Simple Lie subalgebras of locally nite associative algebras

Simple Lie subalgebras of locally nite associative algebras Y.A. Bahturin Department of Mathematics and Statistics Memorial University of Newfoundland St. John's, NL, A1C5S7, Canada A.A. Baranov Department of Mathematics University of Leicester Leicester, LE1 7RH, UK A.E. Zalesski School of Mathematics University of East Anglia Norwich, NR4 7TJ, UK Abstract We prove that any simple Lie subalgebra of a locally nite associative algebra is either nite dimensional or isomorphic to the commutator algebra of the Lie algebra of skew symmetric elements of some involution simple locally nite associative algebra. The ground eld is assumed to be algebraically closed of characteristic 0. This result can be viewed as a classication theorem for simple Lie algebras that can be embedded in locally nite associative algebras. We also establish a link between this class of Lie algebras and that of Lie algebras graded by nite root systems. Key words: Locally nite Lie algebra, simple Lie algebra, involution Email addresses: yuri@math.mun.ca (Y.A. Bahturin), baranov@mcs.le.ac.uk (A.A. Baranov), A.Zalesskii@uea.ac.uk (A.E. Zalesski). 1 The rst author acknowledge a support of an NSERC grant # 227060-00 and URP grant of Memorial University of Newfoundland. 2 The second author acknowledges a support of the Nueld Foundation (NAL/00515/G). Preprint submitted to Elsevier Science 25 September 2005

1 Introduction Throughout the paper we x an algebraically closed eld F of characteristic 0. We study simple innite-dimensional Lie subalgebras of locally nite associative algebras. Our main result describes such algebras in terms of algebras of skew-symmetric elements in simple locally nite associative algebras with involution. Recall that an algebra A (associative, Lie, etc) is called locally nite if any nite set M A is contained in a nite-dimensional subalgebra. Locally nite algebras can be alternatively described as direct limits of nite-dimensional algebras. A well-known theorem of Ado says that any nite-dimensional Lie algebra is isomorphic to a Lie subalgebra of a matrix algebra or, equivalently, to a Lie subalgebra of a nite-dimensional associative algebra. It would be wrong to think that this result remains true if one replaces \nite-dimensional" by \locally nite". In fact, for a Lie algebra the condition of being a subalgebra of a locally nite associative algebra is fairly restrictive. A step in understanding this phenomenon was made by the second author who proved in [5] the following \version of Ado's Theorem" for locally nite Lie algebras. Theorem 1.1 Let L be a simple innite-dimensional Lie algebra. Then L is embeded in a locally nite associative algebra if and only if L is isomorphic to a diagonal direct limit of nite-dimensional Lie algebras (see denition below in Section 2). This theorem was conjectured by the third author who also introduced the term \diagonal direct limit" ([12]). Theorem 1.1 can be viewed as a local characterization of simple Lie subalgebras of locally nite associative algebras because the diagonal direct limit is dened in terms of embeddings of nite dimensional subalgebras.this denition mimics certain properties of embeddings of nite dimensional associative algebras. Our aim in this paper is to obtain a global characterization, without any reference to nite-dimensional subalgebras. At this point we turn our reader's attention to Lie subalgebras of associative algebras with involution. Let A be an associative algebra over a eld F which for this paragraph can be arbitrary of characteristic dierent from 2. Suppose that A has an involution (which will be always denoted by \ "), that is, a linear transformation of A such that (a ) = a and (ab) = b a for all a; b 2 A. Then the set u (A) = fa 2 A j a = ag of all skew-symmetric elements of A is a Lie subalgebra of A. Let su (A) = [u (A); u (A)] denote the commutator subalgebra of u (A). It is well-known (see [11]) that if A is involution simple and the dimension of A is greater than 16, then su (A) is an extension of a simple Lie algebra by an ideal contained in the center of A. This construction 2

yields classical nite-dimensional Lie algebras of types B n, C n, and D n, when A is a matrix algebra over F with appropriate involution (in this case, already u (A) is simple). One can adapt the construction to obtain simple Lie algebras of type A n. Let B be a simple associative over F with center Z(B). Then L = [B; B]=([B; B] \ Z(B)) is a simple Lie algebra. However, taking B B op for A and endowing A with the involution (a; b) = (b; a) for a; b 2 B, one has L = su (A)=(su (A) \ Z(A)). Finite dimensional simple Lie algebras of exceptional types or simple Lie algebras of Cartan type do not appear in this way. Recall that the ground eld is assumed to be algebraically closed and of characteristic 0. One can show that each locally nite Lie algebra constructed as [A; A] or su (A) for a (involution) simple locally nite associative algebra A is simple and diagonal (see Theorem 2.12). It follows from one of our main results (see Theorem 1.3(2) below) that the converse statement is also true. Combining this with Theorem 1.1, we get the following characterization of simple Lie algebras that are embeddable into locally nite associative algebras. Theorem 1.2 Let L be a simple innite-dimensional Lie algebra. Then L can be embedded into a locally nite associative algebra if and only if L is isomorphic to su (A) where A is an involution simple locally nite associative algebra. It has to be emphasized that A cannot be in general obtained as the associative envelope of L in the original embedding. As in the nite-dimensional case, there are many \non-equivalent" embeddings of L into associative locally nite algebras. Another point to warn the reader is that simple locally nite associative algebras as well as their simple Lie subalgebras are not always expressible as direct limits of simple or even semisimple nite dimensional subalgebras. Some examples of this kind have been provided in Bahturin and Strade [4]. Theorem 1.2 considerably sharpens Theorem 1.1 and can be viewed as a classication theorem for simple Lie algebras that are embeddable into locally nite associative algebras. We do not see any straightforward way to prove Theorem 1.2. We rst use Theorem 1.1 to reformulate the problem in terms of diagonal embeddings of nite dimensional (but not necessarily semisimple) subalgebras L i. Next we use the results of [8,9] that under certain conditions diagonal Lie subalgebras of a matrix algebra can be obtained as su (A i ) where A i is now the envelope of L i. Finally, roughly speaking, A is obtained as a direct limit of A i. Thus, the proof is heavily dependent on Theorem 1.1 and the papers [8,9] where the second and the third authors develop a theory of diagonal and plain representations of nite-dimensional Lie algebras (see all the denitions below). 3

