Inhaltsbereich
Yorck Sommerhäuser
Conference talks:
- Workshop `Lie-Algebren und Quantengruppen'
- Title: Deformed enveloping algebras
- Location: Munich, Germany
- Date: Friday, July 28, 1995
- Summary: We construct deformed enveloping algebras by a generalized semidirect product construction from the parts of the triangular decomposition. This gives an alternative description of these objects that is not based on generators and relations.
- Ring theory conference
- Title: Hopf algebras with triangular decomposition
- Location: Miscolc, Hungary
- Date: Friday, July 19, 1996
- Summary: We construct semisimple Lie algebras and their deformations using a generalized
semidirect product construction. This gives an alternative description of these objects which is not based on generators and relations. We discuss related results concerning the modules and the integrals of these Hopf algebras.
- Conference `Interaction between Ring theory and Representations of Algebras'
- Title: On Kaplansky's conjectures
- Location: Murcia, Spain
- Date: Tuesday, January 13, 1998
- Summary:
In 1975, I. Kaplansky set up 10 conjectures on Hopf algebras during his
lectures at the University of Chicago. Since then, these conjectures have been
a great challenge to several researchers in the area. The theory of Hopf algebras now seems to be in a stage of development that makes it possible
to cope with these conjectures, and therefore recently several results on these
conjectures have been established. The talk will review this recent progress,
as well as some work on generalizations, but also some open problems connected
with these conjectures.
- Conference `Hopf algebras and quantum groups'
- Title: Yetter-Drinfel'd Hopf algebras over groups of prime order
- Location: Brussels, Belgium
- Date: Thursday, June 18, 1998
- Summary:
Yetter-Drinfel'd Hopf algebras are Hopf algebras inside a quasisymmetric
category that are the ingredient for the Radford biproduct construction.
We discuss commutative semisimple Yetter-Drinfel'd Hopf algebras over groups of prime order and apply the results to the classification programme for semisimple Hopf algebras.
- International Conference on Algebra and its Applications
- Title: Structure theory for Yetter-Drinfel'd Hopf algebras
- Location: Athens, USA
- Date: Sunday, March 28, 1999
- Summary:
Yetter-Drinfel'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 Yetter-Drinfel'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: They can be decomposed into a tensor product of two group rings. The coalgebra structure is the usual tensor product coalgebra structure, whereas the algebra structure is a crossed product multiplication. This theorem has several applications in the classification program for semisimple Hopf algebras.
- Colloquium on quantum groups and Hopf algebras
- Title: The structure of Yetter-Drinfel'd Hopf algebras over groups
of prime order
- Location: Córdoba, Argentina
- Date: Friday, August 13, 1999
- Summary: Yetter-Drinfel'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 Yetter-Drinfel'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 Yetter-Drinfel'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 Yetter-Drinfel'd Hopf
algebra is divisible by the prime under consideration. This theorem has several applications in the classification program for semisimple Hopf algebras.
- MSRI Workshop `Hopf algebras'
- Title: On central character rings
- Location: Berkeley, USA
- Date: Monday, October 25, 1999
- Summary: We prove that, for a semisimple Hopf algebra over an algebraically closed field of characteristic zero, the dimensions of the simple modules divide the dimension of the Hopf algebra, provided that the character ring of the Hopf algebra is contained in the center of the dual Hopf algebra and the character ring of the dual Hopf algebra is contained in the center of the original Hopf algebra. Afterwards, we discuss possible generalizations of this result.
- Workshop: Operads and their applications
- Title: Vertex operator algebras
- Location: Oberwolfach, Germany
- Date: Friday, October 13, 2000
- Summary: We review the work of Y. Z. Huang on the geometric description of vertex operator algebras. Vertex operator algebras are defined via a graded vector
space, called the state space, and a map, called the state-field correspondence, from the state space to the space of fields, which are formal distributions with coefficients in the endomorphism algebra of the state space.
