Lie properties in associative algebras

TL;DR

This paper explores Lie properties in associative algebras, providing explicit generators, studying Lie centralizers, and offering constructive proofs for key theorems to deepen understanding of Lie-nilpotent subalgebras and automorphisms.

Contribution

It introduces explicit matrix generators for M_{n}(K) as a Lie algebra, studies Lie centralizers, and offers a constructive proof of the Skolem-Noether theorem.

Findings

Explicit generators for M_{n}(K) as a Lie algebra.

Results on Lie centralizers and nilpotent subalgebras.

Constructive proof of the Skolem-Noether theorem.

Abstract

Let K be a field, then we exhibit two matrices in the full nxn matrix algebra M_{n}(K) which generate M_{n}(K) as a Lie K-algebra with the commutator Lie product. We also study Lie centralizers of a not necessarily commutative unitary algebra and obtain results which we hope will eventually be a step in the direction of, firstly, proving that a Lie-nilpotent K-subspace (or a sub Lie K-algebra) of a finite-dimensional associative algebra over K of index k (say) generates a Lie-nilpotent associative subalgebra of much higher nilpotency index, and secondly, in the light of the sharp upper bound for the maximum (K-)dimension of a Lie-nilpotent K-subalgebra of M_{n}(K) of index k (obtained earlier), finding an upper bound for the maximum dimension of a Lie-nilpotent (of index k) sub Lie K-algebra of M_{n}(K). Finally, the constructive elementary proof of the Skolem-Noether theorem for the matrix algebra M_{n}(K) (appeared in the American Math. Monthly), in conjunction with the well-known characteization of Lie automorphisms of M_{n}(K) (if the characteristic of K is different from 2 and 3) in terms of, amongst others, automorphisms and anti-automorhisms of M_{n}(K), leads us to a unifying approach to constructively describe automorphisms and anti-automorphisms of M_{n}(K).