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

Abstract

The class of finitely presented algebras over a field K with a set of generators a 1 , … , a n and defined by homogeneous relations of the form a 1 a 2 ⋯ a n = a σ ( 1 ) a σ ( 2 ) ⋯ a σ ( n ) , where σ runs through an abelian subgroup H of Sym n , the symmetric group, is considered. It is proved that the Jacobson radical of such algebras is zero. Also, it is characterized when the monoid S n ( H ) , with the “same” presentation as the algebra, can be embedded in a group in terms of the stabilizer of 1 and the stabilizer of n in H , where H is an arbitrary subgroup of Sym n . This work is a continuation of earlier work of Cedó, Jespers and Okniński.