Journal of Lie Theory Volume 14 (2004) 371–394 \begin{matrix}c \\ ◯ \end{matrix} 2004 Heldermann Verlag

Root Systems Extended by an Abelian Group and their Lie Algebras

Yoji Yoshii^{∗}

Communicated by K.-H. Neeb

Abstract. We introduce the notion of a root system extended by an abelian group G. This concept generalizes extended affine root systems. We classify them in terms of (translated) reflection spaces of G. Then we see that division (Δ, G)-graded Lie algebras have such root systems. Finally, division (B_{l}, G)-graded Lie algebras and as a special case, Lie G-tori of type B_{l}, are classified for l ≥ 3. 2000 MSC: Primary 17B65; secondary 17C50 Keywords: extended affine root systems; Jordan tori

1. Introduction

We have two purposes in the paper. The first one is to introduce the notion of a root system extended by an abelian group G. This concept generalizes the extended affine root systems in the sense of both [A-P] and [Sa]. More precisely, the root systems extended by ℤ^{n} correspond to Saito’s systems in [Sa] and the ‘reduced’ root systems extended by ℤ^{n} correspond to the systems in [A-P]. The classification of such root systems is essentially the same as in [A-P]. But the lattice in a vector space can be any abelian group in their classification, and it seems that our definition and classification are simpler and easier to understand. Moreover, it turns out that the set of certain support sets of a division (Δ, G)- graded Lie algebra becomes a root system extended by G. Thus the purpose here is to acquaint people with a nice and natural class of Lie algebras and their root systems.

The second purpose is to classify division (B_{l}, G)-graded Lie algebras for the root system B_{l}. Division (Δ, G)-graded Lie algebras generalize the core of extended affine Lie algebras (EALAs) when G = ℤ^{n} [Y4] and the finitedimensional isotropic simple Lie algebras when G is trivial (see Example 4.3(3)). Those Lie algebras were classified for Δ = A_{l} (l ≥ 3), D_{l}, E_{l} in [Y2], and A_{2} when G = ℤ^{n} in [Y3] (see Example 4.3(3)). So the cases of type A_{1} , B_{l}, C_{l}, F_{4} ,

∗ Research supported by an NSERC Postdoctoral Fellowship (2002 and 2003)

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag G_{2} , and BC_{l} were open. Using the recognition theorem of B_{l} -graded Lie algebras by Benkart and Zelmanov [BZ], and the approach in [AG] to the classification of the cores of EALAs of type B_{l}, we have classified division (Δ, G)-graded Lie algebras in the case of type B_{l} for l ≥ 3. For type B_{2} , it is more difficult to classify.

As a special case of division (Δ, G)-graded Lie algebras, we define Lie G-tori. This notion is more concrete and there is more hope to obtain a complete classification. Lie G-tori again generalize the core of EALAs when G = ℤ^{n}. We state the classification of Lie G-tori of type B_{l} for l ≥ 3, using the notion of a Clifford G-torus (which appears in several papers [A-P], [AG], [T], [Y1] and [NY]), as a corollary of the result above.

The organization of the paper is as follows. In §2 we define a root system extended by an abelian group, and in §3 they are classified. In §4 a division (Δ, G)-graded Lie algebra and a Lie G-torus are defined, and in §5 we show that these Lie algebras have root systems extended by G. In §6 we obtain the classification of division (B_{l}, G)-graded Lie algebras for l ≥ 3. In the final section we specialize the result in §6 to the case of Lie G-tori.

Finally, I would like to thank Professors Bruce Allison, Georgia Benkart and Erhard Neher for their encouragement and suggestions. Also, I would like to thank the referee for pointing out some mistakes and Professor George Seligman for answering several questions.

2. Definition of a root system extended by an abelian group

Let Δ be a finite irreducible root system, i.e., Δ = A_{l} (l ≥ 1), B_{l} (l ≥ 1, B_{1} = A_{1} ), C_{l} (l ≥ 2, C_{2} = B_{2} ), D_{l} (l ≥ 4), E_{l} (l = 6, 7, 8), F_{4} , G_{2} or BC_{l} (l ≥ 1). Let

Δ^{red} := \{ \begin{matrix}Δ & if Δ is reduced \\ \{μ ∈ Δ | μ is reduced, i.e., \frac{1}{2}μ ∉ Δ\} & otherwise, i.e., if Δ = BC_{l}. \end{matrix}

Note that Δ^{red} = B_{l} if Δ = BC_{l}. Recall the Weyl group W acting on Δ, i.e.,

W = ⟨σ_{μ} | μ ∈ Δ⟩,

where

σ_{μ}(ν) = ν− < ν, μ > μ

is the reflection on Δ with Cartan integer < ν, μ > for ν ∈ Δ. Let G be an abelian group and

R = {S_{μ}}_{μ∈Δ}

a family of nonempty subsets S_{μ} of G indexed by μ ∈ Δ. We define a subset S_{μ} · S_{ν} of G by

S_{μ} · S_{ν} : = S_{ν}− < ν, μ > S_{μ} = {s− < ν, μ > s′ | s ∈ S_{ν}, s′ ∈ S_{μ}}.

Then R is called a root system of type Δ extended by G if it satisfies the following 3 axioms:

(R1) ⋃_{μ∈Δ} S_{μ} generates G;

(R2) 0 ∈ S_{μ} for all μ ∈ Δ^{red} ;

(R3) S_{μ} · S_{ν} ⊂ S_{σ_{μ}(ν)} for all μ, ν ∈ Δ.

Furthermore, when Δ is nonreduced, (R4) μ, 2μ ∈ Δ ⇒ S_{2μ} ∩ 2S_{μ} = Ø, then R is called reduced.

Remark 2.1. If G is trivial, i.e., G = {0}, then the root system {S_{μ}}_{μ∈Δ} extended by {0} is given by the sets S_{μ} = {0} for all μ ∈ Δ, which can be viewed as Δ itself. If Δ is nonreduced, there does not exist a reduced root system extended by {0} since (R4) cannot be satisfied.

Remark 2.2. When G = ℤ^{n}, the root system {S_{μ}}_{μ∈Δ} naturally corresponds to an extended affine root system (see the last part of §3), and each S_{μ} is an isotropic part of it.

3. Classification of root systems extended by G

Let R = {S_{μ}}_{μ∈Δ} be a root system extended by G. By (R2) and (R3), we have

S_{ν} ⊂ S_{μ} · S_{ν} ⊂ S_{σ_{μ}(ν)}

for all μ ∈ Δ^{red} and ν ∈ Δ. Since W is generated by σ_{μ} for μ ∈ Δ^{red} ,

(1)  S_{w(ν)} = S_{ν} for all w ∈ W and ν ∈ Δ.

In particular, for μ, ν ∈ Δ,

(2)  S_{μ} = S_{ν} if μ and ν have the same length,

and S_{−μ} = S_{μ} for all μ ∈ Δ. Also, by (R3) and (1), we have

(3)  S_{ν}− < ν, μ > S_{μ} ⊂ S_{ν} for all μ, ν ∈ Δ.

In particular, taking ν = μ,

(4)  S_{μ} − 2S_{μ} ⊂ S_{μ} for all μ ∈ Δ,

which implies −S_{μ} = S_{μ} for all μ ∈ Δ.

We next partition the root system Δ according to length. Roots of Δ of minimal length are called short. Roots of Δ which are two times a short root of Δ are called extra long. Finally, roots of Δ which are neither short nor extra long are are called long. We denote the roots of short, long and extra long roots of Δ by Δ_{sh}, Δ_{lg} and Δ_{ex} respectively. Thus

Δ = Δ_{sh} ⊔ Δ_{lg} ⊔ Δ_{ex}.

Of course the last two terms in this union may be empty. Indeed,

Δ_{lg} = Ø ⟺ Δ has simply laced type or type BC_{1} ,

and

Δ_{ex} = Ø ⟺ Δ = Δ^{red}.

In view of (2), we can simplify the notation. We define

S = S_{μ} for μ ∈ Δ_{sh},
L = S_{μ} for μ ∈ Δ_{lg}, provided that Δ_{lg} ≠ Ø, and
E = S_{μ} for μ ∈ Δ_{ex}, provided that Δ_{ex} ≠ Ø.

Thus our root system R = {S_{μ}}_{μ∈Δ} can be written as

R = R(S, L, E)_{Δ}.

Definition 3.1. Let G be an abelian group.

(i) A subset E of G is called a translated reflection space if E − 2E ⊂ E. (ii) A translated reflection space E of G is called full if E generates G. (iii) A translated reflection space E of G is called a reflection space if 0 ∈ E.

Remark 3.2. (1) The notion of a reflection space was introduced in [L], which is a more general concept than ours. If G = ℤ^{n}, it was called a semilattice in [A-P].

(2) If S is a reflection space of G, then so is S + S. In fact, we have 0 ∈ S ⊂ S + S, and S + S − 2(S + S) = (S − 2S) + (S − 2S) ⊂ S + S.

By (R2) and (4), S and L in our root system R are reflection spaces of G, and E is a translated reflection space of G. If R is reduced, i.e., assuming (R4), then

E ∩ 2S = Ø,

and in particular, 0 ∉ E.

