Yorck Sommerhäuser
Seminar talks:
 The first construction (Die erste Konstruktion)
 Date:
 Location: University of Munich, Germany
 Summary: We exhibit a construction for Hopf algebras with triangular decomposition by gluing two YetterDrinfel'd Hopf algebras with an ordinary Hopf algebra. We prove that this is the general form of a Hopf algebra with triangular decomposition whose Borellike parts are Radford biproducts.
 From f to U (Von f zu U)
 Date: Thursday, December 23, 1994
 Location: University of Munich, Germany
 Summary: We apply the first construction in the situation where the two YetterDrinfel"d Hopf algebras are dual. We explain why the Drinfel'd double construction is a special case of this construction. Finally, we explain how the deformed enveloping algebra U considered by G. Lusztig can be obtained from the algebra f introduced by Lusztig via the second construction.
 Semisimple Hopf algebras (Halbeinfache Hopfalgebren)
 Date: Thursday, November 30, 1995
 Location: University of Munich, Germany
 Summary:
 Hopf algebras with triangular decomposition (Hopfalgebren mit Dreieckszerlegung)
 Date: Thursday, June 20, 1996
 Location: University of Munich, Germany
 Summary:
 KacMoody algebras (KacMoodyAlgebren)
 Date: Thursday, December 12, 1996
 Location: University of Munich, Germany
 Summary: We exhibit the analogue of the first and the second construction for Lie bialgebras. For the first construction, we glue two YetterDrinfel'd Lie algebras and an ordinary Lie algebra to obtain a Lie algebra with a triangular decomposition. For the second construction, we apply the first construction to the situation where the two YetterDrinfel'd Lie algebras are dual. Finally, we explain how all symmetrizable KacMoody algebras can be obtained in this way.
 Natural transformations in the category of YetterDrinfel'd modules (Natürliche Transformationen in der Kategorie der YetterDrinfel'dModuln)
 Date: Thursday, February 6, 1997
 Location: University of Munich, Germany
 Summary: We introduce the notion of a ribbon transformation and compare it to the notion of a monoidal transformation. We exhibit a ribbon transformation on the category of YetterDrinfel'd modules. For a YetterDrinfel'd Hopf algebra, we discuss two monoidal transformations, called the integral transformation and the modular transformation, that are inimately related to the structure of the YetterDrinfel'd Hopf algebra. In particular, we find a formula for the fourth power of the antipode of a YetterDrinfel'd Hopf algebra that involves the integral transformation and the ribbon transformation. We use these methods to determine completely the integrals of Hopf algebras with a triangular decomposition.
 On Kaplansky's fifth conjecture (Zur fünften KaplanskyVermutung)
 Date: Thursday, February 13, 1997
 Location: University of Munich, Germany
 Summary: We prove that the antipode of a semisimple Hopf algebra is an involution if the characteristic of the base field is very large.
 Fusion rings and maximal orders (Fusionsringe und Maximalordnungen)
 Date: Thursday, July 17, 1997
 Location: University of Munich, Germany
 Summary: Over algebraically closed fields of characteristic zero, the character ring of a semisimple Hopf algebra is semisimple.The Grothendieck ring, the Zform of the character ring, is an order of the character ring. By expressing the discriminant of the Grothendieck ring jn terms of the character of the adjoint representation, we determine those primes for which the localization of the Grothendieck ring at that prime is a maximal order of the character ring. We prove that the Grothendieck ring is not a maximal order itself. We compare these results with results about the arithmetic properties of fusion rings obtained in conformal field theory.
 The Drinfel'd double construction and the Jones fundamental construction (Die Doppelkonstruktion von Drinfel'd und die Grundkonstruktion von Jones)
 Date: Thursday, July 24, 1997
 Location: University of Munich, Germany
 Summary: The Jones fundamental construction assigns to an extension of semisimple algebras another extension of semisimple algebras in which the new algebra is the endomorphism ring of the old extension. One example of such an extension is the inclusion of the character ring of a semisimple Hopf algebra in the dual Hopf algebra. By proving that the action of the Drinfel'd double on the dual Hopf algebra precisely centralizes the right multiplication with elements of the character ring, we prove, under certain conditions on the base field, that the endomorphism ring of this extension is a quotient of the Drinfel'd double. As a corollary, we get, under certain conditions on the base field, that the character ring of a semisimple and cosemisimple Hopf algebra is semisimple.
