Quantum groupoids and dynamical categories

Galois Theory, Hopf Algebras, and Semiabelian Categories, 2004

A useful general concept of bialgebroid seems to be resolving itself in recent publications; we give a treatment in terms of modules and enriched categories. We define t h e term "quantum category". The definition of antipode for a bialgebroid is less resolved in t h e literature. Our suggestion is that the kind of dualization occurring in Barr's star-autonomous categories is more suitable than autonomy (= compactness = rigidity). This leads to our definition of quantum groupoid intended as a "Hopf algebra with several objects". 2 ⊗ ⊗ η δ J J A J E

2003

In this work we prove, in a self contained way, that any finite dimensional C *-quantum groupoid can be deformed in order that the square of the antipode is the identity on the base. We also prove that for any C *-quantum groupoid with non abelian base, there is uncountably many C *-quantum groupoids with the same underlying algebra structure but which are not isomorphic to it. In fact, the C *quantum groupoids are closed in an analog of the procedure presented by D.Nikshych ([N] 3.7) in a more general situation.

We establish the equivalence of three approaches to the theory of finite dimensional quantum groupoids. These are the generalized Kac algebras of T. Yamanouchi, the weak Kac algebras, i.e., the weak C * -Hopf algebras introduced by G. Böhm-F. Nill-K. Szlachányi which have an involutive antipode, and the Kac bimodules. The latter are an algebraic version of the Hopf bimodules of J.-M. Vallin. We also study the structure and construct examples of finite dimensional quantum groupoids.

Journal of Physics A: Mathematical and Theoretical, 2017

Topology and its Applications, 2003

We use the categories of representations of finite dimensional quantum groupoids (weak Hopf algebras) to construct ribbon and modular categories that give rise to invariants of knots and 3-manifolds.

Journal of Algebra, 2007

We give a unified description of morphisms and comorphisms of Lie pseudoalgebras, showing that the both types of morphisms can be regarded as subalgebras of a Lie pseudoalgebra, called the $\psi$-sum. We also provide similar descriptions for morphisms and comorphisms of Lie algebroids and groupoids.

Journal of Geometry and Physics, 2015

We systematically study noncommutative and nonassociative algebras A and their bimodules as algebras and bimodules internal to the representation category of a quasitriangular quasi-Hopf algebra. We enlarge the morphisms of the monoidal category of A-bimodules by internal homomorphisms, and describe explicitly their evaluation and composition morphisms. For braided commutative algebras A the full subcategory of symmetric A-bimodule objects is a braided closed monoidal category, from which we obtain an internal tensor product operation on internal homomorphisms. We describe how these structures deform under cochain twisting of the quasi-Hopf algebra, and apply the formalism to the example of deformation quantization of equivariant vector bundles over a smooth manifold. Our constructions set up the basic ingredients for the systematic development of differential geometry internal to the quasi-Hopf representation category, which will be tackled in the sequels to this paper, together with applications to models of noncommutative and nonassociative gravity such as those anticipated from non-geometric string theory.

Appl. Categorical Struct., 2021

Taking advantage of the quantale-theoretic description of etale groupoids we study principal bundles, Hilsum–Skandalis maps, and Morita equivalence by means of modules on inverse quantal frames. The Hilbert module description of quantale sheaves leads naturally to a formulation of Morita equivalence in terms of bimodules that resemble imprimitivity bimodules of C*-algebras.

Demonstratio Mathematica, 2004

We show that the Hopf algebra on a transformation groupoid T = E x G, where G is a finite group acting on the total space of a principal fibre boundle over M = E/G, is the cross product of the algebras C'(E) and CG. We study duality properties of this algebra, and consider quantization on orbit spaces program in this context.

Journal of noncommutative geometry

Dynamical quantum groups were introduced by Etingof and Varchenko in connection with the dynamical quantum Yang-Baxter equation, and measured quantum groupoids were introduced by Enock, Lesieur and Vallin in their study of inclusions of type II_1 factors. In this article, we associate to suitable dynamical quantum groups, which are a purely algebraic objects, Hopf C*-bimodules and measured quantum groupoids on the level of von Neumann algebras. Assuming invariant integrals on the dynamical quantum group, we first construct a fundamental unitary which yields Hopf bimodules on the level of C*-algebras and von Neumann algebras. Next, we assume properness of the dynamical quantum group and lift the integrals to the operator algebras. In a subsequent article, this construction shall be applied to the dynamical SU_q(2) studied by Koelink and Rosengren.