We will see that L and E are always contained in S. Moreover, we can prove certain properties of S, L and E, depending on the type of Δ. If Δ_{lg} ≠ Ø, we use the notation k for the ratio of the long square root length to the short square root length in Δ. Hence,

k = \{ \begin{matrix}2 & if Δ has type B_{l}, C_{l}, F_{4} or BC_{l} for l ≥ 2. \\ 3 & if Δ has type G_{2}. \end{matrix}

Proposition 3.3. Let R = R(S, L, E)_{Δ} be a root system extended by G. (a) If Δ_{lg} ≠ Ø, then

L + kS ⊂ L and S + L ⊂ S.

(b) If Δ_{ex} ≠ Ø, i.e., Δ has type BC_{l}, then

E + 4S ⊂ E and S + E ⊂ S.

Moreover, if Δ_{lg} ≠ Ø, i.e., if l ≥ 2, then

E + 2L ⊂ E and L + E ⊂ L.

(c) E ⊂ L ⊂ S, and S is a full reflection space of G. (S is the biggest!)

(d) If Δ does not have type A_{1} , B_{l} or BC_{l}, then S = G.

(e) If Δ has B_{l}, F_{4} , G_{2} or BC_{l} for l ≥ 3, then L is a subgroup of G.

Proof. If the Dynkin diagram of Δ^{red} contains a subgraph ∘ = ∘ or ∘ ≡ ∘, then there exist μ, ν ∈ Δ with μ short such that < μ, ν >= −2 and < ν, μ >= −1, or < μ, ν >= −3 and < ν, μ >= −1, respectively. So (a) follows from (3).

If Δ has type BC_{l}, then there exists μ ∈ Δ such that −2μ ∈ Δ, and so < −2μ, μ >= −4 and < μ, −2μ >= −1. So the first assertion of (b) follows from (3). The second assertion also follows from (3) since there exist μ ∈ Δ_{lg} and ν ∈ Δ_{ex} such that < μ, ν >= −2 and < μ, ν >= −1.

The fourth inclusion of (b) and the second inclusion of (a) show that E ⊂ L ⊂ S since 0 ∈ L and 0 ∈ S. Hence, by (R1), we obtain (c).

If the Dynkin diagram of Δ^{red} contains a subgraph ∘ − ∘, then there exist μ, ν ∈ Δ of the same length such that < μ, ν >= −1. In this case, by (3), we get S +S ⊂ S if μ and ν are short, and L+L ⊂ L if μ and ν are long. If Δ does not have type A_{1} , B_{l}, BC_{l} or G_{2} , you can find a subgraph ∘ − ∘ in short roots, and so S becomes a subgroup of G. Hence S = G since S generates G. Similarly, for the types in (e) except type G_{2} , you can find a subgraph ∘ − ∘ in long roots, and so L becomes a subgroup of G. So (d) and (e) hold except type G_{2} . But if Δ has type G_{2} , both Δ_{sh} and Δ_{lg} have type A_{2} whose Dynkin diagram is ∘ − ∘. Hence (d) and (e) also hold for type G_{2} .

Theorem 3.4. Let {S_{μ}}_{μ∈Δ} be a root (resp. a reduced root) system extended by G of type Δ. Then S_{μ} = S for all μ ∈ Δ_{sh}, S_{μ} = L for all μ ∈ Δ_{lg} and S_{μ} = E for all μ ∈ Δ_{ex}, where S is a full reflection space, L is a reflection space and E is a translated reflection space satisfying

L + kS ⊂ L and S + L ⊂ S, E + 4S ⊂ E and S + E ⊂ S E + 2L ⊂ E and L + E ⊂ L S = G if Δ ≠ A_{1}, B_{l}, BC_{l}, L is a subgroup if Δ = B_{l} (l ≥ 3), F_{4}, G_{2}, BC_{l} (l ≥ 3) (resp. and furthermore, E ∩ 2S = Ø).

Conversely, let S, L and E be as above, and define S_{μ} = S for all μ ∈ Δ_{sh}, S_{μ} = L for all μ ∈ Δ_{lg} and S_{μ} = E for all μ ∈ Δ_{ex}. Then R(S, L, E)_{Δ} is a root (resp. a reduced root) system extended by G.

Proof. We only need to show the second statement. So we show that the set R(S, L, E)_{Δ} is a (reduced) root system. Thus, we must check the axioms (R1)- (R4). But all except (R3) are clear. Considering the possible Cartan integers < μ, ν > in each type Δ, (R3) also follows from the relations among S, L and E.

Example 3.5. (1) Let G = ℤ_{4} × ℤ_{4} . By Theorem 3.4, a root system extended by G of type B_{l} for l ≥ 3 is determined by a subgroup L and a full reflection space S of G satisfying S + L ⊂ S and L + 2S ⊂ L. Note that the second inclusion is equivalent to 2S ⊂ L since L is a subgroup. This is not the case for type B_{2} . Let L = 2G and

S = L ∪ (L + (1, 0)) ∪ (L + (0, 1)).

Then R(S, L)_{B_{l}} is an example of a root system extended by G.

(2) Let R be a root system extended by ℚ of type B_{l} for l ≥ 3. Then since ℚ = 2ℚ = 2⟨S⟩ ⊂ L, where ⟨S⟩ is the subgroup of ℚ generated by S, we have L = S = ℚ. Hence R = R(ℚ, ℚ)_{B_{l}}.

(3) Let R be a root system extended by ℤ of type B_{l} for l ≥ 2. Then any reflection space of ℤ is a subgroup of ℤ. Hence R = R(ℤ, ℤ)_{B_{l}} or R(ℤ, 2ℤ)_{B_{l}}.

A reduced root system {S_{μ}}_{μ∈Δ} extended by ℤ^{n} can be considered as an extended affine root system (EARS): Let V_{1} be a euclidean space with root system Δ, V_{2} an n-dimensional real vector space and V = V_{1} ⊕ V_{2} the vector space with positive semidefinite symmetric bilinear form (·, ·) which is the natural extension of the form on V_{1} with radical V_{2} . Identify ℤ^{n} with the lattice spanned by a basis of V_{2} . Let

S_{0} = S_{μ} + S_{μ}

for any μ ∈ Δ_{sh} (see Remark 3.2(2)) and

R = ⋃_{μ∈Δ∪{0}} (μ + S_{μ}).

Then the descriptions of S_{μ} for μ ∈ Δ ∪ {0} in Theorem 3.4 are exactly the same as in the classification of EARS (see [A-P, 2.32 and 2.37]), and so (V, R) is an EARS and any EARS can be considered as a root system extended by ℤ^{n}. In particular, a reduced root system extended by ℤ can be thought as an affine root system.

A root system {S_{μ}}_{μ∈Δ} extended by ℤ^{n} can be considered as a Saito’s extended affine root system (SEARS) (see [Sa]): In the same setting as above, let

R_{S} = ⋃_{μ∈Δ} (μ + S_{μ}).

Then by the same argument as Azam did (he showed a one-to-one correspondence between EARS and reduced SEARs in [A]), (V, R_{S}) is a SEARS, and any SEARS can be considered as a root system extended by ℤ^{n}. We summarize these as a corollary.

Corollary 3.6. We have the following one-to-one correspondences:

{root systems extended by ℤ^{n}} ↔ {SEARS of nullity n}
∪  ∪
{reduced root systems extended by ℤ^{n}} ↔ {reduced SEARS of nullity n}

↔ {EARS of nullity n}

4. Division (Δ, G)-graded Lie algebras

Throughout the paper the base field F has characteristic 0. We first recall Δgraded Lie algebras and division (Δ, G)-graded Lie algebras, and as a special case, we define Lie G-tori. Let 𝔤 be a finite-dimensional split simple Lie algebra over F with a split Cartan subalgebra 𝔥 and the root system Δ^{red} so that 𝔤 has the root space decomposition 𝔤 = 𝔥 ⊕ ( ⊕_{μ∈Δ^{red}} 𝔤_{μ}) with 𝔥 = 𝔤_{0} . A Δ-graded Lie algebra L over F with grading subalgebra 𝔤 or grading pair (𝔤, 𝔥) is defined as

(i) L contains 𝔤 as a subalgebra; (ii) L = ⊕_{μ∈Δ∪{0}} L_{μ}, where L_{μ} = {x ∈ L | [h, x] = μ(h)x for all h ∈ 𝔥};

and (iii) L_{0} = ∑_{μ∈Δ} [L_{μ}, L_{−μ}].

Note that the Jacobi identity implies [L_{μ}, L_{ν}] ⊂ L_{μ+ν} for all μ, ν ∈ Δ ∪ {0}, defining L_{μ+ν} = 0 if μ + ν ∉ Δ ∪ {0}. Also, the centre Z = Z(L) of L is contained in L_{0} , and L/Z is again a Δ-graded Lie algebra with Z(L/Z) = 0. A Δ-graded Lie algebra having trivial centre is called centreless.

Remark 4.1. The Δ-graded Lie algebras for a reduced Δ were introduced by Berman and Moody [BM], and have been recently generalized for a nonreduced case by Allison, Benkart and Gao [ABG]. They classified more general BC_{l} - graded Lie algebras in [ABG] (and [BS] for l = 1) than our concept above. We only consider a special class of BC_{l} -Lie algebras in their sense. The reason comes from the theory of EALAs. Namely, the core of an EALA is a Δ-graded Lie algebra in our sense, which was shown in [AG, Proposition 1.16].

Let G be an abelian group. We will consider a G-graded (Lie) algebra L = ⊕_{g∈G} L^{g}, which is a G-graded vector space satisfying L^{g}L^{h} ⊂ L^{g+h} ([L^{g}, L^{h}] ⊂ L^{g+h} if L is Lie) for all g, h ∈ G. For convenience, we always assume that

supp L := {g ∈ G | L^{g} ≠ 0} generates G.