 Quadrilinear Frobenius extensions (Quadrilineare FrobeniusErweiterungen)
 Date: Thursday, December 4, 1997
 Location: University of Munich, Germany
 Summary: We introduce the the notion of a quadrilinear Frobenius extension as a generalization of the notion of a sesquilinear Frobenius extension, i. e., the notion of a Frobenius extension of the second kind resp. a Beta Frobenius extension. In the definition of a quadrilinear Frobenius extension, the inclusion map and the twisting automorphism Beta are treated equally. We explain how the most important properties of sesquilinear Frobenius extensions extend to quadrilinear Frobenius extensions. We show that the character ring inside the dual Hopf algebra is, under certain conditions, a quadrilinear Frobenius extension that is not a sesquilinear Frobenius extension.
 Semisimple Hopf algebras of dimension pq (Halbeinfache Hopfalgebren der Dimension pq)
 Date: Thursday, December 11, 1997
 Location: University of Munich, Germany
 Summary: We prove that a semisimple Hopf algebra of dimension pq for two distinct primes p and q over an algebraically closed field of characteristic zero such that the Hopf algebra and its dual contain nontrivial grouplike elements can be decomposed as a Radford biproduct of a group ring of a cyclic group of prime order and a YetterDrinfel'd Hopf algebra that is commutative semisimple as an algebra and cocommutative cosemisimple as a coalgebra. We find sufficient condition for the existence of nontrivial grouplike elements in the cases where p = 5 and p = 7.
 Commutative YetterDrinfel'd Hopf algebras (Kommutative YetterDrinfel'dHopfalgebren)
 Date: Thursday, January 22, 1998
 Location: University of Munich, Germany
 Summary: We prove that a YetterDrinfel'd Hopf algebra of dimension q that is commutative semisimple as an algebra and cocommutative cosemisimple as a coalgebra over a group ring of a cyclic group of prime order p, where p and q are distinct primes, must contain an invariant primitive idempotent.
 YetterDrinfel'd Hopf algebras over groups of prime order (YetterDrinfel'dHopfalgebren über Gruppen von Primzahlordnung)
 Date: Friday, July 3, 1998
 Location: University of Munich, Germany
 Summary: We prove over algebraically closed fields of characteristic zero that for a YetterDrinfel'd Hopf algebra over a group of prime order that is commutative semisimple as an algebra and cocommutative cosemisimple as a coalgebra, the square of the group order divides the dimension. We discuss applications to the classification of semisimple Hopf algebras.
 YetterDrinfel'd Hopf algebras of dimension p^{2}
(YetterDrinfel'dHopfalgebren der Dimension p^{2})
 Date: Friday, July 10, 1998
 Location: University of Munich, Germany
 Summary: We construct for every finite commutative ring a nontrivial YetterDrinfel'd Hopf algebra. In particular, we construct nontrivial YetterDrinfel'd Hopf algebras of dimension p^{2}.
 Structure theory of YetterDrinfel'd Hopf algebras (Strukturtheorie für YetterDrinfel'dHopfalgebren)
 Date: Tuesday, December 8, 1998
 Location: University of Munich, Germany
 Summary: We prove a structure theorem for YetterDrinfel'd Hopfalgebras over groups of prime order which are commutative and semisimple as algebras and cocommutative and cosemisimple as coalgebras: These YetterDrinfel'd Hopf algebras can be decomposed, if they are nontrivial, as a tensor product of two group rings. Here, the coalgebra structure is the ordinary tensor product coalgebra structure, whereas the algebra structure is a crossed product.
 Clifford theory for Radford biproducts (CliffordTheorie für RadfordBiprodukte)
 Date: Tuesday, December 15, 1998
 Location: University of Munich, Germany
 Summary: We show how Clifford theory for Radford biproducts can be used to obtain coherent proofs of results which are important for the structure theory of semisimple Hopf algebras.
 Semisimple Hopf algebras of dimension pq^{2} (Halbeinfache Hopfalgebren der Dimension pq^{2})
 Date: Tuesday, December 22, 1998
 Location: University of Munich, Germany
 Summary: We discuss applications of the structure theory of YetterDrinfel'd Hopf algebras and of Clifford theory to the classification of semisimple Hopf algebras of dimension pq^{2}.
