TAME AND WILD WORKSHOP List of Participants, Schedule and Abstracts of Talks Department of Mathematics Uppsala University . TAME AND WILD WORKSHOP Department of Mathematics Uppsala University Uppsala, Sweden November 26-28, 2004 Supported by • The Swedish Foundation for International Cooperation in Research and Higher Education (STINT) • The Swedish Research Council Within the framework of the joint research project “Representation theory of algebras and applications” of the • Department of Functional Analysis, Institute of Mathematics of the Ukrainian Academy of Science; • Department of Algebra and Mathematical Logics, Faculty of Mechanics and Mathematics, Kyiv National Taras Shevchenko University; • Department of Mathematics, Gothenburg University and Chalmers University of Technology; • Department of Mathematics, Uppsala University. 1 Organizers: • Volodymyr Mazorchuk, Department of Mathematics, Uppsala University; • Lyudmila Turowska, Department of Mathematics, Gothenburg University and Chalmers University of Technology. 2 List of Participants: 1. Jan Adriaenssens, University of Antwerp, Belgium 2. Raf Bocklandt, University of Antwerp, Belgium 3. Lesya Bodnarchuk, University of Kaiserslautern, FRG 4. Igor Burban, Institut de Mathematiques de Jussieu, France 5. Michael C R Butler, University of Liverpool, UK 6. Paolo Casati, II University of Milano, Italy 7. Ernst Dieterich, Uppsala University, Sweden 8. Vlastimil Dlab, Carleton University, Canada 9. Elena Drozd, University of California at Berkeley, USA 10. Yuriy Drozd, Kyiv University, Ukraine/ Uppsala University, Sweden 11. Karin Erdmann, Oxford University, UK 12. Anders Frisk, Uppsala University, Sweden 13. Gert-Martin Greuel, University of Kaiserslautern, FRG 14. Martin Herschend, Uppsala University, Sweden 15. Josef Jirasko, Czech Technical University, Prague, Czech Republic 16. Peter Jorgensen, Leeds University, UK 17. Ekaterina Jushchenko, Kyiv University, Ukraine 18. Dirk Kussin, University of Paderborn, FRG 19. Steffen K¨onig, University of Leicester, UK 20. Helmut Lenzing, University of Paderborn, FRG 21. Viktor Levandovskiy, University of Kaiserslautern, FRG 22. Lars Lindberg, Uppsala University, Sweden 23. Volodymyr Mazorchuk, Uppsala University, Sweden 3 24. Sergiy Ovsienko, Kyiv University, Ukraine 25. Claus Michael Ringel, Bielefeld University, FRG 26. Sibylle Schroll, Oxford University, UK 27. Vladimir Sergeichuk, Institute of Mathematics, Kyiv, Ukraine 28. Sergei Silvestrov, Lund Institute of Technology, Sweden 29. Daniel Simson, Nicholas Copernicus University, Torun, Polen 30. Sverre Smalø, NTNU Trondheim, Norway 31. Alexander Stolin, Gothenburg University, Sweden 32. Stijn Symens, University of Antwerp, Belgium 33. Lyudmila Turowska, Chalmers University of Technology, Gteborg, Sweden 34. Leonid Vainerman, University of Caen, France 35. Geert Van de Weyer, University of Antwerp, Belgium 36. Adrian Williams, Imperial College, London, United Kingdom 4 Schedule of Talks Saturday, November 27th, 2004. 09.00-09.50 Greuel On the classification of vector bundles on curves of arithmetic genus 1 10.00-10.50 Lenzing Tubular and elliptic curves 11.00-11.30 Coffee-break 11.30-12.20 Erdmann Tame self-injective algebras 12.30-14.00 Lunch 14.00-14.50 Smalø Some local Ext-limits do not exist 15.00-15.50 K¨onig Polynomial functors and Schur algebras 16.00-16.30 Coffee-break 16.30-17.20 Vainerman On operator-algebraic quantum groups of different types 17.30-18.20 Ovsienko On exact categories Sunday, November 28th, 2004. 09.00-09.50 Burban Derived category of an irreducible projective curve of arithmetic genus one 10.00-10.50 Ringel Controlled Wildness 11.00-11.30 Coffee-break 11.30-12.20 Dieterich Some normal form problems arising in the classification theory of real division algebras 12.30-14.00 Lunch 5 ABSTRACTS Derived category of an irreducible projective curve of arithmetic genus one Igor Burban In my talk based on my joint work with Bernd Kreussler I shall discuss properties of the derived category of coherent sheaves on a projective curve of arithmetic genus one, comparing common features and pointing out some principal differences between the case of a smooth and a singular curve and giving in particular some ideas about the behavior of the derived category in families. I am going to describe the group of exact auto-equivalences, the set of all t-structures, the set of spherical objects of the bounded derived category of a singular Weierstrass curve. The technique of Fourier-Mukai transforms will be then applied to obtain a combinatorial description of indecomposable semi-stable sheaves of degree zero on a nodal Weierstrass curve. Moreover, it gives another point of view on the classification of indecomposable objects of the derived category of coherent sheaves on a nodal Weierstrass curve, which was obtained in my joint work with Yuriy Drozd. Institut de Mathematiques de Jussieu, France Some normal form problems arising in the classification theory of real division algebras Ernst Dieterich By a real division algebra we mean a real vector space