The minimal dimension of maximal commutative subalgebras of full matrix algebras
A commutative subalgebra A of the full matrix algebra \(M_ n(F)\) over a field F is said to be maximal if the centralizer of A equals A. The purpose of this paper is to determine a lower bound for the dimension of such a subalgebra of \(M_ n(F)\). The author shows that if A is a maximal commutative subalgebra of \(M_ n(F)\) with (rad A)\({}^ 3=0\), then dim \(A\geq [3n^{2/3}-4]\), where rad A is the radical of A and [ ] denotes the greatest-integer function, and that this is the best possible lower bound by giving the explicit construction of an example attaining it for each n. In general case, it is shown that if A is a commutative subalgebra of \(M_ n(F)\), then dim A\(>(2n)^{2/3}-1\).
scientific article; zbMATH DE number 3925006