Linear associative algebras with finitely many idempotents

Abstract

Let A be a linear (i.e., finite-dimensional) associative algebra with unity defined over K, an algebraically closed field. Then A with respect to its multiplication is an algebraic monoid over k, denoted by A_M , and with respect to the the bracket forms a Lie algebra over K, denoted by A_L . The following theorem is established A_M is nilpotent as an algebraic monoid (equivalentlyA_L is so as a Lie algebra) if and only if the set of idempotents of A is finite if and only if all irreducible closed submonoids of codimension 1 are nilpotent.