Twisted Internal coHom Objects in the Category of Quantum Spaces

Details Twisted Internal coHom Objects in the Category of Quantum Spaces Twisted Internal coHom Objects in the Category of Quantum Spaces

2001-12-20

arxiv.org/abs/math/0112233v4

Mathematics

Quantum Algebra

23 pages; Added references and minor changes (including more accurate terminology)

Scientific paper

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).

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