Kantor-Koecher construction from a Jordan algebra J which contains a set of l strongly connected orthogonal idempotents which sum to the identity of J. If l ≥ 3, it follows from a theorem of Jacobson that J itself can be coordinatized as the Jordan algebra of l × lhermitian matrices over an alternative algebra D (with some further conditions depending on l). We obtain properties of the Λ-grading of J and hence D which allow us to show, using results from [BGKN] on graded alternative algebras, that D is either an associative quantum torus in ν variables or, when l = 3, the octonion torus in ν variables. The case when l = 2 is significantly more difficult to handle. Here Jacobson’s coordinatization theorem does not apply. As a replacement we use two coordinatization theorems due to McCrimmon and Neher [McN]. In fact much of our work for type C_{2} involves showing that J satisfies the hypotheses of one or other of those theorems. The two coordinatization theorems, together with further analysis of the coordinate algebras, allow us to conclude either that J is the Jordan algebra of 2 × 2hermitian matrices over an associative quantum torus in ν variables or that J is the Jordan algebra constructed from a symmetric bilinear form over the ring of Laurent polynomials in ν variables. This sort of dichotomy, which intuitively reflects the fact that type C_{2} can be thought of as the rank 2 case of both type C_{l} and type B_{l}, was encountered previously in the work of Seligman [Se] on simple finite dimensional Lie algebras over nonalgebraically closed fields of characteristic zero.

For types B_{l}(l ≥ 3), F_{4} and G_{2}, it follows from [BZ] that K can be constructed using the generalized Tits construction from two coordinate algebras–one a completely specified finite dimensional simple algebra and the other an infinite dimensional algebra which satisfies some additional conditions depending on the type. It is the second of these coordinate algebras which is Λ-graded. In type B_{l}, analysis of this Λ-graded coordinate algebra shows that it is isomorphic to the Jordan algebra constructed from a symmetric bilinear form over the ring of Laurent polynomials. In types F_{4} and G_{2}, we show that an integer invariant of the root system R, called the twist number of R, is bounded between 0 and 3, and we describe the possibilities for the Λ-graded coordinate algebra in each case. This analysis uses results from [BGKN] on graded alternative algebras and results of Yoshii [Y] on graded Jordan algebras. The most interesting case occurs when the twist number is 3, in which case the Λ-graded coordinate algebra is the octonion torus in ν variables (for type F_{4}) or the Albert torus in ν variables (for type G_{2})–the latter being a 27-dimensional Jordan algebra over the ring of Laurent polynomials.

Our methods then lead to a list of all possible Lie algebras K for each nonsimply laced reduced type. As a byproduct of this analysis of K we obtain a description of the possibilities for the root system R. It turns out that the structure of R is much more subtle than for the types considered in [BGK] and [BGKN]. Indeed, as has already been seen in [AABGP, Chapter II], we have

R = (S + S) ∪ (Ṙ_{sh} + S) ∪ (Ṙ_{lg} + L),

where Ṙ_{sh} and Ṙ_{lg} denote the sets of short and long roots in Ṙ respectively, and S and L are sets of isotropic roots in R. S and L are semilattices in V^{0}, which means that they are discrete spanning sets in V^{0} which contain 0 and are closed under the operation (σ, τ) → σ − 2τ. However, S and L are not in general lattices as they are not closed under addition. Our efforts here produce a list of the semilattices S and L which can occur for each type considered. Thus we obtain a complete description of the possible root systems R.

Our description of the root systems for EALA’s of nonsimply laced reduced type has an interesting consequence which was conjectured in [AABGP]. It was shown there that the root system R of any EALA is an extended affine root system. (Extended affine root systems were introduced abstractly in [Sa1] in the study of singularities). Moreover for simply laced types the converse was also shown to be true. However, it turns out that the converse is not true for nonsimply laced reduced types. Indeed, for types C_{l}(l ≥ 2), F_{4} and G_{2} there exist extended affine root systems which are not root systems of EALA’s.

Throughout the paper we assume that all algebras are algebras over the field of complex numbers ℂ. All algebras except Lie algebras are assumed to be unital unless the contrary is specified.

§1. Basic facts about extended affine Lie algebras

In this section, we describe the basic facts that we will need about the root system and the core of an extended affine Lie algebra. We begin by recalling the definition of an extended affine Lie algebra.

Let L be a Lie algebra over ℂ. Assume that

(EA1) L has a nondegenerate invariant symmetric bilinear form (·, ·).

(EA2) L has a nontrivial finite dimensional self–centralizing ad–diagonalizable abelian subalgebra H.

We will be assuming 3 further axioms about the triple (L, (·, ·), H). To describe these axioms we need some further notation. Using (EA2), we have

L = ⨁_{α∈H^{∗}} L_{α},

and

L_{0} = H,

where

L_{α} = {x ∈ L : [h, x] = α(h)x for all h ∈ H}

and H^{∗} is the complex dual space of H. Let

R = {α ∈ H^{∗} : L_{α} ≠ {0}}.

R is called the root system of L. Note that since H ≠ {0} we have 0 ∈ R. Also,

(1.1)  α, β ∈ R, α + β ≠ 0 ⟹ (L_{α}, L_{β}) = {0}.

Thus, −R = R. Moreover, (·, ·) is nondegenerate on H. So as usual we can transfer (·, ·) to a form on H^{∗}. Let

R^{×} = {α ∈ R : (α, α) ≠ 0} and R^{0} = {α ∈ R : (α, α) = 0}.

The elements of R^{×} (respectively R^{0}) are called nonisotropic (respectively isotropic) roots. We have

R = R^{0} ∪ R^{×}.

We assume that (EA3) α ∈ R^{×}, x_{α} ∈ L_{α} ⟹ adx_{α} acts locally nilpotently on L. (EA4) R is a discrete subset of H^{∗}. (EA5) R is irreducible. That is

(a) R^{×} = R_{1} ∪ R_{2}, (R_{1}, R_{2}) = {0} ⟹ R_{1} = ∅ or R_{2} = ∅

(b) σ ∈ R^{0} ⟹ There exists α ∈ R^{×} such that α + σ ∈ R. If L satisfies (EA1)–(EA5), the triple (L, (·, ·), H), or simply the algebra L itself, is called an extended affine Lie algebra or EALA for short.

Throughout the rest of this paper, we will assume that L is an extended affine Lie algebra with root system R.

Suppose that α ∈ R^{×}. Then,

(1.2)  dim_{ℂ} L_{α} = 1

[AABGP, Theorem 1.29]. Let t_{α} be the unique element of H so that

(t_{α}, h) = α(h)

for h ∈ H, and then put

h_{α} = \frac{2}{(α, α)}t_{α}.

Then, there exist nonzero e_{α} ∈ L_{α} and f_{α} ∈ L_{−α} so that

[h, e_{α}] = 2e_{α}, [h, f_{α}] = −2f_{α} and [e_{α}, f_{α}] = h_{α}.

In other words, {e_{α}, h_{α}, f_{α}} is an 𝔰𝔩_{2}–triplet. (The element e_{α} is determined up to nonzero scalar in which case f_{α} is uniquely determined. We assume such a choice is fixed.)

We also have the following basic fact:

Lemma 1.3. Let α, β ∈ R^{×} and suppose that α + β ∈ R^{×}. Then,

[L_{α}, L_{β}] = L_{α+β}.

Proof. By [AABGP, Theorem 1.29], the α-string through β in R is

β − dα, …, β, β + α, …, β + uα,

where d, u ∈ ℤ_{≥0} and d − u = 2\frac{(β,α)}{(α,α)}. Choose 0 ≠ x ∈ L_{β−dα} and let

N = 2\frac{(β − dα, −α)}{(−α, −α)} = d + u.

Then, [f_{α}, x] = 0. Thus, by [AABGP, Lemma I.1.21], ad(e_{α})^{N}x ≠ 0. But since β + α ∈ R, we have u ≥ 1 and so d + 1 ≤ N. Hence, ad(e_{α})^{d+1}x ≠ 0. But ad(e_{α})^{d}x ∈ L_{β} and so [e_{α}, L_{β}] ≠ {0}. Since dim_{ℂ} L_{α+β} = 1, we have [L_{α}, L_{β}] = L_{α+β}. □

As we have said in the introduction, the first main purpose of this paper is to study the root system R of L. We now recall from [AABGP] some of the properties of R that we will need.

Let V be the real span of R in H^{∗}. We assume that (·, ·) is scaled so that (α, α) > 0 for some α ∈ R^{×}. Then (α, β) ∈ ℝ for all α, β ∈ V and (·, ·) restricts to a positive semidefinite real valued symmetric bilinear form on V [AABGP, Theorem I.2.14]. Let

V^{0} = rad (·, ·)|_{V} = {α ∈ V : (α, β) = 0 for all β ∈ V}.

The nullity of R or of L is defined to be the real dimension ν of V^{0}. Let

V̄ = V/V^{0}

and let ¯ : V → V̄ be the canonical projection. Then (·, ·) induces a positive definite symmetric bilinear form on V̄ so that, relative to this form, the image R̄ of R in V̄ is a finite irreducible root system [AABGP, Proposition I.2.19]. (Note however that 0 ∈ R̄. Contrary to the usual convention we are assuming that finite root systems contain 0.) The type of R or of L is defined to be the type of the finite root system R̄.

We next want to lift R̄ to a finite root system Ṙ inside V. To do this, we fix a choice of a fundamental system \overline{Π} = \{ᾱ_{1}, …, ᾱ_{l}\} for R̄. Also for i = 1, …, l we choose a fixed preimage α̇_{i} ∈ R of ᾱ_{i} under ¯ . Let

Π̇ = {α̇_{1}, …, α̇_{l}} and V̇ = Span_{ℝ}Π̇.

Then,

V = V̇ ⊕ V^{0}

and ¯ restricts to an isometry of V̇ onto V̄. Let

Ṙ = {α̇ ∈ V̇ : α̇ + σ ∈ R for some σ ∈ V^{0}}

In other words, Ṙ is the image of R under the projection of V onto V̇ with kernel V^{0}. Thus, ¯ maps Ṙ bijectively onto R̄. Hence, Ṙ is a finite irreducible root system in V̇ which is isomorphic under ¯ to R̄.

Put Ṙ^{×} = Ṙ \ {0}. For each α̇ ∈ Ṙ^{×}, we define

Sα̇ = {σ ∈ V^{0} : α̇ + σ ∈ R}.

Then, by definition, we have

(1.4)  R = R^{0} ∪ ( ⋃_{α̇∈Ṙ^{×}}(α̇ + Sα̇)) .

If α̇ ∈ Ṙ^{×}, we have

(1.5)  Sα̇ ⊆ R^{0},

and

(1.6)  Sα̇ depends only on the length of α̇

[AABGP, Propositions II.2.11 and II.2.15]. Moreover, if α̇ ∈ Ṙ^{×} has minimum length, then

(1.7)  R^{0} = Sα̇ + Sα̇

[AABGP, Corollary II.2.31]. Also, if α̇ ∈ Ṙ^{×} is reduced in Ṙ (i.e. \frac{1}{2} α̇ ∉ Ṙ), then Sα̇ is a semilattice in V^{0} [AABGP, Corollary II.2.30]; that is Sα̇ is a discrete spanning set in V^{0} so that

(1.8)  0 ∈ Sα̇, −Sα̇ = Sα̇ and Sα̇ + 2Sα̇ ⊆ Sα̇

(or equivalently 0 ∈ Sα̇ and Sα̇ − 2Sα̇ ⊆ Sα̇).

We put

Λ = ⟨R^{0}⟩ := the subgroup of V^{0} generated by R^{0}.

Then, Λ is a lattice in V^{0} [AABGP, Corollary II.2.31]; that is Λ is a discrete subgroup of V^{0} which spans V^{0}, and hence Λ is the ℤ–span of a basis of V^{0}. It follows from (1.4) and (1.5) that

(1.9)  R ⊆ Λ ∪ ( ⋃_{α̇∈Ṙ^{×}}(α̇ + Λ)) = Ṙ + Λ

Thus,

(1.10)  L = ⨁_{α̇∈Ṙ} ⨁_{σ∈Λ} L_{α̇+σ}.

In view of (1.4) and (1.6), the description of R is reduced to the description of the semilattices Sα̇ for each possible root length for α̇ ∈ Ṙ^{×}. In particular if Ṙ^{×} has only one root length, we have

(1.11)  R = R(X, S) := (S + S) ∪ (Ṙ^{×} + S),

where S = Sα̇ for α̇ ∈ Ṙ^{×}. The exact possibilities for S in this case were determined in [AABGP] (see Section 3). On the other hand, if Ṙ is reduced and Ṙ^{×} has exactly two root lengths, we have, by (1.4), (1.6) and (1.7),

(1.12)  R = R(X, S, L) := (S + S) ∪ (Ṙ_{sh} + S) ∪ (Ṙ_{lg} + L).

where X is the type of R (or Ṙ), Ṙ_{sh} is the set of short roots in Ṙ^{×}, Ṙ_{lg} is the set of long roots in Ṙ^{×},

S = Sα̇ for α̇ ∈ Ṙ_{sh} and L = Sα̇ for α̇ ∈ Ṙ_{lg}.

In Sections 4 and 5, we will completely determine the possibilities for the semilattices S and L in this case. (This leaves only the case when Ṙ is not reduced for a future investigation.)

We now turn to a discussion of the core of L. The core of L is defined to be the subalgebra L_{c} of L generated by the spaces L_{α}, α ∈ R^{×}. L_{c} is an ideal of L since if α ∈ R^{×}, δ ∈ R^{0} we have (α + δ, α + δ) = (α, α) ≠ 0 and so [L_{α}, L_{δ}] ⊆ L_{α+δ} ⊆ L_{c}. Moreover,

(1.13)  L_{c} = ∑_{α̇∈Ṙ^{×}} ∑_{σ∈Λ} L_{α̇+σ} + ∑_{α̇∈Ṙ^{×}} ∑_{σ,τ∈Λ}[L_{α̇+σ}, L_{−α̇+τ}],

and hence L_{c} is perfect, i.e. [L_{c}, L_{c}] = L_{c}.

The core L_{c} has two types of grading, a grading by Λ and grading by the finite root system Ṙ. First, to describe the Λ–grading, we put

L^{σ} := ⨁_{α̇∈Ṙ} L_{α̇+σ}

for σ ∈ Λ. Then, by (1.9), we have

L = ⨁_{σ∈Λ} L^{σ} and [L^{σ}, L^{τ}] ⊆ L^{σ+τ} for σ, τ ∈ Λ.

In other words, L is a Λ–graded Lie algebra. Moreover, by (1.13), L_{c} is generated by homogeneous generators and so L_{c} is a Λ–graded subalgebra of L. Thus,

(1.14)  L_{c} = ⨁_{σ∈Λ}(L_{c})^{σ},

where

(1.15)  (L_{c})^{σ} := L_{c} ∩ L^{σ} = ∑_{α̇∈Ṙ^{×}} L_{α̇+σ} + ∑_{α̇∈Ṙ^{×}} ∑_{\begin{matrix}τ,υ∈Λ \\ τ+υ=σ \end{matrix}}[L_{α̇+τ} , L_{−α̇+υ}]

for σ ∈ Λ.

To describe the root grading, we let

Ṙ_{red}^{×} := \{α̇ ∈ Ṙ^{×} : \frac{1}{2}α̇ ∉ Ṙ\} and Ṙ_{red} = Ṙ_{red}^{×} ∪ \{0\}.

Then, Ṙ_{red} is an irreducible reduced finite root system in V̇ with fundamental system Π̇. Also, Ṙ ⊆ Ṙ_{red} ∪ 2Ṙ_{red}; and Ṙ = Ṙ_{red} ⟺ Ṙ is reduced.

We now put

Ġ = subalgebra of L_{c} generated by eα̇_{i}, fα̇_{i}, i = 1, …, l,

and

Ḣ = ∑_{i=1}^{l} ℂhα̇_{i}.

Note that the matrix ((hα̇_{i}, hα̇_{j})) is nonsingular and so we have

H = Ḣ ⊕ Ḣ^{⊥},

where

Ḣ^{⊥} = {h ∈ H : (k, h) = 0 for all k ∈ Ḣ} = {h ∈ H : α̇(h) = 0 for all α̇ ∈ Ṙ}.

We may thus identify

H^{∗} = Ḣ^{∗} ⊕ (Ḣ^{⊥})^{∗}

by regarding elements of Ḣ^{∗} (resp. (Ḣ^{⊥})^{∗}) as zero on Ḣ^{⊥} (resp. Ḣ). Then, we have Ṙ ⊆ Ḣ^{∗} and Λ ⊆ (Ḣ^{⊥})^{∗}. With these definitions and identifications we have:

Proposition 1.16.

(a) Ġ is a finite dimensional simple subalgebra of L_{c} with Cartan subalgebra Ḣ and root system Ṙ_{red} relative to Ḣ. The adjoint action of Ḣ on L_{c} is diagonalizable; and relative to this action we have

(1.17)  L_{c} = ⨁_{α̇∈Ṙ}(L_{c})α̇,

where

(L_{c})α̇ := {x ∈ L_{c} : [h, x] = α̇(h)x for h ∈ Ḣ}

for α̇ ∈ Ḣ^{∗}. L_{c} is generated as an algebra by the spaces (L_{c})α̇, α̇ ∈ Ṙ^{×}.

(b) If α̇ ∈ Ṙ we have

(1.18)  (L_{c})α̇ = \{ \begin{matrix}∑_{β̇∈Ṙ^{×}} ∑_{τ,υ∈Λ}[L_{β̇+τ} , L_{−β̇+υ}] & if α̇ = 0 \\ ⨁_{σ∈Λ} L_{α̇+σ} & if α̇ ≠ 0. \end{matrix}

Proof. For the first statement in (a), it is enough to show that the elements eα̇_{i}, fα̇_{i}, hα̇_{i}, i = 1, …, l, satisfy the Serre relations (see for example [H, Theorem 18.3]). These relations follow from the fact that

α̇_{i} − α̇_{j} ∉ R and α̇_{j} + (−2\frac{(α̇_{j}, α̇_{i})}{(α̇_{i}, α̇_{i})} + 1)α̇_{i} ∉ R

for i ≠ j. This in turn follows from the corresponding fact with R replaced by Ṙ and from (1.9). The second statement in (a) as well as (1.18) in (b) follow from (1.13). Finally, the last statement in (a) follows from (b). □

Remark 1.19. If L has reduced type, then Ṙ = Ṙ_{red} is a finite irreducible reduced root system and Proposition 1.16 (a) says that L_{c} is an Ṙ-graded Lie algebra as defined in [BM] and [BZ]. Thus, in general, we refer to the decomposition described in Proposition 1.16 (a) as an Ṙgrading of L_{c}. (We will shortly be restricting our attention to the case when L has reduced type in any case.)

Finally, we note that the Λ-grading and the Ṙ-grading of L_{c} are compatible. That is, by (1.15) and (1.18), we have

(1.20) (L_{c})α̇ = ⨁_{σ∈Λ}(L_{c})α̇ ∩ (L_{c})^{σ} for α̇ ∈ Ṙ, and (L_{c})^{σ} = ⨁_{α̇∈Ṙ}(L_{c})α̇ ∩ (L_{c})^{σ} for σ ∈ Λ

for α̇ ∈ Ṙ. Moreover, if σ ∈ Λ and α̇ ∈ Ṙ, then

(1.21)  (L_{c})α̇ ∩ (L_{c})^{σ} = \{ \begin{matrix}∑_{β̇∈Ṙ^{×}} ∑_{\begin{matrix}τ,υ∈Λ \\ τ+υ=σ \end{matrix}}[L_{β̇+τ} , L_{−β̇+υ}] & if α̇ = 0 \\ L_{α̇+σ} & if α̇ ≠ 0. \end{matrix}

Hence, if α̇ ∈ Ṙ_{red}^{×}, then

(1.22)  (L_{c})α̇ ∩ (L_{c})^{0} = Lα̇ = Ġα̇,

where Ġα̇ denotes the α̇–root space of Ġ relative to Ḣ. (The last equality holds since we have Ġα̇ ⊆ Lα̇ and both spaces are 1-dimensional).

