TY - JOUR

T1 - Finitely presented monoids and algebras defined by permutation relations of abelian type, II

AU - Cedó, Ferran

AU - Jespers, Eric

AU - Klein, Georg

PY - 2015/1/1

Y1 - 2015/1/1

N2 - © 2014 Elsevier B.V. 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.

AB - © 2014 Elsevier B.V. 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.

U2 - https://doi.org/10.1016/j.jpaa.2014.05.037

DO - https://doi.org/10.1016/j.jpaa.2014.05.037

M3 - Article

VL - 219

SP - 1095

EP - 1102

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

ER -