Finitely presented monoids and algebras defined by permutation relations of abelian type, II

摘要

The class of finitely presented algebras A over a field K with a set of generators defined by homogeneous relations of the form xi(1)xi(2) . . . xi(1) = x(sigma(i1)) x(sigma(i2)) . . . x(sigma)(i(1)), where l >= 2 is a given integer and sigma runs through a subgroup H of Sym(n), is considered. It is shown that the underlying monoid Sn,l(H) = < x(1),x(2), . . . , x(n) X1 X2 " " " xi, = x,(ii)x,(to " " " x,(i,), a E H, E {1,, n}) is cancellative if and only if H is semiregular and abelian. In this case Sii,t(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 Sn,i(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,i(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,- I(H)] is of exponential growth. (C) 2014 Elsevier B.V. All rights reserved.