Natural enveloping algebras
by
Alexander A. Baranov
Institute of Mathematics, National Academy of Sciences of Belarus
Coauthors: Alexander E. Zalesskii

The ground field F is an algebraically closed field of zero characteristic. Let L be a Lie subalgebra of an associative algebra A. The algebra A is called enveloping, if L generates A. An enveloping algebra A of L is called natural, if [A, A] subset or equal L subset or equal A.

Assume that L is finite dimensional simple and A is a natural enveloping of L. Then one can easily show that A is simple, so is isomorphic to a matrix algebra M(n, F), and L=[A, A]=sl(n, F). Therefore there is a bijective correspondence between the set of simple Lie algebras of type A and simple associative algebras of finite dimension greater than 1. We extend this result to simple locally finite dimensional (locally finite, for brevity) algebras. To state our results we need some notation.

It is convenient identify any enveloping algebra A with the relevant quotient of the augmentation ideal A(L) of U(L), i.e. the ideal of codimension 1 generated by L. Let us denote by H_A the kernel of the relevant homomorphism A(L) --> A. We say that A <= B, if H_B subset or equal H_A.

Let L be a finite dimensional perfect (i.e. [L, L]=L) Lie algebra. We say that L is of type A, if each simple component S_k (1 <= k <= m) of L/Rad L coincides with sl(n_k, F) for some n_k. We consider the standard (n_k-dimensional) S_k-module W_k as an L-module. Set \Phi_L={W₁, ..., W_m}. An embedding L --> Q of perfect Lie algebras of type A is called strictly diagonal, if for each W in \Phi_Q every nontrivial composition factor of the restriction of W to L belongs to \Phi_L. A simple locally finite Lie algebra is called strictly diagonal, if it can be represented as a direct limit of strictly diagonal embeddings of perfect Lie algebras of type A.

We say that an ideal H of an algebra A is a null-ideal, if HA=AH=H \cap [A, A]=0.

Theorem. Let L be an infinite dimensional simple strictly diagonal locally finite Lie algebra. Then there are two antiisomorphic (universal) natural enveloping algebras N₊ and N_- of L such that the following conditions hold.

(1) R _+/- =Rad N _+/- is a null-ideal of N _+/- .

(2) M _+/- =N _+/- /R _+/- is a simple natural enveloping of L.

(3) For each natural enveloping algebra A of L one has either M₊ <= A <= N₊ or M_- <= A <= N_-.

Corollary. The map L --> M₊(L) is a 1-1 correspondence between the set of all (up to isomorphism) infinite dimensional simple strictly diagonal locally finite Lie algebras and the set of all (up to isomorphism and antiisomorphism) infinite dimensional simple locally finite associative algebras. (The inverse map is A --> [A, A]).

To prove the theorem, we explicitly describe natural envelopings for perfect finite dimensional Lie algebras.

http://im.bas-net.by/~baranov

Date received: November 13, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabw-21.