Definition 4.2.

(1) A Δ-graded Lie algebra L = ⊕_{μ∈Δ∪{0}} L_{μ} with grading pair (𝔤, 𝔥) is called (Δ, G)-graded if L = ⊕_{g∈G} L^{g} is a G-graded Lie algebra such that 𝔤 ⊂ L^{0} . Then we have

L = ⨁_{μ∈Δ∪{0}} ⨁_{g∈G} L_{μ}^{g},

where L_{μ}^{g} = L_{μ} ∩ L^{g} since L^{g} is an 𝔥-submodule of L. Note that if G is trivial, L is just a Δ-graded Lie algebra.

(2) Let Z(L) be the centre of L and let μ^{∨} ∈ 𝔥 for μ ∈ Δ be the coroot of μ. Then L is called a division (Δ, G)-graded Lie algebra if for any μ ∈ Δ and any 0 ≠ x ∈ L_{μ}^{g}, there exists

y ∈ L_{−μ}^{−g} such that [x, y] ≡ μ^{∨} modulo Z(L). (division property)

(These Lie algebras for a reduced Δ were introduced in [Y2].) If G is trivial, L is called a division Δ-graded Lie algebra.

(3) A division (Δ, G)-graded Lie algebra L = ⊕_{μ∈Δ∪{0}} ⊕_{g∈G} L_{μ}^{g} is called a Lie G-torus of type Δ if

dim_{F} L_{μ}^{g} ≤ 1 for all g ∈ G and μ ∈ Δ. (1-dimensionality)

If G = ℤ^{n}, we call it a Lie n-torus or simply a Lie torus.

Example 4.3.

(1) A finite-dimensional simple isotropic Lie algebra L is a centreless division Δ-graded Lie algebra. In fact, let 𝔥 be a maximal abelian diagonalizable subalgebra of L (a maximal split torus in [Se]), which is nonzero since L is isotropic. Then L has the root space decomposition relative to 𝔥, and the root system Δ is a finite irreducible root system (see [Se, Ch.I, §2]). Thus L = ⊕_{μ∈Δ∪{0}} L_{μ} which is the condition (ii) of Δ-graded Lie algebras. Since ∑_{μ∈Δ} ([L_{μ}, L_{−μ}] + L_{μ}) is a nonzero ideal of L and L is simple, the condition (iii) holds. The division property relative to this 𝔥 is one of the most important and starting properties to develop the theory, which is shown in [Se, Ch.I, Lemma 3]. Thus we need to show the condition (i) of Δ-graded Lie algebras, i.e., the existence of a grading subalgebra 𝔤 so that the grading pair is (𝔤, 𝔥). Let Π = {α_{1}, …, α_{l}} be the set of simple roots of Δ^{red} . As Seligman did in [Se, Ch.III, §1], using the division property, one can choose an sl_{2} -triplet 0 ≠ e_{i} ∈ L_{α_{i}}, 0 ≠ f_{i} ∈ L_{−α_{i}} and α_{i}^{∨} ∈ 𝔥 for each α_{i}. These e_{i}, f_{i}, α_{i}^{∨} for 1 ≤ i ≤ l satisfy the Serre relations, and by Serre’s Theorem, one can show that the Lie algebra 𝔤 generated by those elements is a split simple Lie algebra of type Δ^{red} and 𝔥 is a Cartan subalgebra of 𝔤. Note that for the nonreduced case, he constructed a split simple Lie algebra of type C_{l} instead of type Δ^{red} = B_{l}. See also the remark [ABG, 1.15].

(2) Loop (or twisted loop) algebras, toroidal Lie algebras and the cores of EALAs are all examples of Lie tori.

(3) Let F^{t}[G] be a twisted group algebra. Then sl_{l+1}(F^{t}[G]) is a centreless Lie G-torus of type A_{l}. Moreover, any Lie G-torus of type A_{l} for l ≥ 3 is isomorphic to a central extension of some sl_{l+1}(F^{t}[G]). More generally, division (Δ, G)-graded Lie algebras of type A_{l} (l ≥ 3), D_{l} and E_{l} were, up to central extensions, classified in terms of crossed product algebras, and if G = ℤ^{n}, they were classified in more precise description (see [Y2]). Also, division (A_{2}, ℤ^{n})-graded Lie algebras were, up to central extensions, classified in [Y3]. Centreless division Δ-graded Lie algebras are (possibly infinite-dimen-

sional) isotropic simple Lie algebras by the following:

Lemma 4.4. A centreless division (Δ, G)-graded Lie algebra is G-graded simple. Hence, in particular, a centreless division Δ-graded Lie algebra is simple.

Proof. Suppose L = ⊕_{μ∈Δ∪{0}} ⊕_{g∈G} L_{μ}^{g} is a centreless division (Δ, G)- graded Lie algebra with grading pair (𝔤, 𝔥), and I = ⊕_{g∈G} I^{g} is a nonzero graded ideal. Since I^{g} is an 𝔥-submodule, I^{g} is graded for the Δ-grading, say I^{g} = ⊕_{μ∈Δ∪{0}} I_{μ}^{g} for all g ∈ G. Since I ≠ 0, there exists some k ∈ G such that I^{k} ≠ 0. If I^{k} = I_{0}^{k}, then [I^{k}, L_{μ}^{l}] ⊂ I_{μ}^{k+l} for μ ∈ Δ and l ∈ G. If I_{μ}^{k+l} = 0 for all μ ∈ Δ and l ∈ G, then I^{k} ⊂ Z(L) = 0, contradiction. Hence there exist some μ ∈ Δ and l ∈ G such that I_{μ}^{k+l} ≠ 0. Therefore, without loss of generality, we may assume that I_{μ}^{g} ≠ 0 for some μ ∈ Δ and g ∈ G. Then μ^{∨} ∈ I by the division property. If < ν, μ >≠ 0, then μ^{∨} ∈ I implies L_{ν} ⊂ I. Hence, by the division property, ν^{∨} ∈ I. Repeating this process, we get L_{ν} ⊂ I for all ν ∈ Δ, by the irreducibility of Δ. So I = L. Thus L is graded simple.

5. The root systems of division (Δ, G)-graded Lie algebras

We show that a division (Δ, G)-graded Lie algebra has a root system extended by G of type Δ in the following sense:

Theorem 5.1. Let L = ⊕_{μ∈Δ∪{0}} ⊕_{g∈G} L_{μ}^{g} be a division (Δ, G)-graded Lie algebra with centre Z. For each μ ∈ Δ ∪ {0}, let

S_{μ} := {g ∈ G | L_{μ}^{g} ≠ 0} and R = R(L) := {S_{μ}}_{μ∈Δ}.

Then R = R(L/Z), R is a root system extended by G, and S_{0} = S_{μ} + S_{μ} for μ ∈ Δ_{sh}.

Moreover, if L is a Lie G-torus, then R is reduced. Proof. Let S′_{0} := {g ∈ G | (L/Z)_{0}^{g} ≠ 0}. Clearly S′_{0} ⊂ S_{0} . We claim that

(∗)  S_{0} = ⋃_{μ∈Δ} (S_{μ} + S_{−μ}) = S′_{0}.

Since L is Δ-graded, S_{0} ⊂ ∪_{μ∈Δ} (S_{μ} + S_{−μ}). So we need to show that ∪_{μ∈Δ} (S_{μ} + S_{−μ}) ⊂ S′_{0} . Let g ∈ S_{μ} and k ∈ S_{−μ} and let 0 ≠ x ∈ L_{μ}^{g} and 0 ≠ v ∈ L_{−μ}^{k}. Suppose [x, v] = 0 in L/Z. By the division property, there exists y ∈ L_{−μ}^{−g} so that sl_{2} := ⟨x, y, μ^{∨}⟩ in L/Z is isomorphic to sl_{2}(F). Consider L as an sl_{2} -module and the submodule generated by v. Note that [μ^{∨}, v] = −2v, and so v has the negative weight. Hence 0 ≠ (ad y)^{2}(v) ∈ L_{−3μ}, which cannot happen. Hence 0 ≠ [x, v] ∈ L_{0}^{g+k}, i.e., g +k ∈ S′_{0} , and our claim is settled. Since Z ⊂ L_{0} , L_{μ} = (L/Z)_{μ} for μ ≠ 0, and so R(L) = R(L/Z).

For the second statement, we need to check the axioms (R1)-(R3). (R1) and (R2) follow from the definition of a division (Δ, G)-graded Lie algebra. Thus we show (R3). Notice that ad x, for any g ∈ G, μ ≠ 0 and 0 ≠ x ∈ L_{μ}^{g}, is ad-nilpotent since Δ is finite. Let y ∈ L_{−μ}^{−g} be such that [x, y] ≡ μ^{∨} mod Z. Then one can define the automorphism

θ_{μ}^{g} := exp ad x exp ad (−y) exp ad x

of L. Then by the same way as in [A-P, Prop. 1.27], one can see that

θ_{μ}^{g}(L_{ν}^{k}) = L_{σ_{μ}(ν)}^{k−<ν,μ>g}

for all ν ∈ Δ and k ∈ G. So S_{ν}− < ν, μ > S_{μ} ⊂ S_{σ_{μ}(ν)} , which is (R3). Therefore, R is a root system extended by G. So in particular, S_{ν} ⊂ S_{μ} = S_{−μ} for all ν ∈ Δ and μ ∈ Δ_{sh} (see §3, the equation (2) and Proposition 3.3(c)). Hence by (∗), we get S_{0} = S_{μ} + S_{μ} for μ ∈ Δ_{sh}.