Next we formulate in detail the main results of this paper. Let A be an associative enveloping algebra of a Lie algebra L (i.e. L is a Lie subalgebra of A and A is generated by L as an associative algebra). We say that A is a P-envelope of L if [A; A] = L. We say that A is a P -envelope of L if A has an involution such that su (A) = L. Each enveloping algebra A of a Lie algebra L can be considered as a quotient of the augmentation ideal A(L) (i.e. the ideal of codimension 1 of the universal enveloping algebra U(L)). Thus there is a 1{1 correspondence A! H A between the enveloping algebras A for L and the ideals H A in A(L) such that H A \ L = 0 and A(L)=H A = A. This gives a partial ordering on the set of enveloping algebras of L: we say that A B if and only if H A H B. To formulate our main result we denote by Rad A the Jacobson radical of an associative algebra A. Theorem 1.3 Let L be an innite-dimensional simple diagonal locally nite Lie algebra. Then there is a unique (universal) P -envelope N of L such that the following conditions hold. (1) Set R = Rad N. Then R is the annihilator of N. (2) M = N =R is an involution simple P -envelope of L. (3) For each P -envelope A of L one has M A N. Corollary 1.4 The mapping L 7! M is a 1-1 correspondence between the set of all (up to isomorphism) innite-dimensional simple diagonal locally nite Lie algebras and the set of all (up to isomorphism) innite-dimensional involution simple locally nite associative algebras. (The inverse mapping is given by A 7! su (A)). There is a special class of simple diagonal locally nite Lie algebras, which are called plain (see Section 2 for the denition). These algebras play a role similar to that of nite-dimensional simple Lie algebras of type A (see above). Theorem 1.5 Let L be an innite-dimensional simple plain locally nite Lie algebra. Then there are two (universal) P-envelopes N + and N of L such that the following conditions hold. (1) Put R = Rad N. Then R is the annihilator of N. (2) M = N =R is a simple P-envelope of L. (3) For each P-envelope A of L one has either M + A N + or M A N. 4

(4) The mapping : N +! N dened as (x 1 : : : x k ) = ( x k ) : : : ( x 1 ); (k 2 N; x 1 ; : : : ; x k 2 L) is an antiisomorphism. Corollary 1.6 The mapping L 7! M + (L) is a 1-1 correspondence between the set of all (up to isomorphism) innite-dimensional simple plain locally nite Lie algebras and the set of all (up to isomorphism and antiisomorphism) innite-dimensional simple locally nite associative algebras. (The inverse mapping is A 7! [A; A]). Remark 1.7 In Example 3.8 we construct a simple plain locally nite Lie algebra L such that dim(rad N ) = 1. Considering the regular representation of N, we conclude that there exists a non-split extension of a plain L-module V (i.e. the restriction V #L i is plain for all i) by the trivial one-dimensional module. Since each simple plain locally nite Lie algebra is also diagonal, it has both P-envelope and P -envelope. The following theorem describes the relation between these algebras. Theorem 1.8 Let L be an innite-dimensional simple diagonal locally nite Lie algebra. Let M be as in Theorem 1.3 and M as in Theorem 1.5, in the case where L is plain. Then L is plain if and only if M is not simple. In this case M decomposes into the direct sum of two ideals B + B where B = M and B + = B. In the nal section we link the theory of diagonal locally nite Lie algebras to the theory of root-graded Lie algebras, developing further the results of Bahturin and Benkart [2]. Root-graded Lie algebras have been introduced by Berman and Moody for studying the toroidal algebras and Slodowy's intersection matrix algebras [10] (see also an important monograph [1] by Allison, Benkart, and Gao for more references). We show that each simple root-graded locally nite Lie algebra is diagonal and the converse is also true provided we slightly generalize the notion of root-graded Lie algebras (Theorem 4.3). Actually, we prove that any locally nite diagonal Lie algebra is BC r -graded in the sense of [1] (which was also noticed in [2] in the case of diagonal locally nite-dimensional simple algebras of types B; C; D). From this result and the results of Allison-Benkart-Gao [1] one can obtain another proof of the fact that any simple diagonal locally nite Lie algebra can be obtained as a Lie subalgebra of skew symmetric elements of a suitable asssociative algebra. The above results make sense for innite-dimensional algebras only. Thus, if otherwise is not stated, each locally nite Lie algebra, considered in the paper, is assumed to be innite-dimensional. 5

2 Notation and preliminaries A Lie algebra L is called perfect if L = [L; L]. If L is a perfect nite-dimensional Lie algebra and V a nite dimensional L-module then Irr L (resp. Irr L) stands for the set of all isomorphism classes of irreducible (resp. nontrivial irreducible) nite-dimensional L-modules and by Irr V (resp. Irr V ) the set of all isomorphism classes of composition factors (resp. nontrivial composition factors) of V. In particular, Irr L = Irr L [ ft L g and Irr V = Irr V or Irr V [ ft L g where T L is the trivial 1-dimensional L-module. We denote by U(L) the universal enveloping algebra of L and by A(L) its augmentation ideal, i.e. the ideal of codimension 1 generated by L. In this paper (as well as in [8,9]) we mainly work with A(L) rather than with the whole of U(L), and the notion of Irr is sometimes more suitable for us than that of Irr, used in [5,6]. We need to prove some results from [5], stated in this new setting. To this end, the following lemma will be helpful. Lemma 2.1 Let L be a nite-dimensional perfect Lie algebra and V a nitedimensional L-module. Then T L 2 Irr V if and only if Ann U(L) V A(L) (or equivalently, Ann U(L) V = Ann A(L) V ). Proof. Assume T L 2 Irr V. As [L; L] = L, the algebra L acts trivially on T L. Therefore Ann U(L) V Ann U(L) T L = A(L), as required. Assume T L 62 Irr V. Let E be the image of U(L) in End V, in other words, E = U(L)= Ann U(L) V. Since L is perfect, the dimension of all composition factors of V is greater than 1. In particular, E= Rad E has no quotients of dimension 1. Since dim U(L)=A(L) = 1, we have Ann U(L) V 6 A(L). Denote by F (resp. F) the set of all (two-sided) ideals in U(L) (resp. A(L)) of nite codimension. Clearly, each ideal of A(L) is also an ideal of U(L), so that F F. For any X 2 F the quotient U(L)=X is a nite dimensional L-module under the left regular action, hence the notation Irr(U(L)=X) makes sense. Lemma 2.2 F = fx 2 F j T L 2 Irr(U(L)=X)g. Proof. Let X 2 F. Then obviously T L 2 Irr(U(L)=X). Assume now that X 2 F and T L 2 Irr(U(L)=X). As X = Ann U(L) U(L)=X, by Lemma 2.1 one has X A(L), that is, X 2 F. Let be a nite subset of Irr L. Set = \ Irr L and F( ) = fx 2 F j Irr(U(L)=X) = g; F( ) = fx 2 F j Irr(A(L)=X) = g: 6

