Finitely presented algebras and groups defined by permutation relations

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 $a_1{a}_2...a_n=a_{\sigma (a)}{a}_{\sigma (2)}...a_{\sigma (n)}$, where $\sigma$ runs through a subset $H$ of the symmetric group $\Sym_{n}$ of degree $n$, is introduced. The emphasis is on the case of a cyclic subgroup $H$ of $\Sym_{n}$ of order $n$. A normal form of elements of the algebra is obtained. It is shown that the underlying monoid, defined by the same (monoid) presentation, has a group of fractions and this group is described. Properties of the algebra are derived. In particular, it follows that the algebra is a semiprimitive domain. Problems concerning the groups and algebras defined by arbitrary subgroups $H$ of $\Sym_{n}$ are proposed.

10 pages

References in corpus (1)

• Primitivity of finitely presented monomial algebras