On subalgebras of n× n matrices not satisfying identities of degree 2 n−2

Abstract

The Amitsur–Levitzki theorem asserts that M n ( F) satisfies a polynomial identity of degree 2 n. (Here, F is a field and M n ( F) is the algebra of n× n matrices over F.) It is easy to give examples of subalgebras of M n ( F) that do satisfy an identity of lower degree and subalgebras of M n ( F) that satisfy no polynomial identity of degree ⩽2 n−2. In this paper we prove that the subalgebras of n× n matrices satisfying no nonzero polynomial of degree less than 2 n are, up to F-algebra isomorphisms, the class of full block upper triangular matrix algebras.