Yorck Sommerhäuser

Conference talks:

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

9. 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.

10. 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.

11. 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.

12. 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.

13. 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.

14. 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.

15. 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.

16. 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.

17. 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.

18. The Southern Regional Algebra Conference at the University of Louisiana at Lafayette
    • Title: The exponent of a Hopf algebra
    • Date: Sunday, September 30, 2007
    • Location: Lafayette, USA
    • Summary: A Hopf algebra is an algebra for which it is possible to form the tensor product of two representations. The fact that it is possible to form the tensor product of two group representations therefore can be viewed as a consequence of a Hopf structure on the group algebra. Certain concepts and theorems carry over from groups to Hopf algebras: We explain how the exponent may be defined in this setting and how Cauchy's theorem looks for Hopf algebras. The talk is based on joint work with Y. Kashina and Y. Zhu.

19. AMS Fall Central Sectional Meeting: Special session on Hopf algebras and related areas
    • Title: Frobenius-Schur indicators and their generalizations
    • Location: Chicago, USA
    • Date: Saturday, October 6, 2007
    • Summary: We introduce a class of generalized Frobenius-Schur indicators and discuss their applications. The talk is based on joint work with Yongchang Zhu.

20. First Joint International Meeting between the AMS and the NZMS: Special session on Hopf algebras and quantum groups
    • Title: Hopf algebras and congruence subgroups
    • Date: Saturday, December 15, 2007
    • Location: Wellington, New Zealand
    • Summary: We prove that the kernel of the natural action of the modular group on the center of the Drinfel'd double of a semisimple Hopf algebra is a congruence subgroup. To do this, we introduce a class of generalized Frobenius-Schur indicators and endow it with an action of the modular group that is compatible with the original one. The talk is based on joint work with Yongchang Zhu.

21. AMS Spring Southeastern Sectional Meeting: Special session on actions of quantum algebras
    • Title: The congruence subgroup property for factorizable Hopf algebras
    • Location: Baton Rouge, USA
    • Date: Sunday, March 30, 2008
    • Summary: Consider a factorizable semisimple Hopf algebra over an algebraically closed field of characteristic zero. If the Drinfel'd element and its inverse have the same trace in the regular representation, then the action of the modular group on the center of the Hopf algebra yields not only a projective, but rather an ordinary linear representation. We prove that in this case the kernel of this linear representation is a congruence subgroup of level N, where N is the exponent of the Hopf algebra. To do this, we introduce a generalization of the Jacobi symbol that relates the action of the Galois group to the action of the diagonal matrices in the quotient of the modular group. The talk is based on joint work with Yongchang Zhu.

22. AMS Spring Western Sectional Meeting: Special session on Hopf algebras and quantum groups
    • Title: The Hopf symbol
    • Location: Claremont, USA
    • Date: Saturday, May 3, 2008
    • Summary: It is a classical result that Gaussian sums transform with the Jacobi symbol under the action of the Galois group. We explain in the talk how this fact can be generalized to semisimple factorizable Hopf algebras: If one turns the group ring of a cyclic group into a factorizable Hopf algebra by endowing it with a nontrivial R-matrix, the Gaussian sum occurs as the trace of the inverse Drinfel'd element in the regular representation. It now turns out that also for more general factorizable Hopf algebras the trace of the inverse Drinfel'd element transforms under the action of the Galois group in a similar way, namely with a generalization of the Jacobi symbol that we call the Hopf symbol. The talk is based on joint work with Yongchang Zhu.

23. First Canadian Hopf algebra conference: the role of Hopf algebras in Noncommutative Geometry
    • Title: Frobenius-Schur indicators and congruence subgroups
    • Location: Fredericton, Canada
    • Date: Saturday, September 6, 2008
    • Summary: In a recent joint article with Yongchang Zhu, it was shown that the kernel of the action of the modular group on the center of a semisimple factorizable Hopf algebra, which is possibly sometimes only a projective action, is a congruence subgroup whose level is the exponent of the Hopf algebra. We give a comprehensive overview over this article.

24. The Southern Regional Algebra Conference at the University of Colorado at Colorado Springs
    • Title: Hopf algebras, Frobenius-Schur indicators, and the modular group
    • Location: Colorado Springs, USA
    • Date: Sunday, September 28, 2008
    • Summary: Every factorizable Hopf algebra leads to a projective representation of the modular group. In the semisimple case, the kernel of this representation is in fact congruence subgroup whose level is determined by the exponent of the Hopf algebra. We explain how generalized Frobenius-Schur indicators can be used to establish this fact.

25. Graduate Research Conference in Algebra and Representation Theory
    • Title: A survey on the theory of Hopf algebras
    • Location: Manhattan, USA
    • Date: Sunday, May 24, 2009, and Monday, May 25, 2009
    • Summary: A Hopf algebra is an algebra for which one can form the tensor product of two modules. For this, one needs a new structure element, the so-called coproduct. The theory of these algebras has progressed substantially in the last decade.

      We give an overview of the theory of Hopf algebras, starting from the definition and ending at some of the most recent developments. Throughout, we emphasize the analogy with groups; in particular, we will see what the analogues of Lagrange's theorem and Cauchy's theorem for Hopf algebras are.

26. Colloquium on Hopf Algebras, Quantum Groups and Tensor Categories
    • Title: On the central charge of a factorizable Hopf algebra
    • Location: Córdoba, Argentina
    • Date: Wednesday, September 2, 2009
    • Summary: For a semisimple factorizable Hopf algebra over a field of characteristic zero, we show that the value that an integral takes on the inverse Drinfel'd element differs from the value that it takes on the Drinfel'd element itself at most by a fourth root of unity. This can be reformulated by saying that the central charge of the Hopf algebra is an integer. If the dimension of the Hopf algebra is odd, we show that these two values differ at most by a sign, which can be reformulated by saying that the central charge is even. We give a precise condition on the dimension that determines whether the plus sign or the minus sign occurs. To formulate our results, we use the language of modular data. The talk is based on joint work with Yongchang Zhu.

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