Next let

K := L_{c}/Z(L_{c}),

where Z(L_{c}) denotes the centre of L_{c}. The second main goal of this paper is to study the structure of the Lie algebra K. So we conclude this section by summarizing the facts that we will need about K.

First we need some notation. We denote by π the canonical projection π : L_{c} → K defined by

π(x) = x + Z(L_{c}).

One easily checks that

(1.23)  Z(L_{c}) = rad (·, ·)|_{L_{c}} := {x ∈ L_{c} : (x, y) = 0 for all y ∈ L_{c}}.

Hence, we may define a bilinear form (·, ·) on K by

(π(x), π(y)) := (x, y)

for x, y ∈ L_{c}. Also, we define

(1.24)  K^{σ} := π((L_{c})^{σ})

for σ ∈ Λ.

Next π is not trivial on Ġ. Hence, π is injective on Ġ. Thus, we can and do identify Ġ as a subalgebra of K by identifying x with π(x) for x ∈ Ġ. Thus, we have also identified Ḣ as a subalgebra of K, and so we may define

(1.25)  (K)α̇ := {x ∈ K : [h, x] = α̇(h)x for h ∈ Ḣ}

for α̇ ∈ Ḣ^{∗}. In that case, since π commutes with the action of Ḣ, we have

(1.26)  Kα̇ = π((L_{c})α̇)

for α̇ ∈ Ḣ^{∗}. Finally, for σ ∈ Λ and α̇ ∈ Ḣ^{∗}, we put

Kσ ; α̇ := Kα̇ ∩ K^{σ}.

Then we have

(1.27)  Kσ ; α̇ = π((L_{c})α̇ ∩ (L_{c})^{σ}).

Indeed the inclusion “⊇” is clear, and the inclusion “⊆” follows easily from (1.20) and the fact that ker(π) ⊆ (L_{c})_{0}.

The main proposition of this section is:

Proposition 1.28.

(a) (·, ·) is a nondegenerate invariant symmetric bilinear form on K.

(b) K is perfect and the centre of K is {0}.

(c) K = ⨁_{σ∈Λ} K^{σ} is a Λ-graded Lie algebra.

(d) Ġ is a finite dimensional simple subalgebra of K with Cartan subalgebra Ḣ and root system Ṙ_{red} relative to Ḣ. The adjoint action of Ḣ on K is diagonalizable; and relative to this action we have

(1.29)  K = ⨁_{α̇∈Ṙ} Kα̇.

K is generated as an algebra by the spaces Kα̇, α̇ ∈ Ṙ^{×}.

(e) The gradings in (c) and (d) are compatible:

Kα̇ = ⨁_{σ∈Λ} Kσ ; α̇ for α̇ ∈ Ṙ, and K^{σ} = ⨁_{α̇∈Ṙ} Kσ ; α̇ for σ ∈ Λ.

(f) If α̇ ∈ Ṙ^{×}, σ ∈ Λ, then

dim Kσ ; α̇ = \{ \begin{matrix}1 & if σ ∈ Sα̇ \\ 0 & otherwise \end{matrix} .

(g) If α̇, β̇ ∈ Ḣ^{∗} and σ, τ ∈ Λ, then [Kσ ; α̇, Kτ ; β̇] ⊆ K_{α̇+β̇}^{σ+τ}. Moreover, if α̇, β̇, α̇ + β̇ ∈ Ṙ^{×} and

σ ∈ Sα̇, τ ∈ Sβ̇, σ + τ ∈ S_{α̇+β̇} then

(1.30)  [Kσ ; α̇, Kτ ; β̇] = K_{α̇+β̇}^{σ+τ}.

(h) If α̇, β̇ ∈ Ḣ^{∗} and σ, τ ∈ Λ, then (Kσ ; α̇, Kτ ; β̇) = {0} unless α̇ + β̇ = 0 and σ + τ = 0.

(i) If α̇ ∈ Ṙ_{red}^{×}, then K0 ; α̇ = Ġα̇ where Ġα̇ denotes the α̇–root space of Ġ relative to Ḣ.

Proof.

(a): The nondegeneracy follows from (1.23). The rest is clear.

(b): K is perfect since L_{c} is perfect. This and the invariance and nondegeneracy of (·, ·) imply that the centre of K is {0}.

(c): If σ, τ ∈ Λ and σ + τ ≠ 0, then, by (1.1), (L^{σ}, L^{τ}) = {0} and so (L_{c}^{σ}, L_{c}^{τ}) = {0}. It follows from this and (1.23) that Z(L_{c}) is a Λ–graded ideal of L_{c}. So K = L_{c}/Z(L_{c}) is Λ-graded.

(d) and (e): These follow respectively from Proposition 1.16 (a) and (1.20).

(f): Suppose that α̇ ∈ Ṙ^{×}. Then, by (1.21) and (1.27), we have

(1.31)  Kσ ; α̇ = π(L_{α̇+σ})

for σ ∈ Λ. But π|_{(L_{c})α̇} is injective. Thus, dim Kσ ; α̇ = dim L_{α̇+σ}. The required expression then follows from (1.2).

(g): The first statement follows from (c) and (d). Suppose then that α̇, β̇, α̇ + β̇ ∈ Ṙ^{×} and σ ∈ Sα̇, τ ∈ Sβ̇, σ + τ ∈ S_{α̇+β̇}. Then, we have using (1.31) and Lemma 1.3,

[Kσ ; α̇, Kτ ; β̇] = [π(L_{α̇+σ}), π(L_{β̇+τ})] = π[L_{α̇+σ}, L_{β̇+τ}] = π(L_{α̇+β̇+σ+τ}) = K_{α̇+β̇}^{σ+τ}.

(h): Suppose that α̇, β̇ ∈ Ḣ^{∗} and σ, τ ∈ Λ satisfy α̇ + β̇ ≠ 0 or σ + τ ≠ 0. Then, using (1.27), (1.21) and (1.1), we have

(Kσ ; α̇, Kτ ; β̇) = ((L_{c})α̇ ∩ (L_{c})^{σ}, (L_{c})β̇ ∩ (L_{c})^{τ}) ⊆ (L_{α̇+σ}, L_{β̇+τ}) = {0}.

(i): Let α̇ ∈ Ṙ_{red}^{×}. Then, by (1.31) and (1.22), we have K0 ; α̇ = π(Lα̇) = π(Ġα̇) = Ġα̇ using our identification of Ġ with π(Ġ). □

Remark 1.32. If L has reduced type, then once again Proposition 1.28 (d) says that K has the structure of an Ṙ–graded Lie algebra as defined in [BM] and [BZ]. This will be a basic tool in determining the structure of K.

§2. Coordinate algebras

In this section, we introduce the graded algebras that will be used as coordinate algebras to describe the structure of the Lie algebra K. These include the quantum torus, the octonion torus and the Albert torus (see [BGKN], [AABGP] and [Y]).

Throughout this section, Λ will denote a free abelian group of rank ν, where ν ≥ 0. We fix a basis σ_{1}, …, σ_{ν} of Λ, and so

Λ = ℤσ_{1} ⊕ ⋯ ⊕ ℤσ_{ν}.

Whenever Λ = ℤ^{ν}, we assume that σ_{1}, …, σ_{ν} is the standard basis unless the contrary is stated.

If k = 2 or 3 and 0 ≤ p ≤ ν, we write

(2.1)  Λ_{k}^{(p)} = kℤσ_{1} ⊕ ⋯ ⊕ kℤσ_{p} ⊕ ℤσ_{p+1} ⊕ ⋯ ⊕ ℤσ_{ν}.

Of course, Λ_{k}^{(p)} depends on our choice of basis σ_{1}, …, σ_{ν} for Λ.

We will be interested in algebras D (not necessarily associative) with the following properties:

(G1)  D = ⨁_{σ∈Λ} D^{σ} with 1-dimensional summands

and

(G2)  D^{σ}D^{τ} = D^{σ+τ} for σ, τ ∈ Λ.

We note that it follows easily from (G1) and (G2) that

D^{0} = ℂ1.

Also, given an algebra D satisfying (G1) and (G2), we will be interested in involutions ¯ which satisfy

(G3)  \overline{D^{σ}} ⊆ D^{σ} for σ ∈ Λ.

Here an involution means an antiautomorphism of period 2. Of course any algebra D satisfying (G1) and (G2) is a Λ-graded algebra, and (G3) simply says that ¯ preserves the Λ-grading.

If D_{1} and D_{2} are any Λ-graded algebras, we write D_{1} ≅_{Λ} D_{2} to mean that D_{1} and D_{2} are isomorphic as Λ-graded algebras. Also, if (D, −) is any algebra with involution, we use the notation

H(D, −) := {d ∈ D : d̄= d}.

To recall the first example, suppose that q is a ν × ν-complex matrix which satisfies

(2.2)  q_{ii} = 1, 1 ≤ i ≤ ν, and q_{ij} = q_{ji}^{−1}, 1 ≤ i ≠ j ≤ ν.

Let ℂ_{q} be the associative algebra with generators t_{1}^{±1}, …, t_{ν}^{±1} subject to the relations

(2.3)  t_{i}t_{i}^{−1} = t_{i}^{−1}t_{i} = 1, 1 ≤ i ≤ ν, and t_{i}t_{j} = q_{ij}t_{j}t_{i}, 1 ≤ i ≠ j ≤ ν.

ℂ_{q} is called the quantum torus determined by q (see [M]). We write

t^{(n_{1},...,n_{ν})} := t_{1}^{n_{1}} … t_{ν}^{n_{ν}}

for n_{1}, …, n_{ν} ∈ ℤ. Also, if σ = ∑ n_{i}σ_{i} ∈ Λ, we define

ℂ_{q}^{σ} = ℂt^{(n_{1},...,n_{ν})}.

In other words,

deg t^{(n_{1},...,n_{ν})} = ∑ n_{i}σ_{i}.

Then, ℂ_{q} = ⨁_{σ∈Λ} ℂ_{q}^{σ} satisfies (G1) and (G2).

Note that, if q satisfies (2.2), then ℂ_{q} is commutative if and only if q_{ij} = 1 for all i, j. Moreover, if q_{ij} = 1 for all i, j, we have ℂ_{q} = A_{[ν]}, where

A_{[ν]} := algebra of Laurent polynomials in the commuting variables t_{1}, …, t_{ν}.

So A_{[ν]} = ⨁_{σ∈Λ} A_{[ν]}^{σ} satisfies (G1) and (G2).

The algebras ℂ_{q} and A_{[ν]} are characterized as follows:

Proposition 2.4. [BGKN, Lemma 1.8] Suppose that D is an associative algebra satisfying (G1) and (G2). Then D ≅_{Λ} ℂ_{q} for some q satisfying (2.2). Hence, if D is commutative then D ≅_{Λ} A_{[ν]}.

To describe the involutions on ℂ_{q} which satisfy (G3) we require another definition. Suppose that q is a ν × ν-matrix over ℂ so that

(2.5)  q_{ii} = 1, 1 ≤ i ≤ ν, and q_{ij} = q_{ji} = ±1, 1 ≤ i ≠ j ≤ ν.

Let e be a ν-vector over ℂ so that

(2.6)  e_{i} = ±1, 1 ≤ i ≤ ν.

Then, the elements e_{i}t_{i}^{±1}, 1 ≤ i ≤ ν, of the opposite algebra ℂ_{q}^{op} satisfy the defining relations (2.3) for ℂ_{q}, and hence there exists a unique antihomomorphism ¯ : ℂ_{q} → ℂ_{q} so that

\overline{t_{i}} = e_{i}t_{i}, 1 ≤ i ≤ ν.

It is clear from (G1) for ℂ_{q} that ¯ is an involution of ℂ_{q}. We call ¯ the involution determined by e, and we call (ℂ_{q}, −) the quantum torus with involution determined by e and q. In that case, we have, by [AABGP, Lemma III.3.5],

(2.7)  H(ℂ_{q}, −) = ⨁_{σ∈Z_{e,q}} ℂ_{q}^{σ},

where

(2.8)  Z_{e,q} := {∑ n_{i}σ_{i} ∈ Λ : ∑_{i=1}^{ν} δ_{e_{i},−1}n_{i} + ∑_{1≤i<j≤ν} δ_{q_{ij},−1}n_{i}n_{j} ≡ 0 (mod 2)}.

(Here the δ’s are Kronecker delta’s.) Finally, we note that if e = 1 := (1, …, 1), the involution ¯ satisfies

\overline{t_{1}^{n_{1}} … t_{ν}^{n_{ν}}} = t_{ν}^{n_{ν}} … t_{1}^{n_{1}} for n_{1}, …, n_{ν} ∈ ℤ,

and so we call this involution the reversal involution of ℂ_{q}.

The involutions on ℂ_{q} which satisfy (G3) are described by:

Proposition 2.9. Suppose that q is a ν × ν-matrix satisfying (2.2) and ¯ is an involution of ℂ_{q} satisfying (G3). Then q satisfies (2.5) and ¯ is the involution determined by some e satisfying (2.6).

Proof. Since ¯ satisfies (G3), we have \overline{t_{i}} = e_{i}t_{i} for some e_{i} ∈ ℂ^{×}, 1 ≤ i ≤ n. Then, since ¯ has period 2, e satisfies (2.6). Also, if i ≠ j, we have t_{i}t_{j} = q_{ij}t_{j}t_{i}. Applying ¯ to this equation gives e_{i}e_{j}t_{j}t_{i} = q_{ij}e_{i}e_{j}t_{i}t_{j}, and so t_{j}t_{i} = q_{ij}t_{i}t_{j} = q_{ij}^{2}t_{j}t_{i}. Thus, q_{ij}^{2} = 1 and so we have (2.5). □

For our next examples, we now recall the definition of the alternative algebras

(2.10)  A_{[ν]}, A(A_{[ν]}, t_{1}), A(A_{[ν]}, t_{1}, t_{2}), A(A_{[ν]}, t_{1}, t_{2}, t_{3})

([BGKN, §1]). For convenience of reference here we denote these algebras by A_{0}, A_{1}, A_{2} and A_{3} in order. Each of these algebras is an algebra over A_{[ν]} and hence over ℂ. By definition, A_{0} = A_{[ν]}. Moreover, for p = 1, 2, 3,

A_{p} = A_{p−1} ⊕ x_{p}A_{p−1}

is the algebra obtained from A_{p−1} using the Cayley-Dickson process with

x_{p}^{2} = t_{p}.

If 0 ≤ p ≤ 3, note that we also require p ≤ ν for the algebra A_{p} to be defined. A_{p} has a unique Λ-grading so that

deg(x_{i}) = σ_{i}, i = 1, …, p, and deg(t_{i}) = σ_{i}, i = p + 1, …, ν,

in which case (G1) and (G2) hold. (See [BGKN, Lemma 1.20]. We have made a minor change in notation from [BGKN].) Hence, in A_{p}, we have

deg(t_{i}) = \{ \begin{matrix}2σ_{i}, & i = 1, …, p \\ σ_{i} & i = p + 1, …, ν. \end{matrix}

Of course the Λ-grading on A_{p} depends on our choice of basis σ_{1}, …, σ_{ν} for Λ.

We will be particularly interested in the last algebra A(A_{[ν]}, t_{1}, t_{2}, t_{3}) in the above sequence. Following [Y], we call A(A_{[ν]}, t_{1}, t_{2}, t_{3}) the octonion torus over ℂ. The octonion torus is alternative but not associative.

Recall that if D is any algebra, the nucleus (or associative centre) of D is defined to be

N(D) := {x ∈ D : (x, D, D) = (D, x, D) = (D, D, x) = {0}},

where (x, y, z) := (xy)z − x(yz). Also, the centre of D is defined to be

Z(D) := {x ∈ N(D) : [x, D] = {0}},

where [x, y] := xy − yx. N(D) and Z(D) are subalgebras of D.

If A(A_{[ν]}, t_{1}, t_{2}, t_{3}) is the octonion torus, we have

(2.11)  N(A) = Z(A) = A_{[ν]} = ⨁_{σ∈Λ_{2}^{(3)}} A^{σ}.

(To see the first two equalities, one can extend the base ring from A_{[ν]} to the field of fractions of A_{[ν]} and use the corresponding facts for Cayley-Dickson algebras over fields [ZSSS,§2.4, Theorem 5]).

The octonion torus was characterized in [BGKN] as follows:

Proposition 2.12. Suppose that A is an alternative algebra over ℂ which is not associative, and suppose that A satisfies (G1) and (G2). Then, we may choose the basis σ_{1}, …, σ_{ν} for Λ so that

Z(A) = ⨁_{σ∈Λ_{2}^{(3)}} A^{σ}.

Moreover, if the basis σ_{1}, …, σ_{ν} is chosen in that way, we have A ≅_{Λ} A(A_{[ν]}, t_{1}, t_{2}, t_{3}).

Proof. Only the isomorphism A ≅ A(A_{[ν]}, t_{1}, t_{2}, t_{3}) is actually stated in [BGKN, Corollary 1.26]. The more detailed statement of the proposition is part of the proof of [BGKN, Theorem 1.25]. □

If A = A(A_{[ν]}, t_{1}, t_{2}, t_{3}) is the octonion torus, we have

A = A_{[ν]} ⊕ [A, A].

The standard involution on A is then defined by

\overline{α + x} = α − x

for α ∈ A_{[ν]}, x ∈ [A, A]. The standard involution satisfies (G3) and, by (2.11), we have

(2.13)  H(A, −) = ⨁_{σ∈Λ_{2}^{(3)}} A^{σ}.

The standard involution has the following characterization:

Proposition 2.14. If ¯ is an involution on the octonion torus A(A_{[ν]}, t_{1}, t_{2}, t_{3}) so that the symmetric elements of ¯ are contained in the nucleus, then ¯ is the standard involution.

Proof. This is well known. We include a proof since there doesn’t seem to be a convenient reference. Let A = A(A_{[ν]}, t_{1}, t_{2}, t_{3}). We’ve seen that A = K ⊕ L, where K := N(A) = Z(A) and L = [A, A]. Now ¯ stabilizes this decomposition. Thus, K = K_{+} ⊕ K_{−}, where K_{±} is the ±1-eigenspace of ¯ restricted to K. Similarly, L = L_{+} ⊕ L_{−}. But L_{+} ⊆ H(A, −) ⊆ K (by hypothesis) and so L_{+} = {0}. Then K_{−}L = K_{−}L_{−} ⊆ L_{+} and so K_{−}L = {0}. Since L contains invertible elements of A (e.g. x_{1}), we have K_{−} = {0}. So ¯ is standard. □

To give our last examples, we recall some notation. If E is any algebra, E^{+} will denote the algebra with underlying vector space E and the product x · y = \frac{1}{2}(xy + yx). E^{+} is called the plus algebra of E. If E is associative, then E^{+} is a Jordan algebra [J1, Section I.4].

We now recall the definition of the Jordan algebras

(2.15)  A_{[ν]}, J (A_{[ν]}, t_{1}), J (A_{[ν]}, t_{1}, t_{2}), J (A_{[ν]}, t_{1}, t_{2}, t_{3})

(see [AABGP, Example III.5.10]). In this discussion, we denote these algebras by J_{0}, J_{1}, J_{2} and J_{3} in order. Each of these algebras is an algebra over A_{[ν]} and hence over ℂ; and we use · for the product in each case. By definition J_{0} = A_{[ν]}. Also J_{1} is the commutative associative algebra over A_{[ν]} with generator x_{1} subject to the relation x_{1}^{3} = t_{1}. Next J_{2} is the plus algebra of the associative algebra generated by x_{1}, x_{2} subject to the relations x_{1}^{3} = t_{1}, x_{2}^{3} = t_{2}, x_{1}x_{2} = e^{\frac{2πi}{3}} x_{2}x_{1}. Finally,

J_{3} = J_{2} ⊕ J_{2} · x_{3} ⊕ J_{3} · x_{3}^{2}

is the Jordan algebra obtained from J_{2} using Tits’ first Jordan algebra construction (see [J1, Chapter IX]) with

x_{3}^{3} = t_{3}.

