Finitary Simple Lie Algebras

Finitary simple Lie algebras

A.A. Baranov Institute of Mathematics, Academy of Sciences of Belarus, Surganova 11, Minsk, 220072, Belarus.

Abstract An algebra is called finitary if it consists of finite-rank transformations of a vector space. We classify finitary simple Lie algebras over a field of zero characteristic. We also describe finitary irreducible Lie algebras.

1 Introduction

Let V be a vector space over a field F . An element x ∈ EndF V is called finitary if dim xV < ∞. The finitary transformations of V form an ideal fgl(V ) of the Lie algebra gl(V ) of all linear transformations of V . A Lie algebra is called finitary if it is isomorphic to a subalgebra of fgl(V ) for some V . The aim of this paper is to classify infinite dimensional finitary simple Lie algebras over a field of characteristic 0. We also describe finitary irreducible Lie algebras. Since any simple Lie algebra can be considered as an algebra over its centroid and the latter is central simple, one can restrict ourselves to the case of central simple Lie algebras. We prove the following main theorem.

Theorem 1.1 Let F be a field of characteristic 0. Then any infinite dimensional central finitary simple Lie algebra over F is isomorphic to one of the following algebras: (1) a finitary special linear algebra fsl(V, Π); (2) a finitary special unitary algebra fsu(V, Φσ); (3) a finitary orthogonal algebra fo(V, Ψσ); (4) a finitary symplectic algebra fsp(V, Θσ). Here V is a right vector space over a finite dimensional division F -algebra ∆; ∆ is central, except in the case (2) where the center of ∆ is a quadratic extension of F ; σ is an involution (an antiautomorphism of order 1 or 2) of ∆ such that F is the set of central σ-invariant elements of ∆; Φσ, Ψσ, and Θσ are nondegenerate forms on V ; and Π is a total subspace of the dual V ∗.

The definition of the algebras (1)–(4) (for arbitrary F ) is given in Section 6. They are central finitary simple Lie algebras. Note that Theorem 1.1 is similar to Hall’s classification [9] of simple locally finite groups of finitary linear transformations. Theorem 8.2 describes finitary modules for the algebras (1)–(4). It turns out that any such module is a direct sum of a trivial submodule, modules isomorphic to V , and (for the case of finitary special linear algebras only) modules isomorphic to Π. If F is algebraically closed, then there are no nontrivial finite dimensional division algebras, so we have the following corollary.

1 Corollary 1.2 Let F be an algebraically closed field of characteristic 0. Then any infinite dimensional finitary simple Lie algebra over F is isomorphic to one of the following:

(1) a ﬁnitary special linear algebra fsl(V, Π);

(2) a ﬁnitary orthogonal algebra fo(V, Ψ);

(3) a ﬁnitary symplectic algebra fsp(V, Θ).

Here V is a vector space over F ; Ψ (resp. Θ) is a nondegenerate symmetric (resp. skew-symmetric) form on V ; and Π is a total subspace of the dual V ∗.

Corollary 1.2 have been proven in [5]. Note that there we use the notation t(V, Π) instead of fsl(V, Π). It easily follows from Corollary 1.2 (see also [4, Theorem 1.3]) that any finitary simple Lie algebra of (infinite) countable dimension over an algebraically closed field F of characteristic 0 is isomorphic to one of the following algebras: sl∞(F ), o∞(F ), and sp∞(F ). These algebras can be represented as the direct limits of natural embeddings of the relevant classical simple Lie algebras. Assume now that F = R is the field of real numbers. By Frobenius theorem, there are only three finite dimensional real division algebras: R, the field of complex numbers C, and the algebra of quaternions H.

Theorem 1.3 Any real inﬁnite dimensional central ﬁnitary simple Lie algebra is isomorphic to one of the following algebras.

∆ = R: fsl(V, Π), fo(V, Ψ), fsp(V, Θ).

∆ = C: fsu(V, Φσ).

∆ = H: fsl(V, Π), fo(V, Ψσ), fsp(V, Θσ).

Here V is a right vector space over ∆= R, C, or H; σ is the standard involution in ∆= C, H; Ψ, Θ, Φσ, Ψσ, and Θσ are nondegenerate forms; Π is a total subspace of the dual V ∗.

For the case of countable dimension one can get the complete classification. Denote by Mn(F ) R C H ∞ the algebra of n × n matrices over F = , , or ; and by M∞(F ) = ∪n=1Mn(F ) the algebra of infinite matrices with finite numbers of nonzero entries. Let δ → δ¯ be the standard involution in ∗ t t F = C, H. For X ∈ M∞(F ) set X = X¯ where X is the matrix transpose to X. Set

0 1 0 1 0 1 J = diag( , ,..., ,... ); −1 0 −1 0 −1 0

Ik = diag(−1,..., −1, 1, 1,... ), k = 0, 1, 2,... ; k I∞ = diag(−| 1, 1{z, −1,}1,..., −1, 1,... ).

The algebras

(1) sl∞(C)= {X ∈ M∞(C) | tr X = 0},

t (2) o∞(C)= {X ∈ M∞(C) | X + X = 0},

2 t (3) sp∞(C)= {X ∈ M∞(C) | X J + JX = 0} are ﬁnitary and simple as R-algebras, and any other real ﬁnitary simple Lie algebra of countable dimension is central (see Proposition 7.4).

Theorem 1.4 The algebras

(1) sl∞(R)= {X ∈ M∞(R) | tr X = 0},

t (2) o∞(R,k)= {X ∈ M∞(R) | X Ik + IkX = 0}, k = 0, 1,..., ∞,

t (3) sp∞(R)= {X ∈ M∞(R) | X J + JX = 0},

∗ (4) su∞(C,k)= {X ∈ M∞(C) | tr X = 0, X Ik + IkX = 0}, k = 0, 1,..., ∞,

∗ (5) sl∞(H)= {X ∈ M∞(H) | tr(X + X ) = 0},

∗ (6) o∞(H,k)= {X ∈ M∞(H) | X Ik + IkX = 0}, k = 0, 1,..., ∞,

∗ (7) sp∞(H)= {X ∈ M∞(H) | X J + JX = 0} are exactly all, pairwise nonisomorphic, central real ﬁnitary simple Lie algebras of countable dimension.

All these algebras can be constructed as the direct limits of natural embeddings of relevant classical real simple Lie algebras. Note that the algebras o∞(H,k) and sp∞(H) are often denoted as sp (H,k) = lim sp (H,k) and u∗ (H) = lim u∗ (H), respectively. ∞ −→ n ∞ −→ n The algebras (1)–(7) from Theorem 1.4 as well as the algebras sl∞(C), o∞(C), and sp∞(C) often appear in the literature. For instance, Bahturin and Benkart [1] described their weight and root lattices. Natarajan [16] studied unitary questions for their highest weight modules. Olshanskii [17] investigated representations of the corresponding infinite dimensional classical groups. Finitary simple Lie algebras form a natural subclass of so-called diagonal locally finite Lie algebras. The latter are studied in [3, 4, 6], see also Zalesskii’s survey [18]. Theorem 1.1 is certainly true for any field F of positive odd characteristic. So we would like to state the following conjecture.

Conjecture 1.5 Theorem 1.1 holds for any ﬁeld F of characteristic = 2.

In the last section we classify finitary irreducible Lie algebras. The theorem below holds for any field of characteristic = 2 provided Theorem 1.1 and Theorem 8.2 (for irreducible finitary modules) are valid in the case of positive odd characteristic. In the proof we use recent results of Leinen and Puglisi [13, 14, 15] for finitary irreducible Lie algebras.

Theorem 1.6 Let F be a field of characteristic 0 and and L ⊆ fgl(V ) be an infinite dimensional finitary irreducible Lie algebra over F . Let ∆ be the centralizer of the L-module V . Then ∆ is a finite dimensional division F -algebra, V can be considered as a right vector space over ∆, [L, L] is simple and one of the following holds:

(1) [L, L]= fsl(V, Π) ⊆L⊆ fgl(V, Π);

(2) [L, L]= fsu(V, Φσ) ⊆L⊆ fu(V, Φσ);

3 (3) [L, L]= L = fo(V, Ψσ);

(4) [L, L]= L = fsp(V, Θσ).

Moreover, if [L, L] is central simple, then either L = [L, L] is central simple or L = fgl(V, Π) or fu(V, Φσ).

Notation Throughout the paper F is the ground field; char F = 0 except in Sections 5, 6, and 10 where F is arbitrary; P is a field extension of F ; F¯ is the algebraic closure of F ; ∆ is a finite dimensional division algebra over F ; and Γ is the center of ∆, [Γ : F ] = 1, 2. Let V be a vector space over F . ∼ ¯ ¯ We shall denote VP = V ⊗F P and V = VF¯ = V ⊗F F . We identify V with the subset V ⊗F 1 in VP . Unless otherwise stated, modules considered in the paper are assumed to be left. Let A be op an algebra and M be a (left) A-module. The centralizer ∆ of M is the algebra (EndA M) , so M can be considered as a right ∆-module. In view of this, vector spaces over skew fields are usually assumed to be right in the paper. For a finite dimensional Lie algebra L we denote by L(∞) the smallest member of the derived series of L. Note that L(∞) is perfect, i.e. [L,L] = L. All finitary (or, more generally, locally finite Lie algebras) are assumed to be infinite dimensional. A(L) is the augmentation ideal of the universal enveloping algebra U(L) for a Lie algebra L. This is the ideal of codimension 1 generated by all x ∈ L.

2 Finite dimensional central simple Lie algebras

The aim of this section is to recall Jacobson’s classification of finite dimensional central simple Lie algebras of classical type over a field of characteristic 0 [12, Ch. X] and to prove some auxiliary lemmas. First of all we need to recall the notion of a centroid. Let A be an arbitrary (not necessary associative or Lie) algebra. The algebra of multiplications T (A) is the subalgebra of End A generated by all left and right multiplications la : x → ax and ra : x → xa. The centroid Γ of A is the centralizer of T (A) in the algebra End A. If A is simple, then Γ is a field and A can be represented as a Γ-algebra. Moreover, the Γ-algebra A is central, that is, its centroid coincides with Γ. We shall often use the following well known fact (see [12, Theorem 10.1.3].

Proposition 2.1 Let A be a central simple algebra over F and let P be a ﬁeld extension of F . Then AP is central simple.

