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The Geometric Classification of Four Dimensional Superalgebras
by
Aaron Armour
Victoria University of Wellington
The algebraic classification problem for algebras of a given dimension is to determine which algebra structures form irreducible components in Algn, with Algn being the variety of n-dimensional algebra structures. In this talk we shall briefly review these ideas and state the results in dimension four, before examining the corresponding problem for superalgebras and presenting the current state of the results for the geometric classification problem of superalgebras of dimension four.
Date received: December 2, 2007
A structure theorem for relative Hopf bimodules with applications to Morita equivalences
by
Stefaan Caenepeel
Vrije Universiteit Brussel
Coauthors: S. Crivei (University of Murcia, Spain)
A. Marcus (Babes-Bolyai University, Cluj-Napoca, Romania)
M. Takeuchi (University of Tsukuba, Japan)
Consider two Hopf-Galois extensions A and B. We present a Structure Theorem for Hopf bimodules: the category of Hopf bimodules is equivalent to the category of modules over the cotensor product of A and Bop. As an application, we show that a Morita equivalence between AcoH and BcoH can be lifted to an H-Morita equivalence between A and B if and only if the bimodule structure on one of the connecting modules can be extended to an action of the cotensor product on it. As a second application, we present a Hopf algebra version of an exact sequence due to Beattie and del Rio, connecting the graded Picard group of a strongly graded ring, and the stable part of the Picard group of its part of degree zero.
Date received: October 31, 2007
The Hopf-Schur subgroup
by
Juan Cuadra
University of Almeria (Spain)
A finite dimensional central simple k-algebra A (k a field) is Schur if there exists a finite group G and a surjective algebra morphism p:k[G] → A. Such an algebra is a simple component of the Wedderburn decomposition of k[G] when char(k) does not divide |G |. Those classes in Br(k), the Brauer group of k, represented by a Schur k-algebra form a subgroup, called the Schur subgroup of k.
In this talk we will propose a generalization of this subgroup by replacing in the above definition the group algebra by a Hopf algebra. The algebras so obtained are named Hopf-Schur algebras and the subset of Br(k) consisting of classes represented by a Hopf-Schur algebra is a subgroup, the Hopf-Schur subgroup. The aim of this talk is to prove that this new sugroup is much larger than the Schur group. To do this we will show the existence of a family of central simple k-algebras, for certain fields k, ocurring in the Wedderburn decomposition of a semisimple Hopf algebra but not in the Wedderburn decomposition of any semisimple group algebra. The results to be presented in this talk are part of a joint work with E. Aljadeff, S. Gelaki and E. Meir.
Date received: October 16, 2007
(Co)Representation theoretic approach to fundamental results in Hopf algebras
by
Miodrag C Iovanov
SUNY Buffalo and U Bucharest
Co-Frobenius coalgebras were introduced as dualizations of Frobenius algebras. Recently, it was shown in [M.C. Iovanov, Co-Frobenius Coalgebras, J. Algebra 303 (2006), no. 1, 146-153] that they addmit left-right symmetric characterizations analogue to those of Frobenius algebras: a coalgebra C is co-Frobenius if and only if it is isomorphic to its rational dual. We consider the more general quasi-co-Frobenius (QcF) coalgebras; we show that these also addmit symmetric characterizations: a coalgebra is QcF if it is weakly isomorphic to its (left, or equivalently right) rational dual Rat(C*), in the sense that certain coproduct powers of these objects are isomorphic. These show that QcF coalgebras can be viewed as generalizations of bothe co-Frobenius coalgebras and Frobenius algebras. Surprisingly, these turn out to have many applications to fundamental results of Hopf algebras. The equivalent characterizations of Hopf algebras with left (or right) nonzero integrals as left (or right) co-Frobenius, or QcF, or semiperfect or with nonzero rational dual all follow imediately from these results. Also, the uniqueness of integrals follows at the same time also as an equivalent statement. Moreover, as a by-product of our methods, we observe a short proof for the bijectivity of the antipode of a Hopf algebra with nonzero integral. This gives a purely representation theoretic approach to many of the basic fundamental results in the theory of Hopf algebras.