 Cocommutative YetterDrinfel'd Hopf algebras (Kokommutative YetterDrinfel'dHopfalgebren)
 Date: June 15, 1999
 Location: University of Munich, Germany
 Summary: We describe in detail the applications of Clifford theory to the proof of the structure theorem for cocommutative, cosemisimple YetterDrinfel'd Hopf algebras over groups of prime order.
 Semisimplicity and cosemisimplicity (Halbeinfachheit und Kohalbeinfachheit)
 Date: June 22, 1999
 Location: University of Munich, Germany
 Summary: We discuss the question when a semisimple Hopf algebra over a field of positive characteristic is cosemisimple.
 Biproducts, triangular decompositions, and Hopf algebra extensions (Biprodukte, Dreieckszerlegungen und Hopfalgebrenerweiterungen)
 Date: July 20, 1999
 Location: University of Munich, Germany
 Summary: From a YetterDrinfel'd Hopf algebra, it is possible to construct two ordinary Hopf algebras: The Radford biproduct on the one hand and a Hopf algebra with triangular decomposition on the other hand. We discuss the connection of these constructions with extensions of Hopf algebras.
 YetterDrinfel'd Hopf algebras over groups of prime order
 Location: Mathematical Sciences Research Institute, Berkeley, USA
 Date: September 24, 1999
 Summary: YetterDrinfel'd Hopf algebras are Hopf algebras in a certain quasisymmetric category. They give rise to ordinary Hopf algebras via the Radford biproduct construction. In the talk, we consider YetterDrinfel'd Hopf algebras over groups of prime order that are cocommutative and cosemisimple as coalgebras. For these algebras, we outline the proof of the following structure theorem: Such a YetterDrinfel'd Hopf algebra can be decomposed into a tensor product of two group rings, where one of these the group ring of the corresponding cyclic group of prime order. The coalgebra structure is the usual tensor product coalgebra structure, whereas the algebra structure is a crossed product multiplication. In particular, the dimension of such a YetterDrinfel'd Hopf algebra is divisible by the prime under consideration. This theorem has several applications in the classification program for semisimple Hopf algebras.
 Drinfel'd algebras (Drinfel'dAlgebren)
 Date: December 3, 1999
 Location: University of Munich, Germany
 Summary: We discuss an axiomatic characterization of the Drinfel'd double of a finitedimensional Hopf algebra in terms of a factorization into two subalgebras. The talk is based on joint work with Y. Zhu.
 The modular group and the Drinfel'd double construction
(Die Modulgruppe und die Drinfel'dDoppelkonstruktion) Date: Friday, June 16, 2000
 Location: University of Munich, Germany
 Summary: We explain the action of the modular group on the character ring of the Drinfel'd double of a semisimple Hopf algebra.
 The linkage principle (Das Verbindungsprinzip)
 Date: Thursday, February 8, 2001
 Location: University of Munich, Germany
 Summary: We explain how Clifford theory gives a connection between the modules of a YetterDrinfel'd Hopf algebra and the modules of its Radford biproduct, and which role group cohomology plays for this connection.
 Selfdual modules of semisimple Hopf algebras
(Selbstduale Moduln über halbeinfachen Hopfalgebren) Date: Thursday, July 12, 2001
 Location: University of Munich, Germany
 Summary: We prove that a semisimple cosemisimple Hopf algebra that has a nontrivial selfdual simple module must have even dimension. This generalizes a classical result of W. Burnside on groups of odd order.
 Radford biproducts und group cohomology
(RadfordBiprodukte und Gruppenkohomologie)
 Date: Thursday, January 10, 2002
 Location: University of Munich, Germany
 Summary: For YetterDrinfel'd Hopf algebras over abelian groups, the dual group acts on the character ring of the Radford biproduct. In contrast to the commutative case, there is for noncommutative YetterDrinfel'd Hopf algebras no onetoone correspondence between the isotropy groups of the irreducible characters of the Radford biproduct and the inertia groups of the corresponding twosided ideals of the YetterDrinfel'd Hopf algebra. Instead, the isotropy group of the irreducible character is also determined by the cocycle that the connecting homomorphism in nonabelian cohomology assigns to the action of the inertia group on the corresponding ideal of the YetterDrinfel'd Hopf algebra. We explain how Clifford theory can be used to determine the isotropy group from the inertia group and the cocycle.