Throughout the rest of the section all algebras considered are ﬁnite dimensional. Let L be a central simple Lie algebra over a ﬁeld F . Then by Proposition 2.1, L¯ is a simple Lie algebra over F¯. Since F¯ is algebraically closed, we know all simple Lie algebras over F¯:

An (n ≥ 1), Bn (n ≥ 2), Cn (n ≥ 3), Dn (n ≥ 4), F4, G2, E6, E7, and E8.

If L¯ is in this list, we say that L is of type Xn (X = A,B,... ). Usually the subscript n will be omitted and we shall say that L is of type A, of type B, and so on. It is often convenient do not distinguish types B and D, so we shall say that L is of type BD if it is of type either B or D. The algebras of types A, B, C, and D are called classical. Note that by Jacobson’s classiﬁcation the algebras of type A should be subdivided into two subclasses: AI and AII . To

4 expose this classification we need to recall some properties of finite dimensional central simple associative algebras. Let A be a finite dimensional central simple algebra over a field Γ of characteristic 0, i.e. the center of A coincides with Γ. Then by Wedderburn theorem, A can be identified with an algebra Mn(∆) of all n × n matrices over a skew field ∆. Moreover, ∆ is a finite dimensional central division algebra over Γ. Considering Mn(∆) as a Lie algebra under the usual commutator, we get the algebra gln(∆). Its commutant

sln(∆) = [gln(∆), gln(∆)] = {x ∈ gln(∆) | tr x ∈ [∆, ∆]} is a central simple Lie Γ-algebra of type AI (or, as we shall also denote, AI (∆)). Assume now that the ring A has an involution α, i.e. an antiautomorphism of order 1 or 2. The set F of α-invariant elements of Γ is a subﬁeld. If F = Γ, then α is an involution of the ﬁrst kind. Otherwise, Γ is a quadratic extension of F and α is of the second kind. The set

hα(A)= {a ∈ A | aα = −a} of skew-Hermitian elements of A is a Lie F -algebra under the usual commutator. If α is of the ﬁrst kind, then hα(A) is a central simple Lie algebra of type either B(∆), or C(∆), or D(∆). If α is of the second kind, then the commutant

shα(A)=[hα(A), hα(A)] = {x ∈ hα(A) | tr x ∈ [∆, ∆]} is a central simple Lie F -algebra of type AII (∆). Let L be a central simple Lie algebra. Recall that a field P ⊇ F is called splitting for L if LP is split, i.e. the set of diagonal matrices in the adjoint representation for some choice of a basis in LP is a Cartan subalgebra of LP . Note that the algebraic closure F¯ is a splitting field for L. It is also known that LP has the same root system and representation theory as L¯. Since always there exists a finite normal field extension P of L such that LP is split, we often prefer to work with LP instead of L¯. Let now L be classical and P be a splitting field for L. Let us fix a splitting Cartan subalgebra of LP and a base α1,...,αn of the root system. Let ω1,...,ωn be the relevant fundamental weights. The irreducible LP -module of highest weight ω1 is called standard. We say that an LP -module is natural, if it is either standard or dual to it. One can check that the latter notion does not depend on the choice of a splitting Cartan subalgebra and a base of the root system. We shall always identify L with the subset L ⊗F 1P in LP . So any LP -module W can be regarded as the L-module W ↓L (over the field F ). The following is well known.

Lemma 2.2 Let V be an irreducible LP -module. Then V ↓L is a sum of isomorphic irreducible L-modules.

Proof. Fix any irreducible submodule M of V ↓L. Then FM¯ = V . It remains to note that FM¯ is a sum of L-modules isomorphic to M. In view of Lemma 2.2, one can make the following deﬁnition. An irreducible L-module is called standard (resp. natural) if it is isomorphic to a submodule of the restriction of the standard (resp. natural) L¯-module to L. We have L ⊆ LP ⊆ L¯ for any algebraic ﬁeld extension P of F . Since L is central, LP is simple. One can easily see that a simple L-module V is standard (resp. natural)

5 if and only if it is isomorphic to a submodule of the restriction of the standard (resp. natural) LP -module to L. One can also check that an L-module is natural if and only if it is either standard or dual to standard. Let L be a central simple Lie algebra over F and let P be a ﬁnite normal extension of F such that LP is a split Lie algebra. Assume that LP is of type either An (n ≥ 1), or Bn (n ≥ 2), or Cn (n ≥ 3), or Dn (n ≥ 5) (the case D4 is exceptional). Denote by W the standard LP -module and by E the associative F -algebra generated by L in End W . Note that by Lemma 2.2, W ↓L is a sum of isomorphic irreducible L-modules, so E is simple. The following can be derived from Jacobson’s results. Assume that L is of type A. Then either E is central simple and L =[E,E] (type AI ) or the center Γ ⊆ P of E is a quadratic extension of F , there exists an involution α of E of the second kind, and L = shα(E) (type AII ). If L is of type B, C, or D, then E is central simple, there exists an involution α of E of the ﬁrst kind, and L = hα(E). Let L be a Lie algebra and A be an associative enveloping algebra of L, that is, a quotient of the augmentation ideal A(L) (the ideal generated by all x ∈ L) of the universal enveloping algebra U(L). (We prefer to work with A(L) instead of U(L) and do not require the existence of the identity in A.) Denote by I the kernel of the natural homomorphism A(L) → A. Assume that A has an involution α. We say that α is compatible if xα = −x for all x ∈ L. Since A is generated by L, there exists at least one compatible involution of A. One easily checks that A(L) has a compatible involution (we denote it by α). Assume that A has a compatible involution. Then it is clear that Iα = I and this involution is inherited from A(L). Conversely, if Iα = I, then A inherits a compatible involution. We summarize our results in the following lemma.

Lemma 2.3 Let L be a ﬁnite dimensional central simple Lie algebra over F . Assume that L¯ is classical and rk L¯ > 4. Let V be the standard L¯-module. Denote by E the F -algebra generated by L ⊆ End V and by I the annihilator of V in A(L). Then E is simple and one of the following holds.

(i) L is of type AI , E is central simple, and L =[E,E];

(ii) L is of type AII , Iα = I, the center Γ of E is a quadratic extension of F , F is the set of α-invariant elements of Γ, and L = shα(E);

(iii) L is of type B, C, or D; Iα = I, E is central simple, and L = hα(E).

Since E is simple, by Wedderburn theorem, there exists a (unique) number k and a (unique) skew ﬁeld ∆ = ∆(L) such that E is isomorphic to the matrix algebra Mk(∆). Moreover, the center Γ of E coincides with the center of ∆. Recall that Γ = F if L is not of type AII and Γ is a quadratic extension of F if L is of type AII . Recall also that the dimension of any ﬁnite dimensional central simple associative algebra is a square.

Proposition 2.4 Let L be a classical central simple Lie algebra over F and V be the standard L- module. Let n be the dimension of the standard L¯-module, and l2 = [∆(L):Γ]. Then dim V = 2ln if L is of type AII , and dim V = ln, otherwise.

2 Proof. We have E ⊗Γ F¯ =∼ Mn(F¯), so [E(L):Γ]=[E ⊗Γ F¯ : F¯]= n . On the other hand E(L) =∼ Mk(∆) for some k. Comparing dimensions of these Γ-algebras, we conclude that n = kl. Since V is the standard Mk(∆)-module, V can be considered as a right ∆-module and [V : ∆] = k = n/l. It remains to note that [V : F ]=[V : ∆][∆ : Γ][Γ : F ].

6 We say that an embedding of classical split simple Lie algebras (over the same ﬁeld) is natural if the restriction of the standard Q2-module to Q1 contains a unique nontrivial composition factor and the latter is natural. Clearly, this implies that Q1 and Q2 have the same type A, BD, or C. I II Lemma 2.5 Let L1 ⊂ L2 be classical central simple Lie algebras of the same type (A , A , BD, or C). Assume that L¯1 ⊂ L¯2 is a natural embedding. Then the following conditions are equivalent.

(1) ∆(L1)=∆(L2);

(2) the restriction of a natural L2-module to L1 contains a unique nontrivial composition factor.

Proof. Let P be a ﬁnite normal extension of F such that L1P and L2P are split Lie algebras. Clearly, the embedding L1P ⊂ L2P is natural. Set r = [P : F ]. Let W2 be a natural L2P -module and W1 be a natural submodule of W2↓L1P . Set ni = dimP Wi. Clearly, ni is the dimension of the ¯ 2 standard Li-module. Let li be the dimension of ∆(Li) over its center Γi. Then by Proposition 2.4, the dimension of a natural Li-module is either 2lini or lini, i = 1, 2. It follows that Wi↓Li is a sum of r/2li (or r/li) natural Li-modules. Assume that (1) holds. Then l1 = l2. Therefore the restriction of a natural L2-module to L1 contains exactly one nontrivial composition factor and the latter is natural, so (2) holds. Conversely, assume that (2) holds. Then obviously, l1 = l2. Let V2 be a natural L2-module and V1 be the natural submodule of V2↓L1. Denote by Ei the algebra generated by Li in End Vi. We identify E1 with the corresponding subalgebra in E2. Recall that ∼ Ei = Mki (∆i) for some ki, and ∆i = ∆(Li) is the centralizer of the Ei-module Vi. Therefore Vi can be considered as a right vector space over ∆i. Recall that the action of ∆2 commutes with the action of E2. Therefore for each δ ∈ ∆2, the space V1δ is a E1-submodule of V2 isomorphic to V1. In view of uniqueness, V1∆2 = V1. It follows that ∆2 centralizes E1 in End V1, so ∆2 ⊆ ∆1. Now since dim∆2 = dim∆1, we have ∆2 =∆1, as required.

Deﬁnition 2.6 An embedding L1 ⊆ L2 of classical central simple Lie algebras of the same type I II (A , A , BD, or C) is called natural if the embedding L¯1 ⊆ L¯2 is natural and one of the equivalent conditions (1) and (2) of Lemma 2.5 holds. II Lemma 2.7 Let L1 ⊆ L2 be central simple Lie algebras of type A. Assume that L1 has type A , L¯1 ⊆ L¯2 is a natural embedding, and the restriction of a natural L2-module to L1 contains a unique II nontrivial composition factor. Then L2 is of type A . I Proof. We use the notation in the proof of Lemma 2.5. Assume that L2 is of type A . Then we have r/2l1 = r/l2, so l2 = 2l1 > l1. On the other hand, we have as above ∆2 ⊆ ∆1, i.e. l2 ≤ l1. The contradiction obtained proves the theorem.

