CONTEMPORARY MATHEMATICS

229

Trends in the Representation Theory of

Finite Dimensional Algebras 1997 Joint Summer Research Conference on

Trends in the Representation Theory of Finite Dimensional Algebras

July 2Q-24, 1997 Seattle, Washington

Edward L. Green Birge Huisgen-Zimmermann

Editors

Selected Titles in This Series

229 Edward L. Green and Birge Huisgen-Zimmermann, Editors, Trends in the representation theory of finite dimensional algebras, 1998

228 Liming Ge, Huaxin Lin, Zhong-Jin Ruan, Dianzhou Zhang, and Shuang Zhang, Editors, Operator algebras and operator theory, 1999

227 John McCleary, Editor, Higher homotopy structures in topology and mathematical physics, 1999

226 Luis A. Caffarelli and Mario Milman, Editors, Monge Ampere equation: Applications to geometry and optimization, 1999

225 Ronald C. Mullin and Gary L. Mullen, Editors, Finite fields: Theory, applications, and algorithms, 1999

224 Sang Geun Hahn, Hyo Chul Myung, and Efim Zelmanov, Editors, Recent progress in algebra, 1999

223 Bernard Chazelle, Jacob E. Goodman, and Richard Pollack, Editors, Advances in discrete and computational geometry, 1999

222 Kang-Tae Kim and Steven G. Krantz, Editors, Complex geometric analysis in Pohang, 1999

221 J. Robert Dorroh, Gisele Ruiz Goldstein, Jerome A. Goldstein, and Michael Mudi Tom, Editors, Applied analysis, 1999

220 Mark Mahowald and Stewart Priddy, Editors, Homotopy theory via algebraic geometry and group representations, 1998

219 Marc Henneaux, Joseph Krasil'shchik, and Alexandre Vinogradov, Editors, Secondary calculus and cohomological physics, 1998

218 Jan Mandel, Charbel Farhat, and Xiao-Chuan Cai, Editors, Domain decomposition methods 10, 1998

217 Eric Carlen, Evans M. Harrell, and Michael Loss, Editors, Advances in differential equations and mathematical physics, 1998

216 Akram Aldroubi and EnBing Lin, Editors, Wavelets, multiwavelets, and their applications, 1998

215 M. G. Nerurkar, D.P. Dokken, and D. B. Ellis, Editors, Topological dynamics and applications, 1998

214 Lewis A. Coburn and Marc A. Rieffel, Editors, Perspectives on quantization, 1998 213 Farhad Jafari, Barbara D. MacCluer, Carl C. Cowen, and A. Duane Porter,

Editors, Studies on composition operators, 1998 212 E. Ramirez de Arellano, N. Salinas, M. V. Shapiro, and N. L. Vasilevski,

Editors, Operator theory for complex and hypercomplex analysis, 1998 211 Jozef Dodziuk and Linda Keen, Editors, Lipa's legacy: Proceedings from the Bers

Colloquium, 1997 210 V. Kumar Murty and Michel Waldschmidt, Editors, Number theory, 1998 209 Steven Cox and Irena Lasiecka, Editors, Optimization methods in partial differential

equations, 1997 208 Michel L. Lapidus, Lawrence H. Harper, and Adolfo J. Rumbos, Editors,

Harmonic analysis and nonlinear differential equations: A volume in honor of Victor L. Shapiro, 1997

207 Yujiro Kawamata and Vyacheslav V. Shokurov, Editors, Birational algebraic geometry: A conference on algebraic geometry in memory of Wei-Liang Chow (1911-1995), 1997

206 Adam Koranyi, Editor, Harmonic functions on trees and buildings, 1997 205 Paulo D. Cordaro and Howard Jacobowitz, Editors, Multidimensional complex

analysis and partial differential equations: A collection of papers in honor of Fran<;ois Treves, 1997

