Marcelo Aguiar, Texas A&M University

        `` The  Hopf algebra of permutations of Malvenuto and Reutenauer'' (joint work with Frank Sottile).

Abstract: Consider the union of all symmetric groups and the vector space with this basis. The operations of splitting and shuffling permutations  give rise to a graded Hopf algebra structure on this space, which was first defined by Malvenuto and Reutenauer in their work on Solomon's descent algebra and quasi-symmetric functions. This Hopf algebra is one of particular importance among several ``combinatorial Hopf algebras'' that have arisen recently in different contexts, like Gessel's Hopf algebra of quasi-symmetric functions and Loday-Ronco's Hopf algebra of planar binary trees.
In this work, we introduce a second linear basis of the Hopf algebra of permutations, by means of the weak Bruhat order on the symmetric groups. This allows us to obtain explicit combinatorial descriptions of the structure constants (which turn out to be non-negative) and a complete understanding of the Hopf algebra structure.

 Nicolas Andruskiewitsch, University of Cordoba, Argentina

        ``A characterization of quantum groups'' (joint with H.-J. Schneider)

Abstract: We classify pointed Hopf algebras with finite Gelfand-Kirillov dimension, which are domains, whose groups of group-like elements are abelian, and whose infinitesimal braidings are positive. The paper can be downloaded from either
< http://arXiv.org/abs/math/0201095> or <www.mathematik.uni-muenchen.de/~hanssch/Publications.html>

Yuri Bahturin, Memorial University of Newfoundland, Canada, and Moscow State University

        ``Generalized Lie Solvability of Associative Algebras"

Abstract: The talk contains results from a recent joint paper with S. Montgomery and M. Zaicev. The main idea is to apply recent results about the classification of G-gradings on simple associative algebras, where G is a finite abelian group, to describing generalized solvable associative algebras defined by a multiplicative bi-character on the group G.

Margaret Beattie, Mount Allison University, Canada

        ``Duals of Pointed Hopf Algebras"

Abstract: This talk will discuss the  duals of some finite dimensional pointed Hopf algebras over an algebraically closed field of characteristic 0, which are either bicrossed products  $H= {\cal B}(V) \# k[\Gamma]$ or else nontrivial liftings of such a
bicrossed product, with $\Gamma$ a finite abelian group, $V \in ^{k[\Gamma]}_{k[\Gamma]}{\cal YD}$ and ${\cal B}(V)$ the Nichols algebra of $V$. For $V$ such that ${\cal B}(V)$ is finite dimensional, the dual of ${\cal B}(V) \# k[\Gamma]$ is isomorphic to ${\cal B}(W) \#k \hat{[\Gamma]}$, where $\hat{\Gamma}$ is the character group of $\Gamma$.  The dual of a nontrivial lifting of ${\cal B}(V) \# k[\Gamma]$ contains matrix coalgebras in its coradical. For $V$ a quantum linear space of dimension 1 or 2,  the duals of some liftings of ${\cal B}(V) \# k[\Gamma]$ were described by Radford in 1975; we discuss these duals from a coalgebraic point of view. We conclude with some examples where we compute the matrix coalgebras in the coradical of the dual explicitly including the duals of an infinite family of non-isomorphic Hopf algebras of dimension 32.

William Chin, DePaul University

        ```Spectra of quantized hyperalgebras''

Abstract: In this joint work with S. Catoiu and L. Krop, we study ideals in quantized hyperalgebras U at roots of unity over a field of characteristic zero. These Hopf algebras are also known as the restricted specializations of quantized enveloping algebras associated to the semisimple Lie algebra algebra g. We use a series of known results about crossed products and
actions of enveloping algebras to show that Spec(U) homeomorphic to Spec(u) x Spec(U(g)). Concerning primitive ideals, we show that an analog of Duflo's theorem holds:  Primitive ideals of U are annihilators of simple highest weight modules.