Lemma 2.8 Let L1 ⊆ M ⊆ L2 be ﬁnite dimensional simple Lie algebras. Assume that L1 and L2 I II are central simple Lie algebras of the same type X (= A , A , BD, or C), the embedding L1 ⊆ L2 is natural, and rk L¯1 > 10. Then M is a central simple Lie algebra of type X and the embeddings L1 ⊆ M and M ⊆ L2 are natural. Proof. The central simplicity of M follows from Lemma 3.3 below, so M¯ is simple. By [4, Lemma 5.2], the embeddings L¯1 ⊆ M¯ and M¯ ⊆ L¯2 are natural, so M¯ has the same type as L¯1 and L¯2 (A, BD, or C). Observe that the restrictions of a natural L2-module to M and of a natural M-module to L1 have unique nontrivial composition factors. Since L1 and L2 are of the same type X, it follows from Lemma 2.7 that M is also of type X. Therefore the embeddings L1 ⊆ M and M ⊆ L2 are natural.

7 3 Some lemmas on embeddings

In this section we prove some auxiliary lemmas on embeddings of Lie algebras. All Lie algebras considered are ﬁnite dimensional. The following is well known (see, for instance, [7, 1.5.6].

Lemma 3.1 Let L be a Lie algebra over F and let S be a Levi subalgebra of L. Then Rad L¯ = Rad L and S¯ is a Levi subalgebra of L¯.

Lemma 3.2 Let M be a simple Lie algebra over F . Then

(i) M¯ is semisimple;

(ii) M¯ is simple if and only if M is central simple;

(iii) the projection of any nonzero element x ∈ M to each simple component of M¯ is nonzero.

Proof. (i) By Lemma 3.1, Rad M¯ = Rad M = 0, so M¯ is semisimple. (ii) This follows from [12, Theorem 10.1.3]. (iii) Let M1,...,Mk be the simple components of M¯ . Assume that the projection of x to M1 ′ ′ ′ is zero. Then x ∈ M¯ = M2 ⊕⊕ Mk. Note that M¯ is an ideal in M¯ , so M¯ ∩ M is a nontrivial ideal in M. Since M is simple, we have M ⊆ M¯ ′, so M¯ ⊆ M¯ ′. The contradiction obtained proves the lemma.

Let L be a perfect Lie algebra over F . Let R = Rad L and S = S1 ⊕⊕Sk be a Levi subalgebra of L where S1,...,Sk are the simple components of S.

Lemma 3.3 Assume that there exist simple Lie algebras Q1 and Q2 of the same type over F¯ such that Q1 ⊆ L¯ ⊆ Q2 and the embedding Q1 ⊆ Q2 is natural. Then there exists i such that Q1 ⊆ S¯i⊕R¯. Moreover, if Q1 ∩ L = 0, then Si is central simple.

Proof. By Levi-Malcev theorem, there exists a Levi subalgebra T of L¯ such that Q1 ⊆ T =∼ S¯. Let T1,...,Tn be the simple components of T . Note that Q1 is a simple submodule of the Q1-module Q2 under the adjoint action. Moreover, since the embedding Q1 ⊆ Q2 is natural, Q2 has a unique composition factor isomorphic to Q1. Since Q1 is a submodule of T = T1 ⊕⊕ Tn ⊆ Q2, we have Q1 ⊆ Tj for some j. It follows that Q1 ⊆ Tj ⊕ R¯ = Q ⊕ R¯ where Q is a simple component of some S¯i. Assume now that there is a nonzero x ∈ Q1 ∩ L. Since Q1 is simple, x ∈ Rad L, so x has a nonzero projection x1 to Si. Therefore by Lemma 3.2(iii), x1 (and x) has a nonzero projection to each simple component of S¯i. Since x ∈ Q ⊕ R¯, we have Q = S¯i, so by Lemma 3.2(ii), Si is central simple. The following lemma for the case of algebraically closed ﬁelds have been proved in [4, Lemma 4.6].

Lemma 3.4 Let L1 ⊆ L2 be perfect subalgebras of a simple Lie algebra S3 such that L1 ∩Rad L2 = 0. Let also S1 ⊆ S2 be Levi subalgebras of L1, L2, respectively. Assume that the algebras S1, S2, and S3 are central simple and S¯1 ⊆ S¯2 ⊆ S¯3 are natural embeddings of classical simple Lie algebras ′ ′ of the same type (A, B, C, or D). Then there exists a Levi subalgebra S2 of L2 such that L1 ⊆ S2.

8 ′ Proof. Set Ri = Rad Li and Ri = [Ri,Ri] for i = 1, 2. One can assume that R2 = 0. Recall that for each element r ∈ R2 one can deﬁne a (special) automorphism θr of L2 via 1 1 θ = ead r =1+ad r + (ad r)2 + + (ad r)n + ... r 2 n!

Clearly, θr permutes Levi subalgebras. Moreover, by Levi-Malcev theorem, any Levi subalgebra of L2 has the form θr(S2) for some r ∈ R2. We shall prove that there exists a special automorphism θ ′ ′ (∞) of L2 such that θ(L1) ⊆ S2⊕R2. This will imply our assertion. Indeed, note that Rad(S2⊕R2) ⊆ ′ R2 ⊂ R2. Therefore applying induction on dim R2 to the sequence

′ (∞) θ(L1) ⊆ (S2 ⊕ R2) ⊆ S3,

′ −1 −1 we ﬁnd θ1 such that θ1(θ(L1)) ⊆ S2. Hence L1 ⊆ S2 = θ (θ1 (S2)), as required. ¯ ¯ ¯ ¯ ¯ ¯′ ¯ ¯ We have Li = Si ⊕ Ri. Note also that [R2, R2] = R2 and L1 ∩ R2 = 0. Since the embeddings S¯1 ⊆ S¯2 ⊆ S¯3 are natural, by [4, Lemma 4.6], there exists r ∈ R¯2 such that θr(L¯1) ⊆ S¯2. In particular, θr(L1) ⊆ S¯2. Each element l ∈ L1 can be uniquely represented in the form l = sl + nl where sl ∈ S2 and nl ∈ R2. We have

′ θr(l)= sl + nl +[r, sl]+ r ′ ¯′ ¯′ where r ∈ R2. It follows that nl + [r, sl] ∈ R2 for all l ∈ L1. Represent r in the form r = r1 + α2r2 + + αnrn where r1,...,rn ∈ R2 and {1,α2,...,αn} are linearly independent over F ¯ ′ elements of F . Then we have nl +[r1,sl] ∈ R2 for all l ∈ L1. Therefore

′ θr1 (l)= sl + nl +[r1,sl]+ ∈ sl + R2 for all l ∈ L1,

′ so θr1 (L1) ⊆ S2 + R2, as required. The lemma follows.

Lemma 3.5 Let L1 ⊂ L2 be a natural embedding of classical simple Lie algebras of the same type of ranks m ≥ 4 and n, respectively, over an algebraically closed ﬁeld. Let V be a nontrivial irreducible L2-module diﬀerent from the standard one and dual to it. Then dim L1V ≥ [(n − m)/2].

Proof. Let α1,...,αn and ω1,...,ωn be the simple roots and the fundamental weights of L2; αn−m+1,...,αn and ωn−m+1,...,ωn be those of L1. Let λ = a1ω1 + + anωn be the highest weight of V . By assumptions, λ = ω1 and for type A, λ = ω1,ωn. For a root α denote by Xα the corresponding root element of L2. Set s = [(n − m)/2] − 1. We shall prove that there exist weights µ0,...,µs of V and i ≥ n−m+1 such that µk,αi > 0 for 0 ≤ k ≤ s. (Here β,α = 2(β,α)/(α,α) where ( , ) is the standard nondegenerate symmetric bilinear form on the weight system associated with the Killing form.) This will imply the lemma. Indeed, ﬁx any nonzero vector vk of weight µk.

Since µk,αi > 0, we have wk = X−αi vk = 0 (see, for instance, [8, Proposition 8.7.3(iii)]). As all wk are in diﬀerent weight spaces, they are linearly independent, so dim L1V ≥ s +1=[(n − m)/2], as required. We have use the following fact (see [8, Corollary 8.7.1]): if µ is a weight of V and µ,αj > 0, then µ−αj is a weight of V . Set αii = αi, αij = αi +αi+1 ++αj for ij. It easily follows from the above fact that if ai = 0, then λ − αij is a weight of V for any 1 ≤ j ≤ n, except possibly in the cases when L2 is of type Dn and either i = n or j = n. Consider the following cases.

9 Case 1: ai = 0 for some n − m + 1 ≤ i 0 for all k, as required.

Case 2: an = 0. Set µk = λ − αn,n−3−k, 0 ≤ k ≤ s. Then for L of types Bn, Cn, and Dn for all k we have µk,αn > 0, µk,αn−1 > 0, and µk,αn−2 > 0, respectively, as required. Assume that L is of type An. If an ≥ 2, then we have µk,αn > 0, as desired. So one can suppose that an = 1. Then there exists i 0. For k = 0, 1,...,s set µk = λ − αi,i−k if i>s, and µk = λ − αi,i+k, otherwise. (Note that in the latter case i + k ≤ 2s 0 for all k, as required.

Case 3: ai = 0 for some i 1. Then µ = λ + ωi−1 − ωi − ωn−m + ωn−m+1. In particular, µ,αi−1 > 0. Set µk = µ − αi−1,i−1−k if i>s + 1, and µk = µ − αi−1,i−1+k, otherwise. (Note that in the latter case i − 1+ k ≤ 2s 0 for all k, as required. Assume now that i = 1. Then a1 ≥ 2 and µ = λ − ω1 − ωn−m + ωn−m+1. Since µ,α1 > 0, we have that µk = µ − α1,1+k, 0 ≤ k ≤ s, are weights of V . It remains to note that µk,αn−m+1 > 0 for all k, as required.

4 Natural local systems

Note that any finitary Lie algebra is locally finite, that is, each finitely generated subalgebra is finite dimensional. Let L be a locally finite Lie algebra. A set A of finite dimensional subalgebras of L is called a local system if L = ∪L∈AL and for each pair L,M ∈ A there is Q ∈ A such that L,M ⊆ Q. The latter property says that A is a directed set. Observe that L is isomorphic to the corresponding direct limit of algebras from A. In this section we construct ”natural” local systems for finitary simple Lie algebras. First of all we recall some well known facts on local systems (see, for example, [2]).