A, endowed with a bilinear multiplication A × A → A, (x, y) 7→ xy, such that 0 < dim A < ∞ and xy = 0 only if x = 0 or y = 0. Famous theorems assert that (R, C, H) classifies all associative real division algebras (Frobenius 1878), (R, C, H, O) classifies all alternative real division algebras (Zorn 6 1931), and that each real division algebra has dimension 1,2,4 or 8 (Hopf 1940, Kervaire, Bott and Milnor 1958). The problem of classifying all real division algebras of fixed dimension d ∈ {1, 2, 4, 8} is trivial in case d = 1, has lately been solved in case d = 2 by various mathematicians (Burdujan 1985, Gottschling 1998, H¨ubnerand Petersson 2002, Dokovi´cand Zhao 2004, Dieterich 2004), and attracted increasing interest in the cases d = 4 and d = 8, where to date only partial solutions are known. In my talk I will indicate how some parts of this classification problem conceptually can be reduced to normal form problems which are strongly reminiscent of wild matrix problems, and yet admit explicit solutions in terms of n-parameter families, where n = 4, 9, 12,... Uppsala universitet, Matematiska institutionen, Box 480, SE-751 06 Uppsala, Sweden Tame self-injective algebras Karin Erdmann Let A be a finite-dimensional tame self-injective algebra over an algebraically closed field. We discuss the structure of the stable Auslander-Reiten quiver and consequences for classifying some families of such algebras. As a new result, we deduce from existing litera- ture that the stable Auslander-Reiten quiver of A does not have a component isomorphic to ZA∞. Oxford University, UK 7 On the classification of vector bundles on curves of arithmetic genus 1 Gert-Martin Greuel I shall reviev some results about the classification of vector bundles on reduced pro- jective curves. This problem becomes immedeately wild. It is however tame for a bigger class of curves if we restrict ourself to simple vector bundles. We report on new results about the classification of vector bundles on curves of arithmetic genus 1, mainly obtained by Lesya Bodnarchuk, a joint student of Yuriy Drozd and myself. University of Kaiserslautern, Germany Polynomial functors and Schur algebras Steffen K¨onig Polynomial functors have been used in topology, in K-theory and in group cohomology of (finite) general linear groups. They are closely related to polynomial modules of (infinite) general linear groups, that is, to modules over Schur algebras. Few of the rings, whose module categories are categories of polynomial functors, have been described explicitly. Drozd was the first to give a complete description of non-trivial examples. He conjectured a similar description to be true for a larger class of examples, and this conjecture has been verified (in joint work with Alexander Zimmermann). University of Leicester, UK 8 Tubular and elliptic curves Helmut Lenzing Let k be an algebraically closed field. Let T be a tubular curve, that is, a weighted projective line of weight type (2,2,2,2), (3,3,3), (4,2,2) or (6,3,2). It is known for a long time that the category of coherent sheaves over T is very similar to the category of coherent sheaves over an elliptic curve. Theorem. The category of coherent sheaves on a tubular or an elliptic curve is a heredi- tary, noetherian, Hom-finite k-category with Serre duality such that the Auslander-Reiten translation τ has finite period. Conversely, each such category is equivalent to the category of coherent sheaves over a tubular or an elliptic curve, where the period of τ decides which of the cases happens. Moreover, we establish a natural map from (isomorphism classes) of tubular curves to (isomorphism classes) of elliptic curves. This map is surjective and generically bijective, and we relate the categories of coherent sheaves for corresponding curves. University of Paderborn, Germany On exact categories Serge Ovsienko In our talk we present a technique of calculation in some class of exact categories. Under some condition on the exact category C we construct the derived categories D±(C) and endow it with a structure of an A(∞)-category. The constructed derived categories are connected with some dualities, which generalize Ringel duality. We show, how to extend this notion to the wide class of associative algebras. As an application we consider exact categories, associated with stratifying systems of modules. In the talk we emphasize an important role of ideas of Yuriy Drozd for the development of our theory. [BS] P. Balmer M. Sclichtling, Idempotent completion of triangulated categories, J. of Al- gebra, 236, No. 2 (2001), pp.819-834. [BB] W. B. Burt, M. C. R. Butler, Almost split sequences for Bocses, Canadian Math. Soc., Conference Proceedings, vol.11, 1991, 89–121. 9 [K] B. Keller, A(∞)-algebras in rerpesentation theory. Contribution to the Proceedings of ICRA IX, Beijing. [O1] S. A. Ovsienko, Bimodule and matrix problems, Progress in Mathematics, Vol. 173, Birkhaeuser, 1999, 325-357. [O2] S. A. Ovsienko, Boxes and quasi-hereditary algebras, Third international Algebraic Conference in Ukraine. Sumy, July 2-8, 2001, 84-87.