Finitely presented monoids and algebras defined by permutation relations of abelian type (original) (raw)

Finitely presented algebras defined by permutation relations of dihedral type

International Journal of Algebra and Computation, 2016

The class of finitely presented algebras over a field [Formula: see text] with a set of generators [Formula: see text] and defined by homogeneous relations of the form [Formula: see text], where [Formula: see text] runs through a subset [Formula: see text] of the symmetric group [Formula: see text] of degree [Formula: see text], is investigated. Groups [Formula: see text] in which the cyclic group [Formula: see text] is a normal subgroup of index [Formula: see text] are considered. Certain representations by permutations of the dihedral and semidihedral groups belong to this class of groups. A normal form for the elements of the underlying monoid [Formula: see text] with the same presentation as the algebra is obtained. Properties of the algebra are derived, it follows that it is an automaton algebra in the sense of Ufnarovskij. The universal group [Formula: see text] of [Formula: see text] is a unique product group, and it is the central localization of a cancellative subsemigroup ...

Group algebras and semigroup algebras defined by permutation relations of fixed length

Journal of Algebra and Its Applications, 2015

Let H be a subgroup of Sym n, the symmetric group of degree n. For a fixed integer l ≥ 2, the group G presented with generators x1, x2,…,xn and with relations xi1xi2⋯xil = xσ(i1) xσ(i2)⋯xσ(il), where σ runs through H, is considered. It is shown that G has a free subgroup of finite index. For a field K, properties of the algebra K[G] are derived. In particular, the Jacobson radical 𝒥(K[G]) is always nilpotent, and in many cases the algebra K[G] is semiprimitive. Results on the growth and the Gelfand–Kirillov dimension of K[G] are given. Further properties of the semigroup S and the semigroup algebra K[S] with the same presentation are obtained, in case S is cancellative. The Jacobson radical is nilpotent in this case as well, and sufficient conditions for the algebra to be semiprimitive are given.