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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)

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YURI BAHTURIN, Memorial University of Newfoundland
Group gradings on Lie algebras and Casimir elements

We apply so-called "fine" gradings of certain classical finite-dimensional 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 Morita-Takeuchi equivalent to the path coalgebra of its Ext-quiver.

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 Yetter-Drinfeld 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 left-right Yetter-Drinfeld 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 lambda-ring 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 áx₁, ..., x_n : R ñ, where R consists of \binomn2 square-free 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 Gateva-Ivanova 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 non-degenerate conditions and they yield a set theoretical solution to the quantum Yang-Baxter 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 Gelfand-Kirillov dimension and prime ideals. In particular, it follows that there exist examples on 4ⁿ generators so that the algebra has Gelfand-Kirillov 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 Frobenius-Schur Indicators for Hopf Algebras

Frobenius-Schur 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 Frobenius-Schur 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 Frobenius-Schur 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 Yang-Baxter operator or order two to the virtual crossing (so that it satisfies appropriate conditions with respect to the usual Yang-Baxter 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, FES-C, UNAM, 1-ro de Mayo s/n, Campo 1, CIT
Braided version of Shirshov-Witt 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 Shirshov-Witt 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_iÄ1+1 Äx_i, is free and it is freely generated by some Freidrichs-primitive elements. If we consider the free algebra as a braided Hopf algebra with a very special braiding, we get a reformulation of the Mikhalev-Shtern 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, PÈ-P=R, and supp P:=PÇ-P 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 Frobenius-Lusztig 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 Frobenius-Lusztig 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 Curtis-Lusztig for the simple modules for the Frobenius-Lusztig 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 G-adapted 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 G-adapted ring. In this talk, we consider for which finite groups G and G-adapted rings R, RG has only trivial units. Since G-adapted 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 2-groups. For such groups, we establish ring-theoretic 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 well-known 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 Yetter-Drinfel'd modules over semisimple Hopf algebras

This is joint work with L. Grunenfelder. We study the structure of Yetter Drinfel’d 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 anti-symmetric bicharacter, then we describe explicitly some interesting examples of Yetter-Drinfel’d modules over (kN)^* \rtimes_c kT and their liftings.

AKIRA MASUOKA, Institute of Mathematics, University of Tsukuba, Ibaraki 305-8571, JAPAN
More Hopf-algebraic approach to affine (super)group basics

We will make a (more) Hopf-algebraic approach to the basics of affine (super)group theory, discussing Hopf modules, crossed products, equivariant smoothness, and duality (including Takeuchi's hyperalgebras). Non-commutative 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, 05311-970, 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 U₂(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 finite-dimensional Hopf algebra over a field k and R an H-module algebra. We consider when the Jacobson radical J(R) is H-stable. 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 R-modules are finite-dimensional. 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
Frobenius-Schur Indicators for Semi-simple Quasi-Hopf Algebras

In this talk, we will discuss the Frobenius-Schur (FS) indicator c_V for an irreducible representation V of a semi-simple quasi-Hopf 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 non-zero if, and only if, V is self-dual. Moreover, the set of all FS indicators for H is an invariant of the tensor category H-mod. The tensor category H-mod 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^d-1 W_f is a submodule of A, where W_f is the set of zeros of f.  Among all Z-submodules 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 non-torsion 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 R-algebra 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 b-commutative if [a₁, a₂]_b = 0 for all a₁, a₂ in A. Similarly the usual notions of Lie nilpotence and Lie solvability can be extended to b-nilpotence and b-solvability 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 2-cocyle twists of tensor products of pointed Hopf algebras with relatively simple structures. Examples of Hopf algebras obtained by 2-cocycle twists are the finite-dimensional 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 ring-theoretic 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 Lie-theoretic 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 conjectured-and proved in interesting special cases-that the exponent exp(H), defined as the smallest number n such that all n-th 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 examples-bismash products and Drinfeld doubles of groups. In particular, we ask for which n there exist nontrivial elements whose n-th Hopf power is trivial, or, more specifically, elements of Hopf order n; we also investigate whether the answers to these questions are twisting-invariant. 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 H-simple right H-comodule algebra. It is shown that under certain hypotheses every (H,A)-Hopf module is either projective or free as an A-module and A is either a quasi-Frobenius 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 H-simple H-module 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.