Endomorphisms and null subalgebras of finite dimensional algebras

Abstract

In [5] it was shown that a finite dimensional binary algebra without nilpotents over an algebraically closed field of characteristic zero has only the trivial derivation and, consequently, a finite automorphism group. It is shown here that an m-ary algebra without nilpotents has only finitely many endomorphisms, with no assumptions on the characteristic of the base field. An upper bound on the dimension of the algebraic set of endomorphisms of a finite dimensional m-ary algebra, depending on the dimension of a maximal null subalgebra, is obtained if the base field is the complex field. The null subalgebras arising from a single derivation are examined for the case of an algebraically closed base field of characteristic zero.