(Continued in the back of this publication)

http://dx.doi.org/10.1090/conm/229

Trends in the Representation Theory of

Finite Dimensional Algebras

CoNTEMPORARY MATHEMATICS

229

Trends in the Representation Theory of

Finite Dimensional Algebras 1997 Joint Summer Research Conference on

Trends in the Representation Theory of Finite Dimensional Algebras

July 20-24, 1997 Seattle, Washington

Edward L. Green Birge Huisgen-Zimmermann

Editors

American Mathematical Society Providence, Rhode Island

Editorial Board Dennis DeTurck, managing editor

Andy R. Magid Michael Vogelius Clark Robinson Peter M. Winkler

The 1997 AMS-IMS-SIAM Joint Summer Research Conference in the Mathematical Sciences on "Trends in the Representation Theory of Finite Dimensional Algebras" was held at the University of Washington on July 2Q-24, 1997, with support from the Na-tional Science Foundation, Grant DMS-9618514, and the National Security Agency, Grant MDA904-97-1-0071.

1991 Mathematics Subject Classification. Primary 14D15, 14130, 16D30, 16D60, 16E10, 16E30, 16E40, 16G10, 16G20, 16G30, 16G50, 16G60, 16G70, 16P10, 16P20, 16P40,

16S80, 16W30, 17B37, 81R50.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Library of Congress Cataloging-in-Publication Data Trends in the representation theory of finite dimensional algebras : proceedings of the AMS-IMS-SIAM joint summer research conference, trends in the representation theory of finite dimen-sional algebras, July 20-24, 1997, University of Washington / Edward L. Green, Birge Huisgen-Zimmermann, editors.

p. em. - (Contemporary mathematics, ISSN 0271-4132 ; 229) Includes bibliographical references. ISBN 0-8218-0928-8 (soft : alk. paper) 1. Associative algebras-Congresses. 2. Representations of algebras-Congresses. I. Green,

Edward L., [DATE]. II. Huisgen-Zimmermann, Birge, [DATE]. III. Series: Contemporary mathematics (American Mathematical Society) : v. 229. QA251.5.T74 1998 512'.24--dc21 98-44526

CIP

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Contents

Preface IX

List of Talks xi

Postprojective partitions for tilting torsion pairs I. ASS EM AND F. U. COELHO 1

Derived canonical algebras as one-point extensions M. BAROT AND H. LENZING 7

Special biserial algebras and their automorphisms F. M. BLEHER 17

Wild subquivers of the Auslander-Reiten quiver of a tame algebra S. BRENNER AND M. C. R. BUTLER 29

Representation theory of noetherian Hopf algebras satisfying a polynomial identity K. A. BROWN 49

Finite representation type and periodic Hochschild (co-)homology R.-0. BUCHWEITZ 81

The syzygy theorem for monomial algebras M. C. R. BUTLER 111

Algebras whose derived category is tame J. A. DELAPENA 117

Circular biextensions of tame concealed algebras P. DRAXLER 129

On the distribution of AR-components of restricted Lie algebras R. FARNSTEINER 139

Compatible deformations M. GERSTENHABER AND A. GIAQUINTO 159

Directing objects in hereditary categories D. HAPPEL AND I. REITEN 169

On subcategories associated with tilting modules D. HAPPEL AND L. UNGER 181

vii

viii CONTENTS

Modules of the highest homological dimension over a Gorenstein ring Y. lWANAGA AND J. MIYACHI 193

Derived equivalence of graph algebras M. KAUER 201

Basic results on wild hereditary algebras 0. KERNER 215

On minimal approximations of modules H. KRAUSE AND M. SAORIN 227

Serre duality for generalized Auslander regular algebras R. MARTINEZ-VILLA 237

Classifying finite-dimensional semisimple Hopf algebras S. MONTGOMERY 265

Geometry of modules: Degenerations CH. RIEDTMANN 281

The preprojective algebra of a tame quiver: The irreducible components of the module varieties C. M. RINGEL 293

Representation types, Tits reduced quadratic forms and orbit problems for lattices over orders D. SIMSON 307

Degenerations in module varieties with finitely many orbits A. SKOWRONSKI AND G. ZWARA 343

Preface

The American Mathematical Society, the Institute of Mathematical Statis-tics, and the Society for Industrial and Applied Mathematics sponsored a joint summer research conference at the University of Washington, from July 2D-24, 1997, on Trends in the Representation Theory of Finite Dimensional Algebras. The conference was organized by E. L. Green (Virginia Polytechnic Institute and State University) and B. Huisgen-Zimmermann (University of California at Santa Barbara).

