Pure ideals quotient categories and infinite group-graded rings

Abstract

Adapting the idea of twisted tensor products to the category of conic algebras (CA), i.e., finitely generated graded algebras, we define a family of objects hom ^??, 𝒜] there, one for each pair 𝒜, ? ∈ CA, with analogous properties to its internal coHom objects hom ?, 𝒜], but representing spaces of transformations whose coordinate rings and the ones of their respective domains do not commute among themselves. They give rise to a CA ^op -based category different from that defined by the function (𝒜, ?) ?  hom ?, 𝒜]. The mentioned non commutativity is controlled by a collection of twisting maps τ_{𝒜, ?}. We show, under certain circumstances, that (bi)algebras end ^?𝒜] ?  hom ^?𝒜, 𝒜] are counital 2-cocycle twistings of the corresponding coEnd objects end 𝒜]. This fact generalizes the twist equivalence (at a semigroup level) between, for instance, the quantum groups G L _q (n) and their multiparametric versions.