Simple Lie subalgebras of locally finite associative algebras

Abstract

We prove that any simple Lie subalgebra of a locally finite associative algebra is either finitedimensional or isomorphic to the commutator algebra of the Lie algebra of skew symmetric elements of some involution simple locally finite associative algebra. The ground field is assumed to be algebraically closed of characteristic 0. This result can be viewed as a classification theorem for simple Lie algebras that can be embedded in locally finite associative algebras. We also establish a link between this class of Lie algebras and that of Lie algebras graded by finite root systems.  2004 Elsevier Inc. All rights reserved.

Simple Locally Finite Lie Algebras of Diagonal Type

“…Similar results hold for locally finite associative algebras. In particular, every (involution) simple locally finite associative algebra A has a conical ( * -invariant) local system of subalgebras, see [2,Proposition 2.9]. Moreover, this system will be semisimple if A is locally semisimple.…”

“…A full classification of simple diagonal locally finite Lie algebras was obtained in [2]. We need some notation to state the result.…”

Simple, locally finite dimensional Lie algebras in positive characteristic

“…When the ground field has zero characteristic, this is always true. Thus, the proofs of theorems 1 and 2 give a new proof of the main result of [3]: Suppose F is an algebrically closed field of characteristic zero and suppose L is a simple, locally finite Lie algebra over F which embeds into a locally finite associative algebra. Then L is uniformly ad-integrable, and the proof of theorem 2 shows that L contains an ad-nilpotent element.…”

“…Suppose L is a simple, infinite dimensional Lie algebra over an algebraically closed field of characteristic zero. In [3], Bahturin, Baranov, and Zalesskii prove that L embeds into a locally finite associative algebra if and only if L is isomorphic to [K(R, * ), K(R, * )] where * is an involution and R is an involution simple locally finite associative algebra. This utilizes and extends earlier work of Baranov in [2].…”

“…For a ∈ L, let µ a (t) be the minimal polynomial of ad(a) and define d(L) = min{deg µ a (t)|a is not ad-nilpotent}. The condition that p > 2d(L) − 2 is trivial when F has characteristic zero, and so Theorem 2 also reduces to the result of [3] in this case.…”