 YetterDrinfel'd algebras (YetterDrinfel'dAlgebren)
 Date: Thursday, January 17, 2002
 Location: University of Munich, Germany
 Summary: Isomorphism classes of YetterDrinfel'd algebras over abelian groups are in some situations completely determined by a cocycle and a group automorphism. This leads to great restrictions for YetterDrinfel'd algebra homomorphisms into tensor products. We explain how these restrictions arise.
 YetterDrinfel'd Hopf algebras over abelian groups
(YetterDrinfel'dHopfalgebren über abelschen Gruppen) Date: Thursday, January 24, 2002
 Location: University of Munich, Germany
 Summary: A YetterDrinfel'd Hopf algebra is called stable if its twosided ideals are also submodules. We show that stability imposes a very big restriction on the possible structures of a YetterDrinfel'd Hopf algebra.
 Hopf algebras  Historical origins and recent developments
 Date: Wednesday, January 30, 2002
 Location: Syracuse University, USA
 Summary: Hopf algebras are algebras for which one can form the tensor product of two modules. They arise in a number of areas in mathematics, and even in physics: In algebraic topology, in the theory of algebraic groups, in the duality theory for topological groups, in various places in Lie theory, in the theory of exactly solvable models in statistical mechanics, and in conformal field theory, this notion has been employed to understand and to formalize the principles behind several specific phenomena. Today, we witness first beginnings of a general structure theory for these algebras. In the talk, we will consider the role of Hopf algebras in the various mathematical theories from the historical beginnings until the recent first elements of a structure theory.
 Heisenberg systems (HeisenbergSysteme)
 Date: Tuesday, July 11, 2002
 Location: University of Munich, Germany
 Summary: We give an introduction to the theory of Heisenberg systems and their basic combinatorial properties.
 Heisenberg systems
 Date: Wednesday, October 2, 2002
 Location: Hong Kong University of Science and Technology, Hong Kong, China
 Summary: We give an introduction to the theory of Heisenberg systems and their basic combinatorial properties.
 Introduction to YetterDrinfel'd Hopf algebras I
 Date: Friday, October 4, 2002
 Location: Hong Kong University of Science and Technology, Hong Kong, China
 Summary: We discuss basic properties of YetterDrinfel'd Hopf algebras as well as the theory of their integrals.

Introduction to YetterDrinfel'd Hopf algebras II
 Date: Wednesday, October 9, 2002
 Location: Hong Kong University of Science and Technology, Hong Kong, China
 Summary: We discuss the application of YetterDrinfel'd Hopf algebras to the construction of deformed enveloping algebras as well as in the classification of semisimple Hopf algebras.
 Sweedler powers (SweedlerPotenzen)
 Date: Thursday, January 9, 2003
 Location: University of Munich, Germany
 Summary: Applying the comultiplication to an element of a Hopf algebra several times, permuting the arising tensorands, and then multiplying, you get a Sweedler power. In this way, the action of a certain group on the Hopf algebra arises. We discuss the basic formalism that underlies these operations.
 Higher FrobeniusSchur indicators (Höhere FrobeniusSchurIndikatoren)
 Date: Thursday, January 16, 2003
 Location: University of Munich, Germany
 Summary: We explain how to define higher FrobeniusSchur indicators in the case of semisimple Hopf algebras, and present an analogue of the FrobeniusSchur theorem for higher FrobeniusSchur indicators. Furthermore, we present a divisibility result that generalizes the theorem that a semisimple Hopf algebra that has a nontrivial selfdual simple module must have even dimension.

On higher FrobeniusSchur indicators
 Date: Friday, March 14, 2003
 Location: DePaul University, Chicago, USA
 Summary: For a finite group, one can evaluate an irreducible character against the sum of the powers of the group elements. In the case of the sum of the squares of the group elements, the resulting number is called the FrobeniusSchur indicator of the character; the higher powers lead to the higher FrobeniusSchur indicators. All these notions can be generalized to Hopf algebras. We present an analogue of the FrobeniusSchur theorem for higher FrobeniusSchur indicators. Furthermore, we present a divisibility result that generalizes the theorem that a semisimple Hopf algebra that has a nontrivial selfdual simple module must have even dimension.
