Lie Algebras with Prescribed sl3 Decomposition

In this work, we consider Lie algebras L containing a subalgebra isomorphic to sl3 and such that L decomposes as a module for that sl3 subalgebra into copies of the adjoint module, the natural 3-dimensional module and its dual, and the trivial one-dimensional module. We determine the multiplication in L and establish connections with structurable algebras by exploiting symmetry relative to the symmetric group S4.


Introduction
The Lie algebra gl n+k of (n + k) × (n + k) matrices over a field F of characteristic 0 under the commutator product [x, y] = xy−yx, when viewed as a module for the copy of gl n in its northwest corner, decomposes into k copies of the natural n-dimensional gl n -module V = F n , k copies of the dual module V * = Hom(V, F), a copy of the Lie algebra gl k in its southeast corner, and the copy of gl n : gl n+k = gl n ⊕ V ⊕k ⊕ (V * ) ⊕k ⊕ gl k .
As a result, we may write where B = C = F k . This second expression reflects the decomposition of gl n+k as a module for gl n ⊕ gl k . When restricted to sl n , the gl n -modules V and V * remain irreducible, while gl n decomposes into a copy of the adjoint module and a trivial sl n -module spanned by the identity matrix: gl n = sl n ⊕ FI n . Thus, we have the sl n decomposition of gl n+k , where gl k ⊕FI n is the sum of the trivial sl n -modules in gl n+k . Decompositions such as (1.1) also arise in the study of direct limits of simple Lie algebras and give insight into their structure. Indeed, suppose we have a chain of homomorphisms, where g (i) = sl(V (i) ). Assume that sl(V) is a fixed term in the chain for some V = V (j) , and dim V = n. We identify sl(V) with sl n by choosing a basis for V and assume that V (i) = V ⊕k i ⊕ F ⊕z i as a module for sl n for i ≥ j. Then the limit Lie algebra L = lim −→ g (i) admits a decomposition relative to sl n , where s is the sum of the trivial sl n -modules (see [3,Sec. 5]). Bahturin and Benkart in [3,Sec. 4] study Lie algebras having such a decomposition and describe the multiplication in L and the possibilities for A, B, C, s when dim V ≥ 4. When dim V = 2, then V * is isomorphic to V as a module for sl 2 = sl(V). In this case, a Lie algebra having a decomposition, L = (sl 2 ⊗ A) ⊕ (V ⊗ B) ⊕ s is graded by the root system BC 1 , and its structure has been described in [4].
In this paper, we investigate the missing case when dim V = 3, which presents very distinctive features. For direct limit Lie algebras of the type considered above, we could, of course, choose a larger space V (j) having dim V (j) ≥ 4 and apply the results of [3]. However, there are many examples of Lie algebras which admit very interesting decompositions as in (1.3) for n = 3. The exceptional simple Lie algebras provide examples of this phenomenon.
Example 1.1. Each exceptional simple Lie algebra L over an algebraically closed field of characteristic 0 has an automorphism ψ of order 3 that corresponds to a certain node in the Dynkin diagram of the associated affine Lie algebra. The node is marked with a "3" in [10, TABLE Aff 1]. Removing that node gives the Dynkin diagram of a finite-dimensional semisimple Lie algebra sl 3 ⊕ s, which is the subalgebra of fixed points of the automorphism ψ. The Lie algebra s is the centralizer of sl 3 in L; hence, is the sum of trivial sl 3 -modules under the adjoint action. In this table we display the Lie algebra s: For the Lie algebra G 2 we have the well-known decomposition (see [9,Prop. 3]) G 2 ∼ = sl 3 ⊕ V ⊕ V * relative to sl 3 (where sl 3 corresponds to the long roots of G 2 and V = F 3 ). This decomposition can be viewed as the decomposition into eigenspaces relative to ψ, where V corresponds to the eigenvalue ω (a primitive cube root of 1); V * to the eigenvalue ω 2 ; and sl 3 to the eigenvalue 1.
For the other exceptional Lie algebras, where B and C can be identified with H 3 (C), the algebra of 3 × 3 hermitian matrices over a composition algebra C under the product h where α, β, γ ∈ F, a, b, c ∈ C, and "¯" is the standard involution in C. The composition algebra C is displayed below, Here V ⊗ B is the ω-eigenspace of ψ, V * ⊗ C the ω 2 -eigenspace, and sl 3 ⊕ s the 1-eigenspace. For example, when C = O, it is well known that B = H 3 (O) is the 27dimensional exceptional simple Jordan algebra, and its structure algebra s is a simple Lie algebra of type E 6 (see for example, [11, Chap. IV, Sec. 4]). As a module for E 6 , B is irreducible, and relative to a certain Cartan subalgebra, it has highest weight the first fundamental weight. The module C is an irreducible E 6 -module, (the dual module of B) which has highest weight the last fundamental weight. Thus, Reading right to left, we see the decomposition of E 8 as a module for the subalgebra of type E 6 , and reading left to right, its decomposition as an sl 3 -module.
Recently, Lie algebras with a decomposition (1.5) have been considered by Faulkner [8,Lem. 22] in connection with his classification of structurable superalgebras of classical type. (Structurable algebras, which were introduced and studied in [1], form a certain variety of algebras generalizing associative algebras with involution and Jordan algebras.) In this work, we examine Lie algebras L such that L has a subalgebra sl 3 and such that L admits a decomposition as in (1.3) into copies of sl 3 , V = F 3 , V * , and trivial modules relative to the action of sl 3 . Applying results in [3] and [5], we determine that A is an alternative algebra, B a left A-module, and C a right A-module, and we describe s and the multiplication in L.
Using the fact that V can be given the structure of a module for the symmetric group S 4 , we obtain an action of S 4 by automorphisms on L. The elements τ 1 = (1 2)(3 4) and τ 2 = (1 4)(2 3) generate a normal subgroup of S 4 which is a Klein 4-subgroup. Results of Elduque and Okubo [7] enable us to deduce that L 0 = {X ∈ L | τ 1 X = X, τ 2 X = −X} is a structurable algebra under a certain multiplication. We identify the structurable algebra L 0 with the space of 2 × 2 matrices under a suitable multiplication. When L is the exceptional Lie algebra E 8 , . This is a simple structurable algebra (see [1, Secs. 8 and 9]).

