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arXiv · 0810.0352

Finitely presented algebras and groups defined by permutation relations

F. Cedo·E. Jespers·J. Okninksi
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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 $a_{1}a_{2}... a_{n} =a_{σ(a)} a_{σ(2)} ... a_{σ(n)}$, where $σ$ 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.

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F. Cedo, E. Jespers, J. Okninksi. 2008-10-02. Finitely presented algebras and groups defined by permutation relations. https://arxiv.org/abs/0810.0352

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