CLASSICAL SIMPLE LIE 2-ALGEBRAS OF ODD TORAL RANK AND A CONTRAGREDIENT LIE 2-ALGEBRA OF TORAL RANK 4

. After the classiﬁcation of simple Lie algebras over a ﬁeld of characteristic p > 3, the main problem not yet solved in the theory of ﬁnite dimensional Lie algebras is the classiﬁcation of simple Lie algebras over a ﬁeld of characteristic 2. The ﬁrst result for this classiﬁcation problem ensures that all ﬁnite dimensional Lie algebras of absolute toral rank 1 over an algebraically closed ﬁeld of characteristic 2 are soluble. Describing simple Lie algebras (respectively, Lie 2-algebras) of ﬁnite dimension of absolute toral rank (respectively, toral rank) 3 over an algebraically closed ﬁeld of characteristic 2 is still an open problem. In this paper we show that there are no classical type simple Lie 2-algebras with toral rank odd and furthermore that the sim- ple contragredient Lie 2-algebra G ( F 4 ,a ) of dimension 34 has toral rank 4. Additionally, we give the Cartan decomposition of G ( F 4 ,a ).


Introduction
The classification of the simple Lie algebras over an algebraically closed field of characteristic p with p ∈ {2, 3} is still an open problem. In characteristic 2, S. Skryabin showed in [10] that all simple Lie algebras on an algebraically closed field of characteristic 2 have absolute toral rank greater than or equal to 2 (see also [3]). Later, A. Premet and A. Grishkov classified Lie algebras of absolute toral rank 2. They announced in [2] (work in progress) the following result: All finite dimensional simple Lie algebras over an algebraically closed field of characteristic 2 of absolute toral rank 2 are classical of dimension 3,8,14 or 26. In particular, every finite dimensional simple Lie 2-algebra over a field of characteristic 2 of (relative) toral rank 2 is isomorphic to A 2 , G 2 or D 4 . When the absolute rank is greater than or equal to 3 the problem of classification is still open. The main obstacle in this problem is the lack of examples.
In this paper we calculate the toral rank of the classical simple Lie 2-algebras of type X l ∈ {A l , B l , C l , D l , g 2 , f 4 , e 6 , e 7 , e 8 }, i.e., quotients of Chevalley algebras over a field of characteristic 2, modulo the center. As a consequence, we obtain our first main result: Theorem 1. There are no classical type simple Lie 2-algebras of odd toral rank. In particular, there are no classical type simple Lie 2-algebras of toral rank 3.
V. Kac in [7] showed that for p > 3 every simple finite dimensional contragredient Lie algebra is isomorphic to one of the simple Lie algebras of the classical type. If p = 2, this is no longer true and the classification of simple finite dimensional contragredient Lie algebras is still considered an open problem. In the last section we prove that the simple contragredient Lie 2-algebra of dimension 34 constructed by V. Kac and V. Veȋsfeȋler in [13] has toral rank 4 and we calculate the dimension of the root spaces of this contragredient Lie algebra. More specifically, we have: where G := α, β, γ, λ is an elementary abelian group of order 16 and dim K (g ξ ) = 2, for all ξ ∈ G.
(The two theorems above are presented later as Theorems 5.7 and 6.2, respectively.) The only classical type simple Lie 2-algebras of toral rank 4 over an algebraically closed field of characteristic 2 are the following: sl 5 (K), psl 6 (K), sp 10 (K) (2) , and sp 12 (K) (2) /z(gl 12 (K)) (see Corollary 5.6). Theorem 2 gives us an example of a non-classical simple Lie 2-algebra, which should be taken into account in future investigations related to the problem of classifying the simple Lie 2-algebras of toral rank 4.
In section 1 we present some basic definitions and well-known results that will be used throughout the work. In Sections 2 and 3 we show that the linear special Lie algebra sl n+1 (K) and the symplectic Lie algebra sp 2m (K) over an algebraically closed field of characteristic 2 are Lie 2-algebras, and we study the simplicity of these algebras (Theorem 2.2 and 3.4). In section 4 we show that the orthogonal Lie algebra o n (K) (1) is not a Lie 2-algebra. In section 5 we list all classical type simple Lie 2-algebras, we calculate their toral rank, and we conclude that there are no classical type simple Lie 2-algebras with odd toral rank. Finally, in the last section we show that the simple Lie 2-algebra of dimension 34 constructed by V. Kac and V. Veȋsfeȋler in [13] has toral rank 4, and we also give the Cartan decomposition of this algebra.

