Monoids of IG-type and maximal orders

Abstract

Let G be a finite group that acts on an abelian monoid A. If ϕ : A → G is a map so that ϕ ( a ϕ ( a ) ( b ) ) = ϕ ( a ) ϕ ( b ) , for all a , b ∈ A , then the submonoid S = { ( a , ϕ ( a ) ) | a ∈ A } of the associated semidirect product A ⋊ G is said to be a monoid of IG-type. If A is a finitely generated free abelian monoid of rank n and G is a subgroup of the symmetric group Sym n of degree n, then these monoids first appeared in the work of Gateva-Ivanova and Van den Bergh (they are called monoids of I-type) and later in the work of Jespers and Okniński. It turns out that their associated semigroup algebras share many properties with polynomial algebras in finitely many commuting variables. In this paper we first note that finitely generated monoids S of IG-type are epimorphic images of monoids of I-type and their algebras K [ S ] are Noetherian and satisfy a polynomial identity. In case the group of fractions S S −1 of S is torsion-free abelian then it is characterized when K [ S ] also is a maximal order. It turns out that they often are, and hence these algebras again share arithmetical properties with natural classes of commutative algebras. The characterization is in terms of prime ideals of S, in particular G-orbits of minimal prime ideals in A play a crucial role. Hence, we first describe the prime ideals of S. It also is described when the group S S −1 is torsion-free.