Department Mathematik
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Inhaltsbereich

Yorck Sommerhäuser

Seminar talks:

  1. 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 Yetter-Drinfel'd Hopf algebras with an ordinary Hopf algebra. We prove that this is the general form of a Hopf algebra with triangular decomposition whose Borel-like parts are Radford biproducts.

  2. 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 Yetter-Drinfel"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.

  3. Semisimple Hopf algebras (Halbeinfache Hopfalgebren)
    • Date: Thursday, November 30, 1995
    • Location: University of Munich, Germany
    • Summary:

  4. Hopf algebras with triangular decomposition (Hopfalgebren mit Dreieckszerlegung)
    • Date: Thursday, June 20, 1996
    • Location: University of Munich, Germany
    • Summary:

  5. Kac-Moody algebras (Kac-Moody-Algebren)
    • 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 Yetter-Drinfel'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 Yetter-Drinfel'd Lie algebras are dual. Finally, we explain how all symmetrizable Kac-Moody algebras can be obtained in this way.

  6. Natural transformations in the category of Yetter-Drinfel'd modules (Natürliche Transformationen in der Kategorie der Yetter-Drinfel'd-Moduln)
    • 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 Yetter-Drinfel'd modules. For a Yetter-Drinfel'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 Yetter-Drinfel'd Hopf algebra. In particular, we find a formula for the fourth power of the antipode of a Yetter-Drinfel'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.

  7. On Kaplansky's fifth conjecture (Zur fünften Kaplansky-Vermutung)
    • 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.

  8. 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 Z-form 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.

  9. 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.

  10. Quadrilinear Frobenius extensions (Quadrilineare Frobenius-Erweiterungen)
    • 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.

  11. 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 Yetter-Drinfel'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.

  12. Commutative Yetter-Drinfel'd Hopf algebras (Kommutative Yetter-Drinfel'd-Hopfalgebren)
    • Date: Thursday, January 22, 1998
    • Location: University of Munich, Germany
    • Summary: We prove that a Yetter-Drinfel'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.

  13. Yetter-Drinfel'd Hopf algebras over groups of prime order (Yetter-Drinfel'd-Hopfalgebren 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 Yetter-Drinfel'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.

  14. Yetter-Drinfel'd Hopf algebras of dimension p2 (Yetter-Drinfel'd-Hopfalgebren der Dimension p2)
    • Date: Friday, July 10, 1998
    • Location: University of Munich, Germany
    • Summary: We construct for every finite commutative ring a nontrivial Yetter-Drinfel'd Hopf algebra. In particular, we construct nontrivial Yetter-Drinfel'd Hopf algebras of dimension p2.

  15. Structure theory of Yetter-Drinfel'd Hopf algebras (Strukturtheorie für Yetter-Drinfel'd-Hopfalgebren)
    • Date: Tuesday, December 8, 1998
    • Location: University of Munich, Germany
    • Summary: We prove a structure theorem for Yetter-Drinfel'd Hopfalgebras over groups of prime order which are commutative and semisimple as algebras and cocommutative and cosemisimple as coalgebras: These Yetter-Drinfel'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.

  16. Clifford theory for Radford biproducts (Clifford-Theorie für Radford-Biprodukte)
    • 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.

  17. Semisimple Hopf algebras of dimension pq2 (Halbeinfache Hopfalgebren der Dimension pq2)
    • Date: Tuesday, December 22, 1998
    • Location: University of Munich, Germany
    • Summary: We discuss applications of the structure theory of Yetter-Drinfel'd Hopf algebras and of Clifford theory to the classification of semisimple Hopf algebras of dimension pq2.

  18. Cocommutative Yetter-Drinfel'd Hopf algebras (Kokommutative Yetter-Drinfel'd-Hopfalgebren)
    • 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 Yetter-Drinfel'd Hopf algebras over groups of prime order.

  19. 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.

  20. Biproducts, triangular decompositions, and Hopf algebra extensions (Biprodukte, Dreieckszerlegungen und Hopfalgebrenerweiterungen)
    • Date: July 20, 1999
    • Location: University of Munich, Germany
    • Summary: From a Yetter-Drinfel'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.

