


Hopf Algebras and Related Topics / Algèbres de Hopf et sujets connexes (Org: Yuri Bahturin, Memorial University, Margaret Beattie, Mount Allison University, Luzius Grunenfelder, Dalhousie University, Susan Montgomery, University of Southern California, and/et Earl Taft, Rutgers University)
 YURI BAHTURIN, Memorial University of Newfoundland
Group gradings on Lie algebras and Casimir elements

We apply socalled "fine" gradings of certain classical
finitedimensional Lie algebras and superalgebras to the determination
of the center of the universal enveloping algebra. We then extend the
same approach to much wider class of Lie algebras and superalgebras
possessing linear bases labeled by the elements of a finite abelian
group, with the commutator determined in terms of a fixed bicharacter
on this group. These results are joint with Alexander Molev.
 SEBASTIAN BURCIU, Syracuse University
On some representations of the Drinfeld double of a Hopf algebra

For H a finite dimensional bisemisimple Hopf algebra over an
algebraically closed field, the induced representations from H and
H^{*} to the Drinfeld double D(H) are studied. The product of two
such representations is a sum of copies of the regular representation
of D(H). The action of certain irreducible central characters of
H^{*} on the simple modules of H is considered. The modules that
receive trivial action from each such irreducible central character
are precisely the constituents of the tensor powers of the adjoint
representation of H.
 WILLIAM CHIN, DePaul University
Hereditary Coalgebras

A coalgebra is said to be hereditary if every homomorphic image of
every injective comodule is injective. Path coalgebras of arbitrary
quivers are hereditary, and it turns out that these are the only
pointed examples. Thus, over an algebraically closed field, every
hereditary coalgebra is MoritaTakeuchi equivalent to the path
coalgebra of its Extquiver.
 LUZIUS GRUNENFELDER, Dalhousie University, Halifax
On Hopf algebras with radical a Hopf ideal

This report is about Hopf algebras with Jacobson radical a Hopf ideal,
joint work with Mitja Mastnak. We consider in particular the case when
the quotient by the radical is cosemisimple, and we explore how these
Hopf algebras dualize. Such Hopf algebras occur naturally, for example
as duals of certain pointed Hopf algebras.
 THOMAS GUEDENON, Mount Allison University, Sackville, New Brunswick, Canada
Semisimplicity of the categories of YetterDrinfeld modules
and Long dimodules

Let k be a field, and H a Hopf algebra with bijective antipode. If
H is commutative, noetherian, semisimple and cosemisimple, then the
category of leftright YetterDrinfeld modules is semisimple. We also
prove a similar statement for the category of Long dimodules, without
the assumption that H is commutative.
 MICHIEL HAZEWINKEL, CWI, PO Box 94079, 1090 Amsterdam, The Netherlands
Hopf algebras of endomorphisms of Hopf algebras

More or less recently two important generalizations appeared of the
Hopf algebra of symmetric functions. They are the Hopf algebra of
noncommutative symmetric functions, NSymm, and its graded dual, the
Hopf algebra of quasisymmetric functions. The Hopf algebra of
permutations, MPR, is a selfdual generalization of both. It can be
interpreted, in a certain way, as a Hopf algebra of (module)
endomorphisms of the Shuffle Hopf algebra. This way of looking at MPR
permits generalizations such as the word Hopf algebra WHA and the
double word Hopf algebra dWHA (and many others). These may have room
enough for such extra structures as Frobenius and Verschiebung
endomorphisms and lambdaring structures (which is not the case for
MPR).
 ERIK JESPERS, Vrije Universiteit Brussel, Department of Mathematics,
Pleinlaan 2, 1050 Brussel, Belgium
Quadratic algebras of skew type

We consider algebras over a field K with a presentation K áx_{1}, ..., x_{n} : R ñ, where R consists of \binomn2
squarefree relations of the form x_{i} x_{j} = x_{k} x_{l} with
every monomial x_{i} x_{j}, i ¹ j, appearing in one of the
relations (or briefly, quadratic algebras of skew type). Such
algebras first appeared in the work of GatevaIvanova and Van den
Bergh, inspired by earlier work of Tate and Van den Bergh. In this
case the monoids and groups with the same presentation satisfy some
natural nondegenerate conditions and they yield a set theoretical
solution to the quantum YangBaxter equation.
In this talk we present some recent results on the structure of such
algebras; this is joint work with F. Cedo and J. Okninski. First we
describe the algebraic structure of the monoids and groups determined
by such and related presentations. Several interesting group theoretic
results and open problems will be discussed. Second we discuss the
algebraic structure of the algebra. Special attention is given to the
GelfandKirillov dimension and prime ideals. In particular, it
follows that there exist examples on 4^{n} generators so that the
algebra has GelfandKirillov dimension one while the algebra is
noetherian PI and semiprime in case the field K has characteristic
zero.
 YEVGENIYA KASHINA, DePaul University, Chicago, IL 60614
Higher FrobeniusSchur Indicators for Hopf Algebras