Preliminaries
Throughout this paper all algebras are defined over a fixed algebraically closed field K of characteristic 2 containing the prime field K 2 and g is any Lie algebra of finite dimension on K. We start with some basic definitions and known facts.
For our purposes, it will be useful to have the matricial version of g(V, b). Given A ∈ gl n (K), we consider . . , v n } be a basis of V and assume that A ∈ gl n (K) is the Gram matrix of b with respect to the basis Θ, that is, Then, the Lie isomorphism T Θ : gl(V ) → gl n (K) (which sends every f ∈ gl(V ) to its matrix T Θ (f ) relative to Θ) maps g(V, b) onto g(A).
Two matrices A, B ∈ gl n (K) are said to be congruent if there is S ∈ GL n (K) such that In this case, the map g(A) → g(B) given by X → S −1 XS is a Lie 2-isomorphism.

Maximal tori and toral rank.
Definition 1.4. Let (g, [2]) be a Lie 2-algebra. An element t ∈ g is said to be a toral element if t [2] = t. A 2-subalgebra t of (g, [2]) is called toral (or a torus of g) if the 2-mapping is invertible on t.
Any toral subalgebra of g is abelian and admits a basis consisting of toral elements (see e.g. [5,Theorem 13,p. 192]). A torus t of g is called maximal if the inclusion t ⊆ t with t toral implies t = t.
Let (g, [2]) be a simple Lie 2-algebra over an algebraically closed field K and let h be a Cartan subalgebra. The set of toroidal elements in h generates a torus. We denote this torus by the symbol T (h). The torus T (h) is maximal in (g, [2]) (see [1,Lemma 4]). [12]). The toral rank of a Lie 2-algebra (g, [2]) is M T (g) := max{dim K (t) : t is a torus in g}.