For the last statement, we need to check (R4). Let μ, 2μ ∈ Δ and g ∈ S_{μ}. Since R = R(L/Z), we may assume that L is centreless. Thus again let x ∈ L_{μ}^{g} and y ∈ L_{−μ}^{−g} be so that sl_{2} := ⟨x, y, μ^{∨}⟩ ≅ sl_{2}(F). By 1-dimensionality, we have [L_{μ}^{g}, L_{−μ}^{−g}] + [L_{2μ}^{2g}, L_{−2μ}^{−2g}] ⊂ F μ^{∨}. Hence,

M := L_{2μ}^{2g} ⊕ L_{μ}^{g} ⊕ F μ^{∨} ⊕ L_{−μ}^{−g} ⊕ L_{−2μ}^{−2g}

is a finite-dimensional sl_{2} -module. Since the 0-weight subspace of M which is F μ^{∨} is 1-dimensional, M is irreducible. Since M contain sl_{2} as a submodule, M = sl_{2} . So in particular, L_{2μ}^{2g} = 0. Hence S_{2μ} ∩ 2S_{μ} = Ø.

The 1-dimensionality forces the root system to be reduced. The following shows that 2-dimensionality is enough to have a nonreduced example.

Example 5.2. (An example of a Lie algebra which has a nonreduced root system extended by ℤ^{n}, or a nonreduced SEARS of nullity n.) Let θ = (θ_{ij}) be an n × n skew symmetric matrix over ℝ, and let T := ℂ_{θ}[t_{1}^{±1}, …, t_{n}^{±1}] be the noncommutative torus determined by θ, i.e., the associative algebra over ℂ generated by t_{1}^{±1}, …, t_{n}^{±1} with relations

t_{j}t_{i} = e^{\sqrt{−1}θ_{ij}}t_{i}t_{j} for all 1 ≤ i, j ≤ n.

Let be the involution of T over ℝ determined by

\overline{t}_{1} = t_{1}, …, \overline{t}_{n} = t_{n},

which extends the complex conjugation of ℂ. So (T, ) is an associative algebra with involution over ℝ. Also, T has the natural ℤ^{n} -grading, i.e., T = ⊕_{α∈ℤ^{n}} ℂt_{α}, where t_{α} := t_{1}^{α_{1}} ⋯ t_{n}^{α_{n}} for α = (α_{1}, …, α_{n}) ∈ ℤ^{n}. (All the homogeneous spaces are 2-dimensional over ℝ.) Thus it is easy to show that the Kantor construction K := K(T, ), which is a BC_{1} -graded Lie algebra over ℝ, is a division (BC_{1}, ℤ^{n})-graded Lie algebra over ℝ. So R(K) = R(S, E)_{BC_{1}} is a root system extended by ℤ^{n}, S = supp T = ℤ^{n}, and

E = {α ∈ ℤ^{n} | ℂt_{α} ∩ T_{−} ≠ 0},

where T_{−} is the set of skew elements of (T, ). It is easy to see that E = ℤ^{n}. Thus R(K) is nonreduced.

6. Division (Δ, G)-graded Lie algebras of type B

In this section we will classify division (B_{l}, G)-graded Lie algebras for l ≥ 3, and see that there exist such Lie algebras for each root system extended by G of type B_{l} for l ≥ 3.

First we recall a Jordan algebra of a symmetric bilinear form. Let Z be a commutative associative algebra, W a Z -module, and f a symmetric Z -bilinear form on W. Then

J = Z ⊕ W

is called a Jordan algebra of f if the multiplication is defined as

(z + w) · (z′ + w′) = zz′ + f(w, w′) + zw′ + z′w

for z, z′ ∈ Z and w, w′ ∈ W, which is a Jordan algebra over Z.

Next we need a concrete realization of a finite-dimensional split simple Lie algebra 𝔤 over F of type B_{l} (l ≥ 2). Let V be a (2l + 1)-dimensional space over F with basis v_{1}, …, v_{2l+1} . Let u be the symmetric bilinear form on V so that

u(v_{i}, v_{l+i}) = 1 for 1 ≤ i ≤ l, u(v_{2l+1}, v_{2l+1}) = 1 and u(v_{i}, v_{j}) = 0 otherwise.

We may identify 𝔤 as the Lie algebra of endomorphisms of V which are skew relative to u. We may also identify the Cartan subalgebra 𝔥 of 𝔤 with the Lie algebra of elements of 𝔤 which are diagonal relative to the basis v_{1}, …, v_{2l+1} . Let {e_{1}, …, e_{l}} be a basis for 𝔥 defined by

e_{i}(∑_{j=1}^{2l+1} a_{j}v_{j}) = a_{i}v_{i} − a_{l+i}v_{l+i}

for a_{j} ∈ F. Let {ϵ_{1}, …, ϵ_{l}} be the dual basis of {e_{1}, …, e_{l}} in 𝔥^{∗}. Then

V_{ϵ_{i}} = F v_{i}, V_{−ϵ_{i}} = F v_{l+i}, for i = 1, …, l, and V_{0} = F v_{2l+1}.

In this case,

(1) Δ_{sh} = {±ϵ_{i} | 1 ≤ i ≤ l} and Δ_{lg} = {±(ϵ_{i} ± ϵ_{j}) | 1 ≤ i ≠ j ≤ l}.

In particular, the weights of the module V are the elements of Δ_{sh} ∪ {0}. Let

J = F ⊕ V

be the Jordan algebra of u. Then we may identify

𝔤 = D_{V,V} := span{D_{v,v′} | v, v′ ∈ V },

where D_{v,v′} = [L_{v}, L_{v′}] and L is the left multiplication operator, by extending the action of elements of 𝔤 to J so that they kill F. We use the following recognition theorem by Benkart and Zelmanov [BZ]:

Theorem 6.1. Let L be a centreless B_{l} -graded Lie algebra for l ≥ 3 with grading pair (𝔤, 𝔥). Then there exists a Jordan algebra J = Z ⊕ W of a symmetric bilinear form f over a commutative associative algebra Z so that L = 𝔅(J), where

𝔅(J) = (𝔤 ⊗_{F} Z) ⊕ (V ⊗_{F} W) ⊕ D_{W,W} with 𝔤 ⊗ 1 = 𝔤 = D_{V,V} ,
D_{W,W} := span{D_{w,w′} | w, w′ ∈ W}

with D_{w,w′} = [L_{w}, L_{w′}] as above, and the Lie bracket is defined by

[x ⊗ z, x′ ⊗ z′] = [x, x′] ⊗ zz′,
[x ⊗ z, v ⊗ w] = xv ⊗ zw,
(2)  [x ⊗ z, D] = 0,
[v ⊗ w, v′ ⊗ w′] = D_{v,v′} ⊗ f(w, w′) + u(v, v′)D_{w,w′},
[D, v ⊗ w] = v ⊗ Dw

for x, x′ ∈ 𝔤, z, z′ ∈ Z, v, v′ ∈ V , w, w′ ∈ W and D ∈ D_{W,W}, containing D_{W,W} as a subalgebra.

Conversely, for any such Jordan algebra J , 𝔅(J) is a centreless B_{l} - graded Lie algebra for l ≥ 2.

Remark 6.2. If W = 0, i.e., J = Z is a commutative associative algebra, then 𝔅(J) = 𝔤 ⊗_{F} Z is called untwisted.

Remark 6.3. In [BZ] or [AG], D_{v,v′} = −[L_{v}, L_{v′}] and D_{w,w′} = −[L_{w}, L_{w′}] were chosen. Let 𝔅^{−}(J) be the Lie algebra constructed by a Jordan algebra J = Z ⊕ W of f as above, using their choice. Let J^{−} = Z ⊕ W be the Jordan algebra of −f. Then our Lie algebra 𝔅(J) is isomorphic to 𝔅^{−}(J^{−}) via the identity map. We will explain in Remark 6.5 that our choice is more natural.

Also, Seligman [Se] originally defined the Lie bracket on the same space

𝔅_{S}(J) = (𝔤 ⊗_{F} Z) ⊕ (V ⊗_{F} W) ⊕ D_{W,W}

as 𝔅(J), using the natural right action of 𝔤 on V and identifying

𝔤 = span{[R_{v}, R_{v′}] | v, v′ ∈ V },

where R is the right multiplication operator, as

[x ⊗ z, x′ ⊗ z′] = [x, x′] ⊗ zz′,
[v ⊗ w, x ⊗ z] = v.x ⊗ wz,
[x ⊗ z, D] = 0,
[v ⊗ w, v′ ⊗ w′] = [R_{v}, R_{v′}] ⊗ f(w, w′) + u(v, v′)[R_{w}, R_{w′}],
[v ⊗ w, D] = v ⊗ wD

for x, x′ ∈ 𝔤, z, z′ ∈ Z, v, v′ ∈ V , w, w′ ∈ W and

D ∈ D_{W,W} = span{[R_{w}, R_{w′}] | w, w′ ∈ W},

containing D_{W,W} as a subalgebra. One can see that the natural right action is equivalent to the action defined by

v.x := −xv

for x ∈ 𝔤 and v ∈ V (the right-hand side is the natural left action defined above). Then since v′′[R_{v}, R_{v′}] = −[L_{v}, L_{v′}]v′′ for v′′ ∈ V , we have v′′.([L_{v}, L_{v′}]) = −[L_{v}, L_{v′}]v′′ = v′′[R_{v}, R_{v′}]. Thus, the map from 𝔅(J) onto 𝔅_{S}(J) defined by

x ⊗ z + v ⊗ w + [L_{w′}, L_{w′′}] ↦ x ⊗ z + v ⊗ w + [R_{w′}, R_{w′′}]

is an isomorphism of Lie algebras.

