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

Abstract

The class of finitely presented algebras A over a field K with a set of generators x_{1},...,x_{n} and defined by homogeneous relations of the form x_{i_1}x_{i_2}...x_{i_l}=x_{sigma(i_1)}x_{sigma(i_2)}...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)= <x_1,x_2,...,x_n|x_{i_1}x_{i_2}...x_{i_l}=x_{sigma(i_1)}x_{sigma(i_2)}...x_{\sigma (i_l)}, sigma in H, i_1,...,i_l in {1,...,n}> 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.