  21. Yetter-Drinfel'd Hopf algebras over groups of prime order
    • Location: Mathematical Sciences Research Institute, Berkeley, USA
    • Date: September 24, 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.

  22. Drinfel'd algebras (Drinfel'd-Algebren)
    • Date: December 3, 1999
    • Location: University of Munich, Germany
    • Summary: We discuss an axiomatic characterization of the Drinfel'd double of a finite-dimensional Hopf algebra in terms of a factorization into two subalgebras. The talk is based on joint work with Y. Zhu.

  23. The modular group and the Drinfel'd double construction
    (Die Modulgruppe und die Drinfel'd-Doppelkonstruktion)
    • 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.

  24. 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 Yetter-Drinfel'd Hopf algebra and the modules of its Radford biproduct, and which role group cohomology plays for this connection.

  25. Self-dual 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 self-dual simple module must have even dimension. This generalizes a classical result of W. Burnside on groups of odd order.

  26. Radford biproducts und group cohomology (Radford-Biprodukte und Gruppenkohomologie)
    • Date: Thursday, January 10, 2002
    • Location: University of Munich, Germany
    • Summary: For Yetter-Drinfel'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 Yetter-Drinfel'd Hopf algebras no one-to-one correspondence between the isotropy groups of the irreducible characters of the Radford biproduct and the inertia groups of the corresponding two-sided ideals of the Yetter-Drinfel'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 Yetter-Drinfel'd Hopf algebra. We explain how Clifford theory can be used to determine the isotropy group from the inertia group and the cocycle.

  27. Yetter-Drinfel'd algebras (Yetter-Drinfel'd-Algebren)
    • Date: Thursday, January 17, 2002
    • Location: University of Munich, Germany
    • Summary: Isomorphism classes of Yetter-Drinfel'd algebras over abelian groups are in some situations completely determined by a cocycle and a group automorphism. This leads to great restrictions for Yetter-Drinfel'd algebra homomorphisms into tensor products. We explain how these restrictions arise.

  28. Yetter-Drinfel'd Hopf algebras over abelian groups
    (Yetter-Drinfel'd-Hopfalgebren über abelschen Gruppen)
    • Date: Thursday, January 24, 2002
    • Location: University of Munich, Germany
    • Summary: A Yetter-Drinfel'd Hopf algebra is called stable if its two-sided ideals are also submodules. We show that stability imposes a very big restriction on the possible structures of a Yetter-Drinfel'd Hopf algebra.

  29. 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.

  30. Heisenberg systems (Heisenberg-Systeme)
    • 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.

  31. 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.

  32. Introduction to Yetter-Drinfel'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 Yetter-Drinfel'd Hopf algebras as well as the theory of their integrals.

  33. Introduction to Yetter-Drinfel'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 Yetter-Drinfel'd Hopf algebras to the construction of deformed enveloping algebras as well as in the classification of semisimple Hopf algebras.

  34. Sweedler powers (Sweedler-Potenzen)
    • 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.

  35. Higher Frobenius-Schur indicators (Höhere Frobenius-Schur-Indikatoren)
    • Date: Thursday, January 16, 2003
    • Location: University of Munich, Germany
    • Summary: We explain how to define higher Frobenius-Schur indicators in the case of semisimple Hopf algebras, and 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.

  36. On higher Frobenius-Schur 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 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.

  37. On higher Frobenius-Schur 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 Frobenius-Schur indicators of the Drinfel'd double of a finite group means for the group itself, and from this viewpoint discuss various examples.

  38. On higher Frobenius-Schur 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 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.

  39. Higher Frobenius-Schur indicators: Examples
    • Date: Wednesday, March 26, 2003
    • Location: University of Southern California, Los Angeles, USA
    • Summary: We calculate the higher Frobenius-Schur 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.

  40. 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.

  41. Introduction to Yetter-Drinfel'd Hopf algebras I
    • Date: Friday, October 21, 2005
    • Location: University of Munich, Germany
    • Summary: We give an introduction to the theory of Yetter-Drinfel'd modules and Yetter-Drinfel'd Hopf algebras, discuss the connection with the Drinfel'd double construction, and show how via the Radford projection theorem Yetter-Drinfel'd Hopf algebras appear in the theory of ordinary Hopf algebras.