Lemma 4.1 Let A be a local system of a simple locally ﬁnite Lie algebra L and let Q be a ﬁnite dimensional subspace of L. Then the following hold. (i) {L(∞) | L ∈ A} is a local system of L consisting of perfect Lie algebras.

(ii) {L | L ∈ A, Q ⊆ L} is a local system of L.

(iii) There exists L ∈ A and a maximal ideal M of L such that Q ⊆ L and Q ∩ M = 0.

(iv) If A = N1 ∪∪ Nk, then one of the Ni is a local system of L.

Deﬁnition 4.2 A local system N of a ﬁnitary central simple Lie algebra L is called natural (of type X(∆) = AI (∆), AII (∆),BD(∆), or C(∆)) if all L ∈ N are classical central simple Lie algebras of the same type X(∆) naturally embedded each into another. We say that L is of type X(∆) if L has a natural local system of type X(∆).

The following proposition shows that the type of a central ﬁnitary simple Lie algebra is deter- mined uniquely.

Proposition 4.3 Let L be a central ﬁnitary simple Lie algebra having natural local systems N1 and N2 of types X1(∆1) and X2(∆2), respectively. Then X1 = X2 and ∆1 =∆2.

10 ¯ ′ ′ Proof. Fixany N1 ∈ N1 such that rk N1 > 10. Find N2 ∈ N2 and N1 ∈ N1 such that N1 ⊂ N2 ⊂ N1. Then by Lemma 2.8, X1 = X2 and the embedding N1 ⊂ N2 is natural. By deﬁnition, this implies that ∆1 =∆2, as required.

Theorem 4.4 Let L⊂ fgl(V ) be a ﬁnitary central simple Lie algebra. Then L possesses a natural local system N. Moreover, there exist submodules W 0, W 1, and W 2 of V such that (1) V = W 0 ⊕ W 1 ⊕ W 2;

(2) LW 0 = 0, LW 1 = W 1, and LW 2 = W 2;

(3) for each L ∈ N, LW i (i = 1, 2) is a ﬁnite direct some of isomorphic natural L-modules;

(4) if W 2 = 0, then there exists L ∈ N such that the natural submodules of LW 1 are not isomorphic to those of LW 2; in particular, W 2 = 0 whenever L is not of type AI .

Proof. Let A be a local system of L. Set L¯ = L⊗F F¯ and V¯ = V ⊗F F¯. Clearly, V¯ is a ﬁnitary L¯-module, so L¯ is ﬁnitary. Since L is central, by Proposition 2.1, L¯ is simple. Therefore by [4, Theorem 5.1], there exists a natural local system T of L¯. Moreover, all algebras in T are of type either A, or B, or C. Fix any nonzero element x ∈ L. By Lemma 4.1(ii), one can assume that all algebras from A and T contain x. Fix any Q ∈ T. Since A¯ = {L¯ | L ∈ A} and T are local ′ ′ systems of L¯, there exist LQ ∈ A and Q ∈ T such that Q ⊆ L¯Q ⊆ Q . As Q ∩ LQ ∋ x = 0, by Lemma 3.3, there exists a subalgebra MQ of LQ such that Q ⊆ M¯ Q and MQ/ Rad MQ is central (∞) simple. Replacing MQ by MQ , one can assume that MQ is perfect. Note that for each L ∈ A there exists Q ∈ T such that L ⊆ Q ⊆ M¯ Q, so L ⊆ MQ. Therefore M = {MQ | Q ∈ T} is a local system of L. Fix any M1 ∈ M. By Lemma 4.1(iii), there exists M2,M3 ∈ M such that M1 ⊆ M2 ⊆ M3 and Mi ∩ Rad Mi+1 = 0 for i = 1, 2. We have

′ Q ⊆ M¯ 1 ⊆ M¯ 2 ⊆ M¯ 3 ⊆ Q

′ for some Q,Q ∈ T. By Levi-Malcev theorem, there exists a Levi subalgebra P1 of M¯ 1 such that Q ⊆ P1. Similarly, one can choose Levi subalgebras P2 of M¯ 2 and P3 of M¯ 3 such that we have ′ Q ⊆ P1 ⊆ P2 ⊆ P3 ⊆ Q . Note that P1, P2, and P3 are simple. By Lemma 2.8, the algebras Q, ′ Q , P1, P2, and P3 are of the same type (A, BD, or C) and all the embeddings are natural. Now ﬁx Levi subalgebras Si of Mi such that S1 ⊆ S2 ⊆ S3. Note that the corresponding embeddings S¯1 ⊆ S¯2 ⊆ S¯3 are natural. Identifying M1 and M2 with their images in M3/ Rad M3 =∼ S3, and ′ applying Lemma 3.4, one can ﬁnd a Levi subalgebra S2 of M2 containing M1. It follows that each M ∈ M is contained in a central simple subalgebra SM of L. Therefore S = {SM | M ∈ M} is a local system of L. Note that all algebras from S are central simple. Moreover, by Lemma 4.1(iv), one can assume that they have the same type: AI , AII , BD, or C. Note also that for all L,L′ ∈ S with L ⊆ L′ the embedding L¯ ⊆ L¯′ is natural. Fix any L ∈ S. Let m be the rank of L¯ and s = dim L¯V¯ . Find Q ∈ S such that L ⊂ Q and the rank of Q¯ is greater than 2(s +1)+ m. Then it follows from the Lemma 3.5 that all nontrivial composition factors of the Q¯-module Q¯V¯ are natural. Since the embedding L¯ ⊂ Q¯ is natural, all nontrivial composition factors of the L¯-module L¯V¯ ⊆ Q¯V¯ are natural. We now show that L¯V¯ = L¯2V¯ . Indeed, otherwise there exists v ∈ V¯ such that Lv¯ ⊆ L¯2V¯ . Therefore (Fv¯ + L¯V¯ )/L¯2V¯ is a nontrivial L¯-module. But this is not the case because this module is completely reducible and all composition factors are trivial. It follows that the L¯-module L¯V¯ = L¯2V¯ has no trivial

11 composition factors, so L¯V¯ is a sum of natural L¯-modules. Note that LV is an L-submodule of L¯V¯ . Therefore LV is a sum of natural L-modules. It follows that LV is a sum of natural L-modules for each L ∈ S. Let L = L1 ⊂ L2 ⊂⊂ Lk be a chain of algebras from S such that ∆(Li) = ∆(Li+1) for i = 1,...,k − 1. Then by Lemma 2.5, the restriction of a natural Li+1-module to Li contains at least two nontrivial (natural) composition factors. It follows that the restriction of a natural k−1 Lk-module to L contains a direct sum of at least 2 natural L-modules. Since dim LV is fixed, we conclude that k is bounded. Therefore one can assume that our chain is maximal. Hence for each N ∈ S with N ⊇ Lk we have ∆(N)=∆(Lk), so the embedding Lk ⊆ N is natural. Now by Lemma 4.1(ii), N = {N ∈ S | Lk ⊆ N} is a natural local system of L. Fix N ∈ N such that there are nonisomorphic natural submodules in NV . If such an algebra 0 1 2 does not exist, fix arbitrary N ∈ N. We have a unique decomposition V = WN ⊕ WN ⊕ WN where 0 1 2 i NWN = 0, WN ⊕ WN = NV , WN is a finite direct sum of isomorphic natural N-modules for 2 1 2 i = 1, 2, and if WN = 0 then the irreducible submodules of WN are not isomorphic to those of WN . 0 1 2 For each algebra M ⊃ N in N we have a similar decomposition V = WM ⊕ WM ⊕ WM . Moreover, i i i 0 i 0 0 one can assume that WN ⊆ WM ; WM ↓N ⊆ WN ⊕ WN for i = 1, 2; and WM ⊆ WN . Note that i i WM ⊆ WL, i = 1, 2, for all M,L ∈ N with N ⊆ M ⊆ L. Set

i i W = ∪M⊇N WM ⊆ V, (i = 1, 2), 0 0 W = ∩M⊇N WM ⊂ V.

One easily checks that W 0, W 1, and W 2 satisfy the required properties.

5 Simple associative rings with minimal left ideals

In this section we recall the structure of simple associative rings with minimal left ideals. Observe that any such ring possesses a faithful irreducible module, i.e. it is primitive. Primitive rings with minimal left ideals are described in [11, ch. IV] and we shall expose this description at the end of the section. First of all we recall some well known simple facts which will be used in Section 7. Throughout the section F is an arbitrary ﬁeld.

Lemma 5.1 Let V be a vector space over a ﬁeld F and let A be a simple associative subalgebra of End V containing an element of ﬁnite rank. Then A is primitive.

Proof. Observe that A is finitary and, in particular, locally finite dimensional. Let A be a non- nilpotent finite dimensional subalgebra of A. Then there exist nontrivial composition factors in the A-module AV . Denote by M a minimal submodule of AV containing a nontrivial composition factor. Let U be a maximal submodule of the A-module W = AM. Set N = U ∩ M. Since the A-module M/N contains a nontrivial composition factor, W/U is a faithful irreducible A-module.

Lemma 5.2 Let V be a vector space over a ﬁeld F and A be an irreducible associative subalgebra of End V . Assume that A has an element of ﬁnite rank. Then A has a minimal left ideal.

Proof. Since V is an irreducible A-module, the centralizer ∆ of V is a division F -algebra and V can be considered as a right vector space over ∆. Let a ∈A be an element of minimal rank n over ∆. We

12 shall prove that the left ideal Aa is minimal. Fix any b = ca ∈Aa. It suﬃces to show that xb = a for some x ∈ A. Let {e1,...,en} be a ∆-basis of aV . Observe that bV = caV = ce1,...,cen∆. Since rk b ≥ rk a, the elements ce1,...,cen are linearly independent over ∆. Hence by Jacobson’s Density Theorem, there exists x ∈A such that xcei = ei for 1 ≤ i ≤ n. Therefore (xca − a)V = 0, so xb = xca = a, as required.

Proposition 5.3 Let V be a vector space over a ﬁeld F and let A be a simple associative subalgebra of End V containing an element of ﬁnite rank. Then A is primitive with minimal left ideals.