A G-graded Jordan algebra is called division graded if all nonzero homogeneous elements are invertible. Now we are ready to show one of our main results.

Theorem 6.4. Let L be a centreless (B_{l}, G)-graded Lie algebra for l ≥ 3. Then there exists a G-graded Jordan algebra J = Z ⊕ W = ⊕_{g∈G} (Z_{g} ⊕ W_{g}) of a symmetric bilinear form f over a commutative associative algebra Z so that L = 𝔅(J).

Moreover, if L is division graded, then so is J . In this case supp J = S and Z is division L-graded with supp Z = L, where S and L are defined in the system R(L) = R(S, L)_{B_{l}}.

Conversely, for a root system R(S, L)_{B_{l}} for l ≥ 2 so that a division G-graded Jordan algebra J = ⊕_{g∈G} (Z_{g} ⊕ W_{g}) of a symmetric bilinear form satisfies supp J = S and supp Z = L, 𝔅(J) is a centreless division (B_{l}, G)- graded Lie algebra whose root system is R(S, L)_{B_{l}}.

Proof. Let

L = ⨁_{μ∈Δ∪{0}} ⨁_{g∈G} L_{μ}^{g}

be a centreless division (B_{l}, G)-graded Lie algebra for l ≥ 3 with grading pair (𝔤, 𝔥). Then, by Theorem 6.1, we have

(3)  L_{μ} = \{ \begin{matrix}𝔤_{μ} ⊗ Z & if μ ∈ Δ_{lg} \\ (𝔤_{μ} ⊗ Z) ⊕ (V_{μ} ⊗ W) & if μ ∈ Δ_{sh}. \end{matrix}

Let L = {g ∈ G | L_{μ}^{g} ≠ 0, μ ∈ Δ_{lg}}. For all μ ∈ Δ_{lg} and g ∈ G, we define Z_{μ}^{g} as

L_{μ}^{g} = 𝔤_{μ} ⊗ Z_{μ}^{g},

and so Z = ⊕_{g∈G} Z_{μ}^{g}, and in particular, Z_{μ}^{g} = 0 if g ∉ L. If μ, ν ∈ Δ_{lg} and μ − ν ∈ Δ, then

𝔤_{μ} ⊗ Z_{μ}^{g} = [𝔤_{ν} ⊗ Z_{ν}^{g}, 𝔤_{μ−ν} ⊗ 1] = 𝔤_{μ} ⊗ Z_{ν}^{g}.

Thus by the same argument in [AG, (5.11)], we get Z_{μ}^{g} = Z_{ν}^{g} for any μ, ν ∈ Δ_{lg}. So for g ∈ G we put

Z_{g} := Z_{μ}^{g} for any choice of μ ∈ Δ_{lg}.

Then

(4) Z = ⨁_{g∈G} Z_{g} = ⨁_{l∈L} Z_{l}, with L_{μ}^{l} = 𝔤_{μ} ⊗ Z_{l} for all μ ∈ Δ_{lg} and l ∈ L,

is an L-graded space. Moreover, there exist μ, ν, μ + ν ∈ Δ_{lg} from (1) (l ≥ 3), and so for l, k ∈ L, we have

[𝔤_{μ} ⊗ Z_{l}, 𝔤_{ν} ⊗ Z_{k}] = 𝔤_{μ+ν} ⊗ Z_{l}Z_{k} ⊂ L_{μ+ν}^{l+k} = 𝔤_{μ+ν} ⊗ Z_{l+k}.

Hence Z_{l}Z_{k} ⊂ Z_{l+k} Thus Z is a ⟨L⟩-graded algebra with 1 ∈ Z_{0} , where ⟨L⟩ is the subgroup of G generated by L. (L = supp Z is not necessarily a subgroup here.)

Let μ ∈ Δ_{sh} and g ∈ G. We define Z_{μ}^{g} and W_{μ}^{g} as
𝔤_{μ} ⊗ Z_{μ}^{g} = (𝔤_{μ} ⊗ Z) ∩ L_{μ}^{g} and 𝔤_{μ} ⊗ W_{μ}^{g} = (V_{μ} ⊗ W) ∩ L_{μ}^{g}.

We claim that

L_{μ}^{g} = (𝔤_{μ} ⊗ Z_{μ}^{g}) ⊕ (V_{μ} ⊗ W_{μ}^{g}) and Z_{μ}^{g} = Z_{g}.

Let x ∈ L_{μ}^{g}. Then by (3), x = t + v for some t ∈ 𝔤_{μ} ⊗ Z and v ∈ V_{μ} ⊗ W. So we need to show that t, v ∈ L_{μ}^{g}. By (4), we have t = ∑_{l∈L} e_{μ} ⊗ z_{l} for some 0 ≠ e_{μ} ∈ 𝔤_{μ} and z_{l} ∈ Z_{l}. From (1), one can see that there exists ν ∈ Δ such that μ + ν ∈ Δ_{lg}. Let 0 ≠ e_{ν} ∈ 𝔤_{ν} = 𝔤_{ν} ⊗ 1 ⊂ L_{ν}^{0} . Then

(5)  [e_{ν}, x] ∈ L_{μ+ν}^{g} = \{ \begin{matrix}0 & if g ∉ L \\ 𝔤_{μ+ν} ⊗ Z_{g} & if g ∈ L. \end{matrix}

Also, [e_{ν}, v] ∈ [𝔤_{ν} ⊗ Z_{0}, V_{μ} ⊗ W] ⊂ 𝔤_{ν}.V_{μ} ⊗ W, but 𝔤_{ν}.V_{μ} = 0 since μ + ν ∈ Δ_{lg}. Hence [e_{ν}, v] = 0. Thus we obtain [e_{ν}, t] = [e_{ν}, x].

If g ∉ L, then ∑_{l∈L} [e_{ν}, e_{μ} ⊗ z_{l}] = 0 by (5), and so [e_{ν}, e_{μ} ⊗ z_{l}] = 0 for all l ∈ L. Since [𝔤_{ν}, 𝔤_{μ}] ≠ 0, we get z_{l} = 0 for all l ∈ L, i.e., t = 0. Therefore, x = v ∈ L_{μ}^{g}.

If g ∈ L, then [e_{ν}, t] ∈ L_{μ+ν}^{g}, and so t = e_{μ} ⊗ z_{g} ∈ 𝔤_{μ} ⊗ Z_{g}. ¿From (1), there exists ξ ∈ Δ_{lg} such that μ − ξ ∈ Δ. So by (4), we have

𝔤_{μ} ⊗ Z_{g} = [𝔤_{ξ} ⊗ Z_{g}, 𝔤_{μ−ξ} ⊗ 1] ⊂ [L_{ξ}^{g}, L_{μ−ξ}^{0}] ⊂ L_{μ}^{g}.

Therefore, t ∈ L_{μ}^{g} and v = x − t ∈ L_{μ}^{g}. Finally, since μ + ν ∈ Δ_{lg}, we get

[𝔤_{μ} ⊗ Z_{μ}^{g}, 𝔤_{ν} ⊗ 1] = 𝔤_{μ+ν} ⊗ Z_{μ}^{g} ⊂ 𝔤_{μ+ν} ⊗ Z_{g}.

Hence Z_{μ}^{g} ⊂ Z^{g}. Also,

[𝔤_{μ+ν} ⊗ Z^{g}, 𝔤_{−ν} ⊗ 1] = 𝔤_{μ} ⊗ Z^{g} ⊂ 𝔤_{μ} ⊗ Z_{μ}^{g},

and hence Z^{g} ⊂ Z_{μ}^{g}. Thus our claim is settled.

Now, W = ⊕_{g∈G} W_{μ}^{g}, and V_{μ} ⊗ W_{μ}^{g} = L_{μ}^{g} if g ∉ L since Z_{g} = 0. If μ, ν ∈ Δ_{sh} and μ − ν ∈ Δ, then

V_{μ} ⊗ W_{μ}^{g} = [V_{ν} ⊗ W_{ν}^{g}, 𝔤_{μ−ν} ⊗ 1] = V_{μ} ⊗ W_{ν}^{g} for all g ∈ G.

Therefore, W_{μ}^{g} = W_{ν}^{g}. Thus by the same argument in [AG, (5.11)], we get W_{μ}^{g} = W_{ν}^{g} for any μ, ν ∈ Δ_{sh} and all g ∈ G. So for g ∈ G we put

W_{g} := W_{μ}^{g} for any choice of μ ∈ Δ_{sh}.

Consequently,

W = ⨁_{g∈G} W_{g}, with

L_{μ}^{g} = V_{μ} ⊗ W_{g} for all μ ∈ Δ_{sh} and g ∉ L,

and

(6)  L_{μ}^{g} = (𝔤_{μ} ⊗ Z_{g}) ⊕ (V_{μ} ⊗ W_{g}) for all μ ∈ Δ_{sh} and g ∈ L.