Again, if 0 ≤ p ≤ 3, we also require p ≤ ν for J_{p} to be defined. J_{p} has a unique Λ-grading so that

deg(x_{i}) = 0, i = 1, …, p, and deg(t_{i}) = σ_{i}, i = p + 1, …, ν

[AABGP, Proposition III.5.22]. Hence, in J_{p}, we have

deg(t_{i}) = \{ \begin{matrix}3σ_{i}, & i = 1, …, p \\ σ_{i}, & i = p + 1, …, ν. \end{matrix}

Again, the Λ-grading on J_{p} depends on our choice of basis σ_{1}, …, σ_{ν} for Λ. The final algebra J (A_{[ν]}, t_{1}, t_{2}, t_{3}) in the above sequence has been called the Albert torus in [Y].

To conclude this section, we need to describe a characterization of the algebras J (A_{[ν]}, t_{1}, t_{2}) and J (A_{[ν]}, t_{1}, t_{2}, t_{3}) which is due to Y. Yoshii [Y]. For this purpose note that if J is any Jordan algebra satisfying (G1) and (G2), then the centre Z := Z(J) of J is an integral domain [Y, Lemma 1], and so we can form the field of fractions (Z^{∗})^{−1}Z of Z and put

(Z^{∗})^{−1}J := (Z^{∗})^{−1}Z ⊗_{Z} J .

If J = J (A_{[ν]}, t_{1}, t_{2}) or J (A_{[ν]}, t_{1}, t_{2}, t_{3}), then (Z^{∗})^{−1}J is a central simple Jordan algebra of dimension 9 or 27 respectively over (Z^{∗})^{−1}Z, and (Z^{∗})^{−1}J has generic degree 3 over (Z^{∗})^{−1}Z. (See [AABGP, Example III.5.5]. See also [J1, Chapter VI] for facts about generic degree.) Hence,

(2.16)  Z(J) = A_{[ν]} = ⨁_{σ∈Λ_{3}^{(2)}} J^{σ} or ⨁_{σ∈Λ_{3}^{(3)}} J^{σ}

respectively.

Conversely, we have:

Proposition 2.17. (Yoshii) Suppose that J is a Jordan algebra which satisfies (G1) and (G2) and suppose that (Z^{∗})^{−1}J is finite dimensional of generic degree 3 over (Z^{∗})^{−1}Z, where Z = Z(J). Then, we may choose the basis σ_{1}, …, σ_{ν} for Λ so that

Z(J) = ⨁_{σ∈Λ_{2}^{(p)}} J^{σ},

where p = 2 or 3. Moreover, if the basis σ_{1}, …, σ_{ν} is chosen in that way, we have

(2.18)  J ≅_{Λ} \{ \begin{matrix}J (A_{[ν]}, t_{1}, t_{2}) & if p = 2 \\ J (A_{[ν]}, t_{1}, t_{2}, t_{3}) & if p = 3. \end{matrix}

Proof. Only the isomorphism ≅ in (2.18) is actually announced in [Y, Corollary 2]. The more detailed statement of the proposition is part of Yoshii’s proof (not yet published). □

§3. Simply laced types

In this section, we briefly recall for completeness and later reference the known results about the structure of K and R when L has simply laced type. The description of R was proved in [AABGP, Theorem II.2.3]. The description of K for rank ≥ 3 was proved in [BGK, Theorems 1.37 and 2.65], while for rank 2 it was proved in [BGKN, Lemma 3.25].

Theorem 3.1. Suppose that L is an EALA of type X and nullity ν, where X = A_{l}(l ≥ 1), D_{l}(l ≥ 4) or E_{l}(l = 6, 7, 8). Then

(3.2)  R ≅ R(X, S) := (S + S) ∪ (Ṙ^{×} + S),

where S is a semilattice in ℝ^{ν} if l = 1 and S = ℤ^{ν} otherwise. If X = D_{l} or E_{l}, then

(3.3)  K ≅ Ġ ⊗ A_{[ν]},

where Ġ is the complex simple Lie algebra of type X. If X = A_{l}(l ≥ 2), then

(3.4)  K ≅ psl_{l+1}(A),

where either A = ℂ_{q} for some ν × ν-matrix q satisfying q_{ii} = 1, q_{ij} = q_{ji}^{−1}, or l = 2, ν ≥ 3 and A is the octonion torus.

Remark 3.5. If A is an alternative algebra, psl_{3}(A) is defined in [F, Appendix] (where it is denoted pe_{3}(A)), and in a different way in [BGKN, Definition 2.5]. If A is an associative algebra, psl_{l+1}(A) is defined to be sl_{l+1}(A)/Z(sl_{l+1}(A)), where sl_{l+1}(A) is the Lie algebra generated by the (l + 1) × (l + 1) elementary matrices ae_{ij}, a ∈ A, i ≠ j. Note that when A = ℂ_{q}, it follows easily from the fact that Z(A) ∩ [A, A] = {0} [BGK, Proposition 2.44] that Z(sl_{l+1}(A)) = {0} and hence psl_{l+1}(A) = sl_{l+1}(A).

Remark 3.6. The structure of K when L has type A_{1} is not covered by the above theorem. This topic is being studied by Y. Yoshii as part of his dissertation research at the U. of Ottawa. (See [Y] for some related results.)

Remark 3.7. Each of the root systems appearing on the right hand side of (3.2) and each of the Lie algebras appearing on the right hand side of (3.3) or (3.4) actually occur for some EALA. The examples that support this statement can be found in [AABGP, Theorem III.2.9] for type A_{1}, [HT, §III.1] or [AABGP, Example III.1.29] for type D_{l} or E_{l}, and [BGK, Proposition 2.63] and [BGKN, Example 3.27] for type A_{l}(l ≥ 2). Thus, Theorem 3.1 is an exact description of the root systems R and the Lie algebras K (with the exception noted in Remark 3.6) which can arise from an EALA of simply laced type. The rest of the paper will be devoted to obtaining similar descriptions for the nonsimply laced reduced types.

§4. Type C_{l}, l ≥ 2

In this section, we describe the structure of R and K in the case when L has type C_{l}, l ≥ 2. This includes type B_{2} = C_{2} which turns out to be the most difficult case to handle.

Throughout the section we assume that l ≥ 2 and L is an EALA of type C_{l}. We use all of the notation introduced in Section 1.

In particular, by (1.12), we have

(4.1)  R = R(C_{l}, S, L) := (S + S) ∪ (Ṙ_{sh} + S) ∪ (Ṙ_{lg} + L),

where S = Sα̇ for α̇ ∈ Ṙ_{sh} and L = Sα̇ for α̇ ∈ Ṙ_{lg}. Moreover, by [AABGP, Theorem II.2.37], S and L are semilattices in V^{0} so that

(4.2)  L + 2S ⊆ L and S + L ⊆ S

Also, by the same theorem, if l ≥ 3 then S is a lattice in V^{0}.

We now look at the Lie algebra K. Our description will involve Jordan algebras, and so we first need to recall some standard notation from Jordan theory. If J is a Jordan algebra with product ·, we put

K(J) := J ⊕ Instl(J) ⊕ J̃,

where Instl(J) = L_{J} ⊕[L_{J} , L_{J}] is the Lie algebra generated by the set L_{J} of left multiplication operators for J and J̃ = {ã : a ∈ J } is a linear copy of J. K(J) is a Lie algebra under the product [ , ] which satisfies [J , J] = {0}, [J̃, J̃] = {0}, [A, B] := AB − BA for A, B in Instl(J),

[L_{a} + D, b] = a · b + Db, [L_{a} + D, b̃] = (−a · b + Db)˜ and [a, b̃] = L_{a·b} + [L_{a}, L_{b}]

for a, b ∈ J, D ∈ [L_{J} , L_{J}]. K(J) is called the Tits-Kantor-Koecher Lie algebra constructed from J. (For details on the TKK construction see [J1, Section VIII.5].) We note that the Lie algebra T = K(J) contains an 𝔰𝔩_{2}-triplet e = 1, h = 2L_{1}, f = 2˜1 with the following properties: T is the direct sum of the 2, 0 and −2 eigenspaces relative to h, T is generated as a Lie algebra by the 2 and −2 eigenspaces, and the only ideal of T which centralizes e, h and f is {0}. Conversely, it follows from a theorem of Tits’ [T] (see also [BZ, §1]) that if T is any Lie algebra containing an 𝔰𝔩_{2}-triplet e, h, f with the indicated properties then T ≅ K(J), where J is the Jordan algebra with underlying vector space equal to the 2-eigenspace of h and product given by x · y = \frac{1}{2}[[x, f], y].

Also, if (E, −) is an algebra with involution and l ≥ 1, we let

H_{l}(E, −) := {A ∈ Mat_{l}(E) : Ā^{t} = A}.

H_{l}(E, −) is a subalgebra of Mat_{l}(E)^{+} and hence H_{l}(E, −) is an algebra under the product x · y := \frac{1}{2}(xy + yx). If E is associative or if l = 3 and E is alternative with ¯ -symmetric elements in the nucleus, then H_{l}(E, −) is a Jordan algebra [J1, Theorem III.1]. Finally, if E is associative and commutative and ¯ = I, we write H_{l}(E) := H_{l}(E, −) for the Jordan algebra of symmetric l × l-matrices over E.

We are now ready to look at the Lie algebra K. First, the subalgebra Ġ of K is a finite dimensional simple Lie algebra of type C_{l}. Hence, we may identify

Ġ = sp_{2l}(ℂ) := \{[ \begin{matrix}A & B \\ C & −A^{t} \end{matrix} ] : A, B, C ∈ Mat_{l}(ℂ), B^{t} = B, C^{t} = C\},

and

Ḣ = {∑_{i=1}^{l} a_{i}(e_{ii} − e_{l+i,l+i}) : a_{i} ∈ ℂ},

where e_{ij} denotes the (ij)^{th}-matrix unit for 1 ≤ i, j ≤ 2l. In that case,

Ṙ = {0} ∪ {±2ϵ_{i}} ∪ {±(ϵ_{i} ± ϵ_{j}) : i < j},

where ϵ_{i} ∈ Ḣ^{∗} is defined by

(4.3)  ϵ_{i}(∑_{i=1}^{l} a_{i}(e_{ii} − e_{l+i,l+i})) = a_{i}

for 1 ≤ i ≤ l. Moreover, we choose the elements eα̇ ∈ Ġα̇ for α̇ ∈ Ṙ^{×} as follows:

(4.4)  \begin{matrix}e_{2ϵ_{i}} = e_{i,l+i}, e_{−2ϵ_{i}} = e_{l+i,i}, 1 ≤ i ≤ l \\ e_{ϵ_{i}−ϵ_{j}} = e_{i,j} − e_{l+j,l+i}, 1 ≤ i ≠ j ≤ l \\ e_{ϵ_{i}+ϵ_{j}} = e_{i,l+j} + e_{j,l+i}, e_{−ϵ_{i}−ϵ_{j}} = e_{l+i,j} + e_{l+j,i} 1 ≤ i ≤ l, \end{matrix}

We then define fα̇ ∈ Ġ_{−α̇} and hα̇ ∈ Ḣ by

(4.5)  fα̇ = e_{−α̇} and hα̇ = [eα̇, fα̇]

for α̇ ∈ Ṙ^{×}. In that case.

(4.6)  h_{2ϵ_{i}} = e_{ii} − e_{l+i,l+i}, 1 ≤ i ≤ l.

We now follow [BZ, §1] (see also [N, §4]) to obtain a Jordan algebra J so that K ≅ K(J). We put

(4.7) e := [ \begin{matrix}0 & I \\ 0 & 0 \end{matrix} ] = ∑_{i=1}^{l} e_{2ϵ_{i}}, f := [ \begin{matrix}0 & 0 \\ I & 0 \end{matrix} ] = ∑_{i=1}^{l} f_{2ϵ_{i}}, and h := [ \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ] = ∑_{i=1}^{l} h_{2ϵ_{i}}.

In that case {e, h, f} is an 𝔰𝔩_{2}-triplet. Moreover, we have

ϵ_{i}(h) = 1, 1 ≤ i ≤ l.

Thus, by (4.3) and (1.29), the endomorphism ad(h) of K is diagonalizable with eigenvalues 2, 0 and −2; and the corresponding eigenspaces are, in order,

∑_{i=1}^{l} K_{2ϵ_{i}} + ∑_{1≤i<j≤l} K_{ϵ_{i}+ϵ_{j}}, K_{0} + ∑_{1≤i≠j≤l} K_{ϵ_{i}−ϵ_{j}} and ∑_{i=1}^{l} K_{−2ϵ_{i}} + ∑_{1≤i<j≤l} K_{−ϵ_{i}−ϵ_{j}}.

Also, from 𝔰𝔩_{2}-theory for the 𝔰𝔩_{2}-triplet {e_{2ϵ_{i}}, h_{2ϵ_{i}}, e_{−2ϵ_{i}}}, it follows that [K_{ϵ_{i}+ϵ_{j}}, f_{2ϵ_{i}}] = K_{−ϵ_{i}+ϵ_{j}} for 1 ≤ i ≠ j ≤ l. Thus, the 2 and −2 eigenspaces for ad(h) generate K as a Lie algebra. It follows that any ideal of K which centralizes h must centralize K and hence is {0} (by Proposition 1.28 (b)). Thus, by Tits’ theorem, we have

(4.8)  K ≅ K(J),

where J is the Jordan algebra with unit element e defined as follows:

(4.9)  J = 2-eigenspace of ad(h) in K = ∑_{i=1}^{l} K_{2ϵ_{i}} + ∑_{1≤i<j≤l} K_{ϵ_{i}+ϵ_{j}}

with product

(4.10)  x · y = \frac{1}{2}[[x, f], y]

for x, y ∈ J.

The rest of the section will be devoted to the description of the Jordan algebra J. As a byproduct, we will obtain a description of the semilattices S and L. At the end of the section, we will summarize the consequences for the structure of L.

First put

M = Ġ ∩ J .

Then, by (4.9),

M = \{[ \begin{matrix}0 & B \\ 0 & 0 \end{matrix} ] : B ∈ H_{l}(ℂ)\}.

Moreover, as noted in [BZ,§1], M is a subalgebra of J and

(4.11)  M ≅ H_{l}(ℂ) under the isomorphism [ \begin{matrix}0 & B \\ 0 & 0 \end{matrix} ] → B.

Next we consider a Peirce decomposition of J. For this purpose, put

c_{i} := e_{2ϵ_{i}} = e_{i,l+i}

for 1 ≤ i ≤ l. Then, by (4.11), c_{1}, …, c_{l} is a set of orthogonal idempotents in M which sum to 1 (see [J1, §III.1]). Thus, J has a Peirce decomposition

(4.12)  J = ⨁_{1≤i≤j≤l} J_{ij},

where

J_{ii} = {x ∈ J : c_{i} · x = x}, 1 ≤ i ≤ l,

and

J_{ij} = \{x ∈ J : c_{i} · x = c_{j} · x = \frac{1}{2}x\}, 1 ≤ i ≠ j ≤ l.

But [c_{i}, f] = h_{2ϵ_{i}}, 1 ≤ i ≤ l; and hence the left multiplication operator L_{c_{i}} for the algebra J is given by

L_{c_{i}} = \frac{1}{2} ad(h_{2ϵ_{i}})|_{J} , 1 ≤ i ≤ l.

Thus,

(4.13)  J_{ii} = K_{2ϵ_{i}}, 1 ≤ i ≤ l, and J_{ij} = K_{ϵ_{i}+ϵ_{j}}, 1 ≤ i ≠ j ≤ l.

We will use the standard Peirce relations (see [J1,§III.1])

(4.14)

J_{ii} · J_{ii} ⊆ J_{ii},
J_{ii} · J_{ij} ⊆ J_{ij}, J_{ij} · J_{ij} ⊆ J_{ii} + J_{jj}, J_{ii} · J_{jj} = 0 if i ≠ j,
J_{ij} · J_{jk} ⊆ J_{ik}, J_{ij} · J_{kk} = 0 if i, j, k ≠, and
J_{ij} · J_{km} = 0 if i, j, k, m ≠.

Next recall from Section 1 that

Λ := ⟨R^{0}⟩ = ⟨S⟩

(see (1.7)). We can give the Jordan algebra J a Λ-grading. Indeed, we put J_{ii}^{σ} := K_{2ϵ_{i}}^{σ}, 1 ≤ i ≤ l, and J_{ij}^{σ} := K_{ϵ_{i}+ϵ_{j}}^{σ}, 1 ≤ i ≠ j ≤ l,

for σ ∈ Λ. By Proposition 1.28 (f), if σ ∈ Λ, we have

(4.15)  dim J_{ii}^{σ} = \{ \begin{matrix}1 & if σ ∈ L \\ 0 & otherwise \end{matrix}

for 1 ≤ i ≤ l; and

(4.16)  dim J_{ij}^{σ} = \{ \begin{matrix}1 & if σ ∈ S \\ 0 & otherwise \end{matrix}

for 1 ≤ i ≠ j ≤ l. Also, we note that by Proposition 1.28 (i), we have

(4.17)  J_{ij}^{0} = M ∩ J_{ij}

for 1 ≤ i, j ≤ l. We now put

(4.18)  J^{σ} := ⨁_{1≤i≤j≤l} J_{ij}^{σ} = J ∩ K^{σ}

for σ ∈ Λ. (The last equation follows from Proposition 1.28 (e).) Thus, by Proposition 1.28 (i), J^{σ} · J^{τ} = [[J^{σ}, f], J^{τ}] ⊆ [[K^{σ}, K^{0}], K^{τ}] ⊆ K^{σ+τ} and so

(4.19)  J^{σ} · J^{τ} ⊆ J^{σ+τ}

for σ, τ ∈ Λ. Thus J = ⨁_{σ∈Λ} J^{σ} is a Λ-graded Jordan algebra as claimed.

The Λ-grading of J has the following strong property:

Lemma 4.20. If σ ∈ L and τ ∈ S, then

(4.21)  J_{ii}^{σ} · J_{ij}^{τ} = J_{ij}^{σ+τ} for i ≠ j.

Also, if σ, τ ∈ S, then

(4.22)  J_{ij}^{σ} · J_{jk}^{τ} = J_{ik}^{σ+τ} for i, j, k ≠ .