Lie algebras with prescribed sl 3 decomposition
Let L be a Lie algebra over a field F of characteristic = 2, 3 (this assumption on the underlying field will be kept throughout), which contains a subalgebra isomorphic to sl(V), for a vector space V of dimension 3, so that L decomposes, as a module for sl(V) into a direct sum of copies of the adjoint module, the natural module V, its dual V * , and the trivial one-dimensional module. Thus, we write as in (1.3): for suitable vector spaces A, B, C, and for a Lie subalgebra s, which is the subalgebra of elements of L annihilated by the elements in sl(V). The vector space A contains a distinguished element 1 ∈ A such that sl(V) ⊗ 1 is the subalgebra isomorphic to sl(V) we have started with. Fix a nonzero linear map det : 3 V → F. This determines another such form det : This allows us to identify 2 V with V * : u 1 ∧ u 2 ↔ det(u 1 ∧ u 2 ∧ ) and, in the same vein, 2 V * with V.
The invariance of the bracket in L relative to the subalgebra sl(V) gives equations as in [3, (19) and c, c 1 , c 2 ∈ C, and for bilinear maps: and representations s → gl(A), gl(B), gl(C), whose action is denoted by da, db, and dc for d ∈ s and a ∈ A, b ∈ B and c ∈ C; where, as in [3, (17)], for x, y ∈ sl(V), and I 3 denotes the identity map. The difference with [3, (19)] lies in the appearance of the symmetric maps b 1 × b 2 and c 1 × c 2 when V has dimension 3. This slight difference has a huge impact. The distinguished element 1 ∈ A satisfies 1 • a = a, [1, a] = 0, D 1,a = 0, 1b = b, c1 = c, and d1 = 0 for any a ∈ A, b ∈ B, c ∈ C and d ∈ s.
for any a 1 , a 2 , a 3 ∈ A.
(3) For any a ∈ A, b ∈ B and c ∈ C, (4) For any a ∈ A, b 1 , b 2 ∈ B and c 1 , c 2 ∈ C, (5) D b,b×b = 0 for any b ∈ B and D c×c,c = 0 for any c ∈ C. In addition, the trilinear maps Proof. If L is a Lie algebra under the bracket defined in (2.2), then it is clear that s is a Lie subalgebra and all the conditions in item (0) are satisfied. Moreover, sl(V) ⊗ A) ⊕ s is a Lie subalgebra, and the arguments in [5,Sec. 3] show that A is an alternative algebra, and the conditions in item (1) are satisfied. (25) and (27) in [3], work here and give the conditions in item (2). (Note that there is a minus sign missing in [3, (27)].) Now, equations (30)-(33) in [3] establish the identitites in (3). The Jacobi identity applied to elements x ⊗ a, u 1 ⊗ b 1 and u 2 ⊗ b 2 , for x ∈ sl(V), u 1 , u 2 ∈ V, a ∈ A and b 1 , b 2 ∈ B, give the first equation in item (4), the second one being similar.

The arguments in Propositions 4.3 and 4.4, and Equations
The Jacobi identity for elements u i ⊗ b i , i = 1, 2, 3, for u i ∈ V and b i ∈ B gives: which, in view of the symmetry of the bilinear map b 1 × b 2 , proves half of the assertions in item (5); the other half being implied by the Jacobi identity Hence the Jacobi identity here is equivalent to the first condition in item (6); the second condition can be proven in a similar way. The converse follows from straightforward computations.
Given an alternative algebra A, the ideal E(A) generated by the associators (a 1 , a 2 , a 3 ) = (a 1 a 2 )a 3 − a 1 (a 2 a 3 )

is E(A) = (A, A, A) + (A, A, A)A = (A, A, A) + A(A, A, A). The associative nucleus of
Corollary 2.2. Let L be a Lie algebra which contains a subalgebra isomorphic to sl(V) for a vector space V of dimension 3, so that L decomposes, as a module for sl(V), as in (2.1). Then, with the notation used so far, the alternative algebra A is unital (the distinguished element 1 being its unit element), with 1 acting as the identity on both B and C, and the following conditions hold:

• E(A)B = 0 = CE(A), so that B (respectively C) is a left (resp. right) module for the associative algebra A/E(A). • T (B, C) is an ideal of A contained in its associative nucleus N(A),
and T (B, B × B) and T (C × C, C) are ideals of A contained in Z(A). • For any b, b 1 , b 2 ∈ B and any c, c 1 , c 2 ∈ C, the following conditions hold: • If the Lie algebra L is simple, then either the algebra A is associative, or else A = E(A) and B = C = 0. Moreover, if B = 0, then C coincides with B × B, and A coincides with T (B, B × B), and A is a commutative and associative algebra.
Proof. For any a 1 , a 2 , a 3 ∈ A and b ∈ B, is an ideal of A contained in the center Z(A). By similar arguments, T (C × C, C) is shown to be contained in Z(A) too.
For any b 1 , b 2 ∈ B and c ∈ C, the previous theorem gives, We permute b 1 and b 2 and use the fact that × is symmetric to get With the same arguments we prove for any b ∈ B, and c 1 , c 2 ∈ C.  T (B, B × B), which is commutative and associative.

Structurable algebras
This section is devoted to establishing a relationship between the Lie algebras with prescribed sl 3 decomposition considered above with a class of structurable algebras. This will be done by exploiting the action of a subgroup of the group of automorphisms of the Lie algebra isomorphic to the symmetric group S 4 . Theorem 3.1. Let L be a Lie algebra which contains a subalgebra isomorphic to sl(V) for a vector space V of dimension 3, so that L decomposes, as a module for sl(V), as in (2.1). Then, with the notation used so far, the vector space with the multiplication is a structurable algebra.
Proof. Take  (Thus V is the tensor product of the sign module and the standard irreducible 3-dimensional module for S 4 , and in this way, S 4 embeds in the special linear group SL(V).) The inner product given by (e i |e j ) = δ ij for any i, j ∈ {1, 2, 3} is invariant under the action of S 4 , so V is selfdual as an S 4 -module, and the action of S 4 on V * (where σv * = v * σ −1 ) is given by the "same formulas": Since S 4 acts by elements in SL(V), this action of S 4 on V and on V * extends to an action by automorphisms on the whole algebra L. Then the subspace L 0 = {X ∈ L | τ 1 X = X, τ 2 X = −X} becomes a structurable algebra [7,Thm. 7.5] with involution and multiplication given by the following formulas, for any X, Y ∈ L 0 .
But we easily deduce that .
we determine that the structurable product and the involution become as required.
Items (5) and (6) of Theorem 2.1 show that for any b ∈ B and c ∈ C, cT (c × c, c).

(3.4)
Also, using Theorem 2.1 and Corollary 2.2 we compute that and an analogous result with the roles of b and c interchanged. So we conclude that the equations Therefore, when considering simple algebras, we are dealing exactly with the situation considered by Allison and Faulkner in [2]. Theorem 3.1 shows that the restrictions on the bilinear maps involved are sufficient to ensure that the algebra A in (3.1), with multiplication (3.2) and involution (3.3) is a structurable algebra.
A natural question to ask is whether these conditions are also necessary. More precisely, does any structurable algebra of the form A as in (3.1) with multiplication (3.2) and involution (3.3), constructed from a unital alternative algebra A, left and right unital "associative" modules B and C, and bilinear maps T (b, c), b 1 × b 2 , and c 1 × c 2 , coordinatize a Lie algebra L with a subalgebra isomorphic to sl(V) for a vector space V of dimension 3 and with decomposition as in (2.1)? (We do not impose any further conditions on these bilinear maps besides requiring that the resulting algebra A be structurable.) Our last result answers this question in the affirmative.
where T I is the span of the triples T = (T 1 , T 2 , T 3 ) with for x, y ∈ A and (i, j, k) a cyclic permutation of (1, 2, 3). Here L x y = xy = R y x. The subspace T I is a Lie algebra with componentwise bracket, and the Lie bracket in L is given by extending the bracket in T I by setting  for x, y ∈ A, where (i, j, k) is a cyclic permutation of (1, 2, 3), and T = (T 1 , T 2 , T 3 ) is as in (3.6). Theorems 4.1 and 5.5 in [2] show that L is indeed a Lie algebra. Since we are assuming that the characteristic of the field is = 2, 3, Corollary 3.5 of [2] shows that T I = {(D, D, D) | D ∈ Der(A, ·, −)} ⊕ {(L s 2 − R s 3 , L s 3 − R s 1 , L s 1 − R s 2 ) | s i ∈ A,s i = −s i , s 1 + s 2 + s 3 = 0}.
In this way we recover the decomposition in (1.3) with bracket as in (2.2) for suitable maps D .,. , as required.