Algebras and groups defined by permutation relations of alternating type

TL;DR

This paper studies a class of finitely presented algebras and groups defined by permutation relations from the alternating group, describing their structure, radicals, and associated properties.

Contribution

It introduces a new class of algebras and groups defined by alternating group relations and characterizes their radicals and structural properties.

Findings

The radical is finitely generated and nilpotent.

The radical is determined by a specific congruence on the monoid.

The associated group presentation is explicitly described.

Abstract

The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},..., a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}... a_{n} =a_{\sigma (1)} a_{\sigma (2)} ... a_{\sigma (n)}$, where $\sigma$ runs through $\Alt_{n}$, the alternating group, is considered. The associated group, defined by the same (group) presentation, is described. A description of the radical of the algebra is found. It turns out that the radical is a finitely generated ideal that is nilpotent and it is determined by a congruence on the underlying monoid, defined by the same presentation.