Proof. Suppose that σ ∈ L, τ ∈ S and i ≠ j. Then,

J_{ii}^{σ} · J_{ij}^{τ} = [[f, K_{2ϵ_{i}}^{σ}], K_{ϵ_{i}+ϵ_{j}}^{τ}] = [[f_{2ϵ_{i}}, K_{2ϵ_{i}}^{σ}], K_{ϵ_{i}+ϵ_{j}}^{τ}] = [K_{2ϵ_{i}}^{σ}, [f_{2ϵ_{i}}, K_{ϵ_{i}+ϵ_{j}}^{τ}]]
= [K_{2ϵ_{i}}^{σ}, [K_{−2ϵ_{i}}^{0}, K_{ϵ_{i}+ϵ_{j}}^{τ}]] by Proposition 1.28 (i)
= [K_{2ϵ_{i}}^{σ}, K_{−ϵ_{i}+ϵ_{j}}^{τ}] by Proposition 1.28 (g)
= K_{ϵ_{i}+ϵ_{j}}^{σ+τ} = J_{ij}^{σ+τ} by Proposition 1.28 (g)

This proves (4.21). For (4.22), suppose that σ, τ ∈ S and i, j, k ≠. Then, l ≥ 3 and so S + S ⊆ S. Thus, σ + τ ∈ S. Therefore,

J_{ij}^{σ} · J_{jk}^{τ} = [[f, K_{ϵ_{i}+ϵ_{j}}^{σ}], K_{ϵ_{j}+ϵ_{k}}^{τ}] = [[f_{2ϵ_{i}} + f_{2ϵ_{j}}, K_{ϵ_{i}+ϵ_{j}}^{σ}], K_{ϵ_{j}+ϵ_{k}}^{τ}]
= [K_{ϵ_{i}+ϵ_{j}}^{σ}, [f_{2ϵ_{i}} + f_{2ϵ_{j}}, K_{ϵ_{j}+ϵ_{k}}^{τ}]] = [K_{ϵ_{i}+ϵ_{j}}^{σ}, [f_{2ϵ_{j}}, K_{ϵ_{j}+ϵ_{k}}^{τ}]]
= [K_{ϵ_{i}+ϵ_{j}}^{σ}, [K_{−2ϵ_{j}}^{0}, K_{ϵ_{j}+ϵ_{k}}^{τ}]] = [K_{ϵ_{i}+ϵ_{j}}^{σ}, K_{−ϵ_{j}+ϵ_{k}}^{τ}] = K_{ϵ_{i}+ϵ_{k}}^{σ+τ} = J_{ik}^{σ+τ} ,

again using Proposition 1.28 (i) and (g). □

We next define a bilinear form (·, ·)_{J} on J by putting

(x, y)_{J} = ([x, f], [y, f]) = −(x, [[y, f], f])

for x, y ∈ J. Then we have:

Lemma 4.23. (·, ·)_{J} is a nondegenerate invariant symmetric bilinear form on J. Moreover if 1 ≤ i, j, k, m ≤ l and σ, τ ∈ Λ, we have

(4.24)  (J_{ij}^{σ}, J_{km}^{τ})_{J} = {0} unless {i, j} = {k, l} and σ + τ = 0.

Proof. It is clear that (·, ·)_{J} is symmetric and bilinear. Nondegeneracy follows from the fact that the 2-eigenspace of ad(h) and the −2-eigenspace of ad(h) are paired under the form (·, ·) and the fact that ad(f)^{2} maps the former space onto the latter. For invariance, let F = ad(f). Observe that 0 = x · y − y · x = −\frac{1}{2}[F x, y] + \frac{1}{2}[F y, x] and so

[F x, y] + [x, F y] = 0

for x, y ∈ J. Applying F^{2} to both sides of this equation and using the fact that F^{3}J = {0}, we get

[F^{2}x, F y] + [F x, F^{2}y] = 0

for x, y ∈ J. Thus,

(x, y · z)_{J} = −(x, F^{2}(y · z)) = \frac{1}{2}(x, F^{2}[F y, z]) = \frac{1}{2}(x, 2[F^{2}y, F z] + [F y, F^{2}z])

= −\frac{1}{2}(x, [F y, F^{2}z]) = \frac{1}{2}([F y, x], F^{2}z) = (y · x, z)_{J} = (x · y, z)_{J}

for x, y, z ∈ J.

It remains to prove (4.24). Indeed, since (·, ·)_{J} is invariant, we have (J_{ij}, J_{km}) = {0} unless {i, j} = {k, m}. Thus, we can assume that {i, j} = {k, m} and σ + τ ≠ 0. But then

(J_{ij}^{σ}, J_{ij}^{τ})_{J} ⊆ (J^{σ}, J^{τ})_{J} ⊆ (F K^{σ}, K^{τ}) ⊆ (K^{σ}, K^{τ}) = {0}

by Proposition 1.28 (h). □

The next result describes the structure of J in the case when l ≥ 3. Note that in this case S is a lattice and so Λ = S.

Proposition 4.25. Suppose that l ≥ 3. Then we can identify Λ = ℤ^{ν} so that one of the following hold:

(i) J ≅_{Λ} H_{l}(ℂ_{q}, −), S = ℤ^{ν} and L = Z_{e,q}, where e is a ν-vector satisfying e_{i} = ±1, q is a ν × ν-matrix satisfying q_{ii} = 1, q_{ij} = q_{ji} = ±1, and (ℂ_{q}, −) is the quantum torus with involution determined by e and q; or

(ii) l = 3, ν ≥ 3, J ≅_{Λ} H_{3}(A, −), S = ℤ^{ν} and L = Λ_{2}^{(3)}, where (A, −) is the octonion torus with standard involution.

Proof. Now 1 ∈ J is the sum of l ≥ 3 nonzero orthogonal idempotents c_{1}, …, c_{l}. Moreover, for j = 2, …, l, there exists an element u_{1j} ∈ J_{1j} such that u_{1j}^{2} = c_{1} + c_{j} (by (4.11)). In other words, the idempotents c_{1}, …, c_{l} are strongly connected. Thus, as noted in [BZ,§1], Jacobson’s Strong Coordinatization Theorem [J1, Theorem III.5], tells us that there is an isomorphism

(4.26)  J → H_{l}(D, −),

where (D, −) is an algebra with involution so that either D is associative; or l = 3 and D is alternative and not associative with symmetric elements in the nucleus of D. To avoid confusion with the matrix units e_{ij} in Mat_{2l}(ℂ) used earlier in the description of Ġ, we denote the matrix units in H_{l}(D, −) by g_{ij}. We then put

a[ij] := ag_{ij} + āg_{ji} ∈ H_{l}(D, −)

for a ∈ D, 1 ≤ i, j ≤ l. With this notation, Jacobson’s theorem also tells us that the isomorphism (4.26) can be chosen so that c_{i} → \frac{1}{2}1[ii] for 1 ≤ i ≤ l and M → H_{l}(ℂ). We regard the isomorphism (4.26) chosen in this way as an identification and so we have

J = H_{l}(D, −) with c_{i} = \frac{1}{2}1[ii], 1 ≤ i ≤ l, and M = H_{l}(ℂ).

If follows that

J_{ij} = D[ij] := {a[ij] : a ∈ D}

for 1 ≤ i, j ≤ l.

We next want to introduce a Λ-grading on D. First suppose that 1 ≤ i ≠ j ≤ l. We define

D_{ij}^{σ} := {d ∈ D : d[ij] ∈ J_{ij}^{σ}} for σ ∈ Λ.

(We will shortly see that this definition is independent of i, j.) Then,

D = ⨁_{σ∈Λ} D_{ij}^{σ} and (D_{ij}^{σ})[ij] = J_{ij}^{σ} for σ ∈ Λ.

Observe also that (D_{ij}^{0})[ij] = J_{ij}^{0} = J_{ij} ∩ M (by (4.17)) = (ℂ1)[ij] and so

(4.27)  D_{ij}^{0} = ℂ1

Now if i, j, k ≠ and σ, τ ∈ Λ, we have

(D_{ij}^{σ}D_{jk}^{τ}) [ik] = ((D_{ij}^{σ})[ij]) · ((D_{jk}^{τ})[jk]) = J_{ij}^{σ} · J_{jk}^{τ} = J_{ik}^{σ+τ} = D_{ik}^{σ+τ} [ik],

using (4.22), as so

D_{ij}^{σ}D_{jk}^{τ} = D_{ik}^{σ+τ} .

But this together with (4.27) easily implies that D_{ij}^{σ} is independent of i, j for σ ∈ Λ. Hence, we may define

D^{σ} := D_{ij}^{σ}

for σ ∈ Λ and any i ≠ j. Then

(4.28)  J_{ij}^{σ} = (D^{σ})[ij]

for i ≠ j and σ ∈ Λ. Moreover

(4.29)  D = ⨁_{σ∈Λ} D^{σ} and D^{σ}D^{τ} = D^{σ+τ} for σ, τ ∈ Λ.

In particular, D is a Λ-graded algebra. Also, if σ ∈ Λ, then (\overline{D^{σ}}) [ij] = (D^{σ})[ji] = J_{ji}^{σ} = J_{ij}^{σ} = (D^{σ})[ij] for i ≠ j. Hence,

(4.30)  \overline{D^{σ}} = D^{σ} for σ ∈ Λ.

Moreover, by (4.16) and (4.28), we have

(4.31)  dim D^{σ} = 1 for σ ∈ Λ.

Thus, (D, −) satisfies assumptions (G1), (G2) and (G3) of Section 2.

Next J has the Λ-grading with which we are working, and J = H_{l}(D, −) also has the Λ-grading induced from the Λ-grading on D. From (4.28) and the fact that c_{i} ∈ J_{ii}^{0} for all i (see (4.17)), it follows that these two gradings agree on the set \{c_{1}, …, c_{l}\} ∪ (⋃_{{i<j \atop σ∈Λ}} J_{ij}^{σ}) which generates the algebra J. Hence these two gradings are the same. Thus J_{ii}^{σ} = (D^{σ})[ii] for 1 ≤ i ≤ l and σ ∈ Λ. So by (4.15), J_{ii} = ⨁_{σ∈L} J_{ii}^{σ} = ⨁_{σ∈L}(D^{σ})[ii]. Hence, by (4.30) and (4.31), we have

(4.32)  H(D, −) = ⨁_{σ∈L} D^{σ}.

We can now draw our conclusions. Since (D, −) satisfies assumptions (G1), (G2) and (G3) of Section 2, it follows from Propositions 2.4, 2.9, 2.12 and 2.14 that we know the structure of (D, −). Indeed, we may choose a basis σ_{1}, …, σ_{ν} for the lattice Λ = S and thereby identify Λ = S = ℤ^{ν} with standard basis σ_{1}, …, σ_{ν}. Moreover, we may do this so that either (D, −) is a quantum torus with involution determined by a ν-vector e satisfying e_{i} = ±1 and a ν × ν matrix q satisfying q_{ii} = 1, q_{ij} = q_{ji} = ±1; or l = 3 and (D, −) is the octonion torus with its standard involution. Moreover, the last case can occur only if ν ≥ 3. Also, by (2.7) and (2.11), we have H(D, −) = ⨁_{σ∈Z_{e,q}} D^{σ} in the first case, and H(D, −) = ⨁_{σ∈Λ_{2}^{(3)}} D^{σ} in the

last case. Hence, correspondingly, L = Z_{e,q} or L = Λ_{2}^{(3)} (by (4.32)). □

It remains to consider the structure of J in the case l = 2. Thus, for the rest of the section (except in the final summary theorem), we assume that l = 2. Then the Peirce decomposition of J relative to c_{1} and c_{2} is

J = J_{11} ⊕ J_{22} ⊕ J_{12}.

Moreover, by (4.15) and (4.16), we have

(4.33)  J_{ii} = ⨁_{σ∈L} J_{ii}^{σ} with one dimensional summands

for i = 1, 2, and

(4.34)  J_{12} = ⨁_{σ∈S} J_{12}^{σ} with one dimensional summands.

Finally, by (4.21),

(4.35)  J_{ii}^{σ} · J_{12}^{τ} = J_{12}^{σ+τ}

for σ ∈ L, τ ∈ S, i = 1, 2.

Now J_{ii} is a subalgebra of J and we define ρ_{i} : J_{ii} → End_{ℂ}(J_{12}) by

ρ_{i}(a) = 2L_{a}|_{J_{12}}, i = 1, 2.

For i = 1, 2, we put

D_{i} := ⟨ρ_{i}(J_{ii})⟩_{assoc},

where ⟨ρ_{i}(J_{ii})⟩_{assoc} denotes the associative algebra generated by ρ(J_{ii}). By [J1, PD2, p.121], ρ_{1}(J_{11}) and ρ_{2}(J_{22}) commute and so

(4.36)  [D_{1}, D_{2}] = {0}.

Also since J_{ii}^{σ} · J_{12}^{τ} ⊆ J_{12}^{σ+τ} for σ, τ ∈ Λ, it follows that

D_{i} = ⨁_{σ∈Λ} D_{i}^{σ}

is a Λ-graded algebra, where

D_{i}^{σ} := {d ∈ D_{i} : dJ_{12}^{τ} ⊆ J_{12}^{τ+σ} for τ ∈ Λ}.

Moreover, J_{12} is a Λ-graded module for D_{i}.

Lemma 4.37. Suppose that 1 ≤ i ≤ 2. Then, ρ_{i}(J_{ii}) is a Λ-graded subalgebra of the plus algebra D_{i}^{+} of D_{i}; and

ρ_{i} : J_{ii} → ρ_{i}(J_{ii})

is an isomorphism of Λ-graded Jordan algebras. In fact, if 0 ≠ x ∈ J_{12}^{τ}, where τ ∈ S, the map a → ρ_{i}(a)x is injective on J_{ii}.

Proof. By [J1, PD3, p.121], ρ_{i} : J_{ii} → D_{i}^{+} is a homomorphism of Jordan algebras. This map clearly preserves the Λ-grading. Thus, it remains to show the last statement. Since the map a → ρ_{i}(a)x preserves the Λ-grading and the summands in (4.33) are 1-dimensional, it is enough to show that ρ_{i}(J_{ii}^{σ})x ≠ {0} for σ ∈ L. But if σ ∈ L, we have ρ_{i}(J_{ii}^{σ})x = J_{12}^{σ+τ} ≠ {0} by (4.34) and (4.35) (since L + S ⊆ S). □

Lemma 4.38. Suppose that 1 ≤ i ≤ 2. Then, D_{i} has a unique involution ¯ so that

\overline{ρ_{i}(a)} = ρ_{i}(a)

for a ∈ J_{ii}. Moreover, the involution ¯ preserves the Λ-grading, and we have

(dx, y)_{J} = (x, d̄y)_{J}

for x, y ∈ J_{12}, d ∈ D_{i}.

Proof. To prove the first statement, only the existence of ¯ needs to be shown. If d ∈ D_{i}, we can write d = ∑_{j} ρ_{i}(a_{j1}) … ρ_{i}(a_{jr_{j}}), where a_{j1}, …, a_{jr_{j}} ∈ J_{ii}. We then define d̄ = ∑_{j} ρ_{i}(a_{jr_{j}}) … ρ_{i}(a_{j1}). To see that ¯ is well-defined, note that if d = 0, then

0 = (dx, y)_{J} = (∑_{j} ρ_{i}(a_{j1}) … ρ_{i}(a_{jr_{j}})x, y)_{J} = (x, ∑_{j} ρ_{i}(a_{jr_{j}}) … ρ_{i}(a_{j1})y)_{J} = (x, d̄y)_{J}

for x, y ∈ J_{12}, and so d̄= 0 as needed. The last statement is clear. □

Observe next that since L + S ⊆ S, we have

(4.39)  ⟨L⟩ + S ⊆ S,

where ⟨L⟩ is the additive group generated by L.

Lemma 4.40. Suppose that 1 ≤ i ≤ 2. If σ ∈ ⟨L⟩ and τ ∈ S, then

(4.41)  dim(J_{12}^{σ+τ}) = 1 and D_{i}^{σ}J_{12}^{τ} = J_{12}^{σ+τ} .

Proof. By (4.39), σ + τ ∈ S and so, by (4.34), dim(J_{12}^{σ+τ}) = 1. Next σ = σ_{1} + ⋯ + σ_{p} for some σ_{1}, …, σ_{p} ∈ L. Choose 0 ≠ a_{j} ∈ J_{ii}^{σ_{j}}, j = 1, …, p. Then

ρ_{i}(a_{1}) … ρ_{i}(a_{p})J_{12}^{τ} = J_{12}^{σ_{1}+···+σ_{p}+τ} = J_{12}^{σ+τ}

by (4.35). Hence, J_{12}^{σ+τ} ⊆ D_{i}^{σ}J_{12}^{τ}. The reverse inclusion is clear. □

Corollary 4.42. If 1 ≤ i ≤ 2, then

(4.43)  D_{i} = ⨁_{σ∈⟨L⟩} D_{i}^{σ},

(4.44)  D_{i}^{σ}D_{i}^{τ} ≠ {0} for σ, τ ∈ ⟨L⟩,

and

(4.45)  D_{i}^{σ} ≠ {0} for σ ∈ ⟨L⟩.

Moreover,

(4.46)  D_{1}J_{12}^{τ} = D_{2}J_{12}^{τ} = ⨁_{σ∈⟨L⟩} J_{12}^{σ+τ}

for τ ∈ S.

Proof. Since D_{i} is generated by ρ_{i}(J_{ii}), we have (4.43). (4.44), (4.45) and (4.46) then follow from (4.41). □

Next recall that by (4.11) the subalgebra M of J is isomorphic to H_{2}(ℂ) under an isomor-

phism which sends c_{1} → [ \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ] and c_{2} → [ \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ]. Thus, J_{12} ∩ M is spanned by an element

v so that v^{2} = 1. But J_{12} ∩ M = J_{12}^{0} by (4.17). Hence,

(4.47)  J_{12}^{0} = ℂv, where v^{2} = 1.

By (4.39), we have ⟨L⟩ ⊆ S. We will see that J has very different structure depending on whether ⟨L⟩ = S or ⟨L⟩ ≠ S. We first study the case when ⟨L⟩ = S. We note that in that case S is a lattice and so Λ = S = ⟨L⟩.

Lemma 4.48. Suppose that ⟨L⟩ = S. Then,

(4.49)  J_{12} = D_{1}v = D_{2}v.

Moreover, for i = 1, 2, the map

d → dv

from D_{i} to J_{12} is a linear bijection.

Proof. (4.49) follows from (4.46) and (4.47). To show that the map d → dv is injective, we choose j so that {i, j} = {1, 2}. Then, if d ∈ D_{i} and dv = 0, we have dJ_{12} = dD_{j}v = D_{j}dv (by (4.36)) = {0} and so d = 0. □

Lemma 4.50. Suppose that ⟨L⟩ = S. Then, for i = 1, 2, we have

(4.51)  dim(D_{i}^{σ}) = 1 for σ ∈ Λ

and

(4.52)  D_{i}^{σ}D_{i}^{τ} = D_{i}^{σ+τ} for σ, τ ∈ Λ.

Proof. If σ ∈ Λ = ⟨L⟩, we have

dim(D_{i}^{σ}) = dim(D_{i}^{σ}v) by Lemma 4.48
= dim(D_{i}^{σ}J_{12}^{0}) by (4.47)
= dim(J_{12}^{σ}) = 1 by (4.41).

This proves (4.51). (4.52) follows from (4.51) and (4.44). □

The following theorem describes the structure of J in the case when ⟨L⟩ = S. The proof uses the results above and a 2 × 2-hermitian coordinatization theorem due to K. McCrimmon and E. Neher. The coordinatization theorem is stated and proved in the language of Jordan triples systems in [McN]. We will refer to the theorem as described in McCrimmon’s forthcoming book [Mc], since it is described in the language of Jordan algebras there.

Proposition 4.53. Suppose that l = 2 and ⟨L⟩ = S. Then, we can identify Λ = ℤ^{ν} so that

J ≅_{Λ} H_{2}(ℂ_{q}, −), S = ℤ^{ν} and L = Z_{1,q},

where ℂ_{q} is the quantum torus determined by a ν × ν-matrix satisfying the conditions q_{ii} = 1, q_{ij} = ±1, and ¯ is the reversal involution on ℂ_{q}.

Proof. By (4.47) and (4.49), we have

• c_{2} and c_{2} are supplementary orthogonal idempotents in J with Peirce decomposition J = J_{11} ⊕ J_{22} ⊕ J_{12}

• v is a cyclic generator of J_{12} as a J_{11}-module, i.e. J_{12} = D_{1}v

• v^{2} = 1. In the terminology of McCrimmon, this says that the set {c_{1}, v, c_{2}} is a strong hermitian 2frame for J. McCrimmon and Neher’s strong 2 × 2-hermitian coordinatization theorem [Mc, Book 3, Theorem 10.4] says that any Jordan algebra which possesses a strong hermitian 2frame is a Jordan algebra of 2 × 2-hermitian matrices. More precisely, the theorem tells us that there exists an isomorphism ϕ : J → H_{2}(D_{1}, −) so that

ϕ(a_{1} + a_{2} + dv) = [ \begin{matrix}ρ_{1}(a_{1}) & d \\ d̄ & ρ_{1}(ζa_{2}) \end{matrix} ] ,

where ζ = U_{v}|_{J_{22}} maps J_{22} bijectively onto J_{11} and U_{v} : J → J is defined by U_{v} := 2L_{v}^{2} −L_{v^{2}}. It is clear that ϕ preserves the Λ-grading.