  42. Introduction to Yetter-Drinfel'd Hopf algebras II
    • Date: Friday, October 28, 2005
    • Location: University of Munich, Germany
    • Summary: We discuss the theory of integrals in Yetter-Drinfel'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 Yetter-Drinfel'd Hopf algebras over groups of prime order.

  43. Self-centralizing subalgebras
    • Date: Thursday, November 24, 2005
    • Location: University of Munich, Germany
    • Summary: A semisimple self-centralizing 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 self-centralizing subalgebra is stable under a canonical action of the group. We discuss how the group may act on the idempotents of the algebra.

  44. Self-centralizing subalgebras of Yetter-Drinfel'd Hopf algebras
    • Date: Thursday, December 1, 2005
    • Location: University of Munich, Germany
    • Summary: We apply the theory of self-centralizing subalgebras developed so far to describe the coproduct in a Yetter-Drinfel'd Hopf algebra.

  45. 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 Frobenius-Schur indicators.

  46. 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 Frobenius-Schur indicators.

  47. A survey of Yetter-Drinfel'd Hopf algebras
    • Date: Friday, May 19, 2006
    • Location: Hong Kong University of Science and Technology, Hong Kong, China
    • Summary: Yetter-Drinfel'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 Yetter-Drinfel'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, Yetter-Drinfel'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 Yetter-Drinfel'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, Yetter-Drinfel'd Hopf algebras have become important ingredients of these theories.

      In the talk, we will first introduce Yetter-Drinfel'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 Yetter-Drinfel'd Hopf algebras.

  48. 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.

  49. 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 Frobenius-Schur indicators. The talk also offers a good opportunity to learn about the research of a new faculty member.

  50. Cauchy's theorem for Hopf algebras II
    • Date: Monday, October 9, 2006
    • Location: University of Cincinnati, USA

  51. 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.

  52. Introduction to Yetter-Drinfel'd Hopf algebras I
    • Date: Friday, January 12, 2007
    • Location: University of Cincinnati, USA
    • Summary: Yetter-Drinfel'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 Yetter-Drinfel'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 Yetter-Drinfel'd Hopf algebras and then explain the difference by comparing the symmetric algebra and the exterior algebra. We will then discuss elementary properties of Yetter-Drinfel'd Hopf algebras. No prerequisites beyond linear algebra will be needed.

  53. Introduction to Yetter-Drinfel'd Hopf algebras II
    • Date: Friday, February 16, 2007
    • Location: University of Cincinnati, USA
    • Summary: In recent years, Yetter-Drinfel'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 Yetter-Drinfel'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, Yetter-Drinfel'd Hopf algebras have become important ingredients of these theories.

      In the talk, we will first review the definition of Yetter-Drinfel'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 Yetter-Drinfel'd Hopf algebras.

  54. Yetter-Drinfel'd Hopf algebras over groups of prime order
    • Date: Wednesday, March 14, 2007
    • Location: DePaul University, Chicago, USA
    • Summary: We give an introduction to Yetter-Drinfel'd Hopf algebras and then explain the structure theorem for nontrivial semisimple cocommutative Yetter-Drinfel'd Hopf algebras over groups of prime order. The talk is accessible to mathematicians with a general background in Hopf algebra theory.

  55. Yetter-Drinfel'd Hopf algebras over abelian groups
    • Date: Thursday, March 15, 2007
    • Location: DePaul University, Chicago, USA
    • Summary: If one considers semisimple cocommutative Yetter-Drinfel'd Hopf algebras over an arbitrary finite abelian group instead of of a group of prime order, the so-called core of a purely unstable grouplike element is no longer a group, but rather a Yetter-Drinfel'd Hopf subalgebra, because the action of the finite abelian group on the core is in general nontrivial. We propose a construction of those Yetter-Drinfel'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.

  56. 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.

  57. 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.

  58. 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 Frobenius-Schur indicators in this context.

  59. 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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