Proof. This follows from Lemmas 5.1 and 5.2.

Lemma 5.4 Let A be a simple associative algebra with minimal left ideals. Then A is a simple ring with minimal left ideals.

Proof. This is obvious (see, for instance, [11, Proposition 1.9.2 and §5.5]). Let V be a right vector space over a division F -algebra ∆. Denote by V ∗ the set of all ∆-linear functions ϕ : V → ∆. Then V ∗ is a left vector space over ∆ under the operations:

(ϕ + ψ)v = ϕv + ψv, (δϕ)v = δ(ϕv) for all v ∈ V , ϕ,ψ ∈ V ∗, and δ ∈ ∆.

∗ Deﬁnition 5.5 A subspace Π of V is called total if AnnV Π= {v ∈ V | ϕv = 0 for all ϕ ∈ Π} = 0.

Let Π be a total subspace of V ∗. Denote by F(V, Π) the ring of all transformations x ∈ End V of the form v → e1(ϕ1v)+ + en(ϕnv) ∗ where n is an integer, e1,...,en ∈ V , and ϕ1,...,ϕn ∈ Π. Set F(V ) = F(V,V ). Note that F(V ) is the ring of all ﬁnite-rank (over ∆) transformations of V .

Theorem 5.6 ([11, §4.9 and §4.15]) A ring A is isomorphic to a ring F(V, Π) for some vector space V and total subspace Π of V ∗ if and only if A is a simple ring with minimal left ideals.

We say that ﬁnite dimensional subspaces V ′ ⊂ V and Π′ ⊂ Π are compatible if dim V ′ = dimΠ′ ′ and AnnV ′ Π = 0.

∗ Lemma 5.7 Let Π be a total subspace of V . Then for any ﬁnite dimensional subspaces V1 ⊂ V ′ ′ ′ ′ and Π1 ⊂ Π there exists compatible subspaces V ⊂ V and Π ⊂ Π such that V1 ⊆ V and Π1 ⊆ Π .

′ Proof. Choose any V2 ⊂ V compatible with Π1. Set V = V1 + V2. Since AnnV Π = 0, there exists ′ ′ a subspace Π ⊇ Π1 of Π compatible with V . Assume that V ′ and Π′ are compatible, k = dim V ′ = dimΠ′. Denote by F (V ′, Π′) the set of n ′ ′ ′ ′ all transformations x : v → i=1 ei(ϕiv) with ei ∈ V and ϕi ∈ Π . Then F (V , Π ) is a subalgebra of F(V, Π) isomorphic to theP algebra Mk(∆) of all k × k matrices over ∆. We obtain the following corollary from Lemma 5.7.

13 Corollary 5.8 Let Π be a total subspace of V ∗. Then F(V, Π) is the direct limit of its subalgebras ∼ F (Vτ , Πτ ) = Mkτ (∆) (kτ = dim Vτ ) where Vτ and Πτ run over all compatible pairs of subspaces in V and Π.

Remark 5.9 Observe that F (V ′, Π′) ⊆ F (V ′′, Π′′) if and only if V ′ ⊆ V ′′ and Π′ ⊆ Π′′. Moreover, the corresponding embedding is natural, that is, one can identify these algebras with the matrix algebras Mk(∆) and Ml(∆) in such a way that any matrix M ∈ Mk(∆) maps to diag(M, 0,..., 0) ∈ Ml(∆).

Assume that ∆ has an involution δ → δ¯. A function Φ : V × V → ∆ is called a Hermitian form on V if

Φ(v1 + v2,w) = Φ(v1,w)+Φ(v2,w), Φ(vδ,w) = Φ(v,w)δ, Φ(w,v) = Φ(v,w) for all v,w ∈ V and δ ∈ ∆. If we have Φ(w,v)= −Φ(v,w) instead of Φ(w,v)= Φ(v,w), then Φ is called skew-Hermitian. We say that Φ is of the ﬁrst kind, if the involution δ → δ¯ acts trivially on the center Γ of ∆, otherwise we say that Φ is of the second kind. We shall call Hermitian forms of the ﬁrst kind orthogonal and skew-Hermitian those symplectic. We say that Φ is unitary if it is a Hermitian form of the second kind. Assume that Φ is nondegenerate. Denote by ΠΦ the set of all functions ϕu : v → Φ(v,u). ∗ One easily checks that ΠΦ is a total subspace of V . Moreover, the map u → ϕu produces an isomorphism between the left ∆-spaces V and ΠΦ (V can be considered as a left vector space setting δv = vδ¯ for δ ∈ ∆ and v ∈ V ), so we shall denote the algebra F(V, ΠΦ) by FΦ(V,V ). For ∗ ∗ each x ∈ FΦ(V,V ) there exists a unique x ∈ FΦ(V,V ) such that Φ(xv,w) = Φ(v,x w) for all v,w ∈ V . The map x → x∗ is called the adjoint map with respect to Φ.

Theorem 5.10 ([11, §4.12]) Let Φ be a nondegenerate Hermitian or skew-Hermitian form on a vector space V . Then the adjoint map x → x∗ with respect to Φ is an involution of the ring FΦ(V,V ). Conversely, let A be a simple ring with minimal left ideals and an involution. Then there exists a vector space V with a nondegenerate Hermitian or skew-Hermitian form Φ such that A can be realized as the ring FΦ(V,V ) with involution given by the adjoint map with respect to Φ.

Let A be a ring with involution α. Then one can easily check that the set hα(A) = {a ∈A| aα = −a} is a Lie ring under the usual commutator. Let now we have a vector space V with a Hermitian or skew-Hermitian form Φ. Then we deﬁne

fh(V, Φ) = {x ∈ F(V ) | Φ(xv,w)+Φ(v,xw)=0 for all v,w ∈ V } fsh(V, Φ) = [fh(V, Φ), fh(V, Φ)].

The following is obvious.

Proposition 5.11 Let V be a vector space with a nondegenerate Hermitian or skew-Hermitian form Φ and α be the involution of FΦ(V,V ) given by the adjoint map with respect to Φ. Then α h (FΦ(V,V )) = fh(V, Φ).

14 6 Finitary simple Lie algebras

In this section we introduce certain classes of finitary simple Lie algebras. Throughout below Γ ⊆ F are arbitrary fields with [Γ : F ] = 1, 2; ∆ is a finite dimensional central division Γ-algebra; and V is a right vector space over ∆. Finitary special linear algebras. Let Γ = F and let Π be a total subspace of V ∗. The finitary general linear algebra fgl(V, Π) is the F -algebra F(V, Π) under the usual Lie commutator. If Π= V ∗, then we write fgl(V ) instead of fgl(V,V ∗). For each x ∈ F(V ) one can define its trace tr x ∈ ∆ as the trace of x on the finite dimensional ∆-subspace xV . The finitary special linear algebra fsl(V, Π) is the set of all transformations x ∈ fgl(V, Π) with tr x ∈ [∆, ∆]. Clearly, both fgl(V, Π) and fsl(V, Π) are finitary Lie algebras over F . ∗ Assume that k = dim V is finite. Then Π = V , fgl(V, Π) =∼ glk(∆) = Mk(∆), and fsl(V, Π) =∼ I slk(∆) = [Mk(∆),Mk(∆)] is a classical Lie algebra over F of type A (∆). Since F(V, Π) = lim F (V , Π ) where V and Π run over all compatible pairs of subspaces in V and Π, we have −→ τ τ τ τ fsl(V, Π) = lim sl(V , Π ) with each sl(V , Π ) isomorphic to sl (∆), k = dim V . Moreover, in −→ τ τ τ τ kτ τ τ view of Remark 5.9, the algebras sl(Vτ , Πτ ) form a natural local system (we shall call it standard). We have fsl(V, Π) ⊗ F¯ = lim L(V , Π ) with F −→ τ τ ¯ ∼ ¯ L(Vτ , Πτ )= sl(Vτ , Πτ ) ⊗F F = slnτ (F ). ¯ ¯ Recall that slnτ (F ) is simple provided char F = 0 or char F = p and p|nτ . If p|nτ , then slnτ (F ) ¯ ¯ contains an ideal Inτ = F diag(1,..., 1) and slnτ (F )/Inτ is simple except in the case p = nτ = 2 [10, Theorem 7]. Assume that L(Vτ , Πτ ) ⊂ L(Vρ, Πρ). Since this embedding is natural, we have ¯ nτ

Proposition 6.1 Let F be a field, ∆ be a finite dimensional central division algebra over F , and let V be an infinite dimensional vector space over ∆. Let Π be a total subspace of V ∗. Then

(i) fsl(V, Π)=[F(V, Π), F(V, Π)];

(ii) fsl(V, Π) is a central ﬁnitary simple Lie algebra over F of type AI (∆).

When V has infinite dimension then V ∗ has uncountably infinite dimension; and since fsl(V )= ∗ fsl(V,V ) contains all transformations txϕ : v → x(ϕv) with ϕv = 0, the dimension of fsl(V ) is uncountably infinite. There is another finitary counterpart to the finitary special linear algebra which remains to be countably dimensional for countably dimensional V ; the stable special linear algebra sl∞(∆). This is best introduced in terms of matrices. Every n × n matrix M can be extended to (n + 1) × (n +1) matrix M ′ by placing M in the upper lefthand corner of M ′ and then bordering M with 0’s. This gives us natural embeddings

sl2(∆) → sl3(∆) →→ sln(∆) → ... the union of these algebras is then the stable special linear algebra sl∞(∆) and is countably dimensional. Let V be a countably dimensional space with a basis E = {e1,e2,... } and Π be a subspace ∗ ∗ ∗ ∗ ∼ of V spanned by E = {e1,e2,... }, the dual of the basis E. Then, clearly, fsl(V, Π) = sl∞(∆).

15 Proposition 6.2 The algebra fsl(V, Π) has countable dimension if and only if it is isomorphic to sl∞(∆).

Proof. Set L = fsl(V, Π). Let {sl(Vτ , Πτ )} be the standard local system of L. Since dim L is countable, one can choose a chain of subalgebras sl(V1, Π1) ⊂ sl(V2, Π2) ⊂ ... such that ∞ ∞ L = ∪i=1sl(Vi, Πi). Observe that we have V1 ⊂ V2 ⊂ ... . Therefore V = ∪i=1Vi has countable dimension. Now it is clear that one can choose a basis E = {e1,e2,... } of V and a basis ∗ ∗ ∗ ∗ ∼ E = {e1,e2,... } of Π such that E is the dual basis of E. Therefore L = sl∞(∆), as required. The converse statement is obvious.

