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 x1,…,xn defined by homogeneous relations of the form xi1xi2⋯xil=xσ(i1)xσ(i2)⋯xσ(il), where l≥2 is a given integer and σ runs through a subgroup H of Symn, is considered. It is shown that the underlying monoid Sn,l(H)=〈x1,x2,…,xn|xi1xi2⋯xil=xσ(i1)xσ(i2)⋯xσ(il),σ∈H,i1,…,il∈{1,…,n}〉 is cancellative if and only if H is semiregular and abelian. In this case Sn,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 Sn,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[Sn,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[Sn,l(H)] is of exponential growth.