The lectures focused on the following three topics: (1) Interactions between representation theory and algebraic geometry; (2) Homological methods; (3) Applications of representation theory to the study of quantum groups. All three lines were strongly represented in the conference by lectures providing keynote reports and talks addressing current research, as well as by contributions to this volume.

(1) The analysis of representations via algebraic varieties is comparatively young. It goes back to the 1970's, in particular to Gabriel, Bernstein, I. M. Gel'fand, S. I. Gel'fand, and Ponomarev. Their representation-theoretic work, based heavily on geometric argumentation, proved the varieties of modules of fixed dimension to store an enormous amount of accessible algebraic information. Another triggering event was the appearance of two pivotal papers by Kac in the early 1980's, broadening insights of Gabriel and Nazarova among others and completing, via invariant theory, the description of the dimension vectors of the finite dimensional indecomposable representations of quivers. Since then, numerous far-reaching geometric paths have been cut through the subject, with a great deal of movement having occurred in the recent past.

(2) Homological methods constitute one of the most time-honored lines within representation theory. Among the pioneers in this direction were Cartan, Eilen-berg, Nakayama, MacLane, Nagata, Auslander, Buchsbaum, Serre. One would have to list many others, in particular if one were to go back to the commu-tative or topological roots of the subject. As for the commutative origins, let us just point to Hilbert, whose Syzygy Theorem, proved last century, was one of the first ring-theoretic results of an unmistakably homological nature. In the

ix

X PREFACE

meantime, a stately list of fundamental conjectures has been accumulated in the noncommutative situation, and an even more extensive list of partial and full solutions which have furnished strong structural building blocks for the present understanding of important classes of algebras.

(3) Much of the interest in quantum groups revolves around algebras such as Uq(g), a deformation of the universal enveloping algebra of a semisimple Lie algebra g, and 0 q (G), a deformation of the coordinate ring of a semisimple Lie group G. In the case when the parameter q is a root of unity, the irreducible rep-resentations of these algebras are finite dimensional, and so the finite dimensional representation theory of the algebras is of key importance. As a consequence, classification techniques specifically developed for the finite dimensional situa-tion provide powerful tools. For example, Ringel has brought Auslander-Reiten theory to bear in studying the positive part of Uq(g). Similarly, many open problems concerning Sklyanin algebras could be solved if their finite dimensional Koszul duals were better understood.

Many of the talks, especially those in connection with topics (2) and (3), were implicit homages to Maurice Auslander, one of the field's most prominent leaders up to his death in November of 1994.

We would like to express our thanks to the AMS for its generous support, as well as to all participants who contributed to the success of the conference. In particular, our .thanks go to Dr. James Maxwell, the director of the Joint Summer Research Conference program, Romy Cascella, the on-site AMS confer-ence manager, and Christine Thivierge, the coordinator of this volume, for their investment of energy, which went far beyond routine involvement.

Edward L. Green Birge Huisgen-Zimmermann

List of Talks

Ibrahim Assem, Universite de Sherbrooke, Endomorphism algebras of excep-tional sequences over path algebras of type An Michael Bardzell, Salisbury State University, Computational techniques for Hoch-schild cohomology Michael Barot, Universidad Nacional Aut6noma de Mexico, Lifting derived equi-valences to one-point extensions Raymundo Bautista, Universidad Nacional Aut6noma de Mexico, Representa-tions of coalgebras and algebras Frauke Bleher, University of Pennsylvania, Group algebras, Heeke algebras, and special biserial algebras Sheila Brenner, University of Liverpool, Wild sectional subquivers of Auslander-Reiten quivers of tame algebras Kenneth Brown, University of Glasgow, Representation theory of noetherian P.l. Hopf algebras Thomas Briistle, Universitat Bielefeld, Parabolic subgroups acting with a finite number of orbits on their unipotent radicals Ragnar-Olaf Buchweitz, University of Toronto, Simple singularities, liaison of lines, and Hochschild cohomology Michael Butler, University of Liverpool, Minimal resolutions of algebras William Chin, DePaul University, Representations of SLq(2) at a root of 1 Riccardo Colpi, Universita di Padova, Tilting in Grothendieck categories William Crawley-Boevey, University of Leeds, Preprojective algebras, simple representations, and singularities Andrew Dean, Bishops University, The concept of rank for modules over finite dimensional algebras Jose Antonio de la Pefia, Universidad Nacional Aut6noma de Mexico, Tame algebras: some geometrical and combinatorial aspects Peter Draxler, Universitat Bielefeld, Tame algebras, matrix problems, and de-rived categories