FrobeniusSchur indicators appear as important invariants of finite
groups. Using generalized power map we can extend this notion to
semisimple Hopf algebras. However it turns out that unlike in the
group theory case, higher FrobeniusSchur indicators may not be
integers. In this talk we are going to construct an example of a Hopf
algebra, obtained as a certain abelian extension, for which the
higher FrobeniusSchur indicators are not real.
This talk is a part of joint work with Yorck Sommerhäuser and
Yongchang Zhu.
 LOUIS KAUFFMAN, University of Illinois at Chicago
Quantum Invariants of Virtual Knots and Links

This talk will discuss the structure of quantum invariants of virtual
knots and links. Formally, virtual knots are obtained from classical
knots by allowing a crossing that is geometrically a detour from one
point to another. One associates a YangBaxter operator or order two
to the virtual crossing (so that it satisfies appropriate conditions
with respect to the usual YangBaxter operator assigned to the
classical crossings). This talk will review work of the presenter with
David Radford on bioriented quantum algebras, and will discuss new
invariants of virtual knots recently discovered by the presenter.
 VLADISLAV KHARCHENKO, FESC, UNAM, 1ro de Mayo s/n, Campo 1, CIT
Braided version of ShirshovWitt theorem

A. I. Shirshov [1953] and E. Witt [1956] proved that every subalgebra
of a free Lie algebra is free. This result has been generalized to
colored Lie superalgebras (A. A. Mikhalev [1985], A. S. Shtern
[1986]).
The ShirshovWitt theorem admits a formulation it terms a free
associative algebra: Every subHopfalgebra of a free algebra káx_{i} ñ with the diagonal coproduct, d(x_{i}) = x_{i}Ä1+1 Äx_{i}, is free and it is freely generated by some
Freidrichsprimitive elements. If we consider the free algebra as a
braided Hopf algebra with a very special braiding, we get a
reformulation of the MikhalevShtern generalization as well. Our aim
is to extend these results to free algebras with arbitrary braidings.
 MIKHAIL KOCHETOV, Carleton University, 1125 Colonel By Drive, Ottawa, ON
K1S 5B6, Canada
Orderability of Hopf algebras

Let R be a ring. A subset P Ì R is called an ordering
if P+P Ì P, P·P Ì P, PÈP=R, and supp
P:=PÇP is a prime ideal of R. The set of all orderings of R
is called the real spectrum of R. The study of real spectra of
noncommutative rings is known as noncommutative real algebraic
geometry. An ordering with zero support gives rise to a total
order relation on R.
Many important examples of Hopf algebras, such as group algebras,
(quantized) enveloping algebras, and (quantized) function algebras,
often admit a zero support ordering. This observation motivates the
problem of finding criteria of orderability for an arbitrary Hopf
algebra (viewed as a ring). We will investigate this problem for
cocommutative Hopf algebras.
In the context of rings with involution, there is a notion of the so
called *orderings. Many Hopf algebras have a natural involution. We
will consider the problem of existence of a zero support *ordering
for cocommutative Hopf algebras.
Joint work with J. Cimpric and M. Marshall.
 LEONID KROP, DePaul University, 2320 N. Kenmore Ave., Chicago, IL 60614
Simple Modules for the Quantum Double of the
FrobeniusLusztig Kernels

Let
k
be a field of characteristic 0 containing a primitive
root of unity z of an odd order. We denote by u_{z}
the FrobeniusLusztig kernel for the simple Lie algebra of rank 1. We
let D(u_{z}) stand for the quantum (= Drinfel'd) double
of u_{z}.
We present a complete description of the simple D( u_{z})modules. This is based on the notion of a primitive
weight vector in a D(u_{z})module. Our results show that
the simple modules are classified by the weights of their primitive
vectors, an analog of the theorem of CurtisLusztig for the simple
modules for the FrobeniusLusztig kernels of any type and rank.
The talk is part of joint work with D. Radford.
 YUANLIN LI, Brock University, St. Catharines, Ontario, L2S 3A1
Trivial Units For Group Rings With Gadapted Coefficient Rings

For each finite group G, an integral domain R of characteristic
0 with the property that no prime divisor of the order of G is
invertible is called a Gadapted ring. In this talk, we consider for
which finite groups G and Gadapted rings R, RG has only
trivial units. Since Gadapted rings contain Z, this can
only occur if ZG contains only trivial units, and such
groups were classified by Higman. These groups are abelian groups of
exponent dividing 4 or 6, and Hamiltonian 2groups. For such groups,
we establish ringtheoretic conditions under which the group ring RG
has nontrivial units. Several examples of rings satisfying the
conditions and rings not satisfying the conditions are given. In
addition, we extend a wellknown result for fields by showing that if
R is a commutative ring of finite characteristic and RG has only
trivial units, then G has order at most 3.
 MITJA MASTNAK, Dalhousie University
About YetterDrinfel'd modules over semisimple Hopf algebras