Assume T L 2. Then it follows from Lemma 2.2 that F( ) = F( ). The sets F( ) have been described in [5, Theorem 3.4]. Our argument immediately yields a similar description for F( ). Theorem 2.3 Let L be a perfect nite-dimensional Lie algebra, a nite subset of Irr L, and F( ) = fx 2 F j Irr(A(L)=X) = g. Then F( ) is nonempty and has the smallest element N( ) and the largest element M( ), such that N( ) X M( ) for all X 2 F( ). The algebra A(L)=M( ) is semisimple, while M( )=N( ) is nilpotent. Recall, in a slightly dierent form, that a set fl i g i2i of nite-dimensional subalgebras of a locally nite Lie algebra L is a local system of L if L = S i2i L i and for each pair i; j 2 I there exists k 2 I such that L i ; L j L k. Set i j if L i L j. Then I becomes a directed set, i.e. a partially ordered set such that for each pair i; j 2 I there exists k 2 I satisfying i; j k. It is clear that L is the direct limit of the algebras L i, that is, L = lim! L i. Assume that L is simple. Then by [3, Theorem 3.2], all L i can be chosen perfect. We shall call such local systems perfect. Locally nite Lie algebras admitting perfect local systems are called locally perfect. We shall consider only perfect local systems for simple locally nite Lie algebras. So the notation L = lim! L i always means that fl i g i2i is a perfect local system of L. In order to dene diagonal locally nite Lie algebras we rst need the notion of a diagonal module. Let L be a nite-dimensional perfect Lie algebra such that L= Rad L = S 1 S n is the sum of simple components S i. We x a Cartan subalgebra of each S i and a base of the root system. Denote by V i the standard S i -module (i.e. the irreducible S i -module of highest weight 1, which is the rst fundamental weight with respect to the standard labeling). Note that our denition depends on the choice of a base of the root system. We can change it in such a way that the dual module V i becomes standard. However, up to duality, V i doesn't depend on the choice of a Cartan subalgebra and a root system of S i if rank of S i is not too small, which will be our typical situation. Indeed, in this case we do not need to consider the components of exceptional types, so each S i can be always assumed classical, i.e it can be identied with sl(v i ), o(v i ), or sp(v i ), and the standard S i -module is precisely the unique (up to duality) nontrivial irreducible S i -module of minimal dimension. Each V i can be considered as an L-module. The L-modules V 1 ; : : : ; V n are called standard. An L-module V is called diagonal if each nontrivial composition factor of V is either standard or dual to it. An L-module V is called plain if each S i is of type A (i.e. S i = sl(vi )) and each nontrivial composition factor of V is standard. The denition of a plain module slightly diers from that in [8]. Remark 2.4 By changing the base of the root system (or by relabeling the 7

simple roots) we can turn V i into a standard L i -module. This gives us some freedom in the choice of a standard module, which we are going to use in the future. Assume we have another perfect Lie algebra L 0 containing L. Let V1; 0 : : : ; V 0 k be the standard L 0 -modules. The embedding L L 0 is called diagonal (resp. plain) if the restriction of the direct sum V 0 1 V 0 k to L is a diagonal (resp. plain) L-module. We illustrate this denition by the following example. An embedding sl(v )! sl(w ) is diagonal if and only if one can choose a basis of W such that A 7! diag(a; : : : ; A; A t ; : : : ; A t ; 0; : : : ; 0) {z } l {z } r {z } z for any matrix A 2 sl(v ) where l; r; z do not depend on A, z + (l + r) dim V = dim W. This embedding is plain if r = 0. By the rank of a perfect nite-dimensional Lie algebra L we mean the smallest rank of the simple components of L= Rad L. We need the following. Lemma 2.5 Let L 1 L 2 L 3 be three perfect nite-dimensional Lie algebras. Assume that the ranks of L 1 and L 3 are greater than 10 and the embedding L 1 L 3 is diagonal. Then the embedding L 1 L 2 is diagonal. Moreover, if the restriction of each standard L 2 -module to L 1 embeddings L 1 L 2 and L 2 L 3 are diagonal. is nontrivial, then both Proof. By choosing Levi subalgebras of the L i, each embedded into another, one can assume that the algebras L i are semisimple. Moreover, replacing L 2 by the ideal of L 2 generated by L 1, one can assume that the restriction of each standard L 2 -module to L 1 is nontrivial. It suces to show that for each standard L 3 -module W the restriction W#L 2 is diagonal. Indeed, in that case all standard L 2 -modules can obtained as composition factors of such restrictions, so their restrictions to L 1 are diagonal. Let M 2 Irr(W#L 2 ). The module M can be represented in the form M = M 1 M k where M i is a nontrivial irreducible module for a simple component S i of L 2. As the restriction of each standard L 2 -module to L 1 is nontrivial and M#L 1 is diagonal, k = 1. It remains to note that M = M 1 can not be non-standard (see [6, Lemma 5.2]). Let L be a locally nite Lie algebra and L = fl i g i2i a local system of L. We say that L is diagonal (resp. plain) if it is perfect and for each pair L i L j the corresponding embedding is diagonal (resp. plain). Note that in [6, Denition 3.7] we use the term \pure diagonal" instead of \diagonal". A locally nite Lie algebra L is called diagonal (resp. plain) if it has a diagonal (resp. plain) local system. 8

Let L = lim! L i be a locally perfect Lie algebra. Associated with the direct limit, there is a local system L = fl i g i2i of perfect subalgebras in L. Let i be a nite nonempty subset of Irr L i. The set = f ig i2i is called an inductive system (of representations) for L with respect to L if [ '2 j Irr('#L i ) = i; for each pair i < j. If L is xed then we simply say that is an inductive system for L. An inductive system = f ig i2i is called diagonal (resp. plain) if for each i, L '2 i ' is a diagonal (resp. plain) L i -module. We say that is selfdual, if = := f i g i2i where i = f' j ' 2 ig. Otherwise, is nonselfdual. If i = ft Li g for all i 2 I, the system is called trivial. Otherwise, is called nontrivial. We shall denote by [ft L g the system f i [ft Li g i2i g. More generally, let = f ig i2i and = f i g i2i be inductive systems. Then the union [ = f i [ i g i2i is an inductive system as well. As before, it is sometimes convenient to \forget" about trivial modules and to dene a reduced inductive system by replacing Irr L by Irr L and Irr( ) by Irr( ) in the denition above. We denote by S (resp. S) the set of inductive systems (resp. reduced inductive systems) of L with respect to L. We have a mapping S! S dened by = f ig i2i 7! = f ig i2i : (1) One can easily check that this mapping is surjective (if 2 S, then = [ ft L g 2 S). Denote by G (resp. G) the set of (two-sided) ideals of U(L) (resp. A(L)) such that the corresponding factor algebra is locally nite. As before G G. Let X 2 G. Then the set (X) = firr(u(l i )=X \ U(L i ))g i2i is an inductive system for L ([5, Lemma 3.8]). Let X 2 G. Set (X) = firr(a(l i )=X \ A(L i ))g i2i : Then (X) is a reduced inductive system for L and (X) = (X) in the sense of (1). Lemma 2.6 G = fx 2 G j T Li 2 (X) i ; for all ig. Proof. Let X be an ideal of U(L). Clearly X A(L) if and only if X \U(L i ) A(L i ) for all i. Thus the result follows from Lemma 2.2. Let be an inductive system for L and be a reduced inductive system for L. Set 9