 On higher FrobeniusSchur indicators of the Drinfel'd double of a finite group
 Date: Monday, March 17, 2003
 Location: DePaul University, Chicago, USA
 Summary: We explain what the integrality for the higher FrobeniusSchur indicators of the Drinfel'd double of a finite group means for the group itself, and from this viewpoint discuss various examples.
 On higher FrobeniusSchur indicators
 Date: Tuesday, March 25, 2003
 Location: University of Southern California, Los Angeles, USA
 Summary: For a finite group, one can evaluate an irreducible character against the sum of the powers of the group elements. In the case of the sum of the squares of the group elements, the resulting number is called the FrobeniusSchur indicator of the character; the higher powers lead to the higher FrobeniusSchur indicators. All these notions can be generalized to Hopf algebras. We present an analogue of the FrobeniusSchur theorem for higher FrobeniusSchur indicators. Furthermore, we present a divisibility result that generalizes the theorem that a semisimple Hopf algebra that has a nontrivial selfdual simple module must have even dimension.
 Higher FrobeniusSchur indicators: Examples
 Date: Wednesday, March 26, 2003
 Location: University of Southern California, Los Angeles, USA
 Summary: We calculate the higher FrobeniusSchur indicators for certain semisimple Hopf algebras that we construct as special central extensions. The Drinfel'd double of a finite group is a special case of this construction.
 The index of a character
 Date: Wednesday, March 16, 2005
 Location: DePaul University, Chicago, USA
 Summary: For a finite group, the largest value that a character can attain on a group element is the degree of the character. The entries of the character table of the same absolute value arise from central elements of the quotient in which the elements that act trivially have been eliminated. In the talk, we explain how this correspondence can be generalized to semisimple Hopf algebras. The talk is based on joint work with Y. Kashina and Y. Zhu.
 Introduction to YetterDrinfel'd Hopf algebras I
 Date: Friday, October 21, 2005
 Location: University of Munich, Germany
 Summary: We give an introduction to the theory of YetterDrinfel'd modules and YetterDrinfel'd Hopf algebras, discuss the connection with the Drinfel'd double construction, and show how via the Radford projection theorem YetterDrinfel'd Hopf algebras appear in the theory of ordinary Hopf algebras.
 Introduction to YetterDrinfel'd Hopf algebras II
 Date: Friday, October 28, 2005
 Location: University of Munich, Germany
 Summary: We discuss the theory of integrals in YetterDrinfel'd Hopf algebras, explain their role in the theory of deformed enveloping algebras of semisimple Lie algebras, and give a structure theorem for commutative semisimple YetterDrinfel'd Hopf algebras over groups of prime order.
 Selfcentralizing subalgebras
 Date: Thursday, November 24, 2005
 Location: University of Munich, Germany
 Summary: A semisimple selfcentralizing subalgebra of a matrix ring is conjugate to the algebra of diagonal matrices. If this matrix ring is the endomorphism ring of a dual group ring, it may happen that this selfcentralizing subalgebra is stable under a canonical action of the group. We discuss how the group may act on the idempotents of the algebra.
 Selfcentralizing subalgebras of YetterDrinfel'd Hopf algebras
 Date: Thursday, December 1, 2005
 Location: University of Munich, Germany
 Summary: We apply the theory of selfcentralizing subalgebras developed so far to describe the coproduct in a YetterDrinfel'd Hopf algebra.
 Cauchy's theorem for Hopf algebras
 Date: Wednesday, April 12, 2006
 Location: Hong Kong University of Science and Technology, Hong Kong, China
 Summary:
Cauchy's theorem states that a finite group contains an element of prime order for every prime that divides the order of the group. Since the exponent of a group is the least common multiple of the orders of all its elements, this can be reformulated by saying that a prime that divides the order of a group also divides its exponent. It was an open conjecture by P. Etingof and S. Gelaki that this result, in this formulation, holds also for semisimple Hopf algebras. In this talk, we present a proof of this conjecture, which is joint work with Y. Kashina and Y. Zhu.
The talk is intended for a general audience; in particular, no knowledge of Hopf algebras will be assumed. We will therefore begin by explaining what a Hopf algebra is and how the exponent of a Hopf algebra can be defined. We will then explain how the analogue of Cauchy's theorem can be deduced from the theory of higher FrobeniusSchur indicators.