Miriam Cohen, Ben Gurion University, Israel

        ``Some interrelations between Hopf algebras  and their duals'' (joint with Sara Westreich)

Abstract: For an automorphism T of a Hopf algebra H we discuss T-normalizing and T*-cocommutative elements. They are related to  inner and coinner autorphisms.Examples of such elements appear in works of Drinfeld, Radford and Cohen-Zhu.

Yevgenia Kashina , Syracuse University

        ``Frobenius-Schur indicators for Hopf algebras" (joint with Geoffrey Mason and Susan Montgomery)

Abstract: In this talk we will show that for an important class of non-trivial Hopf algebras, the Frobenius-Schur indicator is a computable invariant. The Hopf algebras we consider are all abelian extensions; as a special case, they include the Drinfeld double of a group algebra.

Mikhail Kotchetov, Memorial University of Newfoundland, Canada

    ``Polynomial Identities in Hopf Algebras: Passman's Theorem and Its Dual"

Abstract: We will give necessary and sufficient conditions when a cocommutative Hopf algebra of characteristic 0 satisfies an identity as an algebra, which can be considered as a generalization of Passman's theorem on group algebras. Then we will look at the dual situation: when a commutative (or nearly commutative) Hopf algebra of characteristic 0 satisfies an identity as a coalgebra. Time permitting, we will also discuss some partial results in prime characteristic.

Jean-Louis Loday, CNRS and Universite Louis Pasteur, Strasbourg, France (visiting Northwestern University)

        ``Hopf algebras on planar trees''

Abstract. Different kinds of problems ranging from homological algebra to renormalization theory led to the study of a Hopf algebra based on planar binary trees. In fact this Hopf algebra is the  degree zero part of a larger one which is based on all the planar trees. It is related with a new kind of algebras called dendriform trialgebras.

Susan Montgomery, University of Southern California

        "Skew derivations of some finite-dimensional algebras, with applications
        to actions of u_q(sl2) and the double of the Taft Hopf algebra"(joint with H.-J. Schneider)

Abstract: We discuss some work with H.-J. Schneider. We first give a complete characterization of the skew derivations of some finite-dimensional algebras, in particular the algebra A = k[u], where u^n is in k; this is equivalent to studying the actions of the n^2-dim Taft algebra T. As an application, we show that the actions of u_q(sl2) are determined by the action of a Borel subalgebra; this is the quantum analog of a classical result of Jacobson on the Witt algebra and also extends earlier work on the action of U_q(sl2) when q is not a root of 1. We also determine the actions of D(T) on A. This work recently appeared in the Tsukuba Journal (2001).
We also consider possible extensions of these results to A = k[u,v] and its connection to the Brauer group of the Hopf algebra T, a problem which remains open.

Richard Ng, Towson University

        ``Non-semisimple Hopf Algebras of Dimension $p^2$''

Abstract: Let $H$ be a Hopf algebra of dimension $p^2$ over an algebraically closed field of characteristic 0  where $p$ is a prime. If $H$ is not semisimple, then  $H$ is isomorphic to a Taft algebra. We then complete the classification for the Hopf algebras of dimension $p^2$.

Julia Pevtsova, Northwestern University

        ``Support varieties for Frobenius kernels''

Abstract: We introduce and discuss the geometric notion of support variety for a rational module of a Frobenius kernel ( or, equivalently, for a representation of a certain finite dimensional co-commutative Hopf  algebra over a field of positive characteristic), which carries both cohomological and local representation-theoretic information about the module. The goal of the lecture is to explain how this construction can be generalized to infinite dimensional modules. The generalization leads to examples of infinite dimensional modules whose geometric properties are quite different from those of finite dimensional ones.