Observe that since ϕ is surjective, we have

ρ_{1}(J_{11}) = H(D_{1}, −).

But ρ_{1}(J_{11}) = ⨁_{σ∈L} ρ_{1}(J_{11}^{σ}) with ρ_{1}(J_{11}^{σ}) ≠ {0} for σ ∈ L, by Lemma 4.37. Hence, since dim(D_{1}^{σ}) = 1 for σ ∈ Λ, we have ρ_{1}(J_{11}) = ⨁_{σ∈L} D_{1}^{σ}. Thus,

(4.54)  H(D_{1}, −) = ⨁_{σ∈L} D_{1}^{σ}.

Next since ⟨L⟩ = Λ, we can choose a basis σ_{1}, …, σ_{ν} for Λ consisting of elements of L [AABGP, Proposition II.1.11]. Using this basis we identify Λ = ℤ^{ν} with standard basis σ_{1}, …, σ_{ν}.

Now according to Lemmas 4.38 and 4.50, the associative algebra with involution (D_{1}, −) satisfies the conditions (G1), (G2) and (G3) of Section 2. Hence, by Propositions 2.4 and 2.9, we may identify

D_{1} = ℂ_{q},

where q is a ν × ν-matrix satisfying q_{ii} = 1, q_{ij} = q_{ji} = ±1. But the elements t_{1}, …, t_{ν} of D_{1} = ℂ_{q} have degrees σ_{1}, …, σ_{ν} respectively. Since σ_{1}, …, σ_{ν} ∈ L, it follows from (4.54) that t_{1}, …, t_{ν} ∈ H(ℂ_{q}, −). Thus, the involution on ℂ_{q} is the reversal involution. This proves (a).

Finally, by (2.7), we have

H(ℂ_{q}, −) = ⨁_{σ∈Z_{1,q}} ℂ_{q}^{σ}.

Thus, by (4.54), we have L = Z_{1,q} □

It remains to consider the case when ⟨L⟩ ≠ S. Thus, for the remainder of the section (except in the final summary theorem) we assume that ⟨L⟩ ≠ S. Of course we are still assuming that l = 2.

We begin by fixing some notation. Since 2S ⊆ L ⊆ S by (4.2), we have 2Λ ⊆ ⟨L⟩ ⊆ Λ. Thus, Λ/⟨L⟩ is a finite group. But S ⊆ Λ and, by (4.39), ⟨L⟩ + S ⊆ S. Thus, S is a finite union of cosets of ⟨L⟩ in Λ. So we may chose τ_{1}, …, τ_{m} ∈ S so that

(4.55)  S = ⊎_{j=1}^{m}(⟨L⟩ + τ_{i}),

where ⊎ denotes disjoint union. Since 0 ∈ S, we may choose

τ_{1} = 0.

Note also that since we are assuming that ⟨L⟩ ≠ S, we have

m ≥ 2.

Next we choose 0 ≠ v_{j} ∈ J_{12}^{τ_{j}} in which case we have

(4.56)  J_{12}^{τ_{j}} = ℂv_{j}, j = 1, …, m.

Notice that since τ_{1} = 0, we may choose v_{1} = v (see (4.47)) and so we have

v_{1}^{2} = 1.

Finally we set

X := J_{12}

and

X_{j} := D_{1}v_{j}, j = 1, …, m.

Lemma 4.57. If 1 ≤ j ≤ m, we have

(4.58)  X_{j} = D_{1}v_{j} = D_{2}v_{j} = ⨁_{σ∈⟨L⟩} J_{12}^{σ+τ_{j}} with one dimensional summands.

The form (·, ·)_{J} restricted to X is nondegenerate and, relative to that form, we have

(4.59)  X = X_{1} ⊥ ⋯ ⊥ X_{m}.

Finally, if j ≠ k, we have

(4.60)  X_{j} · X_{k} = {0}.

Proof. (4.58) follows from (4.46). Then, by (4.55) and (4.58), we have

X = X_{1} ⊕ ⋯ ⊕ X_{m}.

The restriction of (·, ·)_{J} to X is nondegenerate by Lemma 4.23. Suppose now for contradiction that (X_{j}, X_{k})_{J} ≠ {0} for some j ≠ k. Then, by (4.58) and Lemma 4.23, we have σ + τ_{j} + λ + τ_{k} = 0 for some σ, λ ∈ ⟨L⟩. Thus, ⟨L⟩ + τ_{j} = ⟨L⟩ − τ_{k} and so ⟨L⟩ + τ_{j} = ⟨L⟩ + τ_{k}, since 2τ_{k} ∈ 2S ⊆ L ⊆ ⟨L⟩. This gives a contradiction and so we have (4.59). Finally suppose j ≠ k, x ∈ X_{j} and y ∈ X_{k}. Then, for i = 1, 2 and a ∈ J_{ii}, we have

(x · y, a)_{J} = (x, a · y)_{J} ∈ (X_{j}, X_{k})_{J} = {0},

by (4.58) and (4.59). Since x · y ∈ J_{11} ⊕ J_{22}, it follows from Lemma 4.23 that x · y = 0. □

Let π_{i} : J_{11} ⊕ J_{22} → J_{ii} be the projection onto the i^{th} factor, i = 1, 2. These maps are homomorphisms of Λ-graded Jordan algebras.

Lemma 4.61. Suppose that 1 ≤ i ≤ 2 and 1 ≤ j ≤ m. Then,

(4.62)  J_{ii} = π_{i}(X_{j} · v_{j}).

Proof. π_{i}(X_{j} · v_{j}) is a Λ-graded subspace of J_{ii} by (4.14) and (4.58). Thus, by (4.33),

π_{i}(X_{j} · v_{j}) = ⨁_{σ∈T} J_{ii}^{σ}

for some T ⊆ L. So, by Lemma 4.23, the orthogonal complement Y of π_{i}(X_{j} ·v_{j}) in J_{ii} satisfies

Y = ⨁_{σ∈U} J_{ii}^{σ},

where U = L \ (−T). But if b ∈ Y then for all x ∈ X_{j}, we have

0 = (b, π_{i}(x · v_{j}))_{J} = (b, x · v_{j})_{J} = (b · v_{j}, x)_{J} ,

and so b · v_{j} = 0. Thus, ρ_{i}(b)v_{j} = 0 and hence, by Lemma 4.37, b = 0. Thus, Y = {0}. So L \ (−T) = ∅. Hence, L ⊆ −T and so L = −L ⊆ T. Thus, L = T and we have (4.62). □

Lemma 4.63. If 1 ≤ j ≠ k ≤ m, x, y ∈ X_{j} and z ∈ X_{k}, we have

(4.64)  (π_{1}(x · y)) · z = (π_{2}(x · y)) · z.

Proof. Since J is a Jordan algebra, we have a^{2} · (a · w) = a · (a^{2} · w) for a, w ∈ J. Linearizing this we get

(a · b) · (c · w) + (b · c) · (a · w) + (c · a) · (b · w) = a · ((b · c) · w) + b · ((c · a) · w) + c · ((a · b) · w)

for a, b, c, w ∈ J. If we set a = x, b = y and c = z, where x, y ∈ X_{j} and z ∈ X_{k} and j ≠ k, we obtain, using (4.60),

(x · y) · (z · w) = z · ((x · y) · w)

for w ∈ J. In particular if we put w = c_{1} in this equation, we get \frac{1}{2}(x · y) · z = (π_{1}(x · y)) · z. Since x · y = π_{1}(x · y) + π_{2}(x · y) this implies (4.64). □

Lemma 4.65. We have

(4.66)  D_{1} = D_{2} = ρ_{1}(J_{11}) = ρ_{2}(J_{22}).

D_{i} and J_{ii} are commutative associative algebras, and the map ρ_{i} : J_{ii} → D_{i} is an isomorphism of Λ-graded associative algebras, i = 1, 2.

Proof. We first prove that

(4.67)  ρ_{1}(J_{11}) = ρ_{2}(J_{22}).

We show that ρ_{1}(J_{11}) ⊆ ρ_{2}(J_{22}), the reverse inclusion being similar. So let a_{1} ∈ J_{11}. Fix j so that 1 < j ≤ m. Then, by (4.62),

a_{1} = π_{1}(x · v_{j})

for some x ∈ X_{j}. Let

a_{2} = π_{2}(x · v_{j}).

Then, by (4.64), we have

a_{1} · v_{1} = a_{2} · v_{1}.

But since v_{1} ∈ J_{12} and v_{1}^{2} = 1, U_{v_{1}} is an automorphism of period 2 of J which stabilizes J_{12} and exchanges J_{11} and J_{22}. (See the argument on p.133 of [J1]. Here again U_{v_{1}} := 2L_{v_{1}}^{2} −L_{v_{1}^{2}}.) Moreover, L_{v_{1}}U_{v_{1}} = L_{v_{1}} (see formula (65) on p.38 of [J1]). Thus, a_{2}·v_{1} = L_{v_{1}}a_{2} = L_{v_{1}}U_{v_{1}}a_{2} = (U_{v_{1}}a_{2}) · v_{1} and so a_{1} · v_{1} = (U_{v_{1}}a_{2}) · v_{1}. Since a_{1} ∈ J_{11} and U_{v_{1}}a_{2} ∈ J_{11}, it follows from Lemma 4.37 that

a_{1} = U_{v_{1}}a_{2}.

Now by (4.64) we have

a_{1} · x = a_{2} · x

for x ∈ X_{k}, k ≠ j. On the other hand, if x ∈ X_{j}, we have

a_{1} · x = (U_{v_{1}}a_{2}) · x = U_{v_{1}}(a_{2} · (U_{v_{1}}x)) = −a_{2} · (−x) = a_{2} · x,

since U_{v_{1}}|_{X_{j}} = −I by (4.60). Thus, a_{1} · x = a_{2} · x for all x ∈ X, and so ρ_{1}(a_{1}) = ρ_{2}(a_{2}). This completes the proof of (4.67).

It follows from (4.67) that D_{1} = D_{2}. Thus, by (4.36), D_{i} is commutative, i = 1, 2. Hence, D_{i} = D_{i}^{+} as algebras. So, by Lemma 4.37, ρ_{i}(J_{ii}) is a subalgebra of D_{i} and therefore

D_{i} = ⟨ρ_{i}(J_{ii})⟩_{assoc} = ρ_{i}(J_{ii}), i = 1, 2.

The rest now follows from Lemma 4.37. □

Corollary 4.68. Each element x ∈ X can be expressed uniquely in the form

x = ∑_{j=1}^{m} ρ_{1}(a^{(j)})v_{j},

where a^{(1)}, …, a^{(m)} ∈ J_{11}. In particular, X is a free D_{1}-module with basis v_{1}, ⋯, v_{m}.

Proof. The first statement follows from (4.59), (4.66) and Lemma 4.37. The second statement follows from the first. □

Corollary 4.69. ⟨L⟩ = L. In other words, L is a lattice.

Proof. If σ ∈ ⟨L⟩, then, by Lemma 4.65 and (4.45), ρ_{1}(J_{11}^{σ}) = ρ_{1}(J_{11})^{σ} = D_{1}^{σ} ≠ {0} and so J_{11}^{σ} ≠ {0}. Thus, ⟨L⟩ ⊆ L. □

Corollary 4.70. We have

(4.71)  D_{1} = ⨁_{σ∈L} D_{1}^{σ},

(4.72)  dim(D_{1}^{σ}) = 1 for σ ∈ L,

and

(4.73)  D_{1}^{σ}D_{1}^{τ} = D_{1}^{σ+τ} for σ, τ ∈ L.

Proof. (4.71) follows from Corollary 4.69 and (4.43). Also, for σ ∈ L,

dim(D_{1}^{σ}) = dim(ρ_{1}(J_{11})^{σ}) = dim(J_{11}^{σ}) = 1,

by Lemma 4.65. Finally, if σ, τ ∈ L, then D_{1}^{σ}D_{1}^{τ} ≠ {0} by (4.44), and so we have (4.73). □

Lemma 4.74. If x, y ∈ X, we have

(4.75)  ρ_{1}(π_{1}(x · y)) = ρ_{2}(π_{2}(x · y)).

Proof. By (4.60), we can assume that x, y ∈ X_{j}, where 1 ≤ j ≤ m. Choose k so that 1 ≤ k ≤ m and k ≠ j. Then, by (4.64), ρ_{1}(π_{1}(x · y))v_{k} = ρ_{2}(π_{2}(x · y))v_{k}. Since D_{1} = D_{2} and X is a free D_{1}-module with basis v_{1}, …, v_{m}, we have (4.75). □

Recall that v_{1} ∈ J_{12} and v_{1}^{2} = 1. This together with (4.75) says, in the terminology of McCrimmon, that the set {c_{1}, v_{1}, c_{2}} is a strong Clifford frame for J. McCrimmon and Neher (see [McN,§3] or [Mc, Book 3, Theorem 9.4]) have shown that any Jordan algebra possessing such a frame is the reduced Clifford Jordan algebra (defined below) of a symmetric bilinear form h. Since we need more information about the form h in our setting and since there is little extra work involved, we use some of McCrimmon and Neher’s arguments as needed (see the proof of Lemma 4.76 (b) below) rather than simply invoking their theorem.

Define h : X × X → D_{1} by

h(x, y) := ρ_{1}(π_{1}(x · y)) = ρ_{2}(π_{2}(x · y))

for x, y ∈ X. Also define

μ_{j} := h(v_{j}, v_{j})

for j = 1, …, m.

Lemma 4.76.

(a) For 1 ≤ j ≤ m, we have 0 ≠ μ_{j} ∈ D_{1}^{2τ_{j}}.

(b) h is the unique symmetric D_{1}-bilinear form on X so that

h(v_{j}, v_{k}) = δ_{jk}μ_{j}, 1 ≤ j, k ≤ m.

Proof. (a): To prove (a), we must show that μ_{j} ≠ 0. Indeed, π_{1}(v_{j}^{2}) ∈ J_{11}^{2τ_{j}} and

(π_{1}(v_{j}^{2}), J_{11}^{−2τ_{j}})_{J} = (v_{j}^{2}, J_{11}^{−2τ_{j}})_{J} = (v_{j}, J_{11}^{−2τ_{j}} · v_{j})_{J}
= (J_{11}^{τ_{j}}, J_{11}^{−τ_{j}})_{J} by (4.56) and (4.35) since − 2τ_{j} ∈ 2S ⊆ L
≠ {0} by Lemma 4.23.

Thus, π_{1}(v_{j}^{2}) ≠ 0 and so μ_{j} ≠ 0.

(b): The only thing that needs to be proved in (b) is that

(4.77)  dh(x, y) = h(dx, y)

for d ∈ D_{1}, x, y ∈ X. For this we need to use the Jordan triple product on J which is defined by {x, y, z} := (x · y) · z + (z · y) · x − (z · x) · y for x, y, z ∈ J. Then,

(4.78)  {x, y, z} = {z, y, x} and {x, y, z} + {y, x, z} = 2(x · y) · z

for x, y, z ∈ J. Since J is a Jordan algebra one has the identify p · {x, y, z} = {p · x, y, z} − {x, p · y, z} + {x, y, p · z} for p, x, y, z ∈ J (see equation (25) in [J1, p.325]). Setting p = c_{1} in this identity and taking into account the definition of the Peirce spaces as eigenspaces for L_{c_{1}}, one obtains

(4.79)  {J_{12}, J_{11}, J_{12}} ⊆ J_{22} and {J_{11}, J_{12}, J_{12}} ⊆ J_{11}.

Now to prove (4.77), observe that d = ρ_{1}(a_{1}) for some a_{1} ∈ J_{11} by (4.66). Thus, for x, y ∈ X, we have

dh(x, y) = ρ_{1}(a_{1})ρ_{1}(π_{1}(x · y)) = ρ_{1}(a_{1} · π_{1}(x · y)) by Lemma 4.65

= ρ_{1}π_{1}(a_{1} · (x · y)) = ρ_{1}π_{1}(−{x, a_{1}, y} + (x · a_{1}) · y + x · (a_{1} · y)) = ρ_{1}π_{1}((x · a_{1}) · y + x · (a_{1} · y)) by (4.79) = ρ_{2}π_{2}((x · a_{1}) · y + x · (a_{1} · y)) by (4.75)

= ρ_{2}π_{2}((x · a_{1}) · y + \frac{1}{2}\{a_{1}, y, x\} + \frac{1}{2}\{y, a_{1}, x\}) by (4.78)

= ρ_{2}π_{2}((x · a_{1}) · y + \frac{1}{2}\{x, a_{1}, y\}) by (4.79) and (4.78)

= ρ_{2}π_{2}((x · a_{1}) · y + \frac{1}{2}\{a_{1}, x, y\} + \frac{1}{2}\{x, a_{1}, y\}) by (4.79)

= 2ρ_{2}π_{2}((x · a_{1}) · y) by (4.78)

= h(dx, y). □

We are now ready to describe the structure of J in the case when ⟨L⟩ ≠ S. For this we require some notation.

Suppose E is a commutative associative algebra, W is a left E-module and g : W × W → E is a symmetric E-bilinear form on W. Then we define

RedCliff(g) := (E, E, W)

with product · defined by

(a_{1}, a_{2}, w) · (b_{1}, b_{2}, x) = (a_{1}b_{1} + g(w, x), a_{2}b_{2} + g(w, x), \frac{1}{2}(a_{1}x + a_{2}x + b_{1}w + b_{2}w)).

RedCliff(g) is a Jordan algebra called the reduced Clifford Jordan algebra of g [Mc, Book 3, §6.8].

Recall that A_{[ν]} denotes the ring of Laurent polynomials in ν variables. Let A_{[ν]}^{m} denote the A_{[ν]}-module of m-tuples over A_{[ν]}.

Proposition 4.80. Suppose that l = 2 and ⟨L⟩ ≠ S. Then, we may identify Λ as a subgroup of ℤ^{ν} so that

(4.81)  J ≅_{Λ} RedCliff(h) = (A_{[ν]}, A_{[ν]}, A_{[ν]}^{m}),

(4.82)  S = ∪_{j=1}^{m}(2ℤ^{ν} + τ_{j}), L = 2ℤ^{ν},

and hence

(4.83)  Λ = ⟨2ℤ^{ν}, τ_{1}, …, τ_{m}⟩,

where h is the symmetric A_{[ν]}-bilinear form on A_{[ν]}^{m} defined by

(4.84)  h((a_{1}, …, a_{m}), (b_{1}, …, b_{m})) = ∑_{j=1}^{m} a_{j}b_{j}t^{τ_{j}},

and τ_{1}, …, τ_{m} ∈ ℤ^{ν} satisfy

(4.85)  m ≥ 2, τ_{1} = 0 and τ_{j} ≢ τ_{k} (mod 2ℤ^{ν}) for j ≠ k.

Here the Λ-grading on RedCliff(h) is defined by

(4.86) deg(t^{σ}, 0, 0) = deg(0, t^{σ}, 0) = 2σ, and deg(0, 0, (0, …, t^{σ}, …, 0)) = 2σ + τ_{j},

where, in the last equation, t^{σ} is in the j-th position of (0, …, t^{σ}, …, 0).

Proof. Since L is a lattice, we can choose a basis for L and thereby identify

L = 2ℤ^{ν} with basis 2σ_{1}, …, 2σ_{ν},

where σ_{1}, …, σ_{ν} is the standard basis of ℤ^{ν}. Observe that since 2S ⊆ L, we have S ⊆ \frac{1}{2}L = ℤ^{ν}, and hence we have (4.82) and (4.83) with τ_{1}, …, τ_{m} ∈ ℤ^{ν} satisfying (4.85).

