Trends in Representation Theory of Algebras and Related Topics
This comprehensive volume contains 15 expository surveys on recent developments and trends in the representation theory of associative algebras, mostly presented at ICRA XII in Torun in August 2007. S. Ariki presents modular representation theory of finite dimensional Hecke algebras (including a reader friendly exposition of quasi-hereditary covers of cyclotomic Hecke algebras in terms of the category O for the rational Cherednik algebra). G.Bobinski, Ch. Riedtmann and A. Skowroński deal with algebras of semi-invariant polynomial functions on the affine varieties of linear representations of quivers of a given dimension vector with respect to conjugate actions of products of general linear groups, and also with the geometry of the sets of common zeros of generating non-constant semi-invariants. The paper by I. Burban and Y. Drozd concerns the category of maximal Cohen-Macaulay modules over surface singularities and its representation type. The focus is on the McKay correspondence for quotient surface singularities and its generalisations.
J. F. Carlson surveys the theory of rank varieties for modules over finite groups and finite group schemes. The article by K. Erdmann and A. Skowroński deals with particular sorts of self-injective algebras (called periodic algebras). Ch. Geiss, B. Leclerc and J. Schröer present the recent connection between representation theory of preprojective algebras of Dynkin type and a class of cluster algebras. The survey by I. A. Gordon is an introduction to Etingov-Ginzburg symplectic reflection algebras. O. Iyama surveys his higher theory of almost split sequences (for which he was awarded the first ICRA Award in 2007). Also, applications to Calabi-Yau triangulated categories and CY-algebras are presented here. Calabi-Yau categories are the major topic of the survey of P. Jørgensen, who studies their links to rational homotopy theory. The main tool here is Auslander-Reiten theory. Calabi-Yau categories are also the subject of the survey by B. Keller, who concentrates on two important classes of examples: the categories arising as orbit categories and those arising as subcategories of derived categories.
The article by S. Kasjan deals with applications of some model theoretic tools in representation theory. S. Koenig's survey provides an elementary introduction to the structure and representation theory of diagram algebras that include Brauer algebras, partition algebras, (affine) Hecke algebras and (affine) Temperly-Lieb algebras. The survey by H. Lenzing and J. A. de la Pena deals with the impact given by the spectrum of the Coxeter transformation of algebras of finite global dimension on the structure and representation theory of some important classes of algebras. M. Reineke's article is an introduction to moduli spaces of representations of quivers, presenting a geometric approach to their classification. The final contribution by A. Skowronski and K. Yamagata surveys representation theory of finite dimensional selfinjective algebras over fields possessing Galois coverings by the repetitive algebras of quasitilted algebras. Each of the fifteen survey papers contains a number of open problems. Thus the volume will not only serve as a remarkable exposition presenting some deep recent ideas in a concise way but also as an important source of problems and directions for research in contemporary representation theory.