By weakening the counit and antipode axioms of a C∗-Hopf algebra and allowing for the coassociative coproduct to be non-unital we obtain a quantum group, that we call a weak C∗Hopf algebra, which is sufficiently general to describe the symmetries of essentially arbitrary fusion rules. This amounts to generalizing the Baaj-Skandalis multiplicative unitaries to multipicative partial isometries. E...

Comodules over Hopf algebroids are of central importance in algebraic topology. It is well-known that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff , the opposite category of commutative rings. We show in this paper that a comodule is the same thing as a quasi-coherent sheaf over this presheaf of groupoids. We prove the general theorem that internal equivalences of preshe...

The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalization endows T (T (B)+), the double tensor algebra of B, with the structure of a noncommutative bialgebra. When the bialgebra B is commutative, renormalization turns S(S(B)+), the double symmetric algebra of B, into a comm...

We study the Ehresmann--Schauenburg bialgebroid of a noncommutative principal bundle as quantization gauge groupoid classical bundle. show that group is isomorphic to bisections bialgebroid, and we give crossed module structure for automorphisms bialgebroid. Examples illustrating these constructions include: Galois objects Taft algebras, monopole over quantum spheres not faithfully flat Hopf--G...

Often A is called H-module algebra. We refer reader to [11, 6] for the basic information concerning Hopf algebras and their actions on associative algebras. Definition 1.2 The invariants of H in A is the set AH of those a ∈ A, that ha = ε(h)a for each h ∈ H. Straightforward computations show, that AH is subalgebra of A. The notion of action of Hopf algebra on associative algebra generalize the ...

Two Hopf algebras are called monoidally Morita equivalent if module categories over them are equivalent as linear monoidal categories. We introduce monoidal Morita invariants for finite-dimensional Hopf algebras based on certain braid group representations arising from the Drinfeld double construction. As an application, we show, for any integer n, the number of elements of order n is a monoida...

We show that the representation category of the quantum group of a non-degenerate bilinear form is monoidally equivalent to the representation category of the quantum group SLq(2) for a well-chosen non-zero parameter q. The key ingredient for the proof of this result is the direct and explicit construction of an appropriate Hopf bigalois extension. Then we get, when the base field is of charact...

Let A be a commutative comodule algebra over a commutative bialgebra H . The group of invertible relative Hopf modules maps to the Picard group of A, and the kernel is described as a quotient group of the group of invertible grouplike elements of the coring A⊗H , or as a Harrison cohomology group. Our methods are based on elementary K-theory. The Hilbert 90 Theorem follows as a corollary. The p...