Finitary special unitary algebras. Assume that ∆ has an involution σ : δ → δ¯ of the second kind, that is, σ acts nontrivially on the center Γ of ∆. Then Γ is a quadratic field extension of the subfield F of σ-invariant elements of Γ. Let Φ be a unitary form on V . The finitary unitary algebra fu(V, Φ) is the set of all transformations x ∈ fgl(V ) such that

Φ(xv,w)+Φ(v,xw)=0 forall v,w ∈ V .

The finitary special unitary algebra fsu(V, Φ) is the set of all x ∈ fu(V, Φ) with tr x ∈ [∆, ∆]. We also have fsu(V, Φ)=[fu(V, Φ), fu(V, Φ)] because this is true for finite dimensional V . Both fu(V, Φ) and fsu(V, Φ) are finitary Lie algebras over F . Assume that dim V is infinite. Let {Vτ } be the set of all finite dimensional subspaces of V with nondegenerate restrictions Φτ = Φ|Vτ . For each Vτ we ⊥ ⊥ have V = Vτ ⊕ Vτ where Vτ is the orthogonal complement to Vτ with respect to Φ. Denote by ⊥ ∼ Lτ the algebra of all elements x ∈ fsu(V, Φ) with xVτ = 0. Clearly, Lτ = su(Vτ , Φτ ). Let L be a finite dimensional subalgebra in fsu(V, Φ). Note that dim LV is finite. Since Φ is ⊥ nondegenerate, there exists Vτ containing LV . Since L ⊂ fsu(V, Φ), L leaves Vτ invariant. As ⊥ LV ⊆ Vτ , we have LVτ = 0, so L ⊆ Lτ . Therefore {Lτ } is a local system of L. We shall call this local system standard. Note that Lτ1 ⊆ Lτ2 if and only if Vτ1 ⊆ Vτ2 . Moreover, if Vτ1 ⊆ Vτ2 , ∼ ∼ then the corresponding embedding of Lτ1 = su(Vτ1 , Φτ1 ) into Lτ2 = su(Vτ2 , Φτ2 ) is natural. As in the case of finitary special linear algebras the algebras su(Vτ , Φτ ) are quasisimple [10, Theorem 7] (central simple if char F = 0), so fsu(V, Φ) is central simple. We obtain the following result.

Proposition 6.3 Let F be a field and let ∆ be a finite dimensional division algebra over F with involution σ of the second kind such that F coincides with the set of central σ-invariant elements of ∆. Let V be an infinite dimensional vector space over ∆ and let Φ be a nondegenerate unitary form on V . Then

(i) fsu(V, Φ)=[fu(V, Φ), fu(V, Φ)];

(ii) fsu(V, Φ) is a central ﬁnitary simple Lie algebra over F of type AII (∆).

Finitary orthogonal and ﬁnitary symplectic algebras. Let Γ = F and let Ψ be a nondegenerate orthogonal form on V . The ﬁnitary orthogonal algebra fo(V, Ψ) is the set of all transformations x ∈ fgl(V ) such that

Ψ(xv,w)+Ψ(v,xw)=0 forall v,w ∈ V .

If we have a nondegenerate symplectic form Θ on V , then we obtain the ﬁnitary symplectic algebra fsp(V, Θ). If char F = 2 and V is ﬁnite dimensional, then fo(V, Ψ) = o(V, Ψ) and fsp(V, Θ) =

16 sp(V, Θ) are classical central simple (except some small dimensions) Lie algebras of types BD(∆) and C(∆), respectively [10, Lemma 7]. Both fo(V, Ψ) and fsp(V, Θ) are finitary Lie algebras over F . As in the case of unitary algebras, it is not difficult to show that fo(V, Ψ) and fsp(V, Θ) are direct limits of finite dimensional central simple subalgebras Lτ of the same type (BD(∆) or C(∆)) naturally embedded each into another. So we have the following proposition.

Proposition 6.4 Let F be a field of characteristic = 2. Let ∆ be a finite dimensional central division algebra over F and let V be an infinite dimensional vector space over ∆. Let Ψ and Θ be nondegenerate orthogonal and symplectic forms on V . Then fo(V, Ψ) and fsp(V, Θ) are central finitary simple Lie algebras over F (of types BD(∆) or C(∆)).

We conclude this section by the following result.

Proposition 6.5 Let V be a vector space with a nondegenerate Hermitian or skew-Hermitian form Φ. Then the algebra fsh(V, Φ) is countably dimensional if and only if V is countably dimensional.

Proof. Let L = fsh(V, Φ) and let {Lτ } be the standard local system of L. Assume that L is countably dimensional. Then one can choose a chain of subalgebras Lτ1 ⊂ Lτ2 ⊂ ... such that ∞ ∞ L = ∪i=1Lτi . We have Vτ1 ⊂ Vτ2 ⊂ ... , so V = ∪i=1Vτi has countable dimension. The converse statement is obvious.

7 The classiﬁcation

In this section we prove the main theorem.

Lemma 7.1 Let L ⊆ fgl(V ) be a ﬁnitary simple Lie algebra. Then V has a faithful composition factor.

Proof. It is well known (see, for instance, [2, 3.1]) that simple locally finite Lie algebras are not locally solvable, so L has a finite dimensional semisimple subalgebra S (one can take a Levi subalgebra of any non-solvable finite dimensional subalgebra). Fix a nontrivial irreducible submodule U of V . Let W be the L-submodule of V generated by U. By Zorn’s lemma, W has a maximal submodule. It remains to note that the relevant factor module is nontrivial, as required. Let Γ ⊇ F be the centroid of L. Since L is simple, by [12, Theorem 10.1.1], Γ is a field. Denote by LΓ the algebra L considered over Γ. Clearly, LΓ is simple.

Lemma 7.2 Let V be a vector space over a field F . Let L⊂ fgl(V ) be a finitary simple Lie algebra and let Γ ⊇ F be the centroid of L. Assume that V is irreducible. Then one can define an action of Γ on V compatible with that of L, i.e. (γx)v = x(γv)= γ(xv) for all γ ∈ Γ, x ∈L, and v ∈ V . Moreover, the corresponding Γ-space V Γ is a finitary irreducible LΓ-module.

Proof. Denote by A(L) the subalgebra in U(L) generated by L (this is an ideal of codimension 1 in U(L)). Clearly, A(L) inherits from L the structure of a Γ-module. Let γ ∈ Γ and v ∈ V . Then the map λγ : uv → (γu)v with u running over A(L) is an endomorphism of V . Moreover, γ → λγ is a homomorphism of the ﬁeld Γ into EndF V . So one can deﬁne an action of Γ on V via

17 γv = λγv. Note that this action commutes with the action of L. Clearly, the relevant Γ-module V Γ is an irreducible ﬁnitary LΓ-module. Observe that for all x ∈L and γ ∈ Γ we have γ(xV )= x(γV )= xV . Therefore [Γ : F ] divides rk x = dim xV . So we have the following corollary.

Corollary 7.3 Let L⊂ fgl(V ) be an irreducible ﬁnitary simple Lie algebra and let Γ be the centroid of L. Then [Γ : F ] is ﬁnite and divides rank of any x ∈L.

Proof of Theorem 1.1. Let L ⊂ fgl(V ) be a finitary simple Lie algebra. By Lemmas 7.1 and 7.2, one can assume that L is central and V is irreducible. Then by Theorem 4.4, there exists a natural local system N for L and for each L ∈ N, LV is a finite direct sum of isomorphic natural L-modules. Fixing an appropriate base in the root system of L¯, one can assume that LV is a sum of standard L-modules. Denote by I and IL the annihilators of V in A(L) and A(L), respectively. Set E = A(L)/I and E = A(L)/I . We have I = I ∩ A(L) and E = lim E . Obviously, I coincides L L L −→ L L with the annihilator of the standard L¯-module in A(L). Therefore we can use Lemma 2.3. By this lemma, EL is simple for each L ∈ N, so E is simple. Moreover, since E is finitary, by Proposition 5.3 and Lemma 5.4, E is a simple ring with minimal left ideals. I Assume that L is of type A . Then L = [EL,EL] for each L, so L = [E,E]. By Theorem 5.6, E is isomorphic to F(U, Π) for some vector space U over a skew field ∆ and total subspace Π of U ∗. Observe that ∆ is a finite dimensional division F -algebra isomorphic to ∆(L). Indeed, take any subalgebra A of F(U, Π) isomorphic to Mk(∆) for sufficiently large k. ∆ is isomorphic to a subalgebra of A, so acts by finitary transformations on U. Since ∆ is a division algebra, this implies that ∆ is finite dimensional. Now note that L contains a subalgebra isomorphic to slk(∆) = [A, A]. Therefore by Lemma 2.8, ∆ = ∆(L). By Proposition 6.1(i),

L =[E,E] =∼ [F(U, Π), F(U, Π)] = fsl(U, Π).

It follows that L and fsl(U, Π) are isomorphic as rings. It remains to note that F is the centroid of L and, consequently, of fsl(U, Π). Therefore L and fsl(U, Π) are isomorphic as F -algebras. Assume that L is not of type AI . Denote by α the standard involution of A(L). Observe that the α α restriction of α to A(L) is the standard involution of A(L). Therefore IL = IL, I = I, and all EL α α α II and E inherit the involution α. Since L = h (EL) (or L =[h (EL), h (EL)] for L of type A ) for all L ∈ N, we have L = hα(E) (or L =[hα(E), hα(E)] for L of type AII ). By Theorem 5.10, there exists a vector space U over a skew field ∆ (as above ∆ = ∆(L)) with a nondegenerate Hermitian or skew-Hermitian form Φ such that E can be realized as the ring FΦ(U, U) with involution given by the adjoint map with respect to Φ. Now, in view of Proposition 5.11, we have L = fh(U, Φ) (or L = fsh(U, Φ) if L is of type AII ). Assume that the form Φ is skew-Hermitian of the second kind, i.e. the relevant involution δ → δ¯ of ∆ acts nontrivially on the center of ∆. Then one can find a nonzero central element δ ∈ ∆ such that δ¯ = −δ. SetΦ′ = δΦ. Then one can check that fh(U, Φ) = fh(U, Φ′) and Φ′ is a Hermitian form on U. So we can exclude skew-Hermitian forms of the second kind from our consideration. Any other form is either unitary, or orthogonal, or symplectic. It follows that L is isomorphic to either fsu(U, Φ), or fo(U, Φ), or fsp(U, Φ), respectively. The theorem follows. The classification of all finitary simple Lie algebras (not necessary central) can be obtained using the following.