G( ) = fx 2 G j (X) = g; G( ) = fx 2 G j (X) = g: Assume T Li 2 i for all i. Then it follows from Lemma 2.6 that, G( ) = G( ). Since the sets G( ) have been described in [5, Theorem 3.9], we immediately derive a similar description for G( ), as follows. The same result can also be directly deduced from Theorem 2.3. Theorem 2.7 Let f : G! S be a mapping dened by f(x) = (X). Then for each reduced inductive system the set G( ) is nonempty and has the smallest element N( ) and the largest element M( ) such that N( ) X M( ) for each X 2 G( ). The algebra A(L)=M( ) is semisimple, while M( )=N( ) is locally nilpotent. Moreover, the mapping f produces a 1-1 correspondence between the set of semiprimitive ideals in G and the set of reduced inductive systems for L (the inverse mapping is given by 7! M( )). To proceed further, we need few facts about simple (and involution simple) locally nite associative algebras. They are similar to those for locally nite Lie algebras and are proved in a similar way. Recall that an associative algebra with involution \ " is called involution simple (or -simple) if it has no - invariant ideals. The following trivial observation is often helpful for studying involution simple algebras. Proposition 2.8 Let A be an associative algebras with involution. Assume that A is involution simple. Then either A is simple or A = B B where B is a simple ideal of A. Proof. Assume that A is not simple and let B be a non-zero proper ideal of A. Then B is an ideal of A and B +B is a -invariant ideal of A, so B +B = A. Let C = B\B. Then C is a proper -invariant ideal of A, so C = 0. Therefore A = B B. Let A be an associative algebra. We denote by A n the linear span of all products x 1 x n, x i 2 A, and call A perfect if A 2 = A. Assume that A has an involution. Then, if A is nite-dimensional, it is well-known (see for example [9, Lemma 2.4]) that A has a -invariant Levi subalgebra Q. Let A be a locally nite associative algebra with involution and A = fa i g i2i a local system of A. Let ^A i be the subalgebra of A generated by A i + A i. Then ^A = f ^A i g i2i is a -invariant local system of A (i.e. ^A i = ^A i for all i 2 I). Moreover ^A is perfect if A is perfect. Now we dene a conical local system A = (A i ) i2i of ( -invariant) perfect subalgebras of an (involution) simple associative algebra A. First of all, we require that I has the smallest element 1 and that 10

(1) A 1 A i for all i 2 I; (2) A 1 is an (involution) simple algebra; (3) the restriction of any nontrivial A i -module to A 1 is nontrivial. Note that (3) implies that (4) for each (involution) simple component T of A i = Rad A i one has dim T dim A 1. Let N be a proper ( -invariant) ideal of A i. Since A i is perfect, the image of N in A i = Rad A i is a proper ideal. Hence the codimension of N in A i is not less than the minimal dimension of (involution) simple components of A i = Rad A i. Combining this with (4), we get the following property of conical systems: (5) for each i 2 I and each proper ( -invariant) ideal N of A i, one has codim N dim A 1. By the rank of a conical system we mean the dimension of A 1. Proposition 2.9 Let A be an (involution) simple locally nite associative algebra and let A = fa i g i2i be a perfect ( -invariant) local system of A. Fix k 2 I. Let S be a (involution) simple component of a ( -invariant) Levi subalgebra of A k. Denote by A S i (i k) the two-sided ideal of A i generated by S and set A S 1 = S. Put I S = fi 2 I j i kg [ f1g. Then A S = fa S i g i2i S is a conical local system of A. Proof. Clearly, A S i is a perfect ( -invariant) ideal of A i and A S i A S j whenever i j. Therefore A S = lim A! S i is a ( -invariant) ideal of A. Since A is (involution) simple, A S = A, so fa S i g i2i S is a perfect ( -invariant) local system of A. The properties (1), (2), and (3) of the denition of a conical system are satised in an obvious manner. Now we can apply the argument in [3, Theorem 3.3] and Proposition 2.8 to easily get the following. Proposition 2.10 Let A and A be as in Proposition 2.9. Then for any i 2 I there exists j i and a maximal ( -invariant) ideal P j of A j such that P j \ A i = 0. In particular, a ( -invariant) Levi subalgebra of A j has a (involution) simple component S with dim S dim A i. Combining this with Proposition 2.9 we derive the following Corollary 2.11 Locally nite (involution) simple associative algebras have conical ( -invariant) local systems of arbitrary large ranks. Finally, we can prove our rst important result. 11