xi

xii LIST OF TALKS

Yuriy Drozd, Kiev Taras Shevchenko University, Vector bundles on projective curves and Cohen-Macaulay modules on surface singularities

Rolf Farnsteiner, University of Wisconsin- Milwaukee, AR-components for re-duced enveloping algebras of reductive Lie algebras

Kent Fuller, University of Iowa, *-Modules over ring extensions

Christof Geiss, Universidad Nacional Aut6noma de Mexico, Deformations and degenerations of finite dimensional algebras

Murray Gerstenhaber, University of Pennsylvania, An overview of algebraic de-formation theory

lain Gordon, University of Glasgow, The cohomology of quantized functional-gebras at roots of unity

Robert Guralnick, University of Southern California, Prehomogeneous spaces for algebraic groups and representations of finite dimensional algebras

Dieter Happel, Technische Universitat Chemnitz, On subcategories associated with tilting modules

Michael Kauer, Universitat Stuttgart, Derived equivalence of graph algebras

Bernhard Keller, Universite de Paris VII, Derived categories and homological invariants

Otto Kerner, Heinrich-Heine-Universitat Dusseldorf, Basic results on wild hered-itary algebras

Mark Kleiner, Syracuse University, Homologically finite subcategories over Artin algebras

Henning Krause, Universitat Bielefeld, Finitistic dimension and Ziegler spectrum Helmut Lenzing, Universitat Paderborn, Weighted projective lines and canonical algebras Edward Letzter, Texas A&M University, Prime ideals in coordinate rings of quantum matrices Lawrence Levy, University of Wisconsin- Madison, Tameness versus wildness of commutative noetherian rings

Shiping Liu, Universite de Sherbrooke, Tilted string algebras

Eduardo Marcos, Universidade de Sao Paolo, Hochschild cohomology of trun-cated quiver algebras Roberto Martinez-Villa, Universidad Nacional Aut6noma de Mexico, Self-injec-tive Koszul algebras and their Yoneda algebras

Hagen Meltzer, Technische Universitat Chemnitz, Exceptional vector bundles and tilting complexes on weighted projective lines Hector Merklen, Universidade de Sao Paolo, On the representation type of some schurian vector space categories

LIST OF TALKS xiii

Susan Montgomery, University of Southern California, Classifying finite-dimen-sional semi-simple Hopf algebras

Daniel Nakano, Utah State University, Representation type of Schur algebras Idun Reiten, Norges Teknisk-Naturvitenskapelige Universitet, Some aspects of derived categories Jeremy Rickard, University of Bristol, Bousfield localization in homological al-gebra Christine Riedtmann, Universitiit Bern, Geometry of modules Claus Ringel, Universitiit Bielefeld, Quiver varieties, preprojective algebras and the canonical bases of quantum groups Manuel Saorin, Universidad de Murcia, Grading techniques in the study of ho-mological properties of algebras Daniel Simson, Uniwersytet Mikolaja Kopernika w Toruniu, Tameness, Tits re-duced quadratic forms, and orbit spaces for lattices over orders Andrzej Skowronski, Uniwersytet Mikolaja Kopernika w Toruniu, Geometric and homological aspects of representation theory of tame algebras Sverre Smal0, Norges Teknisk-Naturvitenskapelige Universitet, Finitistic dimen-sions and contravariantly finite subcategories 0yvind Solberg, Norges Teknisk-Naturvitenskapelige Universitet, Minimal pro-jective resolutions Martha Takane-Imay, Universidad Nacional Aut6noma de Mexico, The directing modules of finite dimensional k-algebras Luise Unger, Universitiit Paderborn, Tilting modules and homological conjec-tures Takayoshi Wakamatsu, Saitama University, On graded Frobenius algebras Dan Zacharia, Syracuse University, Minimal projective resolutions James Zhang, University of Washington, Curves on noncommutative schemes