Our next claim is that

J = ⨁_{g∈G} J_{g}, where J_{g} := \{ \begin{matrix}Z_{g} ⊕ W_{g} & if g ∈ L \\ W_{g} & if g ∉ L, \end{matrix}

satisfies for all g, k ∈ G,

Z_{g}W_{k} ⊂ W_{g+k}, W_{g}W_{k} ⊂ Z_{g+k} and W_{g}W_{k} = 0 if g + k ∉ L.

In fact, there exist μ ∈ Δ_{lg} and ν ∈ Δ_{sh} such that μ + ν ∈ Δ_{sh} by (1). Since [𝔤_{μ}, V_{ν}] = V_{μ+ν}, we have

[𝔤_{μ} ⊗ Z_{g}, V_{ν} ⊗ W_{k}] = V_{μ+ν} ⊗ Z_{g}W_{k} ⊂ L_{μ+ν}^{g+k} = (𝔤_{μ+ν} ⊗ Z_{g+k}) ⊕ (V_{μ+ν} ⊗ W_{g+k}).

Hence Z_{g}W_{k} ⊂ W_{g+k}. Next, one can choose μ, ν ∈ Δ_{sh} such that μ + ν ∈ Δ_{lg}. Let 0 ≠ v_{1} ∈ V_{μ} and 0 ≠ v_{2} ∈ V_{ν}. Then we have

[v_{1} ⊗ W_{g}, v_{2} ⊗ W_{k}] = D_{v_{1},v_{2}} ⊗ f(W_{g}, W_{k}) ⊂ L_{μ+ν}^{g+k} = 𝔤_{μ+ν} ⊗ Z_{g+k}

since u(v_{1}, v_{2}) = 0. Since D_{v_{1},v_{2}} ≠ 0 (e.g. D_{v_{1},v_{2}}V_{−μ} ≠ 0), we get f(W_{g}, W_{k}) = W_{g}W_{k} ⊂ Z_{g+k}. The last equality is clear since Z_{g+k} = 0 for g + k ∉ L. Hence our claim is settled.

Let S := supp J = supp L_{μ} for μ ∈ Δ_{sh}. By (6), we have L ⊂ S, and so S + S ⊃ supp L, which generates G. Hence S generates G, and so J is a G-graded Jordan algebra.

Suppose that L is division graded. Let 0 ≠ z + w ∈ Z_{g} ⊕ W_{g} = J_{g} for z ∈ Z_{g}, w ∈ W_{g} and g ∈ S (z = 0 if g ∉ L). To show the invertibility of z + w, we identify 𝔤 with the matrices relative to the basis {v_{1}, …, v_{2l+1}}. Let e := e_{i,2l+1} − e_{2l+1,l+i} ∈ 𝔤_{ϵ_{i}} (see (1)) for any fixed 1 ≤ i ≤ l, where e_{ij} is the matrix unit, and let e′ := 2e_{2l+1,i}−2e_{l+i,2l+1} ∈ 𝔤_{−ϵ_{i}}. Then [e, e′] = 2e_{ii}−2e_{l+i,l+i} = ϵ_{i}^{∨}. Also, let v := v_{i} and v′ := 2v_{l+i}. Then u(v, v′) = 2, D_{v,v′} = [L_{v}, L_{v′}] = ϵ_{i}^{∨} and ev′ + e′v = −2v_{2l+1} + 2v_{2l+1} = 0.

Now, e ⊗ z + v ⊗ w ∈ L_{ϵ_{i}}^{g}, and by the division property of L, there exists y ∈ L_{−ϵ_{i}}^{−g} such that [e ⊗ z + v ⊗ w, y] = ϵ_{i}^{∨}. Since L_{−ϵ_{i}}^{−g} = (𝔤_{−ϵ_{i}} ⊗ Z_{−g}) ⊕ (V_{−ϵ_{i}} ⊗ W_{−g}), we have y = e′ ⊗ z′ + v′ ⊗ w′ for some z′ ∈ Z_{−g} and w′ ∈ W_{−g}. So

ϵ_{i}^{∨} ⊗ 1 = ϵ_{i}^{∨} = [e ⊗ z + v ⊗ w, e′ ⊗ z′ + v′ ⊗ w′]

= [e, e′] ⊗ zz′ + D_{v,v′} ⊗ f(w, w′) + e.v′ ⊗ zw′ − e′.v ⊗ z′w + u(v, v′)D_{w,w′} = ϵ_{i}^{∨} ⊗ (zz′ + ww′) + e.v′ ⊗ (zw′ + z′w) + 2D_{w,w′}.

Hence zz′ + ww′ = 1 and zw′ + z′w = 0. So (z + w)(z′ + w′) = 1. Also, we have D_{z+w,z′+w′} = D_{w,w′} = 0. Therefore, z + w is invertible. Thus J is a division G-graded Jordan algebra, and in particular L = supp L_{μ} for μ ∈ Δ_{lg} is a subgroup of G, and Z is a division L-graded commutative associative algebra. Also, the root system of L is determined by these S and L.

We show the last statement. For g ∈ G, let L_{μ}^{g} := (𝔤_{μ}⊗Z_{g})⊕(V_{μ}⊗W_{g}) if μ ∈ Δ_{sh}, and L_{μ}^{g} := 𝔤_{μ} ⊗Z_{g} if μ ∈ Δ_{lg}. Then a centreless B_{l} -graded Lie algebra 𝔅(J) admits the compatible G-grading, say 𝔅(J) = ⊕_{μ∈Δ∪{0}} ⊕_{g∈G} L_{μ}^{g}, where L_{0}^{g} = ∑_{μ∈Δ} ∑_{g=p+q} [L_{μ}^{p}, L_{−μ}^{q}], is a centreless (B_{l}, G)-graded Lie algebra.

For μ ∈ Δ_{lg} and g ∈ L, let e ∈ 𝔤_{μ} and e′ ∈ 𝔤_{−μ} such that [e, e′] = μ^{∨} . Then for 0 ≠ x ∈ L_{μ}^{g}, there exists 0 ≠ z ∈ Z_{g} such that x = e ⊗ z. Taking y = e′ ⊗ z^{−1} ∈ L_{−μ}^{−g}, we get [x, y] = μ^{∨}.

For ϵ_{i} ∈ Δ_{sh} and g ∈ S, let e ∈ 𝔤_{ϵ_{i}}, e′ ∈ 𝔤_{−ϵ_{i}}, v = v_{i} and v′ = 2v_{l+i} as before so that [e, e′] = ϵ_{i}^{∨} = D_{v,v′} and ev′ + e′v = 0. Then for 0 ≠ x ∈ L_{μ}^{g}, there exist z ∈ Z_{g} and w ∈ W_{g} such that x = e ⊗ z + v ⊗ w. Taking y = e′ ⊗ z′ + v′ ⊗ w′ ∈ L_{−μ}^{−g}, where z′ and w′ are defined as (z + w)^{−1} = z′ + w′, we get [x, y] = ϵ_{i}^{∨}. Hence 𝔅(J) is division graded.

Remark 6.5. In a similar way, one can show that

𝔅^{−}(J) is division graded ⟺ J^{−} is division graded,

but this does not imply that J is division graded. For example, J^{−} = ℝ ⊕ ℝw with w^{2} = −1 is a division algebra over ℝ, which is ℂ, but J = ℝ ⊕ ℝw with w^{2} = 1 is not even simple. Thus if one considers the division property of the coordinate algebras J for division B_{l} -graded Lie algebras, our 𝔅(J) seems more natural.

However, for the case of Lie G-tori and Clifford G-tori defined in the next section, we have

𝔅^{−}(J) is a Lie G-torus ⟺ J is a Clifford G-torus

⟺ 𝔅(J) is a Lie G-torus since there is no homogeneous element z + w ∈ J with z ≠ 0 and w ≠ 0. So the choice of either 𝔅^{−}(J) or 𝔅(J) does not effect the division property of J .

Now we show the existence of a division G-graded Jordan algebra in Theorem 6.4 for any root system R(S, L)_{B_{l}} extended by G for l ≥ 3. By Theorem 3.4, S is a full reflection space of G and L is a subgroup of G such that

2S ⊂ L and S + L ⊂ S

(see Example 3.5(1)). We choose a collection of coset representatives {u_{t}}_{t∈T} in G/L so that

G = ⨆_{t∈T} (u_{t} + L).

By the second inclusion, S is a union of cosets of G/L containing L, say

S = ⨆_{t∈I⊔{0}⊂T} (u_{t} + L) with u_{0} = 0.

Also, by the first inclusion, we have

l_{i} := 2u_{i} ∈ L for all i ∈ I ⊔ {0}.

Let

Z = ⨁_{l∈L} Z_{l}

be a division L-graded commutative associative algebra over F, i.e., a commutative associative L-graded algebra so that all nonzero homogeneous elements are invertible. So in particular supp Z = L. Let K = Z_{0} which is a field extension of F. For each i ∈ I ⊔ {0},

(i) let (W_{i}, φ_{i}, z_{i}) be a triple consisting of a vector space W_{i} over K, a symmetric anisotropic bilinear form φ_{i} of W_{i}, i.e., φ_{i}(w, w) ≠ 0 for all 0 ≠ w ∈ W_{i}, and 0 ≠ z_{i} ∈ Z_{l_{i}} with z_{0} = 1.

(ii) assume that W_{i} ≠ 0 if i ≠ 0, (iii) assume that φ_{0} does not represent 1 (so K ⊕ W_{0} is a Jordan division

algebra).

We denote the family of triples by W_{I}:

W_{I} = {(W_{i}, φ_{i}, z_{i})}_{i∈I⊔{0}}.

