Finitely Presented Monoids and Algebras Defined by Permutation Relations of Abelian Type, II

The class of finitely presented algebras $A$ over a field $K$ with a
set of generators $x_{1},\ldots ,x_{n}$ and defined by homogeneous
relations of the form
$x_{i_1}x_{i_2}\cdots x_{i_l} =x_{\sigma (i_1)} x_{\sigma (i_2)} \cdots
x_{\sigma (i_l)}$,
where $l\geq 2$ is a given integer and $\sigma$ runs through a subgroup $H$ of $\Sym_{n}$, is
considered. It is shown that the underlying monoid $S_{n,l}(H)=\langle x_1,x_2,\dots ,x_n\mid
x_{i_1}x_{i_2}\cdots x_{i_l} =x_{\sigma (i_1)} x_{\sigma (i_2)}
\cdots
x_{\sigma (i_l)}, \;
\sigma \in H, \; i_1,\ldots ,i_l \in \{ 1, \ldots , n \}
\rangle $ is cancellative if and only if $H$ is semiregular and abelian. In this case $S_{n,l}(H)$ is a submonoid of its universal group $G$.
If, furthermore, $H$ is transitive then the periodic elements $T(G)$ of $G$ form a finite abelian subgroup, $G$ is periodic-by-cyclic and it is a central localization of $S_{n,l}(H)$, and
the Jacobson radical of the algebra $A$ is determined by the Jacobson radical of the group algebra $K[T(G)]$.
Finally, it is shown that if $H$ is an arbitrary group that is transitive then $K[S_{n,l}(H)]$ is a Noetherian PI-algebra of Gelfand-Kirillov dimension one; if furthermore $H$ is abelian then often $K[G]$ is a principal ideal ring. In case $H$ is not transitive then $K[S_{n,l}(H)]$ is of exponential growth.

Jaar van publicatie:2015

Trefwoorden:semigroup ring, finitely presented, semigroup, Jacobson radical, semiprimitive, primitive

CSS-citation score:1

Toegankelijkheid:Open