COLLOQUIUM

 

Friday, October 4, 2002

4:30 P.M.

206 ARTS

 

SPEAKER

Prof. Yuri Bahturin

Memorial University of Newfoundland

Moscow State University

 

TITLE

Locally Finite Simple Lie Algebras

 

Abstract

A Lie algebra L is called locally finite if any finite subset of L generates a finite-dimensional subalgebra. This talk will be devoted to the structure theory of simple locally finite Lie algebras. In general this is a very difficult question but by now there are a whole number of situations where the structure can be completely described. In the case of algebras over algebraically closed field of prime characteristic at least 7 this is possible if the dimensions of classical simple factors are totally bounded. We recall that C is a factor of L if C is isomorphic to A/B where B is an ideal of A, and A is subalgebra of L. So the restriction is imposed on the dimensions of classical simple factors C, for which A is finite-dimensional. Such L can also be distinguished as Lie algebras satisfying a non-trivial polynomial identity. Another case is that of algebraically closed fields of zero characteristic and so-called diagonal direct limits of finite-dimensional classical simple algebras. A locally finite Lie algebra L is called diagonal if it is isomorphic to a Lie subalgebra under the ordinary bracket operation [a,b]=aböba of a locally finite associative algebra A. More recently some structural results became available in the case of algebras, which belong to the classes of root-graded algebras or their immediate generalizations. All necessary definitions will be given during the talk. We are going to report all these cases and discuss possible directions of further development.