On the other hand, geometric vertex operator algebras are mappings from the partial operad of moduli spaces of punctured spheres to the partial endomorphism operad. Huang's work gives a one-to-one correspondence between these two objects by looking at correlation functions of vertex operator algebras.
- Workshop: Groups, Rings, Lie and Hopf algebras
- Title: The Drinfel'd double and the modular group
- Location: St. John's, Canada
- Date: Tuesday, May 29, 2001
- Summary:
As we learn from conformal field theory, it is possible to construct an action
of the modular group SL(2,Z) on the centralizer of H* in the
Drinfel'd double of a semisimple Hopf algebra H. One construction of this
action can be carried out by using the Drinfel'd element together with the
natural evaluation form. We explain the relation of this construction to other
constructions and discuss some consequences.
- International Hopf Algebras Conference
- Title: Self-dual modules of semisimple Hopf algebras
- Location: Chicago, USA
- Date: Saturday, February 2, 2002
- Summary:
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.
- Hopf Algebras in Noncommutative Geometry and Physics
- Title: Self-dual modules of semisimple Hopf algebras
- Location: Brussels, Belgium
- Date: Tuesday, May 28, 2002
- Summary:
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.
- AMS Spring Eastern Sectional Meeting:
Special session on Hopf algebras and quantum groups
- Title: Higher Frobenius-Schur indicators
- Location: New York, USA
- Date: Sunday, April 13, 2003
- 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 Frobenius-Schur indicator of the character; the higher powers lead to the higher Frobenius-Schur indicators. All these notions can be generalized to Hopf algebras. We present an analogue of the Frobenius-Schur theorem for higher Frobenius-Schur indicators. Furthermore, we present a divisibility result that generalizes the theorem that a semisimple Hopf algebra that has a nontrivial self-dual simple module must have even dimension.
- International Conference on Quantum Groups
- Title: The index of a character
- Location: Haifa, Israel
- Date: Friday, July 9, 2004
- 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.
- AMS Spring Southeastern Sectional Meeting: Special session on Hopf algebras and related topics
- Title: Cauchy's theorem for Hopf algebras
- Date: Saturday, March 19, 2005
- Location: Bowling Green (Kentucky), 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. We prove that, in this formulation, Cauchy's theorem also holds for semisimple Hopf algebras: A prime that divides the dimension of a semisimple Hopf algebra also divides its exponent. This result has been conjectured by P. Etingof and S. Gelaki. It is known in the case of the prime 2.
- Second joint meeting of AMS, DMV, and ÖMG: Special session on Hopf algebras and quantum groups
- Title: Yetter-Drinfel'd Hopf algebras over abelian groups
- Date: Friday, June 17, 2005
- Location: Mainz, Germany
- Summary:
We review recent progress on the structure theory of Yetter-Drinfel'd Hopf algebras over abelian groups.
- First Joint International Meeting between the AMS and the PTM: Special session on noncommutative geometry and quantum groups
- Title: Hopf algebras, Cauchy's theorem, and Frobenius-Schur indicators
- Date: Wednesday, August 1, 2007
- Location: Warsaw, Poland
- Summary: In 1906, F. G. Frobenius and I. Schur considered the question
when an irreducible matrix representation of a finite group is conjugate
to a matrix representation in which all matrices are real. They found
that, besides the condition that the character of the representation is real-valued, it is necessary that another number, now called the Frobenius-Schur indicator, is +1. This indicator is actually part of
a family of numbers, called the higher Frobenius-Schur indicators. We explain how this concept can be generalized to Hopf algebras, and then can be used to prove a version of Cauchy's theorem for Hopf algebras.
- Noncommutative Geometry and Quantum Groups 2007
- Title: Anti-Yetter Drinfel'd modules
- Date: Friday, August 3, 2007
- Location: Warsaw, Poland
- Summary: We explain what anti-Yetter Drinfel'd modules are and how they
differ from Yetter-Drinfel'd modules. We then explain the concept of stability and give applications to cyclic homology.
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