18 Proposition 7.4 Let K be a field and F be a finite field extension of K. Let L be one of the central finitary simple Lie F -algebras described in Theorem 1.1. Denote by LK the algebra L considered over the field K. Then LK is finitary and simple. Moreover, any finitary simple Lie algebra over K can be obtained in such a way.

Proof. The simplicity of LK follows from [12, Theorem 10.1.3]. Now if M is a ﬁnitary simple Lie algebra over K, then by Corollary 7.3, the centroid Γ of M is a ﬁnite extension of K, so the proposition follows.

8 Finitary modules

The aim of this section is to describe finitary modules for central finitary simple Lie algebras. Let V be a vector space. Let Π be a total subspace of V ∗. Observe that Π can be viewed as a right vector space over the opposite skew field ∆op. Moreover, V , viewed as a left vector space over ∆op, can be identified with a subspace of Π∗ setting eϕ = ϕe for all e ∈ V and ϕ ∈ Π. One can check that V is total. So one can define the ring F(Π,V ). This is exactly the ring of all op n transformations of the right ∆ -space Π of the form ϕ → i=1 ϕi(eiϕ) where ei ∈ V and ϕi ∈ Π for i = 1,...,n. P

Lemma 8.1 Let V be a vector space, Π be a total subspace of V ∗, and Φ be a nondegenerate Hermitian or skew-Hermitian form on V . Then

(1) the enveloping algebra of fsl(V, Π) in End V is F(V, Π);

(2) the enveloping algebra of fsl(V, Π) in EndΠ is F(Π,V );

(3) the enveloping algebra of fsh(V, Φ) in End V is FΦ(V,V ).

Proof. Let L =∼ fsl(V, Π) or fsh(V, Φ) and let {Lτ } be the standard local system of L. Recall that in the case (1) and (2) Lτ = sl(Vτ , Πτ ) where Vτ and Πτ are compatible subspaces in V and Π. By [12, Lemma 10.4.4], the enveloping algebras of Lτ in V and Π are F (Vτ , Πτ ) and F (Πτ ,Vτ ), respectively, so (1) and (2) follows. For the case (3) we have Lτ = sh(Vτ , Φτ ) with Φτ = Φ|Vτ . ⊥ ⊥ Let Vτ be the orthogonal complement to Vτ , i.e. V = Vτ ⊕ Vτ . One can extend Φτ to V setting ⊥ ∗ ⊥ Φτ (Vτ ,V )=0. Let Πτ be the set of all linear functions ϕ ∈ V with ϕ(Vτ ) = 0. Note that Πτ is ∗ the image of Vτ in V under the canonical map u → ϕu with ϕu(v)=Φ(v,u). We have ∼ Lτ = sh(Vτ , Φτ ) ⊂ F(Vτ , Πτ ) = Mkτ (∆) where kτ = dim Vτ . By [12, Lemma 10.4.4], the enveloping algebra of Lτ is isomorphic to the full matrix algebra Mkτ (∆), so coincide with F(Vτ , Πτ ). Therefore the enveloping algebra of fsh(V, Φ) in End V is FΦ(V,V ).

Theorem 8.2 Let V be a vector space, Π be a total subspace of V ∗, and Φ be a nondegenerate Hermitian or skew-Hermitian form on V . Let L = fsl(V, Π) or fsh(V, Φ) and let W be a finitary L-module, i.e. L⊂ fgl(W ). Then W = W 0 ⊕ W 1 ⊕ W 2 where LW 0 = 0, the module W 1 is a finite direct sum of copies of V , and either W 2 = 0 or L = fsl(V, Π) and W 2 is a finite direct sum of copies of Π.

19 Proof. Let {Lτ } be the standard local system of L. In view of Theorem 4.4, one can assume that for all τ, Lτ W is a finite direct sum of isomorphic natural Lτ -modules. Let A(L) and A(Lτ ) be the augmentation ideals in U(L) and U(L ), respectively. We have A(L) = lim A(L ). Set τ −→ τ I = AnnA(L) W , Iτ = AnnA(Lτ ) W , E = A(L)/I, and Eτ = A(Lτ )/Iτ . Clearly, Iτ = I ∩ A(Lτ ); E and E are the enveloping algebras of L and L in End W ; and E = lim E . Assume that for each τ τ −→ τ τ, Lτ W is a sum of standard Lτ -modules. Then Iτ = AnnA(Lτ ) V , so I = AnnA(L) V . Therefore the enveloping algebras of L in End W and End V are isomorphic. Hence by Lemma 8.1, E =∼ F(V, Π) 0 for L = fsl(V, Π) and E =∼ FΦ(V,V ) for L = fsh(V, Φ). Denote by E the algebra E considered as a ring. In both cases E0 is primitive simple with minimal left ideals. Since E0W = W , by [11, Theorem 4.14.1], the E0-module W is a direct sum of submodules isomorphic to V . By [11, Proposition 1.9.2] (see also similar Lemma 7.2 for the case of Lie algebras), the E0 module V has a unique structure of a E-module. It follows that W is a sum of E-modules isomorphic to V . Since E ⊆ F(W ), this sum is finite. If L = fsl(V, Π) and for all τ, Lτ W is a direct sum of modules dual to standard, then by Lemma 8.1, E = F(Π,V ), so W is a finite direct sum of modules isomorphic to Π. The theorem follows.

9 Real ﬁnitary simple Lie algebras

The aim of this section is to prove Theorems 1.3 and 1.4, which describe real finitary simple Lie algebras. Proof of Theorem 1.3. By Frobenius theorem, there are only three finite dimensional division algebras over R: R, the field of complex numbers C, and the algebra of quaternions H. The algebra R has no nontrivial involutions. The algebras C and H have standard involutions δ = a+bi → δ¯ = a−bi and δ = a + bi + cj + dk → δ¯ = a − bi − cj − dk, respectively. (Here {1,i} and {1,i,j,k} are the standard bases of C and H, respectively). We shall denote both these involutions by σ. Now by Theorem 1.1, any real finitary simple Lie algebra is isomorphic to one of the following algebras: fsl(V, Π) (type AI (∆)), fsu(V, Φα) (type AII (∆)), fo(V, Ψα) (type BD(∆) or C(∆)), fsp(V, Θα) (type BD(∆) or C(∆)), where V is a vector space over ∆ = R, C, H; α is an involution of the R-algebra ∆; Φα, Ψα, and Θα are nondegenerate forms; and Π is a total subspace of V ∗. Consider the following cases. ∆= R. Then we have the algebras fsl(V, Π) (type AI (R)), fo(V, Ψ) (type BD(R)), and fsp(V, Θ) (type C(R)). ∆= C. Since the R-algebra C is not central and has only standard involution (which is of the second kind), we have only AII (C)-type algebras: fsu(V, Φσ). ∆= H. Assume that α is a nonstandard involution of H. Then β : δ → α(δ¯) is an automorphism of the algebra H. Since H is central simple, each automorphism of H is internal, i.e. there is q ∈ H such that α(δ¯)= qδq−1 for all δ ∈ H, or equivalently, α(δ)= qδq¯ −1. Since α2 = 1, one easily checks thatq ¯ = −q. Let now Φα be an orthogonal (resp. symplectic) form on a vector space V over H with respect to the involution α. ThenΦ=Φαq is a symplectic (resp. orthogonal) form on V with respect to the standard involution σ of H. Indeed, we have for all v,w ∈ V

Φ(v,w) = Φα(v,w)q = ±α(Φα(w,v))q = ±qΦα(w,v)= ∓Φ(w,v).

20 Therefore we do not need to consider nonstandard involutions, so we get the following algebras: fsl(V, Π) (type AI (H)), fo(V, Ψσ) (type C(H)), and fsp(V, Θσ) (type D(H)). The theorem follows.

Proof of Theorem 1.4. Let L be a central real ﬁnitary simple Lie algebra of (inﬁnite) countable dimension. We identify L with one of the algebras in Theorem 1.3. Since L has countable dimension, by Propositions 6.2 and 6.5, the underlying space V also has countable dimension. Moreover, sl∞(R) I I and sl∞(H) are unique countably dimensional algebras of types A (R) and A (H), respectively. So one can assume that L preserves a nondegenerate Hermitian or skew-Hermitian form Φ. We shall consider the following cases. (a) ∆ = R, C, or H; and Φ is Hermitian (unitary or orthogonal). (b) ∆ = R and Φ is skew-Hermitian (symplectic). (c) ∆ = H and Φ is skew-Hermitian (symplectic).