Now by Corollary 4.70, the algebra D_{1} satisfies the condition (G1) and (G2) of Section 2 (with Λ replaced by L). Thus, by Proposition 2.4, we can identify

D_{1} = D_{2} = A_{[ν]} = ⨁_{σ∈ℤ^{ν}} ℂt^{σ},

as L-graded algebras if we give A_{[ν]} the L-grading defined by deg(t^{σ}) = 2σ, σ ∈ ℤ^{ν}. Next, by Lemma 4.76(a), we have D_{1}^{2τ_{j}} = ℂμ_{j} and so ℂμ_{j} = ℂt^{τ_{j}}, j = 1, …, m. But then, since ℂ is algebraically closed, we can replace v_{j} by a scalar multiple of itself and assume that

μ_{j} = t^{τ_{j}}, j = 1, …, m.

We now use the A_{[ν]}-basis v_{1}, …, v_{m} of X (see Corollary 2.68) to identify

X = A_{[ν]}^{m},

in which case, by Lemma 4.76 (b), h is the form described by (4.84). We now define ϕ : J → RedCliff(h) by

ϕ(a_{1} + a_{2} + x) = (ρ_{1}(a_{2}), ρ_{2}(a_{2}), x)

for a_{1} ∈ J_{11}, a_{2} ∈ J_{22}, x ∈ X. If we give RedCliff(h) the grading defined by (4.86), then ϕ preserves the grading. So we only need to prove that ϕ preserves the product. For this it is enough to show that ϕ((a_{1} + a_{2} + x)^{2}) = (ϕ(a_{1} + a_{2} + x))^{2} for a_{1} ∈ J_{11}, a_{2} ∈ J_{22}, x ∈ X. This is completely straightforward using only the definitions of ρ and h, Lemma 4.65 and (4.75). □

We can now summarize the consequences for EALA’s of Propositions 4.25, 4.53 and 4.80.

Theorem 4.87. Suppose that L is an EALA of type C_{l}, l ≥ 2, and nullity ν. Then

K ≅ K(J) and R ≅ R(C_{l}, S, L),

where one of the following hold:

(i) J = H_{l}(ℂ_{q}, −), S = Λ and L = Z_{e,q}, where (ℂ_{q}, −) is the quantum torus with involution determined by a ν-vector e satisfying e_{i} = ±1 and a ν×ν-matrix q satisfying q_{ii} = 1, q_{ij} = q_{ji} = ±1, and Λ = ℤ^{ν}. Moreover, e = 1 if l = 2.

(ii) l = 3, ν ≥ 3, J = H_{3}(A, −), S = Λ and L = Λ_{2}^{(3)}, where (A, −) is the octonion torus

with standard involution and Λ = ℤ^{ν}. (iii) l = 2, J = RedCliff(h), S = ∪_{i=1}^{m}(2ℤ^{ν} + τ_{i}) and L = 2ℤ^{ν}, where h is the symmetric

A_{[ν]}-bilinear form on A_{[ν]}^{m} defined by

h((a_{1}, …, a_{m}), (b_{1}, …, b_{m})) = ∑_{j=1}^{m} a_{j}b_{j}t^{τ_{j}}

and τ_{1}, …, τ_{m} ∈ ℤ^{ν} satisfy

m ≥ 2, τ_{1} = 0 and τ_{j} ≢ τ_{k} (mod 2ℤ^{ν}) for j ≠ k.

Remark 4.88. In the three cases of Theorem 4.87 the indicated Jordan algebra J possesses a Λ-grading. The Λ-grading on J = H_{l}(ℂ_{q}, −) in (i) and J = H_{3}(A, −) in (ii) is induced from the Λ-grading on ℂ_{q} and A respectively; and the Λ-grading on J = RedCliff(h) in (iii) is defined by (4.86). Given this Λ-grading on J, one obtains a Λ-grading on K(J) in a natural fashion (see for example [AABGP, Chapter III]). In that case, the isomorphism K = K(J) is an isomorphism of Λ-graded Lie algebras. An understanding of the Λ-grading on K will be important in the further study of the structure of L. (See [BGK] and [BGKN] in the simply lace case.)

Remark 4.89. Each of the Lie algebras K(J) and the root systems R(C_{l}, S, L), where J, S and L are as in (i), (ii) or (iii) of Theorem 4.87, actually occur as K and R for some EALA L. Examples for cases (i) and (iii) can be found in Theorems III.4.7 and III.3.32 respectively of [AABGP]. (In [AABGP] the Lie algebras K are denoted by G and described in a somewhat different way as matrix algebras.) For case (ii), it is not difficult using the methods of [AABGP, §III.1] to construct an EALA L with K ≅ K(H_{3}(A, −)) and R ≅ R(C_{3}, Λ, Λ_{2}^{(3)}), where A is an octonion torus. We leave this to the reader. Thus, Theorem 4.87 gives an exact description of the root systems R and the Lie algebras K that can arise from EALA’s of type C_{l}(l ≥ 2).

§5. Type B_{l}(l ≥ 3), F_{4} and G_{2}

In this section, we describe the structure of R and K in the case when L has type B_{l}(l ≥ 3), F_{4} and G_{2}.

Throughout the section, we assume that L is an EALA of type X, where X = B_{l}(l ≥ 3), F_{4} or G_{2}. Again we use all of the notation from Section 1.

By (1.12), we again have

(5.1)  R = R(X, S, L) := (S + S) ∪ (Ṙ_{sh} + S) ∪ (Ṙ_{lg} + L).

where S = Sα̇ for α̇ ∈ Ṙ_{sh} and L = Sα̇ for α̇ ∈ Ṙ_{lg}. Moreover, by [AABGP, Theorem II.2.37], L is a lattice in V^{0} and S is a semilattice in V^{0} so that

(5.2)  L + kS ⊆ L and S + L ⊆ S,

where

k = \{ \begin{matrix}2 & if X = B_{l} or F_{4} \\ 3 & if X = G_{2}. \end{matrix}

Moreover, by the same theorem, S is a lattice in V^{0} if X = F_{4} or X = G_{2}. In all cases, by (1.7), we have

Λ = ⟨S⟩.

Now, by Proposition 1.28 (d), K is an X-graded Lie algebra as defined in [BM] and [BZ]. Thus, according to the recognition theorems of Benkart and Zelmanov [BZ], K can be constructed using the generalized Tits’ construction. We now recall that construction.

Let A be an associative commutative algebra over ℂ and let X be an algebra over A. Let t : X → A be a normalized trace on X, which means that t is A-linear,

t(xx′) = t(x′x) and t((xx′)x′′) = t(x(x′x′′))

for x, x′, x′′ ∈ X, and t(1) = 1. We then have

X = A1 ⊕ X_{0},

where X_{0} := {x ∈ X : t(x) = 0}; and we have

xx′ = t(xx′)1 + x ∗ x′

for x, x′ ∈ X, where x ∗ x′ ∈ X_{0}. Let Der_{A}^{0}(X) be the Lie algebra of A-derivations which map X_{0} to X_{0}. Let D(X) be a subalgebra of Der_{A}^{0}(X) and let (x, x′) → D_{x,x′} be an A-bilinear map of X_{0} × X_{0} into D(X) which is skew-symmetric and satisfies

[D, D_{x,x′}] = D_{Dx,x′} + D_{x,Dx′}

for D ∈ D(X), x, x′ ∈ X_{0}. Similarly, suppose B is another associative commutative algebra over ℂ, Y is an algebra over B with normalized trace t, D(Y) is a subalgebra of Der_{A}^{0}(X), and (y, y′) → D_{y,y′} is a B-bilinear skew-symmetric map of Y_{0} × Y_{0} into D(Y) so that

[E, D_{y,y′}] = D_{Ey,y′} + D_{y,Dy′}

for E ∈ D(Y), y, y′ ∈ Y_{0}. Put

T (X /A, Y/B) := (D(X) ⊗ B) ⊕ (X_{0} ⊗ Y_{0}) ⊕ (A ⊗ D(Y)).

Here the tensor products ⊗ are over ℂ. Then, T (X /A, Y/B) is an anticommutative algebra with multiplication defined by

(5.3)  \begin{matrix}[D ⊗ β, D′ ⊗ β′] = [D, D′] ⊗ ββ′, \\ [α ⊗ E, α′ ⊗ E] = αα′ ⊗ [E, E′], \\ [D ⊗ β, α ⊗ E] = 0, \\ [D ⊗ β, x ⊗ y] = Dx ⊗ βy = −[x ⊗ y, D ⊗ β], \\ [α ⊗ E, x ⊗ y] = αx ⊗ Ey = −[x ⊗ y, α ⊗ E], \\ [x ⊗ y, x′ ⊗ y′] = D_{x,x′} ⊗ t(yy′) + (x ∗ x′) ⊗ (y ∗ y′) + t(xx′) ⊗ D_{y,y′} \end{matrix}

for D, D′ ∈ D(X), β, β′ ∈ B, α, α′ ∈ A, E, E′ ∈ D(Y), x, x′ ∈ X_{0}, y, y′ ∈ Y_{0}. If X, Y are suitably chosen, T (X /A, Y/B) will be a Lie algebra over ℂ (see Proposition 3.9 of [BZ]).

We will only need a special case of the above construction, namely the case when A = ℂ,

D(X) = D_{X ,X} and D(Y) = D_{Y,Y}.

Thus, in all cases considered we will have

T (X /ℂ, Y/B) := (D_{X ,X} ⊗ B) ⊕ (X_{0} ⊗ Y_{0}) ⊕ D_{Y,Y},

where we are identifying ℂ ⊗ D_{Y,Y} with D_{Y,Y}.

We are now ready to separate into the three cases when X = B_{l}(l ≥ 3), F_{4} and G_{2}.

Type B_{l}(l ≥ 3).

We first assume that L is an EALA of type B_{l}, where l ≥ 3. Note that L is a lattice and, arguing as in Section 4 (see (4.55)), we have

(5.4)  S = ⊎_{j=1}^{m}(L + τ_{i}),

where τ_{1}, …, τ_{m} ∈ S and τ_{1} = 0.

Now Ġ is a simple Lie algebra of type B_{l} over ℂ. We thus may realize Ġ as follows: Let 𝕍 be a 2l + 1-dimensional space over ℂ with basis v_{1}, …, v_{2l+1}. Let f be the symmetric bilinear form on 𝕍 so that

f(v_{i}, v_{l+i}) = 1, 1 ≤ i ≤ l, f(v_{2l+1}, v_{2l+1}) = 1,

and f(v_{i}, v_{j}) = 0 otherwise. We may identify Ġ as the Lie algebra of endomorphisms of 𝕍 which are skew relative to f. We may also identify the Cartan subalgebra Ḣ of Ġ with the Lie algebra of elements of Ġ which are diagonal relative to the basis v_{1}, …, v_{2l+1}.

Next let

X = Cliff(f) = ℂ1 ⊕ 𝕍

be the Clifford Jordan algebra of f with multiplication

(α1 + v) · (α′1 + v′) = αα′1 + f(v, v′)1 + αv′ + α′v.

(This terminology and notation follows [Mc]. X is also often just called the Jordan algebra of f [J1].) Then the algebra X over ℂ has a unique normalized trace t so that X_{0} = 𝕍. Moreover, since X_{0} · X_{0} ⊆ ℂ1, the product ∗ defined on X_{0} is 0. Also for v, v′ ∈ 𝕍, define

D_{v,v′} = −[L_{v}, L′_{v}].

In that case, we may identify

Ġ = D_{𝕍,𝕍},

by extending the action of elements of Ġ to ℂ1 ⊕ 𝕍 so that they kill 1.

Now K is a B_{l}-graded Lie algebra and, by Proposition 1.28 (b), Z(K) = {0}. Thus, by the recognition theorem of Benkart and Zelmanov [BZ, Theorem 3.53], we have

(5.5)  K = T (Cliff(f)/ℂ, Cliff(g)/B) = (Ġ ⊗ B) ⊕ (𝕍 ⊗ W) ⊕ D_{W,W},

where B is a commutative associative algebra over ℂ, W is a B-module and g : W ⊗ W → B is a symmetric B-bilinear form on W. Here

Y = Cliff(g) = B1 ⊕ W

is the Clifford Jordan algebra of g with multiplication

(β1 + w) · (β′1 + w′) = ββ′1 + g(w, w′)1 + βw′ + β′w.

Y has the normalized trace t so that Y_{0} = W, and the product ∗ on Y_{0} is 0. Also

D_{w,w′} = −[L_{w}, L′_{w}]

for w, w′ ∈ W. Finally, we are identifying the subalgebra Ġ of K with Ġ ⊗ 1.

We note that the decomposition (5.5) is an orthogonal decomposition of K relative to (·, ·). To see this observe that under the adjoint action the Ġ-modules Ġ ⊗ B, 𝕍 ⊗ W and D_{W,W} are sums of Ġ-modules which are isomorphic to Ġ, 𝕍 and ℂ respectively, and that no two of the irreducible Ġ-modules Ġ, 𝕍 and ℂ are dual to each other.

We can now deduce the structure of the form g on W.

Proposition 5.6. We may identify Λ as a subgroup of ℤ^{ν} so that

S = ∪_{i=1}^{m}(2ℤ^{ν} + τ_{i}), L = 2ℤ^{ν},

and hence

Λ = ⟨2ℤ^{ν}, τ_{1}, …, τ_{m}⟩.

Also we may identify

B = A_{[ν]} and W = A_{[ν]}^{m−1}

so that

(5.7)  g((a_{2}, …, a_{m}), (b_{2}, …, b_{m})) = ∑_{j=2}^{m} a_{j}b_{j}t^{τ_{j}},

where A_{[ν]} is the ring of Laurent polynomials in ν variables.

Proof. We may choose an orthogonal basis ϵ_{1}, …, ϵ_{l} for Ḣ^{∗} so that the weight spaces for the Ḣ-module 𝕍 are

𝕍_{ϵ_{i}} = ℂv_{i}, 𝕍_{−ϵ_{i}} = ℂv_{l+i}, i = 1, …, l,

and

𝕍_{0} = ℂv_{2l+1}.

In that case,

(5.8)  Ṙ_{sh} = {±ϵ_{i} : 1 ≤ i ≤ l} and Ṙ_{lg} = {±(ϵ_{i} ± ϵ_{j}) : 1 ≤ i < j ≤ l}.

In particular, the weights of the module 𝕍 are the elements of Ṙ_{sh} ∪ {0}. Thus, if α̇ ∈ Ṙ^{×}, the corresponding root space Kα̇ in K is given by

(5.9)  Kα̇ = \{ \begin{matrix}Ġα̇ ⊗ B & if α̇ ∈ Ṙ_{lg} \\ Ġα̇ ⊗ B + 𝕍α̇ ⊗ W & if α̇ ∈ Ṙ_{sh}. \end{matrix}

Recall that if α̇ ∈ Ṙ_{lg}

(5.10)  Kα̇ = ⨁_{σ∈L} Kσ ; α̇ with one dimensional summands,

by Proposition 1.28 (e) and (f). Similarly, if α̇ ∈ Ṙ_{sh}, we have

(5.11)  Kα̇ = ⨁_{σ∈S} Kσ ; α̇ with one dimensional summands.

If follows from (5.9) and (5.10) that if α̇ ∈ Ṙ_{lg}, then B has a decomposition

B = ⨁_{σ∈L} Bσ ; α̇ with one dimensional summands

so that

Kσ ; α̇ = Ġα̇ ⊗ Bσ ; α̇ for σ ∈ L.

Now, if σ ∈ L, α̇, β̇ ∈ Ṙ_{lg} and β̇ − α̇ ∈ Ṙ^{×}, we have

[Kσ ; α̇, K_{β̇−α̇}^{0}] = Kσ ; β̇,

by (1.30), and so

[Ġα̇ ⊗ Bσ ; α̇, Ġ_{β̇−α̇} ⊗ 1] = Ġβ̇ ⊗ Bσ ; β̇

using Proposition 1.28 (i). Thus, if σ ∈ L, α̇, β̇ ∈ Ṙ_{lg} and α̇ − β̇ ∈ Ṙ^{×}, then

(5.12)  Bσ ; α̇ = Bσ ; β̇.

But one checks using (5.8) that:

(5.13)

If α̇, β̇ are distinct elements of Ṙ^{×} having the same length, then there exists a sequence α̇_{1}, …, α̇_{p} of roots in Ṙ^{×} having the same length as α̇ so that α̇_{1} = α̇, α̇_{p} = β̇ and α̇_{i+1} − α̇_{i} ∈ Ṙ^{×} for i = 1, …, p − 1.

This fact together with (5.12) implies that

Bσ ; α̇ = Bσ ; β̇

for σ ∈ L and α̇, β̇ ∈ Ṙ_{lg}. Thus, if σ ∈ L, we may define

B_{σ} := Bσ ; α̇

for any choice of α̇ ∈ Ṙ_{lg}. Then,

(5.14)  B = ⨁_{σ∈L} B^{σ} with one dimensional summands.

Moreover, if α̇ ∈ Ṙ_{lg}, we have

(5.15)  Kσ ; α̇ = Ġα̇ ⊗ B^{σ} for σ ∈ L.

We now look at the grading of Kα̇ when α̇ ∈ Ṙ_{sh}. First if α̇ ∈ Ṙ_{sh}, then there exists β̇ ∈ Ṙ_{lg}

so that α̇ − β̇ ∈ Ṙ^{×}, in which case we have Kσ ; α̇ = [Kσ ; β̇, K_{α̇−β̇}^{0}] = [Ġβ̇ ⊗ B^{σ}, Ġ_{α̇−β̇} ⊗ 1] = Ġα̇ ⊗ B^{σ}

for σ ∈ L (since σ ∈ L ⊆ S). Thus, if α̇ ∈ Ṙ_{sh}, we have

(5.16)  Kσ ; α̇ = Ġα̇ ⊗ B^{σ} for σ ∈ L,

and so

(5.17)  Ġα̇ ⊗ B = ⨁_{σ∈L} Kσ ; α̇.

We next claim that if α̇ ∈ Ṙ_{sh}, we have

(5.18)  𝕍α̇ ⊗ W = ⨁_{σ∈S\L} Kσ ; α̇.

Indeed, by (5.9), (5.11) and (5.17), both sides of (5.18) are complements of Ġα̇ ⊗ B in Kα̇. Thus, it suffices to show the inclusion “⊆”in (5.18). For this let x ∈ 𝕍α̇ ⊗ W. But 𝕍 ⊗ W and Ġ ⊗ B are orthogonal relative to the form (·, ·). Hence, x is orthogonal to Ġ_{−α̇} ⊗ B. Thus, by (5.17), x is orthogonal to ⨁_{σ∈L} K_{−α̇}^{σ}. But by (5.9) and (5.11), x ∈ ⨁_{σ∈S} Kσ ; α̇. Hence, by Proposition 1.28 (h), x ∈ ⨁_{σ∈S\L} Kσ ; α̇. Thus, we have proved (5.18). But then, given α̇ ∈ Ṙ_{sh}, it follows from (5.18) that W has a decomposition

W = ⨁_{σ∈S\L} Wσ ; α̇ with one dimensional summands

so that

Kσ ; α̇ = 𝕍α̇ ⊗ Wσ ; α̇ for σ ∈ S \ L.

Moreover, if σ ∈ S \ L, α̇, β̇ ∈ Ṙ_{sh} and β̇ − α̇ ∈ Ṙ^{×}, we have [Kσ ; α̇, K_{β̇−α̇}^{0}] = Kσ ; β̇ and so

[𝕍α̇ ⊗ Wσ ; α̇, Ġ_{β̇−α̇} ⊗ 1] = 𝕍β̇ ⊗ Wσ ; β̇, which implies that Bσ ; α̇ = Bσ ; β̇. Hence, by (5.13), we have

Wσ ; α̇ = Wσ ; β̇

for σ ∈ S \ L, α̇, β̇ ∈ Ṙ_{sh}. Thus, if σ ∈ S \ L, we may define

W^{σ} := Wσ ; α̇

for any choice of α̇ ∈ Ṙ_{lg}. Then,

(5.19)  W = ⨁_{σ∈S\L} W^{σ} with one dimensional summands.