This is joint work with L. Grunenfelder. We study the structure of
Yetter Drinfeld modules over those Hopf algebras, that are crossed
products of a group algebra and the dual of a group algebra. In
particular, if a group T is acting on a group N and c:N×N® k^{·} is an antisymmetric bicharacter, then we
describe explicitly some interesting examples of YetterDrinfeld
modules over (kN)^{*} \rtimes_{c} kT and their liftings.
 AKIRA MASUOKA, Institute of Mathematics, University of Tsukuba,
Ibaraki 3058571, JAPAN
More Hopfalgebraic approach to affine (super)group basics

We will make a (more) Hopfalgebraic approach to the basics of affine
(super)group theory, discussing Hopf modules, crossed products,
equivariant smoothness, and duality (including Takeuchi's
hyperalgebras). Noncommutative Hopf algebras as well as Hopf
algebras in braided monoidal categories are within our scope.
 FRANCISCO CESAR POLCINO MILIES, Universidade de São Paulo, Caixa Postal 66.281, 05311970,
São Paulo, SO, Brazil
Free groups and involutions in the group of units of a group
algebra

Let U(RG) denote the group ring of a group G over a commutative
ring with unity R. In the case when the coefficient ring is a field
F or a ring of algebraic 1ntegers, the existence of free subgroups
of rank 2 in U(RG) has been studied and explicit constructions of
such groups were given by several authors.
Recently, Gonçalvez and Passman investigated the existence of free
groups in the subgroup of unitary units with respect to the natural
involution of FG induced by g® g^{1}, for all g Î G.
We shall discuss the existence of free groups in another significant
subgroup of U (FG): the subgroup U_{2}(FG) generated by all units of
order 2 of FG. These results were obtained in joint work with
Prof. A. Giambruno.
 SUSAN MONTGOMERY, University of Southern California, Los Angeles, CA 90089, USA
Stability of the Jacobson radical under Hopf algebra actions

Let H be a finitedimensional Hopf algebra over a field k and R
an Hmodule algebra. We consider when the Jacobson radical J(R)
is Hstable. This is true trivially for group actions, and is true
for gradings by a finite group G if G is invertible in k, by a
old result of Cohen and the speaker.
Counterexamples exist when H is not semisimple. We prove it is true
if H is semisimple, k has characteristic 0, and all irreducible
Rmodules are finitedimensional. As a consequence it is true if
R is an affine PI algebra.
This is joint work with Vitaly Linchenko and Lance Small.
 RICHARD NG, Iowa State University, Iowa, USA
FrobeniusSchur Indicators for Semisimple QuasiHopf Algebras

In this talk, we will discuss the FrobeniusSchur (FS) indicator c_{V}
for an irreducible representation V of a semisimple quasiHopf
algebra H. There exists a canonical central element n_{H} in H
which is invariant under gauge transformations. The FS indicator
c_{V} is defined to be c(n_{H}) where c is the character
afforded by V. The scalar c(n_{H}) is nonzero if, and only
if, V is selfdual. Moreover, the set of all FS indicators for H
is an invariant of the tensor category Hmod. The tensor
category Hmod also admits a canonical pivotal structure which
can be described via the character of the regular representation and
the normalized integral of H. The pivotal structure implies that
the FS indicators for H can only be 0, 1 or 1.
 JAMES OSTERBURG, University of Cincinnati
The Final Value Problem

A polynomial form f is a not necessarily linear map, from an
infinite module to a finite abelian group of exponent n. We show
that for a form of degree d then n^{d1} W_{f} is a submodule
of A, where W_{f} is the set of zeros of f. Among all
Zsubmodules of finite index, there is a submodule B such that
f(B) (the order of the subset f(B)) is as small as
possible. f(B) is called the final value of f and Passman
asks if f(B) is necessarily a subgroup of S. This paper shows
that if the degree of f £ 2 then the final value is a subgroup and
if the form f has arbitrary degree from a nontorsion finitely
generated abelian group, then the final value is 0. We will also
discuss the zeros of f.
 MIKE PARMENTER, Memorial University of Newfoundland, St. John's, NL A1C 5S7
Generalized Lie nilpotence and Lie solvability of integral
group rings