Selected Titles in This Series (Continued from the front of this publication)

204 Yair Censor and Simeon Reich, Editors, Recent developments in optimization theory and nonlinear analysis, 1997

203 Hanna Nencka and Jean-Pierre Bourguignon, Editors, Geometry and nature: In memory of W. K. Clifford, 1997

202 Jean-Louis Loday, James D. Stasheff, and Alexander A. Voronov, Editors, Operads: Proceedings of Renaissance Conferences, 1997

201 J. R. Quine and Peter Sarnak, Editors, Extremal Riemann surfaces, 1997 200 F. Dias, J.-M. Ghidaglia, and J.-C. Saut, Editors, M .. "hematical problems in the

theory of water waves, 1996 199 G. Banaszak, W. Gajda, and P. Krason, Editors, Algebraic K-theory, 1996 198 Donald G. Saari and Zhihong Xia, Editors, Hamiltonian dynamics and celestial

mechanics, 1996 197 J. E. Bonin, J. G. Oxley, and B. Servatius, Editors, Matroid theory, 1996 196 David Bao, Shiing-shen Chern, and Zhongmin Shen, Editors, Finsler geometry,

1996 195 Warren Dicks and Enric Ventura, The group fixed by a family of injective

endomorphisms of a free group, 1996 194 Seok-Jin Kang, Myung-Hwan Kim, and Insok Lee, Editors, Lie algebras and their

representations, 1996 193 Chongying Dong and Geoffrey Mason, Editors, Moonshine, the Monster, and

related topics, 1996 192 Tomek Bartoszynski and Marion Scheepers, Editors, Set theory, 1995 191 Tuong Ton-That, Kenneth I. Gross, Donald St. P. Richards, and Paul J. Sally,

Jr., Editors, Representation theory and harmonic analysis, 1995 190 Mourad E. H. Ismail, M. Zuhair Nashed, Ahmed I. Zayed, and Ahmed F.

Ghaleb, Editors, Mathematical analysis, wavelets, and signal processing, 1995 189 S. A.M. Marcantognini, G. A. Mendoza, M.D. Moran, A. Octavio, and W. 0.

Urbina, Editors, Harmonic analysis and operator theory, 1995 188 Alejandro Adem, R. James Milgram, and Douglas C. Ravenel, Editors,

Homotopy theory and its applications, 1995 187 G. W. Brumfiel and H. M. Hilden, SL(2) representations of finitely presented groups,

1995 186 Shreeram S. Abhyankar, Walter Feit, Michael D. Fried, Yasutaka lhara,

and Helmut Voelklein, Editors, Recent developments in the inverse Galois problem, 1995

185 Raul E. Curto, Ronald G. Douglas, Joel D. Pincus, and Norberto Salinas, Editors, Multivariable operator theory, 1995

184 L. A. Bokut', A. I. Kostrikin, and S. S. Kutateladze, Editors, Second International Conference on Algebra, 1995

183 William C. Connett, Marc-Olivier Gebuhrer, and Alan L. Schwartz, Editors, Applications of hypergroups and related measure algebras, 1995

182 Selman Akbulut, Editor, Real algebraic geometry and topology, 1995 181 Mila Cenkl and Haynes Miller, Editors, The Cech Centennial, 1995 180 David E. Keyes and Jinchao Xu, Editors, Domain decomposition methods in

scientific and engineering computing, 1994 179 Yoshiaki Maeda, Hideki Omoro, and Alan Weinstein, Editors, Symplectic

geometry and quantization, 1994 178 HEi!Eme Barcelo and Gil Kalai, Editors, Jerusalem Combinatorics '93, 1994 177 Simon Gindikin, Roe Goodman, Frederick P. Greenleaf, and Paul J. Sally, Jr.,

Editors, Representation theory and analysis on homogeneous spaces, 1994

(See the AMS catalog for earlier titles)

ISBN Q-8218-0928-8

21 809280