Moreover, if α̇ ∈ Ṙ_{sh},

(5.20)  Kσ ; α̇ = \{ \begin{matrix}Ġα̇ ⊗ B^{σ} & if σ ∈ L \\ 𝕍α̇ ⊗ W^{σ} & if σ ∈ S \ L. \end{matrix}

We next claim that

(5.21)  B^{σ}B^{τ} = B^{σ+τ} for σ, τ ∈ L

and

(5.22)  B^{σ}W^{τ} = W^{σ+τ} for σ ∈ L and τ ∈ S \ L.

To prove (5.21), choose α̇, β̇ ∈ Ṙ_{lg} so that α̇ + β̇ ∈ Ṙ_{lg} (e.g. α̇ = ϵ_{1} − ϵ_{2} and β̇ = ϵ_{2} − ϵ_{3}). Then, for σ, τ ∈ L, we have

(5.23)  [Ġσ ; α̇, Ġτ ; β̇] = Ġ_{α̇+β̇}^{σ+τ},

which together with (5.15) implies (5.21). To prove (5.22), choose α̇ ∈ Ṙ_{lg}, β̇ ∈ Ṙ_{sh} so that α̇ + β̇ ∈ Ṙ_{sh} (e.g. α̇ = ϵ_{1} − ϵ_{2}, β̇ = ϵ_{2}). Then, for σ ∈ L, τ ∈ S \ L, we again have (5.23) which together with (5.15) and (5.20) implies (5.22).

We next fix a choice of 0 ≠ w_{i} ∈ W^{τ_{i}}, i = 2, …, m. Thus

(5.24)  W^{τ_{i}} = ℂw_{i}, i = 2, …, m.

Then, since S \ L = ⊎_{i=2}^{m}(L + τ_{i}), it follows from (5.14), (5.19), (5.21) and (5.22) that

(5.25)  W is a free B-module with basis w_{2}, …, w_{m}.

Next suppose that 2 ≤ i, j ≤ m. Then, we have

[v_{1} ⊗ w_{i}, v_{2} ⊗ w_{j}] = D_{v_{1},v_{2}} ⊗ t(w_{i} · w_{j}) + v_{1} ∗ v_{2} ⊗ w_{i} ∗ w_{j} + t(v_{1} · v_{2})D_{w_{i},w_{j}}.

But the products ∗ are 0. Also t(v_{1} · v_{2}) = f(v_{1}, v_{2}) = 0 and t(w_{i} · w_{j}) = g(w_{i}, w_{j}). So

[v_{1} ⊗ w_{i}, v_{2} ⊗ w_{j}] = D_{v_{1},v_{2}} ⊗ g(w_{i}, w_{j}).

But by (5.20), K_{ϵ_{2}}^{τ_{j}} = ℂv_{2} ⊗ w_{j}. Moreover, by Proposition 1.28 (g),

[K_{ϵ_{1}}^{τ_{i}}, K_{ϵ_{2}}^{τ_{j}}] = \{ \begin{matrix}K_{ϵ_{1}+ϵ_{2}}^{2τ_{i}} & if i = j \\ \{0\} & if i ≠ j, \end{matrix}

since 2τ_{i} ∈ 2S ⊆ L and τ_{i} + τ_{j} ∉ L if i ≠ j. Thus,

(5.26)  g(w_{i}, w_{j}) = 0 if i ≠ j

and

(5.27)  ℂg(w_{i}, w_{i}) = B^{2τ_{i}}.

We are now ready to draw the conclusions of the Proposition. First choose a basis for the lattice L and thereby identify L = 2ℤ^{ν} with basis 2σ_{1}, …, 2σ_{ν}, where σ_{1}, …, σ_{ν} is the standard basis for ℤ^{ν}. Next, by (5.14) and (5.21), the algebra B satisfies conditions (G1) and (G2) of Section 2 (with Λ replaced by L). Thus, by Proposition 2.4, we can identify B = A_{[ν]} as L-graded algebras if we give A_{[ν]} the L-grading defined by deg(t^{σ}) = 2σ, σ ∈ ℤ^{ν}. Then, by (5.27), we can replace w_{i} by a scalar multiple of itself and assume that

(5.28)  g(w_{i}, w_{i}) = t^{τ_{i}}, i = 2, …, m.

Finally, by (5.25), we can use the A_{[ν]}-basis w_{2}, …, w_{m} for W to identify W = B^{m−1} = A_{[ν]}^{m−1}. But then by (5.26) and (5.28), we have (5.7). □

It follows from Proposition 5.6 and its preceding discussion that we have:

Theorem 5.29. Suppose that L is an EALA of type B_{l}(l ≥ 3), and nullity ν. Then,

(5.30)  K ≅ T (Cliff(f)/ℂ, Cliff(g)/A_{[ν]})

and

(5.31)  R = R(B_{l}, S, 2ℤ^{ν}),

where f is a nondegenerate symmetric ℂ-bilinear form on a 2l + 1-dimensional space 𝕍 over ℂ, g is the symmetric A_{[ν]}-bilinear form on A_{[ν]}^{m−1} defined by g((a_{2}, …, a_{m}), (b_{2}, ⋯, b_{m})) = ∑_{j=2}^{m} a_{j}b_{j}t^{τ_{j}}, S = ∪_{j=1}^{m}(2ℤ^{ν} + τ_{j}), and τ_{1}, …, τ_{m} ∈ ℤ^{ν} satisfy

m ≥ 1, τ_{1} = 0 and τ_{j} ≅ τ̸_{k} (mod 2ℤ^{ν}) for j ≠ k.

Remark 5.32. If l = 2 and f and g are as described in Theorem 5.29, the Lie algebra T (Cliff(f)/ℂ, Cliff(g)/A_{[ν]}) is isomorphic to K(RedCliff(h)), where h is the symmetric A_{[ν]}- bilinear form on A_{[ν]}^{m}-defined by h((a_{1}, …, a_{m}), (b_{1}, …, b_{m})) = ∑_{j=1}^{m} a_{j}b_{j}t^{τ_{j}}. Thus, the Lie algebras K which occur in Theorem 4.87 (iii) can be regarded as the “l = 2 case”of the Lie algebras occurring in Theorem 5.29.

Type F_{4}.

We next assume that L is an EALA of type F_{4}. Now S and L are lattices and we may choose a basis σ_{1}, …, σ_{ν} for Λ so that

S = Λ = ℤσ_{1} ⊕ ⋯ ⊕ ℤσ_{ν}

and

L = Λ_{2}^{(p)} = 2ℤσ_{1} ⊕ ⋯ ⊕ 2ℤσ_{p} ⊕ ℤσ_{p+1} ⊕ ⋯ ⊕ ℤσ_{ν}

for some 0 ≤ p ≤ ν [AABGP, Lemma II.4.15]. p is called the twist number of L (or of R). We identify Λ = S = ℤ^{ν} with standard basis σ_{1}, …, σ_{ν}.

Next Ġ is a simple Lie algebra of type F_{4} over ℂ. Thus, we may identify

Ġ = Der(𝕁),

where 𝕁 is the Albert algebra over ℂ, i.e. 𝕁 is the 27-dimensional exceptional simple Jordan algebra over ℂ. (See for example [J2] for the facts we will use about 𝕁 and Der(𝕁).) Let t be the normalized trace on 𝕁 obtained by dividing the usual generic trace by 3. Also, for x, x′ ∈ 𝕁_{0}, define

D_{x,x′} = [L_{x}, L_{x′}].

Then,

Ġ = Der(𝕁) = D_{𝕁_{0},𝕁_{0}}.

Now K is an F_{4}-graded Lie algebra and Z(K) = {0}. So by the recognition theorem [BZ, Theorem 3.36] for such algebras, we have

(5.33)  K = T (𝕁/ℂ, C/B) = (Ġ ⊗ B) ⊕ (𝕁_{0} ⊗ C_{0}) ⊕ D_{C_{0},C_{0}},

where B is a commutative associative algebra over ℂ and C is an alternative algebra over B with a normalized trace t which satisfies the trace identity

ch_{2}(y) := y^{2} − 2t(y)y + (2t(y)^{2} − t(y^{2}))1 = 0.

We note that the inner derivations for C are defined by

D_{y,y′} = \frac{1}{4}(L_{[y,y′]} − R_{[y,y′]} − 3[L_{y}, R_{y′}])

for y, y′ ∈ C_{0}. Also, as before Ġ is being identified with Ġ ⊗ 1. Moreover, the decomposition (5.33) is an orthogonal decomposition relative to (·, ·).

We can now describe the structure of C.

Proposition 5.34. We have 0 ≤ p ≤ 3. Moreover, we may identify

B = A_{[ν]}

and

(5.35)  C = \{ \begin{matrix}B & if p = 0 \\ A(B, t_{1}) & if p = 1 \\ A(B, t_{1}, t_{2}) & if p = 2 \\ A(B, t_{1}, t_{2}, t_{3}) & if p = 3. \end{matrix}

Proof. We may choose an orthogonal basis ϵ_{1}, …, ϵ_{4} for Ḣ^{∗} so that

(5.36)  \begin{matrix}Ṙ_{sh} = \{±ϵ_{i}|1 ≤ i ≤ 4\} ∪ \{±\frac{1}{2}(ϵ_{1} ± ϵ_{2} ± ϵ_{3} ± ϵ_{4})\} and \\ Ṙ_{lg} = \{±(ϵ_{i} ± ϵ_{j}) : 1 ≤ i < j ≤ 4\}, \end{matrix}

where the signs can be chosen independently [H, §12.1]. Moreover, the set of weights of the Ġ-module 𝕁_{0} is Ṙ_{sh} ∪ {0} [J2, §6]. Thus, if α̇ ∈ Ṙ^{×}, the corresponding root space Kα̇ in K is given by

Kα̇ = \{ \begin{matrix}Ġα̇ ⊗ B & if α̇ ∈ Ṙ_{lg} \\ Ġα̇ ⊗ B + (𝕁_{0})α̇ ⊗ C_{0} & if α̇ ∈ Ṙ_{sh}. \end{matrix}

Now exactly as in the proof of Proposition 5.6 we obtain the following facts: We have

(5.37)  B = ⨁_{σ∈L} B^{σ} with one dimensional summands

and, if α̇ ∈ Ṙ_{lg}, we have

(5.38)  Kσ ; α̇ = Ġα̇ ⊗ B^{σ} for σ ∈ L.

We also have

(5.39)  C_{0} = ⨁_{σ∈S\L}(C_{0})^{σ} with one dimensional summands.

and, if α̇ ∈ Ṙ_{sh}, we have

(5.40)  Kσ ; α̇ = \{ \begin{matrix}Ġα̇ ⊗ B^{σ} & if σ ∈ L \\ (𝕁_{0})α̇ ⊗ (C_{0})^{σ} & if σ ∈ S \ L, \end{matrix}

Moreover,

(5.41)  B^{σ}B^{τ} = B^{σ+τ} for σ, τ ∈ L

and

(5.42)  B^{σ}(C_{0})^{τ} = (C_{0})^{σ+τ} for σ ∈ L, τ ∈ S \ L.

(When using the argument in the proof of Proposition 5.6 to obtain (5.37)–(5.42), one needs some simple facts such as (5.13) about the root system Ṙ. These facts are all easily verified using (5.36).)

We next prove that if σ, τ ∈ S \ L, then

(5.43)  (C_{0})^{σ}(C_{0})^{τ} = \{ \begin{matrix}B^{σ+τ} & if σ + τ ∈ L \\ (C_{0})^{σ+τ} & if σ + τ ∉ L. \end{matrix}

To prove this we choose α̇, β̇ ∈ Ṙ_{sh} so that α̇ + β̇ ∈ Ṙ_{sh} and choose x ∈ (𝕁_{0})α̇, y ∈ (𝕁_{0})β̇ so that

D_{x,y} ≠ 0 and x ∗ y ≠ 0.

The existence of α̇, β̇, x, y with these properties is easily verified using the explicit weight space decomposition of 𝕁_{0} given for example in [J2, §6]. Let 0 ≠ a ∈ (C_{0})^{σ}, 0 ≠ b ∈ (C_{0})^{τ}, where σ, τ ∈ S \ L. Then,

Kσ ; α̇ = (𝕁_{0})α̇ ⊗ a and Kτ ; β̇ = (𝕁_{0})β̇ ⊗ b.

But

[x ⊗ a, y ⊗ b] = D_{x,y} ⊗ t(ab) + (x ∗ y) ⊗ (a ∗ b) + t(x · y) ⊗ D_{a,b}

and

[Kσ ; α̇, Kτ ; β̇] = K_{α̇+β̇}^{σ+τ} = \{ \begin{matrix}Ġ_{α̇+β̇} ⊗ B^{σ+τ} & if σ + τ ∈ L \\ (𝕁_{0})_{α̇+β̇} ⊗ (C_{0})^{σ+τ} & if σ + τ ∈ S \ L. \end{matrix}

Hence, if σ + τ ∈ L, we have 0 ≠ t(a, b) ∈ B^{σ+τ} and a ∗ b = 0; and, if σ + τ ∈ S \ L, we have t(a, b) = 0 and 0 ≠ a ∗ b ∈ (C_{0})^{σ+τ}. This proves (5.43).

We now define

(5.44)  C^{σ} = \{ \begin{matrix}B^{σ} & if σ ∈ L \\ (C_{0})^{σ} & if σ ∈ S \ L. \end{matrix}

Then, by (5.37) and (5.39), we have

(5.45)  C = ⨁_{σ∈S} C^{σ} with one dimensional summands,

and, by (5.41), (5.42) and (5.43), we have

(5.46)  C^{σ}C^{τ} = C^{σ+τ} for σ, τ ∈ S.

Thus, since Λ = S, C is a Λ-graded algebra which satisfies assumptions (G1) and (G2) of Section 2.

We note next that

(5.47)  σ, τ, σ + τ ∈ Λ \ L, 0 ≠ a ∈ C^{σ}, 0 ≠ b ∈ C^{τ} ⟹ 0 ≠ ab = −ba ∈ C^{σ+τ} .

To see this we only have to show that ab + ba = 0. Indeed, linearizing ch_{2}, we have ab + ba − 2t(a)b − t(b)a + (4t(a)t(b) − 2t(ab)) = 0. But a, b, ab ∈ C_{0}. So ab + ba = 0 as claimed.

Let Z = Z(C). Then Z ⊇ B and Z is a Λ-graded subalgebra of C. Thus, by (5.45),

Z = ⨁_{σ∈M} C^{σ},

where

L ⊆ M ⊆ Λ.

Then, by (5.45), M + M ⊆ M and so, since Λ/L is a finite group, M is a subgroup of L.

We claim next that either

(5.48)  M = L or M = Λ.

Indeed suppose the contrary. Then, we may choose σ ∈ M \ L and τ ∈ Λ \ M. Choose 0 ≠ a ∈ C^{σ} and 0 ≠ b ∈ C^{τ}. Then, ab = ba, since a ∈ Z. Thus, by (5.47), σ + τ ∈ L. Hence, τ ∈ −σ + L ⊆ M, a contradiction. This proves (5.48).

Suppose now that C is not associative. Then, in particular, Z ≠ C and so M ≠ Λ. Thus, by (5.48), M = L and B = Z. Hence, 2^{p} = |Λ/L| = |Λ/M| = 2^{3} by Proposition 2.12. Thus, p = 3. So M = L = Λ_{2}^{(3)} and therefore, again by Proposition 2.12, we may identify C = A(A_{[ν]}, t_{1}, t_{2}, t_{3}). Hence, B = ⨁_{σ∈Λ_{2}^{(3)}} C^{σ} = A_{[ν]}.

Thus, we may assume that C is associative. Then, C is generated by elements

x_{1}^{±1}, …, x_{p}^{±1}, t_{p+1}^{±1}, …, t_{ν}^{±1}

so that

deg(x_{i}) = σ_{i}, i = 1, …, p and deg(t_{i}) = σ_{i}, i = p + 1, …, ν.

Moreover, if we put

t_{i} = x_{i}^{2}, i = 1, …, p,

we have

B = ⨁_{σ∈L} C^{σ} = ℂ[t_{1}^{±1}, …, t_{ν}^{±1}] = A_{[ν]},

where A_{[ν]} is graded by L = Λ_{2}^{(p)}. Also, if 1 ≤ i ≠ j ≤ p, we have σ_{i}, σ_{j}, σ_{i} + σ_{j} ∈ Λ \ L and so, by (5.47),

(5.49)  x_{i}x_{j} = −x_{j}x_{i}

Next note that if p ≥ 3, then σ_{1}, σ_{2} + σ_{3}, σ_{1} + σ_{2} + σ_{3} ∉ L, and so, by (5.47), 0 ≠ x_{1}(x_{2}x_{3}) = −(x_{2}x_{3})x_{1}, which contradicts (5.49). Thus, p ≤ 2. It is then clear that (5.35) holds with p = 0, 1 or 2. □

Proposition 5.34 and its preceding discussion have the following consequence:

Theorem 5.50. Suppose that L is an EALA of type F_{4}, nullity ν and twist number p. Then, p ≤ 3,

(5.51)  K ≅ T (𝕁/ℂ, C/A_{[ν]})

and

(5.52)  R = R(F_{4}, Λ, Λ_{2}^{(p)}),

where 𝕁 is the Albert algebra over ℂ,

C = \{ \begin{matrix}A_{[ν]} & if p = 0 \\ A(A_{[ν]}, t_{1}) & if p = 1 \\ A(A_{[ν]}, t_{1}, t_{2}) & if p = 2 \\ A(A_{[ν]}, t_{1}, t_{2}, t_{3}) & if p = 3, \end{matrix}

and Λ = ℤ^{ν}.

Type G_{2}.

We assume finally that L is an EALA of type G_{2}. Again S and L are lattices and we may choose a basis σ_{1}, …, σ_{ν} for Λ so that

S = Λ = ℤσ_{1} ⊕ ⋯ ⊕ ℤσ_{ν}

and

L = Λ_{3}^{(p)} = 3ℤσ_{1} ⊕ ⋯ ⊕ 3ℤσ_{p} ⊕ ℤσ_{p+1} ⊕ ⋯ ⊕ ℤσ_{ν}

for some 0 ≤ p ≤ ν [AABGP, Lemma II.4.15]. p is called the twist number of L (or of R). We again identify Λ = S = ℤ^{ν} with standard basis σ_{1}, …, σ_{ν}.

Now Ġ is a simple Lie algebra of type G_{2} over ℂ. Thus, we may identify

Ġ = Der(𝔸),

where 𝔸 is the Cayley algebra over ℂ, i.e. 𝔸 is the 8-dimensional simple alternative but not associative algebra over ℂ. (Again see [J2] for facts about 𝔸 and Der(𝔸).) Let t be the normalized trace on 𝔸 obtained by dividing the usual generic trace by 2. Also, for x, x′ ∈ 𝔸_{0}, define

D_{x,x′} = \frac{1}{4}(L_{[x,x′]} − R_{[x,x′]} − 3[L_{x}, R_{x′}]).

Then,

Ġ = Der(𝔸) = D_{𝔸_{0},𝔸_{0}}.

Now K is a G_{2}-graded Lie algebra and Z(K) = {0}. So by the recognition theorem [BZ, Theorem 3.47] for such algebras, we have

(5.53)  K = T (𝔸/ℂ, J /B) = (Ġ ⊗ B) ⊕ (𝔸_{0} ⊗ J_{0}) ⊕ D_{J_{0},J_{0}},

where B is a commutative associative algebra over ℂ and J is a Jordan algebra over B with a normalized trace t which satisfies the trace identity

ch_{3}(y) := y^{3} − 3t(y)y^{2} + (\frac{9}{2}t(y)^{2} − \frac{3}{2}t(y^{2}))y + (t(y^{3}) − \frac{9}{2}t(y^{2})t(y) + \frac{9}{2}t(y)^{3})1 = 0.

