Commutative Nilpotent Subalgebras with Nilpotency Index n-1 in the Algebra of Matrices of Order n

Resumo

The paper establishes the existence of an element with nilpotency index n − 1 in an arbitrary nilpotent commutative subalgebra with nilpotency index n−1 in the algebra of upper niltriangular matrices N_n(????) over a field ???? with at least n elements for all n ≥ 5, and also, as a corollary, in the full matrix algebra M_n(????). The result implies an improvement with respect to the base field of known classification theorems due to D. A. Suprunenko, R. I. Tyshkevich, and I. A. Pavlov for algebras of the class considered.