Theorem 2.12 (1) Let A be a simple locally nite associative algebra. Set L = [A; A]. Then L is a simple plain locally nite Lie algebra and A is a P-envelope of L. (2) Let A be an involution simple locally nite associative algebra. Set L = su (A). Then L is a simple diagonal locally nite Lie algebra and A is a P - envelope of L. Proof. (1) By Corollary 2.11 A has a conical local system fa i g i2i of rank > 4. Set L i = [A i ; A i ]. By property (5) of conical systems, each A i has no proper ideals of codimension 4, i.e. A i is strongly perfect in the sense of [8]. Hence, by [8, Theorem 6.3(1)], L i is perfect, A i is a plain L i -module with respect to the left regular action, and L i generates A i. Therefore L generates A. We also claim that fl i g i2i is a plain local system of L. Indeed, we need to show that for each pair L i L j the corresponding embedding is plain, i.e. the restriction of each standard L j -module V to L i is a plain L i -module. Let A j = A j + F1 be the algebra that is obtained from A by external adjointing the identity. Then A j is a faithful plain L j -module. Thus V is a composition factor of its submodule A j, so it is enough to show that the restriction of A j to L i is a plain L i -module. The latter is obvious as this restriction factors through A i. Denote by Rad A the maximal locally nilpotent ideal of A and by Rad L the maximal locally solvable ideal of L. Clearly Rad A \ A i Rad A i and Rad L \ L i Rad L i for all i. Note that Rad L i Rad A i for all i. Indeed, the image of L i in Q = A i = Rad A i is [Q; Q], which is a direct sum of simple Lie algebras of type A. Fix any x 2 Rad L. Then there exists k such that x 2 Rad L i for all i k. Hence x 2 Rad A i for all i k, so x 2 Rad A. Therefore Rad L Rad A. As Rad A = 0, this implies that Rad L = 0. Now, to prove simplicity of L, one could use the method developed in [6], by showing that the Bratteli diagrams of A and L are identical, and applying a simplicity criterion in terms of Bratteli diagrams (see for example [6, Theorem 3.2]). However it is easier to use a much more general result [11, Theorem 4], which claims that for any simple ring R (of characteristic dierent from 2), each proper ideal of [R; R] is in the center of R. Since Rad L = 0, we get that L is simple. (2) By Corollary 2.11 A has a conical -invariant local system fa i g i2i of rank > 36. Set L i = su (A i ). Then by [9, Theorem 6.3], L i is perfect, A i is a diagonal L i -module, and L i generates A i. Thus L generates A. Moreover, as in (1), we get that fl i g i2i is a diagonal local system of L. It remains to check that L is simple. As in (1), Rad L i Rad A i for all i. (this follows, for example, from [9, Lemmas 2.3 and 2.4]. Thus Rad L = 0, so we could use a simplicity criterion in terms of Bratteli diagrams [6, Theorem 3.2]. The only dierence is that we have to modify a bit the notion 12

of Bratteli diagrams for associative algebras with involution in an obvious way: the nodes of the diagram must correspond to the involution simple components of A i = Rad A i rather than to the ordinary simple components. However it is easier to refer to [11, Theorem 10], which claims that each proper ideal of su (A) must be in the center of A. Since Rad L = 0, we get that L is simple. 3 Locally nite Lie algebras To formulate a Lie algebra analog of Theorem 2.9 (see [6]) we have to dene the Lie algebra counterpart of the notion of a conical system. If L is a Lie algebra then a local system L = (L i ) i2i of nite-dimensional perfect subalgebras is called conical if I contains the smalest element 1 such that (1) L 1 L i for all i 2 I; (2) L 1 is simple; (3) for each i 2 I the restriction of any standard L i -module to L 1 is nontrivial. Note that (3) implies that (4) for each i 2 I and each simple component T of L i = Rad L i, one has dim T dim L 1. Let N be a proper ideal of L i. Since L i is perfect, the image of N in L i = Rad L i is a proper ideal. Hence the codimension of N in L i is not less than the minimal dimension of simple components of L i = Rad L i. Combining this with (4), we get the following property of conical systems: (5) for each i 2 I and each proper ideal N of L i, one has codim N dim L 1. By the rank of the conical system L we mean the rank of L 1. Proposition 3.1 Let L be a simple locally nite Lie algebra and let L = fl i g i2i be a perfect local system for L. Fix k 2 I. Let S be a simple component of a Levi subalgebra of L k. Denote by L S i (i k) the ideal of L i generated by S and set L S 1 = S. Put I S = fi 2 I j i kg [ f1g. Then L S = fl S i g i2i S is a (perfect) conical local system of L with the following additional properties. (6) If L is diagonal (resp. plain), then L S is diagonal (resp. plain). (7) Let i 2 I. Fix any j 2 I such that L i L S j. Assume that the embedding L i L j is diagonal (resp. plain). Then the embedding L i L S j is diagonal (resp. plain). 13

Proof. Since S is perfect, L S i is a perfect ideal of L i. Indeed, let SL k denote [: : : [S; L] : : : L] where L occurs k times (k 0). Then L S i = 1X k=0 SL k = 1X k=0 [S; S]L k = 1X X k=0 i+j=k [SL i ; SL j ] [L S i ; L S i ]: Clearly L S i L S j whenever i j. Therefore L S = lim! L S i is an ideal of L. Since L is simple, L S = L, so fl S i g i2i S is a perfect local system of L. Now the properties (1), (2), and (3) of the denition of the conical system are obvious. The properties (6) and (7) of the Proposition easily follow from the fact that the simple components of L S i = Rad L S i are simple components of L i = Rad L i. For full details see [6, Section 3]. Proposition 3.2 ([3, Theorem 3.3]) Let L and L be as in Proposition 3.1. Then for any i 2 I there exists j i and a maximal ideal P j of L j such that P j \ L i = 0. In particular, a Levi subalgebra of L j has a simple component S with dim S dim L i. Corollary 3.3 Simple locally nite Lie algebras have conical local systems of arbitrary large ranks. Let L = fl i g i2i and M = fm j g j2j be perfect local systems of L, and = f ig i2i be an inductive system with respect to L. For each j 2 J x k(j) 2 I such that M j L k(j). Set j = [ '2 k(j) Irr('#M j ): Then it is not dicult to see that = f j g j2j is an inductive system with respect to M. We shall denote this system by #M. Clearly, ( #M)#L = : Proposition 3.4 Let L be a simple locally nite Lie algebra and let L = fl i g i2i and M = fm j g j2j be two diagonal local systems of L. Let = f jg j2j be a diagonal inductive system with respect to M. Then #L is a diagonal inductive system with respect to L. Proof. Let L S = fl S i g i2i S be a conical diagonal local system of rank > 10 constructed as in Proposition 3.1. Fix i 2 I S and any j(i) 2 J such that L S i M j(i). Fix any i 0 2 I S such that L S i M j(i) L S i 0: Since the embedding L S i L S 0 i is diagonal, by Lemma 2.5, the embedding M j(i) is diagonal. Therefore the restriction of each standard M j(i) -module L S i 14

