27 June 2014, Room 104, Carleton Hall,

University of New Brunswick, Fredericton, Canada

AIM: The goal of this workshop is to communicate state-of-the-art research results involving actions and coactions of Hopf algebras. The main theme is Hopf-cyclic cohomology, and key topics include principal coactions, foliations, monoidal categories, deformation quantization, universal quantum symmetries, and path coalgebras.

ORGANIZERS: Piotr M. Hajac and Bahram Rangipour

10:00 - 11:00   BRAIDED JOIN COMODULE ALGEBRAS OF GALOIS OBJECTS (see arXiv1407.6840, cf. arXiv1407.6020)

We construct the join of noncommutative Galois objects (quantum torsors) over a Hopf algebra H. To ensure that the join algebra enjoys the natural (diagonal) coaction of H, we braid the tensor product of the Galois objects. Then we show that this coaction is principal. Our examples are built from the noncommutative torus with the natural free action of the classical torus, and arbitrary anti-Drinfeld doubles of finite-dimensional Hopf algebras. The former yields a noncommutative deformation of a non-trivial torus bundle, and the latter a finite quantum covering. (Based on joint work with L. Dąbrowski, T. Hadfield and E. Wagner.)

PIOTR M. HAJAC (IMPAN / Warsaw University / UNB)


For any Lie-Hopf algebra, we introduce a commutative DG-algebra which is quasi-isomorphic to the Hopf-cyclic cohomology of the bicrossed product Hopf algebra. We investigate algebraic connection on this algebra in general, and show that all geometric Lie-Hopf algebras admit such a connection. We conclude that all known characteristic classes of foliations can be written explicitly as cyclic cohomology classes of the groupoid action algebra. At the end we give an example that does not quite fit in the picture and needs further developments.



A unified approach to noncommutative geometry is presented in terms of monoidal categories. The classical commutative case is categorified to symmetric monoidal structure, and next symmetry is dropped. Then it is shown how to redefine classical structures to make them resistible to the loss of symmetry while preserving their classical meaning in the commutative case. This is applied to affine morphisms and coordinate rings, flat covers, symmetries, quotients, principal fibrations, classifying maps, global homological invariants, infinitesimals and differential operators. In particular, a categorification of Hopf-cyclic (co)homology is constructed. The categorification has the known algebraic theory as a particular component corresponding to the monoidal unit. Finally, the loss of commutativity under deformation quantization is categorified and a corresponding Gerstenhaber algebra for such categorified deformations is constructed.

TOMASZ MASZCZYK (Warsaw University)


We introduce a Radon-Nikodym differentiable structure on a finitely-dimensional algebra. This allows us to speak about the "quantum fundamental cycle" for finite-dimensional algebras which exists for any Frobenius algebra. For the algebra of functions on a finite set the quantum fundamental cycle coincides with its classical counterpart. For any Hopf algebra H with bijective antipode coacting on a finite-dimensional algebra A with a fundamental cycle, one can ask whether the coaction preserves the fundamental cycle in the same way as classical permutations do. If it is the case, there is a canonical cohomology class in H1(H, A×) which is an obstruction to the existence of an invariant Frobenius structure on A supported on the fundamental class. We call it the modular class. We show that, for every finite-dimensional algebra A with a fundamental cycle, there exists a universal Hopf algebra with bijective antipode and coaction preserving the fundamental class on A. Thus the modular class of the universal coaction becomes an invariant of a Frobenius algebra. We show that the universal Hopf algebra with bijective antipode and coaction on the algebra of functions on a finite set preserves the fundamental cycle and that, for a finite set of cardinality bigger than one, the modular class is non-trivial, although it vanishes on classical permutations.

TOMASZ MASZCZYK (Warsaw University)


Assuming that q is not a root of unity, X. W. Chen and P. Zhang embed the quantized enveloping Hopf algebra Uq(sl2) of the Lie algebra sl2 into the path coalgebra of the Gabriel quiver D of the underlying coalgebra of Uq(sl2). They also give a new basis of Uq(sl2) in terms of combinations of paths in D, and describe the category of Uq(sl2)-comodules in terms of representations of the quiver D. Our ultimate goal is to classify stable anti-Yetter-Drinfeld (SAYD) modules over the Hopf algebra Uq(sl2). In this talk, I will present examples of comodules over Uq(sl2), and show that all two-dimensional Yetter-Drinfeld modules over Uq(sl2) are trivial.