 Cauchy's theorem for Hopf algebras
 Date: Thursday, May 4, 2006
 Location: University of Hong Kong, Hong Kong, China
 Summary:
Cauchy's theorem states that a finite group contains an element of prime order for every prime that divides the order of the group. Since the exponent of a group is the least common multiple of the orders of all its elements, this can be reformulated by saying that a prime that divides the order of a group also divides its exponent. It was an open conjecture by P. Etingof and S. Gelaki that this result, in this formulation, holds also for semisimple Hopf algebras. In this talk, we present a proof of this conjecture, which is joint work with Y. Kashina and Y. Zhu.
The talk is intended for a general audience; in particular, no knowledge of Hopf algebras will be assumed. We will therefore begin by explaining what a Hopf algebra is and how the exponent of a Hopf algebra can be defined. We will then explain how the analogue of Cauchy's theorem can be deduced from the theory of higher FrobeniusSchur indicators.
 A survey of YetterDrinfel'd Hopf algebras
 Date: Friday, May 19, 2006
 Location: Hong Kong University of Science and Technology, Hong Kong, China
 Summary:
YetterDrinfel'd Hopf algebras are a class of generalized Hopf algebras that in applications appear as often as usual Hopf algebras.
The simplest example being the exterior algebra of a vector space, already the original examples considered by H. Hopf were not Hopf algebras in our sense today, but rather YetterDrinfel'd Hopf algebras.
The generalization lies in the fact that the algebra structure on the tensor product is not the canonical one, but rather depends on a more complicated interchange law when the factors pass each other  for example, they may pick up a sign dependent on their degree in a chain complex.
In recent years, YetterDrinfel'd Hopf algebras have become one of the most important tools in the development of the structure theory for ordinary Hopf algebras. This is a consequence of the Radford projection theorem, which asserts that in a semidirect product situation one of the factors is not an ordinary Hopf algebra, but rather a YetterDrinfel'd Hopf algebra. As semidirect product decompositions play a key role in both the structure theory for semisimple Hopf algebras and the structure theory for pointed Hopf algebras, YetterDrinfel'd Hopf algebras have become important ingredients of these theories.
In the talk, we will first introduce YetterDrinfel'd Hopf algebras and develop their elementary properties, and then discuss the Radford projection theorem that links them to ordinary Hopf algebras. We will then focus on the structure theorems for semisimple YetterDrinfel'd Hopf algebras.
 Hopf algebras  Foundations and recent developments
 Date: Monday, June 26, 2006  Tuesday, July 4, 2006
 Location: University of Almeria, Spain
 Summary: We give an introduction to the theory of Hopf algebras with a view toward recent results on semisimple Hopf algebras. Starting from basic notions, we cover the theory of integrals and its applications to semisimplicity questions and the order of the antipode. Particular emphasis will be put on results that are analogues of results for finite groups, like the theorems of Lagrange and Cauchy.
 Cauchy's theorem for Hopf algebras
 Date: Monday, September 25, 2006
 Location: University of Cincinnati, USA
 Summary:
Cauchy's theorem states that a finite group contains an element of prime order for every prime that divides the order of the group. Since the exponent of a group is the least common multiple of the orders of all its elements, this can be reformulated by saying that a prime that divides the order of a group also divides its exponent. It was an open conjecture by P. Etingof and S. Gelaki that this result, in this formulation, holds also for semisimple Hopf algebras. In this talk, we present a proof of this conjecture, which is joint work with Y. Kashina and Y. Zhu.
The talk is intended for a general audience; in particular, no knowledge of Hopf algebras will be assumed. We will therefore begin by explaining what a Hopf algebra is and how the exponent of a Hopf algebra can be defined. We will then explain how the analogue of Cauchy's theorem can be deduced from the theory of higher FrobeniusSchur indicators. The talk also offers a good opportunity to learn about the research of a new faculty member.
 Cauchy's theorem for Hopf algebras II
 Date: Monday, October 9, 2006
 Location: University of Cincinnati, USA
 On Kaplansky's conjectures
 Date: Thursday, November 2, 2006
 Location: University of Cincinnati, USA
 Summary: In 1973, I. Kaplansky taught a course on Hopf algebras at the University of Chicago. In an appendix to the lecture notes that he prepared on this occasion, he lists ten problems in the theory that he considered as important. In this talk, we discuss the current status of these conjectures.