For each i ∈ I ⊔ {0}, we extend φ_{i} to Z ⊗_{K} W_{i} as a Z -bilinear map, denoted f_{i}, as follows:

f_{i}(z ⊗ w, z′ ⊗ w′) = zz′φ_{i}(w, w′)z_{i}

for z, z′ ∈ Z, w, w′ ∈ W_{i}. Obviously f_{i} is symmetric and the image is Zz_{i} for each i ∈ I ⊔ {0}. Let

f := ⨁_{i∈I⊔{0}} f_{i}

be the symmetric bilinear form over Z on

W := ⨁_{i∈I⊔{0}} (Z ⊗_{K} W_{i}).

We claim that the Jordan algebra

J := Z ⊕ W

of f is a division G-graded Jordan algebra over F for some G-grading. First we put

W_{t} := 0 and φ_{t} = 0 for t ∈ T \ (I ⊔ {0}).

For g ∈ G, there exist unique u_{t} ∈ G and l ∈ L such that g = u_{t} + l. We set

J_{g} = J_{u_{t}+l} := \{ \begin{matrix}Z_{l} ⊗_{K} (K ⊕ W_{0}) & if t = 0, \\ Z_{l} ⊗_{K} W_{t} & otherwise. \end{matrix}

Since Z ⊗_{K} W_{t} = ⊕_{l∈L} (Z_{l} ⊗_{K} W_{t}) and

Z ⊕ (Z ⊗_{K} W_{0}) = Z ⊗_{K} (K ⊕ W_{0}) = ⨁_{l∈L} (Z_{l} ⊗_{K} (K ⊕ W_{0}))

(identifying Z with Z ⊗_{K} K), we get

J = Z ⊕ ( ⨁_{i∈I⊔{0}} (Z ⊗_{K} W_{i})) = (Z ⊕ (Z ⊗_{K} W_{0})) ⊕ ( ⨁_{i∈I} Z ⊗_{K} W_{i})

= ( ⨁_{l∈L} (Z_{l} ⊗_{K} (K ⊕ W_{0}))) ⊕ ( ⨁_{(i,l)∈I×L} (Z_{l} ⊗_{K} W_{i})) = ⨁_{g∈G} J_{g}

as K -vector spaces. Let g = l + u_{t}, g′ = l′ + u_{t′} ∈ G (l, l′ ∈ L), z ∈ Z_{l} and z′ ∈ Z_{l′}. If t = t′ = 0, then for a, b ∈ K and x, y ∈ W_{0} ,

(z ⊗ (a + x))(z′ ⊗ (b + y)) = zz′ ⊗ (ab + φ_{0}(x, y) + bx + ay) ∈ J_{g+g′}.

If t = 0 and t′ ≠ 0, then for a ∈ K, x ∈ W_{0} and y ∈ W_{t′},

(z ⊗ (a + x))(z′ ⊗ y) = azz′ ⊗ y ∈ J_{g+g′}

since W_{0}W_{t′} = 0. Finally, if t ≠ 0 and t′ ≠ 0, then for x ∈ W_{t} and y ∈ W_{t′},

(z ⊗ x)(z′ ⊗ y) = \{ \begin{matrix}φ_{t}(x, y)zz′z_{t} ⊗ 1 & if t = t′ ∈ I \\ 0 & otherwise (since W_{t}W_{t′} = 0), \end{matrix}

which is in J_{g+g′}. Therefore, we obtain J_{g}J_{g′} ⊂ J_{g+g′} for all g, g′ ∈ G. Since supp J = S which generates G, J is a G-graded algebra over K. Any 0 ≠ az ⊗ u ∈ J_{l} for a ∈ K, l ∈ L, z ∈ Z_{l} and u ∈ K ⊕ W_{0} has the inverse a^{−1}z^{−1}⊗u^{−1} since K⊕W_{0} is a Jordan division algebra. (Note that dim_{K} Z_{l} = 1 for all l ∈ L.) Also, for any 0 ≠ az ⊗ x ∈ J_{l+u_{i}}, i ∈ I, a ∈ K, l ∈ L, z ∈ Z_{l} and x ∈ W_{i}, we have (az ⊗ x)^{2} = a^{2}φ_{i}(x, x)z^{2}z_{i} ⊗ 1, which is invertible since φ_{i} is anisotropic. Hence az ⊗ x is invertible. So we have shown that

J = J (S, L, K, Z, W_{I})

is a division G-graded Jordan algebra over K, and hence over F. Thus:

Theorem 6.6. For any root system R(S, L)_{B_{l}} extended by G for l ≥ 3, there exists a division G-graded Jordan algebra J = J (S, L, K, Z, W_{I}) of a symmetric bilinear form over an L-graded commutative associative algebra Z with supp J = S and supp Z = L.

Conversely, for any division G-graded Jordan algebra J = Z ⊕ W = ⊕_{g∈G} (Z_{g} ⊕ W_{g}) of a symmetric bilinear form over a commutative associative algebra Z, the pair of S := supp J and L := supp Z satisfies the conditions of a root system R(S, L)_{B_{l}} extended by G for l ≥ 2, and J is graded isomorphic to some J (S, L, K, Z, W_{I}).

Proof. We only need to show the second statement. Note first that L = supp Z is a subgroup of G since Z is associative, and that S = supp J is a full reflection space of G (see [NY, 2.3(b)]). Also, for g ∈ S and 0 ≠ w ∈ W_{g}, we have 0 ≠ w^{2} ∈ W_{g}W_{g} = f(W_{g}, W_{g}) ∈ Z_{2g} and so 2g ∈ L. Hence 2S ⊂ L, and so L + 2S ⊂ L. Moreover, since W is a graded Z -module, we have L + S ⊂ S. Thus the pair of S and L satisfies the conditions of a root system extended by G of type B_{l} (l ≥ 2).

Let S′ := supp W, which is a union of cosets of G/L. So, letting u_{0} = 0,

S′ = ⨆_{i∈Ĩ} (u_{i} + L)

for some index set Ĩ which may contain 0. Let K := Z_{0} which is a field extension of F, I := Ĩ\ {0}, W_{i} = W_{u_{i}} and W_{0} = 0 if 0 ∉ Ĩ. Then

W = ⨁_{s∈S′} W_{s} = ⨁_{i∈Ĩ} ZW_{u_{i}} = ⨁_{i∈I⊔{0}} ZW_{i}.

Let f be the symmetric bilinear form of the Jordan algebra J , and let

f_{i} = f |_{W_{i}×W_{i}}

for all i ∈ I ⊔ {0}. For 0 ≠ w_{i} ∈ W_{u_{i}}, we have 0 ≠ w_{i}^{2} ∈ Z_{2u_{i}}, and so l_{i} := 2u_{i} ∈ L for all i ∈ I. We fix 0 ≠ z_{i} ∈ Z_{l_{i}} for all i ∈ I. Also, let l_{0} = 0 and z_{0} = 1. Then, since

f_{i}(W_{i}, W_{i}) ⊂ Z_{l_{i}} = Kz_{i},

one can define symmetric bilinear forms φ_{i} on W_{i} over K as

f_{i}(w, w′) = φ_{i}(w, w′)z_{i}

for w, w′ ∈ W_{i} and all i ∈ I ⊔ {0}. Thus we get a family of triples of K - vector spaces W_{i}, symmetric anisotropic bilinear forms φ_{i} and nonzero z_{i} ∈ Z_{l_{i}} indexed by i ∈ I ⊔ {0}, satisfying (i), (ii) and (iii) above.

Let u_{i} ≠ u_{j}. If u_{i} + u_{j} + L = L, then u_{i} + u_{j} − 2u_{j} + L = L, and so u_{i} + L = u_{j} + L, which contradicts the fact that {u_{i}}_{i∈Ĩ} is a collection of coset representatives of S′ in G/L. Hence u_{i} + u_{j} + L ≠ L. So we have

W_{i}W_{j} ⊂ f(W_{i}, W_{j}) ⊂ J_{u_{i}+u_{j}} ∩ Z = 0

for all i ∈ I ⊔ {0}. Hence we have

f = ⨁_{i∈I⊔{0}} f_{i}.

Finally, ZW_{i} is a free Z -module with rank_{Z} ZW_{i} = dim_{K} W_{i}, and so one can easily show that there is a natural Z -module isomorphism

ZW_{i} ≅ Z ⊗_{K} W_{i}

via zx ↔ z ⊗ x for z ∈ Z and x ∈ W_{i}. Identifying them, we get J = J (S, L, K, W_{I}).

Combining Theorem 6.4 with Theorem 6.6, we get the complete classification of division (B_{l}, G)-graded Lie algebra for l ≥ 3, up to central extensions.

Theorem 6.7. Let L be a centreless division (B_{l}, G)-graded Lie algebra for l ≥ 3 with root system R(S, L)_{B_{l}}. Then L = 𝔅(J) for some J = J (S, L, K, Z, W_{I}). Conversely, for l ≥ 2, 𝔅(J) is a centreless division (B_{l}, G)- graded Lie algebra for any J = J (S, L, K, Z, W_{I}).

7. Lie G-tori of type B

We now specialize Theorem 6.7 to the case of Lie G-tori. Also, we show that there is a one-to-one correspondence between centreless Lie G-tori of type B_{l} and root systems extended by G of type B_{l} for l ≥ 3 if the base field is algebraically closed and G is free.

