The minimal dimension of maximal commutative subalgebras of full matrix algebras

Abstract

Let F be a field and A a maximal commutative subalgebra of the full matrix algebra M n ( F ). It is shown that dim A > (2 n ) 2 3 − 1. It is also shown that if the radical of A has cube zero, then dim A ⩾ [3 n 2 3 − 4], and that this result is best possible for infinitely many natural numbers n .