David Radford, University of Illinois at Chicago

        ``The Double of the Taft Hopf Algebra as a Source of Link Invariants" (joint with Sara Westreich)

Abstract: Generally quantum doubles are rather complicated Hopf algebras. We study various aspects the rather basic example $D(H_{\omega, n})$, where $H_{\omega, n}$ is a Taft Hopf algebra over a field $k$ with primitive $n^{th}$ root of unity $\omega$. Of particular interest to us is the set of cocommutative elements of $D(H)^*$. We refer to them as trace-like functionals.
Suppose that $H$ is a finite-dimensional Hopf algebra over a field $k$. Then ${\rm End}\,(H)$ can be identified with $D(H)^*$ in a natural way. Under this identification, a certain easily described subspace of endomorphisms of $H$ corresponds to the trace-like functionals of $D(H)^*$.
There are connections between the trace-like functionals and the exponent of $H$. The double $D(H)$ is an example of an oriented quantum algebra. As such it accounts for regular isotopy invariants of oriented links. Preliminary investigation indicates that the invariants, based in part on trace-like functionals, which arise from $D(H_n)$ could be rather interesting.
This talk is based on joint work with Sara Westreich.

Serban Raianu, Syracuse University

        ``Semiprime crossed products" (joint with Declan Quinn).

Abstract: If $H$ is a finite dimensional copointed Hopf algebra, then a left nonsingular crossed product $A\#_{\sigma}H$ is semiprime whenever $A$ is $H$-semiprime. This explores a particular case of the well known open question about the semiprimeness of a crossed product between a semiprime algebra and a semisimple Hopf algebra acting weakly on it.

Axel Schueler, University of Southern California

        ``Laplace operator and Hodge decomposition for quantum groups'' (joint with Istvan Heckenberger)

Abstract: We consider the standard bicovariant differential calculi on the q-deformed function algebras to the classical four series A -- D of Lie groups. We show that the de Rham cohomology of the exterior algebras coincides with the de Rham cohomology of its left-invariant, right-invariant and of its biinvariant subcomplexes. In the cases  GL_q(N) and SL_q(N) the cohomology is isomorphic to the biinvariant exterior algebra and to the space of harmonic forms. We prove a Hodge decomposition theorem in these cases.

Zoran Skoda, University of Wisconsin-Madison

        ``Localized coinvariants"

Abstract: Let H be a Hopf algebra and A an H-comodule algebra. The case when A is a well-behaved extension of the algebra of coinvariants U, e.g. the theory of H-Galois extensions has been widely studied. On the contrary, we study the case when U is too small but the problem can be avoided using localizations. An Ore localization of a H-comodule algebra is compatible with the coaction if there is an induced H-comodule algebra structure on the localization. We consider finite sets of compatible Ore localizations A(s), covering A, where each localization A(s) has a structure of a smash product U(s)#H, and view this setup as an analogue of a locally trivial principal fibre bundle. We study a nontrivial class of natural examples related to quantum linear groups.

Yorck Sommerhaeuser, University of Munich

        ``Self-dual modules of semisimple Hopf algebras''

Abstract: We prove that, over an algebraically closed field of characteristic zero, a semisimple Hopf algebra that has a nontrivial self-dual simple module must have even dimension. This generalizes a classical result of W. Burnside. As an application, we show under the same assumptions that a semisimple Hopf algebra that has a simple module of even dimension must itself have even dimension. The talk is based on joint work with Y. Kashina and Y. Zhu.

Sarah Witherspoon, University of Massachusetts at Amherst and Amherst College

        "Clifford correspondence for Hopf algebras"

Abstract: Classical Clifford correspondence yields a description of how the simple modules for a group arise from those of a normal subgroup and the quotient group.  We will discuss recent generalizations and applications of this theory to Hopf algebras and quantum groups.

Quanshui Wu, Math. Institute, Fudan University, China

        ``Regularity of Involutory PI Hopf Algebras''

Abstract: Let $H$ be an involutory Hopf algebra over a field of characteristic 0. If $H$ is a finite module over an affine commutative subring, then it has finite global dimension. Further $H$ is Auslander regular and Cohen-Macaulay and is a finite direct sum of prime rings of the same Gelfand-Kirillov dimension.

James Zhang, University of Washington

        ``Gorenstein property of noetherian PI Hopf algebras''

Abstract: We prove that every noetherian affine PI Hopf algebra has finite injective dimension, which answers a question of K. A. Brown. Some other ring-theoretic properties of noetherian PI Hopf algebras will be discussed.