Algebras and groups defined by permutation relations of alternating type

The class of finitely presented algebras over a field K with a set of generators a1,…,an and defined by homogeneous relations of the form a1a2⋯an=aσ(1)aσ(2)⋯aσ(n), where σ runs through Altn, the alternating group of degree n, is considered. The associated group, defined by the same (group) presentation, is described. A description of the Jacobson radical of the algebra is found. It turns out that the radical is a finitely generated ideal that is nilpotent and it is determined by a congruence on the underlying monoid, defined by the same presentation.