Assume that an Ralgebra A (where R is a commutative ring with
1) is graded by a finite abelian group H and let b: H×H ® R^{*} be a skew symmetric bicharacter on H with values
in R. If x Î A_{g}, y Î A_{h} are homogeneous elements of A
(g,h Î H) define a generalized Lie bracket by [x,y]_{b} = xy b(g,h) yx and extend this bracket to all of A by linearity.
A is said to be bcommutative if [a_{1}, a_{2}]_{b} = 0 for
all a_{1}, a_{2} in A. Similarly the usual notions of Lie nilpotence
and Lie solvability can be extended to bnilpotence and
bsolvability on A.
In this talk we discuss the above concepts when A is the integral
group ring ZG of a finite group G. The results described
are part of ongoing joint work with Yuri Bahturin.
 DAVID RADFORD, University of Illinois at Chicago, Chicago, IL, USA
Representations of Pointed Hopf Aglebras

There is an extensive class of pointed Hopf algebras which consists of
quotients of 2cocyle twists of tensor products of pointed Hopf algebras
with relatively simple structures. Examples of Hopf algebras obtained by
2cocycle twists are the finitedimensional quantum doubles.
We will describe the general outline of the theory of the irreducible
representations of Hopf algebras which belong to the class. Our results
apply to many of the Hopf algebras described by Andruskiewitsch and
Schneider in their classification program for pointed Hopf algebras. Most
of the work described in this talk is joint with Schneider.
 DAVID RILEY, The University of Western Ontario, London, Ontario, Canada
Group algebras whose augmentation ideal is Jacobson radical

Kurosh was the first to pose certain ringtheoretic analogues of the
famous Burnside problem for groups. Specifically, he asked whether or
not every nil algebra is locally nilpotent. While Golod eventually
constructed a counterexample to the general problem, Kaplansky had
already given a positive solution for all the class of all algebras
satisfying a nontrivial polynomial identity (PI). In my talk, I shall
first discuss why Kaplansky's PI condition can be weaken to
"infinitesimally PI". The proof uses strong Lietheoretic results of
Zelmanov. Applications will then be made to the Kurosh problem for
group algebras: if the augmentation ideal of a group algebra is nil,
is it locally nilpotent? More generally, I shall address the
following problem raised by Kaplansky: if a group algebra has the
property that its augmentation ideal coincides with its Jacobson
radical, is the augmentation ideal locally nilpotent?
 PETER SCHAUENBURG, Mathematisches Institut der Universität München,
Theresienstr. 39, 80333 München, Germany
Hopf powers and Hopf orders in some bismash products

Let H be a Hopf algebra, semisimple over the complex
numbers. Kashina initiated the study of the Hopf power maps on H,
defined by h^{[n]} = h_{(1)} · ¼ ·h_{(2)}. She
conjecturedand proved in interesting special casesthat the
exponent exp(H), defined as the smallest number n such that all
nth powers are trivial, always divides the dimension of H. Such a
conjecture is easily motivated by comparison to the group case.
Etingof and Gelaki proved several general results on the exponent, in
particular that exp(H)  dim(H)^{3}, and that the exponent is
invariant under twists.
We report on computer experiments (using Maple) with the Hopf power
maps in specific examplesbismash products and Drinfeld doubles of
groups. In particular, we ask for which n there exist nontrivial
elements whose nth Hopf power is trivial, or, more specifically,
elements of Hopf order n; we also investigate whether the answers to
these questions are twistinginvariant. Our results show that the
behavior of the Hopf powers of individual elements is much less
predictable than that of the exponent, and deviates further from
expectations one might have from comparing to the group case.
 SERGEY SKRYABIN, Free University of Brussels VUB
Projectivity and Freeness over Comodule Algebras

Let H be a Hopf algebra and A an Hsimple right Hcomodule
algebra. It is shown that under certain hypotheses every (H,A)Hopf
module is either projective or free as an Amodule and A is either
a quasiFrobenius or a semisimple ring. As an application it is proved
that every weakly finite (in particular, every finite dimensional)
Hopf algebra is free both as a left and a right module over its finite
dimensional right coideal subalgebras, and the latter are Frobenius
algebras. Similar results are obtained for Hsimple Hmodule
algebras.
 SARAH WITHERSPOON, Mathematical Sciences Research Institute, 17 Gauss Way,
Berkeley, CA 94709, USA
Deformations arising from quantum group actions on algebras

A module algebra for a bialgebra may be deformed by a twisting element
or a universal deformation formula over the bialgebra. Classically, a
polynomial ring carries the action of its Lie algebra of derivations,
and resulting deformations include quantum space and the Weyl
algebra. More recently crossed products of polynomial rings (and their
deformations) with finite groups have been of interest in relation to
the corresponding geometry. For these crossed product algebras, the
infinitesimal versions of deformations may be described
explicitly. Some of the bialgebras involved are Taft algebras, their
Drinfel'd doubles, and other related finite quantum groups. An example
of a universal deformation formula arising in this context will be
given.

