Algebras and groups defined by permutation relations of alternating type

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.