The inner derivations for J are defined by

D_{y,y′} = [L_{y}, L_{y′}]

for y, y′ ∈ J_{0}. Also, Ġ is being identified with Ġ ⊗ 1, and the decomposition (5.53) is an orthogonal decomposition.

We can now describe the structure of J.

Proposition 5.54. We have 0 ≤ p ≤ 3. Moreover, we may identify

B = A_{[ν]}

and

(5.55)  J = \{ \begin{matrix}B & if p = 0 \\ J (B, t_{1}) & if p = 1 \\ J (B, t_{1}, t_{2}) & if p = 2 \\ J (B, t_{1}, t_{2}, t_{3}) & if p = 3. \end{matrix}

Proof. We may choose a basis α̇_{1}, α̇_{2} for Ḣ^{∗} so that

(5.56)  \begin{matrix}Ṙ_{sh} = \{±α̇_{2}, ±(α̇_{1} + α̇_{2}), ±(α̇_{1} + 2α̇_{2})\} and \\ Ṙ_{lg} = \{±α̇_{1}, ±(α̇_{1} + 3α̇_{2}), ±(2α̇_{1} + 3α̇_{2})\}. \end{matrix}

Moreover, the set of weights of the Ġ-module 𝔸_{0} is Ṙ_{sh} ∪ {0} [J2]. Thus, if α̇ ∈ Ṙ^{×}, we have

Kα̇ = \{ \begin{matrix}Ġα̇ ⊗ B & if α̇ ∈ Ṙ_{lg} \\ Ġα̇ ⊗ B + (𝔸_{0})α̇ ⊗ J_{0} & if α̇ ∈ Ṙ_{sh}. \end{matrix}

Exactly as in the proof of Proposition 5.6 we get decompositions

B = ⨁_{σ∈L} B^{σ} and J_{0} = ⨁_{σ∈S\L}(J_{0})^{σ}

with one-dimensional summands satisfying

B^{σ}B^{τ} = B^{σ+τ} for σ, τ ∈ L

and

B^{σ}(J_{0})^{τ} = (J_{0})^{σ+τ} for σ ∈ L, τ ∈ S \ L.

(Again one needs simple facts about the root system Ṙ which are checked using (5.56).) Moreover, the argument used to prove (5.43) can be applied here to show that if σ, τ ∈ S \ L then

(J_{0})^{σ}(J_{0})^{τ} = \{ \begin{matrix}B^{σ+τ} & if σ + τ ∈ L \\ (J_{0})^{σ+τ} & if σ + τ ∉ L. \end{matrix}

Indeed to use this argument one needs only to know that there exist α̇, β̇ ∈ Ṙ_{sh} so that α̇ + β̇ ∈ Ṙ_{sh}, and there exist x ∈ (𝔸_{0})α̇, y ∈ (𝔸_{0})β̇ so that D_{x,y} ≠ 0 and x ∗ y ≠ 0. This is easily verified using the explicit weight space decomposition of 𝔸_{0} given for example in [J2,§2].

We now define

(5.57)  J^{σ} = \{ \begin{matrix}B^{σ} & if σ ∈ L \\ (J_{0})^{σ} & if σ ∈ S \ L. \end{matrix}

Then

(5.58)  J = ⨁_{σ∈S} J^{σ} with one dimensional summands,

and

(5.59)  J^{σ}J^{τ} = J^{σ+τ} for σ, τ ∈ S.

Thus J is a Λ-graded Jordan algebra satisfying assumptions (G1) and (G2) of Section 2. We note next that

(5.60)  \begin{matrix}σ, τ, 2σ, σ + τ, 2σ + τ ∈ Λ \ L, \\ 0 ≠ a ∈ J^{σ}, 0 ≠ b ∈ J^{τ} \end{matrix} ⟹ 0 ≠ a^{2} · b = −2a · (a · b) ∈ J^{2σ+τ} .

To see this we only have to show that a^{2} · b = −2a · (a · b). But because of the assumptions on σ and τ, it follows from (5.59) that a, b, a^{2}, a · b, a^{2} · b, a · (a · b) ∈ J_{0}. Thus, the λ-coefficient of ch_{3}(a + λb) is a^{2} · b + 2a · (a · b). This proves (5.60).

Let Z = Z(J). Then as in the proof of Proposition 5.34, we have

Z = ⨁_{σ∈M} J^{σ},

where M is a subgroup of Λ so that

L ⊆ M ⊆ Λ.

We claim next that either

(5.61)  M = L or 0 ≤ p ≤ 1.

Indeed suppose the contrary. Then, regarding Λ/L as a p-dimensional vector space over ℤ/3ℤ, we may choose σ ∈ M, τ ∈ Λ so that {σ + L, τ + L} is part of a basis for Λ/L over ℤ/3ℤ. So σ, τ, 2σ, σ + τ, 2σ + τ ∈ Λ \ L. Thus, if we choose 0 ≠ a ∈ J^{σ} and 0 ≠ b ∈ J^{τ}, we have 0 ≠ a^{2} · b = −2a · (a · b) by (5.60). But a ∈ Z and so a^{2} · b = a · (a · b). Hence, a^{2} · b = 0, a contradiction. This proves (5.61).

Suppose first that p ≥ 2. Then, by (5.61), M = L and B = Z. We want to apply Proposition 2.17 to obtain the structure of J. To do this we need to calculate the generic degree of (Z^{∗})^{−1}J over (Z^{∗})^{−1}Z. Choose elements x_{1}, …, x_{p} ∈ J so that x_{i} ∈ J^{σ_{i}}, i = 1, …, p. Then, since M = L = Λ_{3}^{(p)}, it follows from (G1) and (G2) for J that the elements

(5.62)  (… ((x_{1}^{i_{1}} · x_{2}^{i_{2}}) · x_{3}^{i_{3}}) · . . . · x_{p}^{i_{p}}), 0 ≤ i_{1}, …, i_{p} ≤ 2

form a basis for J as a Z-module. Hence, we may identify J with 1 ⊗ J in (Z^{∗})^{−1}J, and the elements (5.62) form a basis for (Z^{∗})^{−1}J as a (Z^{∗})^{−1}Z-space. In particular, 1, x_{1}, x_{1}^{2} are independent over (Z^{∗})^{−1}Z. Thus, the generic degree of (Z^{∗})^{−1}J over (Z^{∗})^{−1}Z is at least 3. But every element of (Z^{∗})^{−1}J satisfies a polynomial of degree 3 over (Z^{∗})^{−1}Z. Hence, (Z^{∗})^{−1}J has generic degree 3 over (Z^{∗})^{−1}Z. Thus, by Proposition 2.17, we have p = 2 or p = 3 and we may identify

J = \{ \begin{matrix}J (A_{[ν]}, t_{1}, t_{2}) & if p = 2 \\ J (A_{[ν]}, t_{1}, t_{2}, t_{3}) & if p = 3, \end{matrix}

and hence B = A_{[ν]}.

Finally suppose that p = 0 or 1. If p = 0, we have J = B and then we can identify B = A_{[ν]} by Proposition 2.4. So we may suppose that p = 1. Then, J is generated by elements x_{1}^{±1}, t_{2}^{±1}, …, t_{ν}^{±1}so that

deg(x_{1}) = σ_{1} and deg(t_{i}) = σ_{i}, i = 2, …, ν.

If we define t_{1} = x_{1}^{3}, we have deg(t_{1}) = 3σ_{1} and so

B = ⨁_{σ∈L} J^{σ} = ℂ[t_{1}^{±1}, …, t_{ν}^{±1}] = A_{[ν]},

where A_{[ν]} is graded by L = Λ_{3}^{(1)}. Hence, J = J (B, t_{1}) as claimed. □

We now have:

Theorem 5.63. Suppose that L is an EALA of type G_{2}, nullity ν and twist number p. Then, p ≤ 3,

(5.64)  K ≅ T (𝔸/ℂ, J /A_{[ν]})

and

(5.65)  R = R(G_{2}, Λ, Λ_{3}^{(p)}),

where 𝔸 is the Cayley algebra over ℂ,

J = \{ \begin{matrix}A_{[ν]} & if p = 0 \\ J (A_{[ν]}, t_{1}) & if p = 1 \\ J (A_{[ν]}, t_{1}, t_{2}) & if p = 2 \\ J (A_{[ν]}, t_{1}, t_{2}, t_{3}) & if p = 3, \end{matrix}

and Λ = ℤ^{ν}.

Remark 5.66. In Theorem 5.29 (resp. 5.50 or 5.63), it is clear from the proof how to give the Lie algebra T (Cliff(f)/ℂ, Cliff(g)/A_{[ν]}) (resp. T (𝕁/ℂ, C/A_{[ν]}) or T (𝔸/ℂ, J /A_{[ν]})) a Λ-grading so that the isomorphism (5.30) (resp. (5.51) or (5.64)) between K and this Lie algebra is an isomorphism of Λ-graded Lie algebras. We leave the details to the reader.

Remark 5.67. As in the case of the theorems in the last two sections, the structure theorems 5.29, 5.50 and 5.63 of this section each give an exact description of the Lie algebras K and the root systems R that arise from an EALA of the corresponding type. Examples of EALA’s with K and R as on the right hand sides of (5.30)-(5.31), (5.51)-(5.52) and (5.54)-(5.65) can be found in Theorems III.3.32, III.5.36 and III.5.31 respectively of [AABGP].

§6. Comparison with extended affine root systems.

The structure theorems discussed in Sections 3, 4 and 5 give an exact description of the root systems that arise as the root system of an EALA of reduced type. (See Theorems 3.1, 4.87, 5.29, 5.50 and 5.63, as well as the remarks following those theorems.) In this section, we compare the list of these root systems with the list of (abstractly defined) extended affine root systems obtained in [AABGP].

According to the structure theorems, the following is a complete list of the root systems R that arise as the root system of an EALA of reduced type and nullity ν. The notation is explained in (1.11), (1.12), (2.1) and (2.8) (or in [AABGP, Chapter II]).

List 6.1. Root systems of EALA’s of reduced type.

Type A_{1}: R(A_{1}, S), where S is a semilattice in ℝ^{ν}

Type A_{l}(l ≥ 2), D_{l}(l ≥ 4), E_{l}(l = 6, 7, 8): R(A_{l}, ℤ^{ν}), R(D_{l}, ℤ^{ν}), R(E_{l}, ℤ^{ν})

Type C_{2}: R(C_{2}, Λ, Z_{1,q}), where Λ = ℤ^{ν} and q satisfies q_{ii} = 1, q_{ij} = q_{ji} = ±1; and R(C_{2}, S, 2ℤ^{ν}), where S is the union of at least two cosets of 2ℤ^{ν} in ℤ^{ν} including the trivial coset 0 + 2ℤ^{ν}.

Type C_{3}: R(C_{3}, Λ, Z_{e,q}), where Λ = ℤ^{ν}, e satisfies e_{i} = ±1 and q satisfies q_{ii} = 1, q_{ij} = q_{ji} = ±1; and R(C_{3}, Λ, Λ_{2}^{(3)}), where Λ = ℤ^{ν} and ν ≥ 3.

Type C_{l}(l ≥ 4): R(C_{l}, Λ, Z_{e,q}), where Λ = ℤ^{ν}, e satisfies e_{i} = ±1 and q satisfies q_{ii} = 1, q_{ij} = q_{ji} = ±1

Type B_{l}(l ≥ 3): R(B_{l}, S, 2ℤ^{ν}) where S is a union of cosets of 2ℤ^{ν} in ℤ^{ν} including the trivial coset.

Type F_{4} and G_{2}: R(F_{4}, Λ, Λ_{2}^{(p)}) and R(G_{2}, Λ, Λ_{3}^{(p)}), where Λ = ℤ^{ν} and 0 ≤ p ≤ min(ν, 3).

In [Sa1], [Sa2] and [AABGP] an abstract notion of extended affine root system (EARS) was studied. By definition an EARS is a subset of a finite dimensional real vector space V satisfying certain natural axioms. In [AABGP, Chapter II], a complete description of EARS was given. The motivation for the study of EARS in [AABGP] was the fact that the root system of an EALA is a EARS. Conversely, if R is an EARS of type A_{l}(l ≥ 1), D_{l}(l ≥ 4), E_{l}(l = 6, 7, 8) or B_{l}(l ≥ 3), then R is isomorphic to the root system of some EALA [AABGP, Theorem III.2.10]. However, we can now see that for types C_{l}(l ≥ 2), F_{4} and G_{2} there exist EARS which are not isomorphic to the root system of an EALA. This was conjectured in [AABGP].

Theorem 6.2. Let Λ = ℤ^{ν} with standard basis σ_{1}, …, σ_{ν}.

(a) If ν ≥ 4,

S = {∑_{i=1}^{ν} n_{i}σ_{i} ∈ Λ : n_{1}n_{2} is even}

and

L = {∑_{i=1}^{ν} n_{i}σ_{i} ∈ Λ : n_{1}, n_{2} and ∏_{i=3}^{ν} n_{i} are even},

then R(C_{2}, S, L) is an EARS which is not isomorphic to the root system of an EALA.

(b) If 4 ≤ p ≤ ν, then

(6.3)  R(C_{l}, Λ, Λ_{2}^{(p)}) (l ≥ 3), R(F_{4}, Λ, Λ_{2}^{(p)}) and R(G_{2}, Λ, Λ_{3}^{(p)})

are EARS which are not isomorphic to the root system of an EALA.

Proof. (a) One easily checks that S and L are semilattices in ℝ^{ν} such that L + 2S ⊆ L and S + L ⊆ S. Thus, by [AABGP, Theorem II.2.37], R(C_{2}, S, L) is an EARS. Moreover neither S nor L is a lattice, and this property is an isomorphism invariant of EARS [AABGP, Theorem 3.1 (b)]. Thus, R(C_{2}, S, L) is not isomorphic to any of the root systems of type C_{2} in List 6.1.

(b) We first need to recall a definition. If R is an EARS of nonsimply laced reduced type and nullity ν, then R ≅ R(X, S, L), where S and L are semilattices in ℝ^{ν} which satisfy kS ⊆ L ⊆ S (and some further properties which we don’t need here) [AABGP, Theorem II.2.37]. Here k = 3 if X = G_{2} and k = 2 otherwise. The twist number of R is then defined to be t = log_{k}(|⟨S⟩/⟨L⟩|). t is an integer which is between 0 and ν, and t is an isomorphism invariant of R [AABGP, Definition II.4.11].

Now the root systems in (6.3) are EARS (by [AABGP, Theorem II.2.37]) with twist number p. By hypothesis, p ≥ 4. Thus, it suffices to show that the root systems of type C_{l}(l ≥ 3), F_{4} and G_{2} in List 6.1 have twist number ≤ 3. This is clear for the root systems of type F_{4} and G_{2} in List 6.1 and for the root system R(C_{3}, Λ, Λ_{2}^{(3)}). So we must compute the twist number t of the root system

R(C_{l}, Λ, Z_{e,q}),

where l ≥ 3, e_{ii} = ±1 and q_{ii} = 1, q_{ij} = q_{ji} = ±1. For this purpose, let Λ̃ = Λ/2Λ and let ˜ : Λ → Λ̃ be reduction mod 2. We regard Λ̃ as a vector space over ℤ/2ℤ. Then, 2^{t} = |Λ/⟨Z_{e,q}⟩| = |Λ̃/⟨Z_{e,q}⟩˜ | = |Λ̃/⟨Z̃_{e,q}⟩| and so

t = dim_{ℤ/2ℤ}(Λ̃/⟨Z̃_{e,q}⟩).

Define a quadratic form Q on Λ̃ by

Q(∑_{i=1}^{ν} a_{i}σ̃_{i}) = ∑_{i=1}^{ν} δ_{e_{i},−1}a_{i} + ∑_{1≤i<j≤ν} δ_{q_{ij},−1}a_{i}a_{j}.

Then, Z_{e,q} = {a ∈ Λ : Q(ã) = 0} (see (2.8)), and so ⟨Z̃_{e,q}⟩ = M_{Q}, where M_{Q} is the subspace of Λ̃ generated by the set of zeroes of Q. Thus,

t = dim_{ℤ/2ℤ}(Λ̃/M_{Q}).

To calculate t, we now appeal to the classification of quadratic forms over ℤ/2ℤ [Di, Chapitre I, §16]. It is straightforward to calculate M_{Q} for each quadratic form Q in the classification list. We leave this to the reader. One finds that in all cases Λ̃/M_{Q} has dimension 0, 1 or 2. □

References

[AABGP] B. Allison, S. Azam, S. Berman, Y. Gao and A. Pianzola, Extended affine Lie algebras and their root systems, Memoirs of the A.M.S. (to appear).

[BZ] G. M. Benkart and E. I. Zelmanov, Lie algebras graded by finite root systems and intersection matrix algebras, Invent. Math. 126 (1996), 1–45.

[BGK] S. Berman, Y. Gao, Y. Krylyuk, Quantum tori and the structure of elliptic quasi-simple Lie algebras, J. Funct. Anal 135 (1996), 339–389.

[BGKN] S. Berman, Y. Gao, Y. Krylyuk, E. Neher, The alternative torus and the structure of elliptic quasisimple Lie algebras of type A_{2}, Trans. Amer. Math. Soc 347 (1995), 4315–4363.

[BM] S. Berman and R. V. Moody, Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy, Invent. Math 108 (1992), 323–347.

[Di] J. Dieudonné, La géométrie des groupes classiques, Troiseiéme Edition, Springer-Verlag, New York, 1971.

[F] J. R. Faulkner, Barbilian planes, Geom. Dedicata 30 (1989), 125–181.

[G] Y. Gao, The degeneracy of extended affine Lie algebras, preprint.

[H] J.E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York, 1972.

[HT] R. Høegh-Krohn and B. Torresani, Classification and construction of quasi-simple Lie algebras, J. Funct. Anal. 89 (1990), 106–136.

[J1] N. Jacobson, Structure and representations of Jordan algebras, Amer. Math. Soc. Colloq. Publ., vol. 39, Amer. Math. Soc., Providence, R.I., 1968.

[J2] N. Jacobson, Exceptional Lie algebras, Lecture notes in pure and applied mathematics, Marcel Dekker, New York, 1971.

[M] Y. I. Manin, Topics in noncommutative geometry, Princeton University Press, 1991.

[Mc] K. McCrimmon, A taste of Jordan algebras, preliminary version of a textbook on Jordan algebras.

[McN] K. McCrimmon and E. Neher, Coordinatization of triangulated Jordan systems, J. of Algebra 114 (1988), 411-451.

[N] E. Neher, Lie algebras graded by 3-graded root systems, Amer. J. Math. 118 (1996), 439–491.

[Sa1] K. Saito, Extended affine root systems 1 (Coxeter transformations), Publ. RIMS., Kyoto Univ. 21 (1985), 75–179.

[Sa2] K. Saito, Extended Affine Root Systems 2 (flat invariants), Publ. RIMS., Kyoto Univ. 26 (1990), 15–78.

[Se] G. B. Seligman, Rational methods in Lie algebras, Lect. Notes Pure Appl. Math., New York: M. Dekker, 1976.

[T] J. Tits, Une classe d’algèbres de Lie en relation avec les algèbres de Jordan, Indag. Math. 24 (1962), 530–535.

[Y] Y. Yoshii, Jordan tori, C.R. Math. Rep. Acad. Sci. Canada 18 (1996), 153–158.

[ZSSS] K.A. Zhevlakov, A.M. Slinko, J.P. Shestakov and A.I. Shirshov, Rings that are nearly associative, Academic Press, New York, 1982.

Department of Mathematics, University of Alberta, Edmonton, Canada T6G 2G1 E-mail address: Bruce.Allison@ualberta.ca

Mathematisches Seminar, Universität Hamburg, D-20146 Hamburg

E-mail address: gao@geomat.math.uni-hamburg.de