to L S i is diagonal. This implies that = #L S is a diagonal inductive system. By Proposition 3.1(7), #L is diagonal. Therefore #L = #L S #L = #L is diagonal, as required. Let L = fl i g i2i be a perfect local system of a locally nite Lie algebra L. Let V i 1 ; : : : ; Vk i i be the standard L i -modules. Set i = fv i 1 ; : : : ; V i k i ; T Li g; i = f(v i 1 ) ; : : : ; (V i k i ) ; T Li g; ^ i = i [ i Irr L i : Lemma 3.5 Let L be a simple locally nite Lie algebra and L = fl i g i2i be a perfect local system of L. Then L is diagonal (resp. plain) if and only if ^ = f^ i g i2i (resp. = f i g i2i ) is an inductive system for L. Proof. This follows from the denition of diagonal and plain embeddings. We shall need the following easy observation from [5, Section 5]. Lemma 3.6 Let L be a simple locally nite Lie algebra, fl i g i2i a perfect local system of L, and = f ig i2i an inductive system. Then for each simple component S of L i = Rad L i there is ' 2 i such that '#S is nontrivial. Lemma 3.7 Let L be a simple locally nite Lie algebra and L = fl i g i2i a perfect local system of L. Suppose that = f ig i2i is a nontrivial diagonal inductive system for L. Then the following are true. (1) If =, then [ft L g = ^ for all i. In particular, L and L are diagonal. (2) If 6=, then L is plain. (3) If 6= and L is plain, then [ ft L g = or. Proof. (1) It follows from Lemma 3.6 that for each i 2 I and each j = 1; : : : ; k i, i contains Vj i or (V i j ). Since i = i, we have i [ ft Li g = ^ i, as required. (2) By changing the bases of the root systems (see Remark 2.4) and by using Lemma 3.6, we can assume that i contains all standard L i -modules. As 6=, there is k 2 I and a simple component S of a Levi subalgebra of L k such that k contains the standard S-module W and W =2 k. Let L S = fl S i g i2i S be the conical local system generated by S (see Proposition 3.1). Set = #L S. Clearly, is a diagonal inductive system, i contains all standard L S i -modules, 15

and 1 = Irr( i#s) = fwg. As L S is conical, for each i 2 I S the restriction of any standard M i -module V to S is nontrivial, so Irr(V #S) = fwg. It follows that i \ i ft Li g, so and L S are plain. Thus L is plain. (3) Assume [ft L g 6=. Set = [. In view of Lemma 3.6, it suces to show that =. Clearly is a diagonal inductive system containing and 6=. Arguing as in (1), we see that there is a conical local system L S of L such that #L S is plain. Therefore #L S = #L S. Hence =, as required. Proof of Theorem 1.2. This follows from Theorems 1.1, 2.12(2), and 1.3(2) (which we prove below). Proof of Theorem 1.3. Let L be a simple diagonal locally nite Lie algebra. By Proposition 3.1 and Corollary 3.3, there is a diagonal conical local system L = fl i g i2i of rank > 10. Then ^ is a diagonal inductive system with respect to L. Let be the reduced inductive system obtained from ^. Then by Lemma 3.7, is the only diagonal reduced inductive system for L. Let the ideals N( ) and M( ) of A(L) be as in Theorem 2.7. Set N = A(L)=N( ); M = A(L)=M( ): By Theorem 2.7, M = N =R where R is the Jacobson radical of N. Since L is simple, M( ) \ L = 0, so N and M are enveloping algebras of L. Consider the left regular action of A(L i ) on V i = U(L i )=(N( ) \ A(L i )): This makes V i into a faithful L i -module with Irr V i = i, so V i is diagonal. Observe that N( ) is invariant under the action of the standard involution of A(L i ). Therefore V i is selfdual (hence -plain in the sense of [9]). Clearly N i = A(L i )=(N( ) \ A(L i )) is an enveloping algebra of L i in End V i with involution inherited from the standard involution of A(L i ). Therefore by [9, Theorem 1.3(2)], N i is a P - envelope of L i, i.e. su (N i ) = L i. Hence su (N ) = L, i.e. N is a P -envelope of L. Since R \ L = 0, we also have that su (M) = L. Let us prove that M is involution simple. Let Q be any proper -invariant ideal of M. As L generates M and L is simple, one has Q \ L = 0. Thus the reduced inductive system corresponding to the preimage of Q in A(L) is a nontrivial -invariant diagonal subsystem of. Hense by Lemma 3.7(1), 16

=. Thus by Theorem 2.7, Q = 0. Therefore M is involution simple, and (2) is proved. Set R i = R \ N i. Then R i is a -invariant ideal of N i with R i \ L i = 0. Therefore N i =R i is a P -envelope of L. Now [9, Theorem 6.5] implies that R i annihilates N i. Therefore R annihilates N, so (1) is proved. Let us prove (3). Represent A as A(L)=H A where H A is an ideal of A(L). In view of Theorem 2.7, it suces to show that the reduced inductive system corresponding to H A coincides with. Let fa i g i2i be a -invariant local system of A. Since L is simple, by Proposition 3.2, L has a nite dimensional simple subalgebra Q of rank greater than 8. Fix any i such that A i contains Q and x any -invariant Levi subalgebra of A i. As Q su (A i ), the Levi subalgebra has a -simple component S of suciently large dimension (we need > 36). Let A S = lim A! S i be the ideal of A generated by S (see proof of Proposition 2.9). Note that A S is -invariant and fa S i g i2i S is a conical local system of A S. Observe that su (A S ) is an ideal of L. Since L is simple and su (A S ) su (S) 6= 0, we get that su (A S ) L. As L generates A, we have A S = A. Thus fa S i g i2i S is a conical local system of A of rank greater than 36. By property (5) of conical systems, each A S i has no proper -invariant ideals of codimension 36. In particular, A S i is admissible in the sense of [9]. Thus, by [9, Theorem 6.3], M i := su (A S i ) is a perfect Lie algebra and A S i is a selfdual diagonal (or equivalently, -plain) M i -module (with respect to the regular action). Clearly, M = fm i g i2i S is a local system of L and = f ig i2i S with i = Irr(A S i ) is a diagonal selfdual reduced inductive system for L with respect to M. By Proposition 3.4, #L is a diagonal selfdual inductive system. Therefore by Lemma 3.7(1) #L =, as required. Let us denote by f 1 and f 2 the mappings L 7! M Proof of Corollary 1.4. and A 7! su (A), respectively. By Theorem 1.3, f 2 f 1 (L) = L. Let A be an involution simple locally nite associative algebra. Then by Theorem 2.12(2) L = su (A) is a diagonal simple Lie algebra and A is a P -envelope of L Therefore by Theorem 1.3, A = M, so f 1 f 2 (A) = A, as required. Proof of Theorem 1.5. Let L be a simple plain locally nite Lie algebra. By Corollary 3.3, there is a plain conical local system L = fl i g i2i of rank > 10. Then is a plain inductive system with respect to L. Let + (resp. ) be the reduced inductive system obtained from (resp. ). Then by Lemma 3.7(3), + and are the only non-selfdual diagonal reduced inductive systems. Let the ideals N( ) and M( ) of A(L) be as in Theorem 2.7. Set 17