 Introduction to YetterDrinfel'd Hopf algebras I
 Date: Friday, January 12, 2007
 Location: University of Cincinnati, USA
 Summary:
YetterDrinfel'd Hopf algebras are a class of generalized Hopf algebras that in applications appear as often as usual Hopf algebras. The simplest example being the exterior algebra of a vector space, already the original examples considered by H. Hopf were not Hopf algebras in our sense today, but rather YetterDrinfel'd Hopf algebras.
The generalization lies in the fact that the algebra structure on the tensor product is not the canonical one, but rather depends on a more complicated interchange law when the factors pass each other  for example, they may pick up a sign dependent on their degree in a chain complex.
In the talk, we will first contrast ordinary Hopf algebras and YetterDrinfel'd Hopf algebras and then explain the difference by comparing the symmetric algebra and the exterior algebra. We will then discuss elementary properties of YetterDrinfel'd Hopf algebras. No prerequisites beyond linear algebra will be needed.
 Introduction to YetterDrinfel'd Hopf algebras II
 Date: Friday, February 16, 2007
 Location: University of Cincinnati, USA
 Summary:
In recent years, YetterDrinfel'd Hopf algebras have become one of the most important tools
in the development of the structure theory for ordinary Hopf algebras. This is a
consequence of the Radford projection theorem, which asserts that in a
semidirect product situation one of the factors is not an ordinary Hopf algebra,
but rather a YetterDrinfel'd Hopf algebra. As semidirect product decompositions
play a key role in both the structure theory for semisimple Hopf algebras and the
structure theory for pointed Hopf algebras, YetterDrinfel'd Hopf algebras have become
important ingredients of these theories.
In the talk, we will first review the definition of YetterDrinfel'd Hopf algebras and then discuss the Radford projection theorem that links them to ordinary Hopf algebras. We will then focus on the structure theorems for semisimple YetterDrinfel'd Hopf algebras.
 YetterDrinfel'd Hopf algebras over groups of prime order
 Date: Wednesday, March 14, 2007
 Location: DePaul University, Chicago, USA
 Summary: We give an introduction to YetterDrinfel'd Hopf algebras and then explain the structure theorem for nontrivial semisimple cocommutative YetterDrinfel'd Hopf algebras over groups of prime order. The talk is accessible to mathematicians with a general background in Hopf algebra theory.
 YetterDrinfel'd Hopf algebras over abelian groups
 Date: Thursday, March 15, 2007
 Location: DePaul University, Chicago, USA
 Summary: If one considers semisimple cocommutative YetterDrinfel'd Hopf algebras over an arbitrary finite abelian group instead of of a group of prime order, the socalled core of a purely unstable grouplike element is no longer a group, but rather a YetterDrinfel'd Hopf subalgebra, because the action of the finite abelian group on the core is in general nontrivial. We propose a construction of those YetterDrinfel'd Hopf algebras that can arise as cores in the above situation. The talk is addressed at specialists in the theory of semisimple Hopf algebras.
 Cauchy's theorem for Hopf algebras
 Date: Friday, March 23, 2007
 Location: University of South Alabama, Mobile, USA
 Summary: Cauchy's theorem states that a finite group contains an element of prime order for every prime that divides the order of the group. Since the exponent of a group is the least common multiple of the orders of all its elements, this can be reformulated by saying that a prime that divides the order of a group also divides its exponent. It was an open conjecture by P. Etingof and S. Gelaki that this result, in this formulation, holds also for semisimple Hopf algebras. In this talk, we present a proof of this conjecture, which is joint work with Y. Kashina and Y. Zhu.
 Modular group actions arising from Hopf algebras I
 Date: Tuesday, May 8, 2007
 Location: University of Cincinnati, Cincinnati, USA
 Summary: We explain how the modular group acts on the center of the Drinfel'd double of a semisimple Hopf algebra.
 Modular group actions arising from Hopf algebras II
 Date: Tuesday, May 22, 2007
 Location: University of Cincinnati, Cincinnati, USA
 Summary: We explain how the modular group acts on the center of the Drinfel'd double of a semisimple Hopf algebra, and discuss FrobeniusSchur indicators in this context.
 Modular group actions arising from Hopf algebras III
 Date: Friday, June 1, 2007
 Location: University of Cincinnati, Cincinnati, USA
 Summary: We explain the orbit theorem and the congruence subgroup theorem.
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