2.
The special linear Lie 2-algebra (sl n+1 (K), [2]) In this section we consider the Lie algebra consisting of matrices of trace zero over K, and we study some properties concerning the simplicity of this algebra.
It is a known fact that the commutator of the Lie general algebra gl n+1 (K) is a Lie subalgebra of gl n+1 (K). This algebra is called the special Lie algebra, and it is denoted by sl n+1 (K). That is, It is easy to prove that sl n+1 (K) = {A ∈ gl n+1 (K) : tr(A) = 0}.
Remark 2.1. The Lie 2-algebra sl 2 (K) is not simple, since In the next theorem we consider the case where n ≥ 2.
Proof. In order to prove (1), it is enough to see that sl n (K) is closed by the 2-map [2] : sl n+1 (K) → sl n+1 (K). But this is an immediate consequence of the fact that tr(A 2 ) = tr(A) 2 , for all A ∈ sl n+1 (K).
Recall some well-known facts about quadratic forms over an algebraically closed field of characteristic 2 and their corresponding Lie algebras. Let V be an ndimensional vector space over K, and let b : V × V → K be a non-degenerate symmetric bilinear form. This means that b(x, y) = b(y, x), for all x, y ∈ V , and b(x, V ) = 0 implies x = 0. A non-degenerate symmetric bilinear form b is called symplectic if b(x, x) = 0 for all x ∈ V . Otherwise, it is called an orthogonal bilinear form.
3. Lie 2-algebras (g(V, b), [2]) with b a symplectic bilinear form In this section we study the simplicity of Lie 2-algebras which preserve a bilinear symplectic form over K.
Let b : V × V → K be a symplectic bilinear form. From Example 1.3, we have that g(V, b) is a Lie 2-algebra. We denote this algebra by sp (V, b), and call it the symplectic Lie 2-algebra. In [8,Theorem 19] it is shown that the dimension of V is even, that is, n = 2m, and there exists a basis β of V in which b has Gram matrix which has dimension 2m 2 + m and a basis consisting of the following elements: The Lie bracket of sp 2m (K) is given in Table 1, where the elements of the diagonal are results of the 2-map in the elements of their rows and corresponding columns. We now calculate the derived algebras of sp 2m (K), and then we show that the second derived algebra is a simple Lie 2-algebra whenever 2 does not divide m and m ≥ 3.  span{d 1 , d 2 , a 12 , a 21 , b 12 , b 1 , b 2 , c 12 , c 1 , c 2 }. By direct computations, we obtain span{d 1 , d 2 , a 12 , a 21 , b 12 , c 12 }, d 2 , a 12 , a 21 , b 12 , c 12 }, Therefore, if m = 1, 2, then sp 2m (K) is a solvable Lie 2-algebra.
Proof. To prove (1), we set Let us consider the linear map ϕ : gl m (K) → Alt m (K) given by a → a + a T . Since Ker(ϕ) = {a ∈ gl m (K) : a is symmetric} and we conclude that Im(ϕ) = Alt m (K). That is, ϕ is a surjective map. Then, given b ∈ Alt m (K) there exists a ∈ gl m (K) such that a + a T = b. Hence, Similarly, we prove that 0 0 c 0 ∈ sp 2m (K) (1) . Therefore, g 1 ⊆ sp 2m (K) (1) .
To prove (2), let g 2 = a b c a T : b, c ∈ Alt m (K), tr(a) = 0 . We will prove that sp 2m (K) (2) = g 2 . From the description of sp 2m (K) (1) in (1), we deduce that the Lie algebra sp 2m (K) (1) is generated by a ij , b ij , c ij , and d i for 1 ≤ i, j ≤ m. Therefore, sp 2m (K) (2) = [sp 2m (K) (1) , sp 2m (K) (1) ] is generated by a ij , b ij , c ij , and d i + d j for 1 ≤ i, j ≤ m. Since all of these elements belong to g 2 , we conclude that sp 2m (K) (2) ⊆ g 2 . The other inclusion is established reasoning as in the proof of (1). Finally, we prove (3). In the proof of (2), it is shown that sp 2m (K) (2) is generated by d i + d j for 1 ≤ i, j ≤ m. Therefore, From Table 1, we conclude that

Lemma 3.3.
Let I be a nontrivial ideal of sp 2m (K) (2) . Then c ij , b ij / ∈ I, for all i, j. (2) , for all l, k with l = k we have that [a il , c ik ] = c lk , [b il , c ik ] = a lk , and [d i + d l , b lk ] = b lk belong to I. Therefore, I = sp 2m (K) (2) , which is a contradiction. Similarly, if we suppose that b ij ∈ I, we arrive at a contradiction. Hence, c ij , b ij / ∈ I for all i, j.
In particular, for where b := e ij + e ji ∈ Alt m (K), we have a ij = 0 for i = j and a ii = a 11 for all i. Then α = a ii I 2m ∈ z(gl 2m (K)). Hence, I = z(gl 2m (K)) and sp 2m (K) 2 /z(gl 2m (K)) is simple.

Lie 2-algebras (g(V, b), [2]) with b an orthogonal bilinear form
In this section we show that the Lie algebra which preserves the orthogonal linear form over K is not a Lie 2-subalgebra of gl n (K).
Suppose that b : V × V → K is an orthogonal bilinear form, and let o(V, b) be the Lie 2-algebra associated to b.
In [8,Theorem 20] it is shown that there exists a basis β of V in which b has Gram matrix D = diag (d 1 , d 2 , . . . , d n ), where 0 = d i ∈ K for all i; therefore, the matricial algebra g(D) corresponding to the Lie 2-algebra o(V, b) is given by Since K is an algebraically closed field, we have that K 2 = K, that is, every element of K is a square. This fact implies that we can take the diagonal matrix D as the identity matrix I n . So, o n (K) := g(I n ) = {A ∈ gl n (K) : A is symmetric} is a Lie 2-algebra with basis {e ii } ∪ {ē ij := e ij + e ji }, 1 ≤ i < j ≤ n, and whose Lie bracket is given by

