HOPF ALGEBRAS IN (CO)ACTION
27 June 2014, Room 104, Carleton Hall,
University of New Brunswick,
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
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.
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)
11:15 - 12:15
LIE-HOPF ALGEBRAS AND THEIR HOPF-CYCLIC COHOMOLOGY
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.
12:30 - 13:30
QUANTIZATION BY CATEGORIFICATION
A unified approach to noncommutative geometry is presented in terms of
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.
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.
15:30 - 16:30
QUANTUM PERMUTATIONS OF TWO ELEMENTS
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
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
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.
16:45 - 17:45
TWO-DIMENSIONAL YETTER-DRINFELD MODULES OVER
Uq(sl2) ARE TRIVIAL
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
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.