Fix a chain V1 ⊂ V2 ⊂ ⊂ Vn ⊂ ... of subspaces in V such that Φ|Vn is nondegenerate ∞ and V = ∪i=1Vi. Set W0 = V1. Let Wi be the orthogonal complement to Vi in Vi+1, that is, ∞ Vi+1 = Vi + Wi and Φ(Vi,Wi) = 0. Then V = ⊕i=0Wi. Note that Φ|Wi is nondegenerate for all i ≥ 0. Let now W be a finite dimensional vector space over ∆ = R, C, or H with a nondegenerate Hermitian or skew-Hermitian form Φ with respect to the standard involution. By classical results (see also [12, §10.7]), there exists a basis {w1,...,wn} of W such that in the cases (a), (b), and (c) above we have, respectively, (a) Φ(wi,wj)= ±δij, 1 ≤ i,j ≤ n; (b) Φ(wi,wj)=0 for all i and j, except Φ(w2i−1,w2i)=1 for 1 ≤ i ≤ n/2 (n is even); (c) Φ(wi,wj)= qδij, 1 ≤ i,j ≤ n, where q is any fixed element in H withq ¯ = −q. Moreover, the numbers of −1 and 1 in the case (a) is an invariant. Choosing such a basis in each Wi and reordering these basic vectors, one can get a basis E = {e1,e2,...,en,... } of V such that the matrix (Φ(ei,ej)) of the form Φ in the case (a) coincides with ±Ik, k = 0, 1, 2,..., ∞, (see Introduction); in the case (b) coincides with J; and in the case (c) coincides with qI. Since the forms Φ and −Φ yield the same algebras, one can assume that the matrix of Φ in the case (a) is Ik for some k. Observe that for ∆ = H all skew-Hermitian forms Φ are equivalent. Therefore in the case (c) there exists a basis of V such that the matrix of Φ is J (since the form given by J is skew-Hermitian). We redenote this basis by E. We identify the algebra M∞(∆) with the corresponding set of finitary transformations of V ⊥ (with respect to the basis E). Set Un = e1,...,e2n∆ and Un = e2n+1,e2n+2,... . We have ⊥ V = Un ⊕ Un for all n ≥ 1. Let now x be a finitary transformation of V preserving the form Φ, that is,

Φ(xv,w)+Φ(v,xw)=0 forall v,w ∈ V . (1)

⊥ ⊥ ⊥ Then there exists n such that xV ⊆ Un. Since x preserves Φ, xUn ⊆ Un . As Un ∩ Un = 0, we ⊥ have xUn = 0. It follows that x ∈ M∞(∆). Let X and B = Ik,J be the matrices of x and Φ with respect to the basis E. Then the condition (1) can be written in the matrix form

X∗B + BX = 0 where X∗ = Xt for ∆ = R, and X∗ = X¯ t for ∆ = C, H. It follows that any real ﬁnitary central simple Lie algebra of countable dimension is isomorphic to one of the algebras (1)–(7) in the theorem.

21 It remains to show that the algebras described in the theorem are pairwise nonisomorphic. In view of Proposition 4.3, it suﬃces to prove that all algebras of the same series (2), (4), or (6) are pairwise nonisomorphic. So assume that L = o∞(R,k) =∼ o∞(R,l) for some k 2k; Qm = om(R,l) for l < ∞; and Qm = om(R, m/2) with m even and greater than 4 for l = ∞. Therefore there are integers m 2l for l < ∞ and m > 2k for l = ∞ such that Qm ⊂ Ln ⊂ Qm′ . Since the embedding Qm ⊂ Qm′ is natural, by Lemma 2.8, the embedding Qm ⊂ Ln is natural. Represent the latter algebra in the form Ln = h(V, Φ) where Φ is a nondegenerate orthogonal form on V with inertia indices (k,n − k) (the numbers of −1 and 1 in the canonical matrix presentation). We have a unique decomposition V = W ⊕U where W is a natural Qm-module and QmU = 0. Assume that Φ|W is degenerate. Then there exists w ∈ W and u ∈ U such that Φ(w,u) = 0. Since QmW = W , there exist x ∈ Qm ⊂ Ln and w′ ∈ W such that w = xw′. Then

Φ(xw′,u)+Φ(w′,xu)=Φ(w,u) = 0.

It follows that x ∈ Ln does not respect Φ, which contradicts the assumption. Therefore Φ|W is nondegenerate. Let (k1, m − k1) be the inertia indices of this form. Clearly, k1 ≤ k. Observe that Qm ⊆ h(W, Φ|W )= om(R,k1). Since the dimensions of these algebras coincide (extending the ﬁeld to C, we obtain the algebras isomorphic to os(C) with s = m(m − 1)/2), we have Qm = h(W, Φ|W ), i.e. either om(R,l) =∼ om(R,k1) with m > 2l > 2k1 for l < ∞, or om(R, m/2) =∼ om(R,k1) with m > 2k1 for l = ∞. It follows from Jacobson’s classiﬁcation that both these isomorphisms are impossible (op(R,q1) =∼ op(R,q2) with p > 4 if and only if either q1 = q2 or q1 = p − q2). The proposition follows.

10 Finitary irreducible Lie algebras

The aim of this section is to classify infinite dimensional finitary irreducible Lie algebras over a field of characteristic 0. In fact the main result (Theorem 1.6) holds for any field of characteristic = 2 provided Theorem 1.1 and Theorem 8.2 (for irreducible finitary modules) are valid in the case of positive odd characteristic. Let ∆ be a finite dimensional division algebra over F and V be a (right) vector space over ∆. An element t ∈ End V is called a transvection if its rank (the dimension of tV ) is 1, i.e. there exists ∗ u ∈ V and ϕ ∈ V such that tv = u(ϕv) for all v ∈ V . We shall write t = tuϕ. Note that we use the term ”transvection” in a nonstandard way (in the standard definition one takes t − 1 rather than t). Recall that for a total subspace Π of V ∗ the ring F(V, Π) can be defined as

F(V, Π) = tuϕ | u ∈ V ϕ ∈ ΠZ.

The following properties of transvections can be checked by the direct calculations.

Lemma 10.1 For all v,u ∈ V , ϕ,ψ ∈ Π, and δ ∈ ∆ we have

(1) tv+u,ϕ = tvϕ + tuϕ;

(2) tu,ϕ+ψ = tuϕ + tuψ;

(3) tuδ,ϕ = tu,δϕ;

22 (4) tuϕtvψ = tu(ϕv),ψ = tu,(ϕv)ψ. In particular tuϕtvψ = 0 if and only if ϕv = 0.

Note that any element x ∈ F(V, Π) can be represented as a sum of n = rk x transvections.

Lemma 10.2 Let x ∈ F(V, Π) and n = rk x. Assume that x is represented as a sum of n transvections x = tu1ϕ1 + + tunϕn . Then

(1) u1,...,un are linearly independent over ∆;

(2) ϕ1,...,ϕn are linearly independent over ∆;

(3) ϕ1,...,ϕn ∈ Π.

Proof. (1) and (2) are obvious. Since x ∈ F(V, Π), it can be represented in the form x = tv1ψ1 +

+ tvmψm with ψ1,...,ψm ∈ Π. Using Lemma 10.1(1–3), one can assume that v1,...,vm and ψ1,...,ψm are linearly independent. Therefore

v1,...,vm∆ = xV = u1,...,un∆.

Representing each vi as a linear combination of the uj and using Lemma 10.1, we have that x = tu1θ1 + + tunθn with θ1,...,θn ∈ Π. Therefore tu1,ϕ1−θ1 + + tun,ϕn−θn = 0. This implies ϕi = θi ∈ Π for all i.

Proposition 10.3 Let V be an inﬁnite dimensional vector space over ∆ and Π be a total subspace of V ∗. Let x be a ﬁnitary endomorphism of V as an F -space. Assume that [x, F(V, Π)] ⊆ F(V, Π). Then x ∈ F(V, Π). Moreover, if [x, F(V, Π)] = 0, then x = 0.

Proof. Assume that x = 0. First of all we show that x is ∆-linear. Since x is ﬁnitary, there exists u ∈ V such that xu = 0. Let ϕ ∈ Π. Set

τ =[x,tuϕ]= xtuϕ − tuϕx = −tuϕx.

By assumption, τ is ∆-linear. Therefore for all v ∈ V and δ ∈ ∆ we have

0= τ(vδ) − (τv)δ = −tuϕx(vδ)+ tuϕ(xv)δ = uϕ((xv)δ − x(vδ)).

Since ϕ can be taken arbitrary in Π and Π is total, (xv)δ = x(vδ), as desired.

Since x is finitary and ∆-linear, x can be represented in the form x = tu1ϕ1 + + tunϕn where all u1,...,un and ϕ1,...,ϕn are linearly independent over ∆. It suffices to show that ϕi ∈ Π for i = 1,...,n. Without loss of generality we can restrict ourselves to the case i = 1. Take u ∈ V and ϕ ∈ Π such that ϕ1u = = ϕnu = 0, ϕu1 = 1, and ϕu2 = = ϕun = 0. Then it is not difficult to check that [x,tuϕ]= −tuϕtu1ϕ1 = −tuϕ1 . Now by Lemma 10.1(3), tuϕ1 ∈ F(V, Π) if and only if ϕ1 ∈ Π, as required. Observe that [x,tuϕ] = 0. Therefore [x, F(V, Π)] = 0 implies x = 0. The proposition follows. Proof of Theorem 1.6. By [14, 15], L has a unique minimal ideal M. Moreover, M is simple and M = [[L(4), L](1), L] (M = [L, L] if char F = 0). Since M is an ideal of L, by [13], V is an irreducible M-module. Let ∆1 be the centralizer of the M-module V . We have ∆1 ⊆ ∆. Let Γ be the centroid of M. Denote by MΓ the Lie ring M considered as a Γ-algebra. The algebra Γ M is central simple. By Lemma 7.2, V respects the action of Γ, so Γ ⊆ ∆1. By Theorem 8.2,

23 Γ V is the standard M -module. Therefore EM = EMΓ = F(V1, Π) where EM and EMΓ are the Γ enveloping rings of M and M , respectively, V1 is the abelian group V considered as a ∆1-space, ∗ and Π is a total subspace of V1 . Since M is an ideal, [x,EM] ⊆ EM for all x ∈ L. Therefore by I Proposition 10.3, L⊆ EM. It follows that ∆1 = ∆ and EL = EM = F(V, Π). If M is of type A , then M = fsl(V, Π), so we get (1). Assume now that M preserves a Hermitian or skew-Hermitian α form Φ, i.e. M = fsh(V, Φ) = sh (EM) where α is the relevant involution of EM. Let x ∈ L and y ∈M. Since y and z =[x,y] lie in M, we have yα = −y and zα = −z. Therefore

[x + xα,y]= xy + xαy − yx − yxα =(xy − yx)+(xy − yx)α = z + zα = 0.

α α α It follows that [x + x , M]=0, so [x + x ,EM] = 0. Now by Proposition 10.3, x + x = 0, so α α x = −x . Therefore L ⊆ fh(V, Φ) = h (EM) and we get (2), (3), or (4). If M = [L, L] is central over F , then the codimensions of fsl(V, Π) in fgl(V, Π) and of fsu(V, Φσ) in fu(V, Φσ) equal to 1, since this is true in the ﬁnite dimensional case. Therefore either L =[L, L] is simple or L = fgl(V, Π) or fu(V, Φσ). The theorem follows.

ACKNOWLEDGMENTS

The author has been supported by INTAS (Project 93-183-Ext), by the Belarus Basic Research Foundation (Project F97M-072), and by the Institute of Mathematics of the National Academy of Sciences of Belarus in the framework of the state program “Mathematical structures”.

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