Classical type simple Lie 2-algebras and their toral rank
W. Killing and E. Cartan showed that all simple Lie algebras over an algebraically closed field of characteristic zero is isomorphic to one of the classical Lie algebras A n (n ≥ 1), B n (n ≥ 2), C n (n ≥ 3), D n (n ≥ 4), or to the exceptional Lie algebras g 2 , f 4 , e 6 , e 7 , e 8 (see [5]). But in characteristic 2, it seems that many new phenomena arise; for instance, these algebras are not necessarily simple, or some of them are isomorphic, and therefore, the classification of simple Lie algebras in characteristic 2 will differ from the classification of such algebras in characteristics 0 and p ≥ 5. In this section, we calculate the toral rank of the simple Lie 2-algebra of the classical type and we conclude that there are no classical type simple Lie 132 C. R. PAYARES GUEVARA AND F. A. ARIAS AMAYA 2-algebras of odd toral rank. In particular, there are no classical type simple Lie 2-algebras of toral rank 3.
Definition 5.1. Given an irreducible root system of type X l and its corresponding Chevalley algebra g(X l , K) over the field K, the quotient g(X l , K) := g(X l , K)/z(g(X l , K)), where z(g(X l , K)) is the center of g(X l , K), is usually called the classical Lie algebra of type X l .

Remark 5.2.
This definition is exactly the same as Steinberg's [11], but Steinberg excluded some types of characteristic 2 and 3.
The simplicity of the classical type Lie algebras in characteristic 2 has been determined by Hogeweij in [4], as indicated in the following proposition.

Proposition 5.3.
Suppose that X l is a Lie algebra which is not of type A 1 , B l , C l , or f 4 . Then g(X l , K) is a simple Lie 2-algebra.
(2) Type D l : if l is even.
(5) Type e 7 : g(e 7 , K). Proof. Let g be a classical type simple Lie 2-algebra. Then from Corollary 5.4 it follows that g = g(X l , K) with X l = A 1 , B l , C l , f 4 . Hence, any quotient of the form where h(X l , K) is a Cartan subalgebra of the Chevalley K-algebra g(X l , K), is a Cartan subalgebra of g. Since h(X l , K) = span{h i ⊗ 1 : h i ∈ h X l } and h X l is the subalgebra of diagonal matrices of sl l+1 (K), we obtain (h i ⊗ 1) [2] K)), and as T (h(X l , K)) ⊆ h(X l , K), we have that h(X l , K) = T (h(X l , K)). Since any pair of Cartan Lie subalgebras of a finitedimensional classical type Lie algebra g over K are conjugate (see [9]), there exists an automorphism σ ∈ Aut(g) such that h = σ(h(X l , K)). Then, from [1, Lemma 5] we obtain T (h) = σ(T (h(X l , K))) = σ(h(X l , K)) = h. Then any Cartan subalgebra of a simple Lie 2-algebra g of classical type is a maximal torus in g, and hence The following is a direct consequence of Theorem 5.5. In this section we show that the contragredient Lie 2-algebra G(F 4,a ) constructed by V. Kac and V. Veȋsfeȋler (see [13]) has toral rank 4, and we obtain the Cartan decomposition of this algebra.
They proved that G(F 4,a ) is a simple Lie 2-algebra of dimension 34 with Cartan matrix F 4,a , with a ∈ K \ {0, 1} (see [13,Proposition 3.6]). We now prove that this Lie 2-algebra has toral rank 4 and, furthermore, we give its Cartan decomposition. From (6.1), we conclude that is a Cartan subalgebra of G(F a,4 ). We now explicitly describe the maximal torus T (h) consisting of toroidal elements in h.