N = A(L)=N( ); M = A(L)=M( ): By Theorem 2.7, M = N =R where R is the Jacobson radical of N. Since L is simple, M( ) \ L = 0, so N and M are enveloping algebras of L. Consider the left regular action of A(L i ) on V i = U(L i )=N( + ) \ A(L i ): This makes V i into a faithful L i -module with Irr V i = ( + ) i, so V i is plain. Clearly N i + = A(L i )=(N( + ) \ A(L i )) is an enveloping algebra of L i in End V i. Therefore by [8, Theorem 1.5], N i + is is a P-envelope of L i, i.e. [N+; i N+] i = L i. Hence [N + ; N + ] = L, i.e. N + a P-envelope of L. Arguing similarly, we conclude that N and M are P- envelopes of L. Let us prove that M is simple. Let Q be any proper ideal of M. As L generates M and L is simple, one has Q\L = 0. Thus the reduced inductive system corresponding to the preimage of Q in A(L) is a nontrivial diagonal subsystem of. Hense by Lemma 3.7(3), =. Thus by Theorem 2.7, Q = 0. Therefore M is simple, and (2) is proved. Set R i = R \ N i. Then R i is an ideal of N i with R i \ L i = 0. Therefore N i =R i is a P-envelope of L. Now [8, Theorem 6.10(6)] implies that R i annihilates N i. Therefore R annihilates N, so (1) is proved. Let us prove (3). Represent A as A(L)=H A where H A is an ideal of A(L). In view of Theorem 2.7, it suces to show that the reduced inductive system corresponding to A coincides with + or. Let fa i g i2i be a local system of A. Since L is simple, by Proposition 3.2, L has a nite dimensional simple subalgebra Q of rank greater than 8. Fix any i such that A i contains Q and x any Levi subalgebra of A i. As Q [A i ; A i ], the Levi subalgebra has a simple component S of suciently large dimension (we need > 4). Let A S = lim! A S i be the ideal of A generated by S (see proof of Proposition 2.9). Note that fa S i g i2i S is a conical local system of A S. Observe that [A S ; A S ] is an ideal of L. Since L is simple and [A S ; A S ] [S; S] 6= 0, we get that [A S ; A S ] L. As L generates A, we have A S = A. Thus fa S i g i2i S is a conical local system of A of rank greater than 4. By property (5) of conical systems, each A S i has no proper ideals of codimension 4, i.e. A S i is strongly perfect in the sense of [8]. Thus, by [8, Theorem 6.3(1)], M i = [A S i ; A S i ] is a perfect plain Lie algebra and A S i is a plain M i - module (with respect to the regular action). Clearly, M = fm i g i2i S is a local system of L and = f ig i2i S with i = Irr(A S i ) is a plain reduced inductive 18

system for L with respect to M. By Proposition 3.4, #L is a diagonal inductive system. Since = ( #L)#M and is non-selfdual, #L is non-selfdual. Therefore #L =, as required. (4) follows from the duality of + and. Indeed, consider the opposite algebra N op +, i.e. the vector space N + with new multiplication dened as a b = ba. Since a b b a = ba ab = [a; b], the mapping x 7! x is a Lie homomorphism of L into N op +, which can be extended to an antiisomorphism of N + onto N op +. Clearly the image of L generates N op + as an associative algebra, so N op + is a locally nite envelope of L. Observe that the corresponding reduced inductive system is. Therefore by Theorem 2.7, N op + = N, as required. Proof of Corollary 1.6. Let us denote by f 1 and f 2 the mappings L 7! M + and A 7! [A; A], respectively. By Theorem 1.5, f 2 f 1 (L) = L. Let A be a simple locally nite associative algebra. Then by Theorem 2.12(1), L = [A; A] is a plain simple Lie algebra and A is a P-envelope of L. Therefore by Theorem 1.5, A = M, so f 1 f 2 (A) = A, as required. Proof of Theorem 1.8. Let L be a simple plain locally nite Lie algebra and be let M be as in Theorem 1.5. Set A = M + M. Let : M +! M as in Theorem 1.5(4). Recall that is an antiisomorphism and (x) = x for all x 2 L. Set (a; b) = ( 1 (b); (a)). Then is an involution of A. Note that A is involution simple. The embedding x 7! (x; x) (x 2 L) turns A into an envelope of L. One can easily check that u (A) = f(a; (a)) j a 2 M + g Therefore L = su (A), so A is a simple P -envelope of L. Theorem 1.3 implies that A = M, as required. Assume now that M is not simple. Then by Theorem 2.8, M = B B. One can easily check that su (M) = [B; B], so B is a simple P-envelope of L. Therefore by Theorem 1.5, B = M. In the example below we construct a simple plain locally nite Lie algebra L such that the radical of N is nonzero. Considering the regular representation of N, we conclude that there exists a non-split extension of a plain L-module V (i.e. the restriction V #L i is plain for all i) by the trivial one-dimensional module. Example 3.8 Let us recall the construction of the algebra L n (n 3) from [8, Example 6.12]. We denote by L n the Lie algebra of (3n + 3) (3n + 3) 19

matrices of the form 0 0 x 2 x 4 1 x = B @ x 0 x 1 0 x 0 x 1 x 3 0 x 2 x 0 C A (2) where x 0 runs over all n n matrices with zero traces; x 1 ; x 2 ; x 3 run over all matrices of sizes n 1, 1 n, n n, respectively; x 4 = tr x 3 (one can see that x 4 is a 1 1 matrix); and all empty spaces are zero matrices. Let x 2 L n and m = 3n + 1. We denote by n (x) the element y of L m such that y 0 = 0 B @ x 0 x 0 x 1 x 3 0 x 2 x 0 1 C A ; y 1 = 0 B @ x 1 0. 0 1 0 ; y 3 = C A B @ x 3 0 y 2 = (x 2 0 : : : 0), and y 4 = x 4. Then the mapping n : L n! L m is an injective homomorphism. We will identify L n with n (L n ). Obviously, Rad L m = fy 2 L m j y 0 = 0g, so L n \ Rad L m = 0. Let A n be the enveloping algebra of L n in our matrix representation (2). It was shown in [8, Example 6.12] that A n is a universal P-envelope of L n and A n consists of all matrices of the form (2) where x 0 ; : : : ; x 4 are arbitrary. Denote by R n the two-sided annihilator of A n in A n. Clearly, R n is the one-dimensional subspace consisting of matrices with x 0 = x 1 = x 2 = x 3 = 0. Note that the associative subalgebra of A m generated by L n is isomorphic to A n. Identifying this subalgebra with A n, we see that R n = R m. Let H n and H n be the kernels of the canonical homomorphisms A(L n )! A n and A(L n )! A n =R n, respectively. It follows from the arguments above that H m \ A(L n ) = H n and H m \ A(L n ) = H n. Let us denote by L the direct limit of the sequence L n! 3n+1 n L3n+1! 9n+4 L 9n+4! : : : Obviously, A(L) has ideals H and H such that H \ A(L n ) = H n and H \ A(L n ) = H n for each algebra L n in the sequence. Since L n \ Rad L m = 0, and L m = Rad L m is simple, the algebra L is simple. One can easily check that A = A(L)=H is a universal P-envelope of L, R = H=H is the Jacobson radical (and the annihilator) of A of dimension 1, and A = A=R is a simple P-envelope of L.... 0 1 C A ; 20

