Local Centrally Essential Subalgebras of Triangular Algebras

TL;DR

This paper investigates the structure of local centrally essential subalgebras within upper triangular matrix algebras, revealing that small matrices only have commutative such subalgebras, while larger matrices contain non-commutative ones.

Contribution

It characterizes the existence and nature of local centrally essential subalgebras in upper triangular matrix algebras of various sizes, highlighting the transition from commutative to non-commutative cases.

Findings

3x3 and 4x4 upper triangular matrix algebras have only commutative local centrally essential subalgebras.

Algebras of size greater than 6 contain non-commutative local centrally essential subalgebras.

The results contribute to understanding the subalgebra structure of triangular matrix algebras.

Abstract

We study local centrally essential subalgebras in the algebra of all upper triangular matrices over a field of characteristic $\ne 2$. It is proved that the algebras of upper triangular $3\times 3$ or $4\times 4$ matrices have only commutative local centrally essential subalgebras. Every algebra of upper triangular matrices of order exceeding $6$ contains a non-commutative local centrally essential subalgebra. The paper will appear in Linear and Multilinear Algebra. The work of O.V. Lyubimtsev is done under the state assignment No~0729-2020-0055. A.A. Tuganbaev is supported by Russian Scientific Foundation, project 16-11-10013P.