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).
Categories: math.QA math-ph math.MP