4 Generalized root-graded locally nite Lie algebras In this section we link diagonal locally nite Lie algebras to root-graded Lie algebras. We show that each simple root-graded locally nite Lie algebra is diagonal and the converse is also true, provided we generalize slightly the notion of root-graded Lie algebras. Denition 4.1 Let be a root system of type X n (X = A; : : : ; G) and let P ( ) be the group of integral weights of. Let be a subset of P ( ) containing and 0. A Lie algebra L is called -graded if ( 1) L contains as a Lie subalgebra a nite-dimensional simple Lie algebra g = h L2 g whose root system is relative to a Cartan subalgebra h = g 0 ; ( 2) L = L 2 L where L = fx 2 L j [h; x] = (h)x for all h 2 hg; ( 3) L 0 = P ;2 nf0g[l ; L ]. The subalgebra g is called the grading subalgebra of L. The assumption ( 3) is included for nondegeneracy (e.g. consider L = g M where g and M are ideals of L). However the following trivial observation can be useful. Lemma 4.2 Let g and be as in Denition 4.1. Assume that a Lie algebra L satises ( 1) and ( 2). Then the space X M X S = ( L ) [L ; L ] 2 nf0g ;2 nf0g is a nonzero -graded ideal of L. In particular, if L is simple then L is - graded. Proof. Using [L ; L ] L +, one can easily check that [x; S] S for all x 2 L, so S is an ideal of L. Clearly S is -graded. Let g and be as in Denition 4.1. Asssume that is classical of type A n, B n, C n, or D n. We denote by all weights of the g-module V = (V V ) (V V ) (V V ) V V T g where V is the standard g-module (of weight 1 ) and T g is the trivial 1- dimensional g-module. If g is of type A n, denote by A all weights of the g-module V A = (V V ) V V T g : 21

Obviously and A contain and 0. Theorem 4.3 Let L be an innite-dimensional simple locally nite Lie algebra. (1) Assume that L is -graded where is nite. Then L is diagonal. (2) Assume that L is diagonal. Then L is -graded for each root system of classical type (A n, B n, C n, or D n ). Moreover, if L is plain then L is A - graded. Proof. (1) Let g be the grading subalgebra of L and let be its root system. Fix any root 2 and pick any non-zero x 2 g and x 2 g. Then S = hx ; x ; [x ; x ]i F is a subalgebra of g isomorphic to sl 2. Let 1 be the restriction of to the Cartan subalgebra F[x ; x ] of S. Then 1 is nite and L is 1 -graded with the grading subalgebra S of type A 1. Indeed the properties ( 1) and ( 2) obviously hold and ( 3) follows from Lemma 4.2. Let L = fl i g i2i be a perfect local system for L containing S. Let L S = fl S i g i2i S be the conical local system generated by S, so L S 1 = S. Put i = Irr(L#L S i ) := [ ji Irr(L S j #L S i ) where L and L S j are considered as L S i -modules with respect to the adjoint action. Then clearly [ '2 j Irr('#L S i ) = i for each pair i < j. Thus = f ig i2i S is an inductive system for L if we show that each i is a nite set. Note that 1 = 1, so 1 is nite. Assume that i is innite for some i. Since S \ Rad L i = 0 and Rad L i annihilates all irreducible L i -modules, without loss of generality we can assume that L i is semisimple. Let L i = Q 1 Q k be the decomposition of L i into a sum of simple components Q j. Then each ' 2 i can be written as ' = ' 1 ' k where ' j is an irreducible Q j -module. Clearly '#S = (' 1 #S 1 ) (' k #S k ) where S j is the projection of S into Q j. By Proposition 3.1(3), each projection S j is nontrivial, so S j = S = sl2. Since i is innite, there exists j such that the set j i = f' j j ' 2 ig Irr Q j is innite. Hence the set Irr( j i#s j ) is also innite (this follows, for example, from [5, Lemma 6.5]). This implies that 1 = 1 = Irr( i#s) = Irr(Irr( 1 i #S 1 ) Irr( k i #S k )) 22

is innite, which contradicts the assumption. Thus each i is nite, so = f ig i2i S is a nontrivial inductive system for L. Thus by [6, Corollary 3.9], L is diagonal. (2) Let be a root system of classical type. By Proposition 3.1 and Corollary 3.3, L has a conical diagonal local system L = fl i g i2i of suciently large rank, so that L 1 contains a diagonally embedded simple Lie algebra g with root system. Fix any i 2 I. By Proposition 3.2, there exists j > i such that the g-module L i is isomorphic to a submodule of the g-module L j =P j where P j is a maximal ideal of L j. Now L j =P j is isomorphic to a submodule of the g-module W W where W is a standard L j -module. Since the embedding g! L j is diagonal, Irr(W#g) fv; V ; T g g: Therefore the weights of the g-module W W belong to, so the weights of the g-module L i belong to. Thus L is -graded. If L is plain, then we can assume that L = fl i g i2i is a plain local system for L. Thus Irr(W#g) fv; T g g; so the weights of the g-module W W belong to A. Therefore L is A - graded. Notice that in the case where = B n, C n, or D n the set reduces to the set of all weights of the g-module V = (V V ) V T g : This means that -graded algebras are actually BC r -graded where r is the rank of g. Therefore, the following is true. Corollary 4.4 Each simple diagonal locally nite Lie algebra is BC r -graded for all r 1. ACKNOWLEDGMENTS The authors thank the Ban International Research Station for the excellent environment created during the International Workshop \Locally Finite Lie Algebras" (August 30 - September 4, 2003), where the nal discussions about this paper took place. The authors acknowledge that their discussions and 23