Titles and Abstracts
Yang Han: Hochschild (co)homology dimension
In 1989 Happel asked the question, whether for a finite-dimensional algebra A over an algebraically closed field k, the global dimension gl.dim.(A) is finite if and only if hch.dim.(A) is finite. Here, the Hochschild cohomology dimension hch.dim.(A) of A is the infimum of all n such that HH^i(A)=0 for i>n. Recently Buchweitz-Green-Madsen-Solberg gave a negative answer to Happel's question. They found a family of pathological algebras A_q for which gl.dim.(A_q) is infinite but hch.dim.(A_q)=2. These algebras are pathological in many aspects, however their Hochschild homology behaviors are not pathological any more, indeed one has that hh.dim.(A_q) is infinite. Here, the Hochschild homology dimension hh.dim.(A) is the infimum of all n such that HH_i(A)=0 for i>n. This suggests to pose a seemingly more reasonable conjecture by replacing Hochschild cohomology dimension in Happel's question with Hochschild homology dimension: gl.dim.(A) is finite if and only if hh.dim.(A) is finite if and only if hh.dim.(A)=0. The conjecture holds for commutative algebras and monomial algebras. In case A is a truncated quiver algebras these conditions are equivalent to that the quiver of A has no oriented cycles. Moreover, an algorithm for computing the Hochschild homology of any monomial algebra is provided. Thus the cyclic homology of any monomial algebra can be read off in case the underlying field is characteristic 0.
Joachim Hilgert: Hecke algebra actions on spaces of period functions
Period functions are the analog of period polynomials, wellknown in the theory of modular forms, for Maass cusp forms. Before they were considered in the context of Maass forms they came up in physicists work on the transfer operator of geodesic flows on hyperbolic surfaces. The structure of these operators suggested the existence of Hecke type operators on spaces of period functions. It turns out that the constructions given in the context of transfer operators can be related to the standard Hecke operators on Maass forms in a precise way.
Andrew Hubery: Ringel-Hall Algebras of Affine Quivers
Over the last ten years there has been much work done on understanding the structure of the Ringel-Hall algebras of affine quivers. In this talk we offer an elementary proof of the structure of the corresponding composition algebras, thereby strengthening the previous results as well as obtaining some new insights.
Steffen Koenig: Schur's thesis and some almost relatively true statements
This talk is about comparing ring structures, representations and cohomology of symmetric groups, diagram algebras and Schur algebras. (Taken from joint works with Changchang Xi / Luca Diracca / Robert Hartmann, Anne Henke and Rowena Paget.)
Dirk Kussin: One-parameter families for finite dimensional algebras
In the study of one-parameter families of modules over a finite dimensional algebra over an arbitrary field the case of a tame bimodule algebra is fundamental. We show how the geometric structure of these families can be explained with the concept of non-commutative unique factorization domains (in the sense of Chatters and Jordan). As application we show that there is a strong relationship between two invariants, which were studied/introduced by C. M. Ringel in 1979: The endomorphism ring of the generic module, which we call the function (skew) field, and the multiplicity function.
Yuming Liu: A new construction of stable equivalence of Morita type
We will present a new construction of stable equivalence of Morita type for finite dimensional algebras by using the technique of adjointness. This is a joint work with Changchang Xi.
Claus Michael Ringel: Semilinear strings and bands
The semilinear strings and bands which we discuss concern the following situation: There is given a quiver with relations as for a string algebra, and a field k. In addition, for every arrow α there is attached an automorphism Σα of k (as a field). The semilinear representations to be considered are given as usual by vector spaces and maps, however the maps are semilinear: the map corresponding to the arrow α has to be Σα-semilinear. We will outline that the usual procedure of determining the indecomposable representations of a string algebra (leading to string modules and bimodules) generalizes to this setting: we obtain semilinear string modules and semilinear band modules. In this way, the combinatorics is the same, what differs is the geometry of the families of band modules: here we deal with finite length modules over a skew polynomial ring. One special case will be considered in more detail, it concerns the quiver with n vertices arranged on a cycle, with arrows α going around clockwise and β going around anti-clockwise, such that αβ=0=βα. This setting occurs in the work of Kottwitz and Rapoport dealing with F-crystal and F-isocrystals (Kottwitz-Rapoport: On the existence of F-crystals.Comment. Math. Helv 78 (2003), 153-184).
Changchang Xi: APR-tiltings, AR-sequences, and Homological dimensions
In this talk I shall report some new results on the relationship between AR-sequences, APR-tilting modules on the one hand and the homological dimensions (including global dimensions, representation dimensions) on the other hand.
Pu Zhang: Quiver Poisson Algebras
Inner Poisson algebras on a given associative algebra are introduced and characterized, which gives a way of constructing non-commutative Poisson structures. Applying these to the finite-dimensional path algebras kQ together with the decomposition into indecomposable Lie ideals of the standard Poisson structure on kQ, we classify all the inner Poisson structures on kQ, which turn out to be the piecewise scale Poisson algebras. We also determine all the finite quivers Q without oriented cycles such that kQ admits outer Poisson structures: this is exactly the finite quivers without oriented cycles such that there exists two non-trivial paths lying in a reduced closed walk, which can not be connected by a sequence of non-trivial paths.
Bin Zhu: BGP-reflection functors in cluster categories and applications
BGP-reflection functors of cluster categories are induced by BGP-reflection functors (or APR-tilting functor) in the category of representations of a quiver. They are triangle equivalent and are proved to be useful in the study of cluster algebras: they are quiver representation's interpretation of "truncated simple reflections" defined by Fomin and Zelevinsky on the one hand and on the other hand they induce some automorphisms of cluster algebras.