Twisted Internal coHom Objects in the Category of Quantum Spaces

TL;DR

This paper introduces a framework for twisted internal Hom objects in the category of quantum spaces, revealing how noncommutative coordinate rings relate to quantum semigroup twistings and generalizing known quantum group equivalences.

Contribution

It defines twisted internal Hom objects in quantum spaces and shows their endomorphism algebras are 2-cocycle twistings of quantum semigroups, extending quantum group twistings.

Findings

End(A) algebras are 2-cocycle twistings of untwisted quantum semigroups.

Generalizes twist equivalence between quantum groups like GL_q(n) and multiparametric versions.

Provides a new perspective on noncommutative coordinate rings and their transformations.

Abstract

Adapting the idea of twisted tensor products to the category of finitely generated algebras, we define on its opposite, the category QLS of quantum linear spaces, a family of objects hom(B,A)^{op}, one for each pair A^{op},B^{op} there, with analogous properties to its internal Hom ones, but representing spaces of transformations whose coordinate rings hom(B,A) and the ones of their respective domains B^{op} do not commute among themselves. The mentioned non commutativity is controlled by a collection of twisting maps \tau_{A,B}. We show that the (bi)algebras end(A)=hom(A,A), under certain circumstances, are 2-cocycle twistings of the quantum semigroups end(A) in the untwisted case. This fact generalizes the twist equivalence (at a semigroup level) between, for instance, the quantum groups GL_{q}(n) and their multiparametric versions GL_{q,\phi}(n).