AN ALGEBRAIC MONOID APPROACH TO LINEAR ASSOCIATIVE ALGEBRAS, II

Abstract

Let A be a linear associative K-algebra with unity (LAA), K an algebraically closed field. A K–subalgebra of A containing the unity of A is referred to as a sub-LAA of A. A with respect to the multiplication (respectively, Lie bracket) is an algebraic monoid (respectively, Lie algebra) over K, denoted by AM (respectively, AL). Let G(A) and J(A) denote the unit group of AM and Jacobson radical of A, respectively. The following are proved in this paper. (i) If BM is a Borel submonoid of AM, the Zariski closure of a Borel subgroup of G(A), then B is a sub-LAA of A and [Formula: see text], where [Formula: see text] is the set of Borel submonoids of AM. (ii) If dim A≠3, then G(A) is nilpotent iff for every proper sub-LAA N of A, G(N) is nilpotent. In addition, a relationship of Borel submonoids of AM and Borel subalgebras of AL is revealed.