A division G-graded Jordan algebra J = Z ⊕W = ⊕_{g∈G} (Z_{g} ⊕W_{g}) of a symmetric bilinear form is called a Clifford G-torus over F if dim_{F} (Z_{g}⊕W_{g}) ≤ 1 for all g ∈ G (1-dimensionality).

For any root system R(S, L)_{B_{l}} extended by G for l ≥ 3, we construct such a Clifford G-torus J = Z ⊕ W with supp J = S and supp Z = L. Although the construction is just a special case of division G-graded Jordan algebras in §6, they can be described in a simpler way. As in §6, we choose a collection of coset representatives {u_{t}}_{t∈T} in G/L so that

G = ⨆_{t∈T} (u_{t} + L),

S = ⨆_{t∈I⊔{0}⊂T} (u_{t} + L) with u_{0} = 0

and

l_{i} := 2u_{i} ∈ L for all i ∈ I ⊔ {0}.

Let

Z = ⨁_{l∈L} Z_{l}

be a commutative associative L-torus over F, i.e., a division L-graded commutative associative algebra with dim_{F} Z_{l} ≤ 1 for all l ∈ L. So supp Z = L and dim_{F} Z_{l} = 1 for all l ∈ L. Let W be a free Z -module with basis {w_{i}}_{i∈I} and choose 0 ≠ z_{i} ∈ Z_{l_{i}}. Define a symmetric bilinear form f on W over Z as

f(w_{i}, w_{j}) = δ_{i,j}z_{i}.

We claim that the Jordan algebra J := Z ⊕ W over Z determined by the symmetric bilinear form f is a Clifford G-torus. In fact we put

w_{0} := 1 and w_{t} := 0 for t ∈ T \ (I ⊔ {0}).

For g ∈ G, there exist unique u_{t} ∈ G and l ∈ L such that g = u_{t} + l, and so we set

J_{g} = J_{u_{t}+l} := Z_{l}w_{t}.

Then

J = Z ⊕ W = ⨁_{(t,l)∈T ×L} Z_{l}w_{t} = ⨁_{g∈G} J_{g}

as F -vector spaces. Let g = l + u_{t}, g′ = l′ + u_{t′} ∈ G (l, l′ ∈ L), z ∈ Z_{l} and z′ ∈ Z_{l′}. Then

(zw_{t})(z′w_{t′}) = \{ \begin{matrix}δ_{t,t′}zz′z_{t} & if t, t′ ∈ I \\ zz′w_{t}w_{t′} & otherwise. \end{matrix}

Since z_{t} ∈ Z_{l_{t}} = Z_{2t}, we obtain J_{g}J_{g′} ⊂ J_{g+g′} for all g, g′ ∈ G. Since supp J = S which generates G, J is a G-graded algebra. For any 0 ≠ zw_{t} ∈ J_{l+u_{t}}, we have (zw_{t})^{2} = z^{2}z_{t}, which is invertible. Hence zw_{t} is invertible. Thus we have shown that

J = J (S, L, Z, {z_{i}}_{i∈I})

is a Clifford G-torus.

Conversely, let J = Z ⊕ W = ⊕_{g∈G} (Z_{g} ⊕ W_{g}) be a Clifford G-torus. Let L = supp Z, S = supp J and S′ = supp W. Then, by the same reason as in §6, the pair of S and L satisfies the conditions of a root system extended by G of type B_{l} for l ≥ 3 (so l ≥ 2), and we have

S = L ⊔ S′ = L ⊔ ( ⨆_{i∈I} (u_{i} + L))

for some 0 ≠ u_{i} ∈ S′ indexed by some set I. Now Z = ⊕_{l∈L} Z_{l}, which is a commutative associative L-torus. Let 0 ≠ w_{i} ∈ W_{u_{i}} for all i ∈ I. Then {w_{i}}_{i∈I} is a Z -basis of W so that

W = ⨁_{s∈S′} W_{s} = ⨁_{i∈I} ZW_{u_{i}} = ⨁_{i∈I} Zw_{i}.

Let l_{i} := 2u_{i} ∈ L and z_{i} := w_{i}^{2} ∈ Z_{l_{i}} for all i ∈ I. Then by the same argument as in §6, we get f(w_{i}, w_{j}) = δ_{i,j}z_{i} for all i, j ∈ I, and so J = J (S, L, Z, {z_{i}}_{i∈I}) constructed above. Thus:

Theorem 7.1. For any root system R(S, L)_{B_{l}} extended by G for l ≥ 3, there exists a Clifford G-torus J = J (S, L, Z, {z_{i}}_{i∈I}).

Conversely, for any Clifford G-torus J = Z ⊕ W, the pair of S := supp J and L := supp Z satisfies the conditions of a root system R(S, L)_{B_{l}} extended by G for l ≥ 2, and J is graded isomorphic to some J (S, L, Z, {z_{i}}_{i∈I}).

Since 1-dimensionality of (B_{l}, G)-graded Lie algebras reflects 1-dimen-sionality of G-graded Jordan algebras, we obtain the following as a corollary of Theorem 6.6:

Theorem 7.2. Let L be a centreless Lie G-torus of type B_{l} for l ≥ 3 with root system R(S, L)_{B_{l}}. Then L = 𝔅(J) for some J = J (S, L, Z, {z_{i}}_{i∈I}). Conversely, for l ≥ 2, 𝔅(J) is a centreless Lie G-torus for any J = J (S, L, Z, {z_{i}}_{i∈I}).

Example 7.3. (1) Let F = ℚ, and let G, S and L be in Example 3.5(1). Let Z = ℚ(\sqrt{2}, \sqrt{3}), and W a vector space over Z with basis \{w_{1}, w_{2}\}. Define a Z -bilinear form f on W by

f(w_{i}, w_{j}) = \{ \begin{matrix}\sqrt{2} & if i = j = 1 \\ \sqrt{3} & if i = j = 2 \\ 0 & otherwise. \end{matrix}

Consider the Jordan algebra J = Z ⊕ W of f. Let ε_{1} = (1, 0) and ε_{2} = (0, 1). Then

J_{ε_{1}} := ℚw_{1}, J_{ε_{2}} := ℚw_{2} and J_{ε_{1}+ε_{2}} := 0

determine a unique G-grading of J so that Z is L-graded. Thus J is a Clifford G-torus with S = supp J and L = supp Z. The Lie G-torus 𝔅(J) is a finitedimensional simple Lie algebra over ℚ since J is a finite-dimensional Jordan division algebra.

(2) Any centreless Lie ℚ-torus L over F of type B_{l} for l ≥ 3 is untwisted, i.e., L = 𝔅(Z) = 𝔤 ⊗_{F} Z, where Z is a commutative associative ℚ-torus over F, by the reason in Example 3.5(2).

(3) If G = ℤ, then for l ≥ 2, we have R = R(ℤ, ℤ)_{B_{l}} or R(ℤ, 2ℤ)_{B_{l}} (see Example 3.5(3)). For the first case, the Clifford ℤ-torus J = Z can be identified with the group algebra F [ℤ] or equivalently the algebra of Laurent polynomials F [t, t^{−1}], and the centreless Lie 1-torus 𝔅(Z) is an untwisted loop algebra 𝔤 ⊗_{F} F [t, t^{−1}] of type B_{l}^{(1)} .

For the second case, the Clifford ℤ-torus J = J (ℤ, 2ℤ, Z, {z}) is written as J = Z ⊕ Zw, where 0 ≠ w ∈ J_{1} and w^{2} = z ∈ Z_{2} . Also, J is associative since J has rank 2 over Z. We identify Z with the group algebra F [2ℤ], say Z = F [2ℤ] = ⊕_{m∈ℤ} F \overline{2m}. Since 2ℤ is a free group generated by 2, we can put z = \overline{2}. Hence J is uniquely determined by the root system R(ℤ, 2ℤ)_{B_{l}}, and J = F [ℤ] = ⊕_{m∈ℤ} F \overline{m} is also a group algebra, where \overline{1} = w. Thus J can be again identified with F [t, t^{−1}], and the centreless Lie 1-torus 𝔅(J) is a twisted loop algebra (𝔤 ⊗_{F} F [t^{±2}]) ⊕ (V ⊗_{F} tF [t^{±2}]) of type D_{l+1}^{(2)} .

In the construction of a Clifford torus J = J (S, L, Z, {z_{i}}_{i∈I}) above, the commutative associative L-torus Z over F is a group algebra

Z = F [L] = ⨁_{l∈L} F \overline{l}

if L is free. Also, if any element of F has a square root in F, then one can make z_{i} = \overline{l}_{i} for all i ∈ I by switching w_{i} to

\frac{1}{\sqrt{a_{i}}} w_{i} where a_{i} = z_{i}\overline{l}_{i}^{−1} ∈ F .

Thus J is uniquely determined by S and L, say J = J (S, L). If G is free, so is L. Hence we have:

Theorem 7.4. If G is free and if any element of F has a square root in F, e.g. F is an algebraically closed field, then there is a one-to-one correspondence between the centreless Lie G-tori of type B_{l} and the root systems extended by G of type B_{l} for l ≥ 3.

Remark 7.5. Division (Δ, G)-graded Lie algebras for the other types, Δ = A_{1} , C_{l}, F_{4} , G_{2} or BC_{l} (see Example 4.3(3)) are not classified yet, even in the case G = ℤ^{n}.

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Yoji Yoshii Department of Mathematics and Statistics University of Saskatchewan 106 Wiggins Rd, Saskatoon, SK., S7N 5E6 Canada yoshii@math.usask.ca

Received December 02, 2002 and in final form April 7, 2003