Twisted Internal coHom for Conic Algebras#

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 ϒ[ℬ, 𝒜] there, one for each pair 𝒜, ℬ ∈ CA, with analogous properties to its internal coHom objects hom [ℬ, 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, ℬ) ↦ hom [ℬ, 𝒜]. The mentioned non commutativity is controlled by a collection of twisting maps τ𝒜, ℬ. We show, under certain circumstances, that (bi)algebras end ϒ[𝒜] ≐ hom ϒ[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.