Date received: October 31, 2007
Classifying Semisimple Hopf Algebras of dimension 2n.
by
Yevgenia Kashina
DePaul University
In this talk we will discuss some recent progress in classification of semisimple Hopf algebras of dimension 2n with large abelian groups of grouplike elements.
Date received: October 30, 2007
On cocycle deformations of pointed Hopf algebras with abelian grouplikes
by
Akira Masuoka
University of Tsukuba
I will discuss cocycle deformations of some Hopf algebras, including the quantized enveloping algebras and the finite-dimensional pointed Hopf algebras due to Andruskiewitsch and Schneider.
Date received: September 25, 2007
Frobenius-Schur indicators for Hopf algebras
by
Susan Montgomery
University of Southern California
Frobenius-Schur indicators were originally defined for simple modules over finite groups, but have been extended to Hopf algebras, where they have proved very useful. A Hopf algebra H is called totally orthogonal if all of its simple modules have indicator +1 (this implies that each module admits a non-degenerate, symmetric, H-invariant bilinear form). In recent work, Guralnick and I have shown that the Drinfel'd double of a finite real reflection group is totally orthogonal, and Jedwab and I have studied this property for two bismash products associated to the symmetric group.
Date received: October 31, 2007
On the classification of Hopf algebras of dimension pq
by
Siu-Hung Ng
Iowa State University, USA
The classification of Hopf algebras of dimension pq, where p and q are distinct primes, is still open in general. It has been widely believed these Hopf algebras are trivial. In this talk, we will talk about some recent development of the problem. In particular, we will discuss a proof for the case when 2 < p < q ≤ 4p+11.
Date received: October 28, 2007
On Crystalline Graded Rings
by
Fred Van Oystaeyen
University of Antwerp, Belgium
We introduce a class of graded rings generalizing crossed product algebras as well as generalized Weyl algebras. For finite grading groups there are problems concerning the determination of the center and related properties like being a maximal order, aan Azumaya algebra, or an hereditary order. Fixing the part of degree zero to be a commutative Dedekind domain we study these properties in some detail. For infinite grading groups there are interesting examples generalizing the Weyl algebra.
Date received: November 7, 2007
On the representations of pointed Hopf algebras
by
David E. Radford
University of Illinois at Chicago
Let H be a Hopf algebra over a field k whose coradical is a sub-Hopf algebra. There is a program, called the Andruskiewitsch-Schneider classification program, to determine the structure of H. First pass to the associated graded Hopf algebra gr(H), secondly determine the structure of gr(H), and thirdly "lift" the relations of gr(H) to determine H.
This program has been carried out by these two authors with great success in particular when H is finite-dimensional, whose coradical is the group algebra of a finite commutative group, and k is algebraically closed of characteristic zero. In many cases H is the quotient of a generalized double, the irreducible modules of the tensor factors of which are one-dimensional.
We discuss the finite-dimensional irreducibles of such doubles and describe a generalized "highest weight" theory for them. The focus of this presentation will be a detailed discussion of applications to the representation theory of quotients of generalized doubles. This is the basis of an article based on joint work with Schneider.
Date received: October 29, 2007
Hopf Algebras and Congruence Subgroups
by
Yorck Sommerhäuser
University of South Alabama
Coauthors: Yongchang Zhu
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.
Date received: October 28, 2007
The Dickson Subcategory Splitting Conjecture for Pseudocompact Algebras
by
Blas Torrecillas
University of Almería, Spain
Coauthors: Miodrag Cristian Iovanov, Constantin Nastasescu
Let A be a pseudocompact (or profinite) algebra, so A = C* where C is a coalgebra. We show that the if the semiartinian part (the ”Dickson” part) of every A-module M splits off in M, then A is semiartinian, giving thus a positive answer in the case of algebras arising as dual of coalgebras (pseudocompact algebras), to a well known conjecture of Faith.
Date received: October 29, 2007