Workshop "Representation Theory in Bielefeld – Past and Future"
Tuesday 24 September to Thursday 26 September 2019
The representation theory group at Bielefeld will celebrate in September 2019 the 50th anniversary of the Faculty of Mathematics with a workshop. Right after the workshop the Faculty of Mathematics will host a two day conference (26–27 September).
From the very beginning in 1969, algebra and group theory were important areas of research at Bielefeld University. In 1978 Claus Michael Ringel moved to Bielefeld and created a research group working on representation theory of associative algebras. The recent activities in representation theory are supported through an Alexander von Humboldt Professorship (see the announcement and a journal article).
We aim to celebrate the achievements in the representation theory of algebras over the last 50 years, and highlight the open problems and interesting directions to keep people working into the future.
Location: Bielefeld University, please see the Programme page for details.
Organisers: Bill Crawley-Boevey, Henning Krause
Schedule: The workshop starts on Tuesday morning and ends on Thursday around noon, please see the Programme page for details. The social programme includes a dinner on Wednesday evening and the opportunity to see the play "Mathematische Spaziergänge mit Emmy Noether" (in German) on Thursday evening.
Funding: Limited support for participants is available. The meeting is supported by the the Alexander von Humboldt Foundation in the framework of an Alexander von Humboldt Professorship endowed by the Federal Ministry of Education and Research.
Workshop Speakers
Invited speakers include:
- Thomas Brüstle (Sherbrooke)
- Igor Burban (Paderborn)
- Ivo Dell'Ambrogio (Lille)
- Karin Erdmann (Oxford)
- Steffen Koenig (Stuttgart)
- Julian Külshammer (Uppsala)
- Rosanna Laking (Verona)
- Gus Lehrer (Sydney)
- Yanan Lin (Xiamen)
- Markus Schmidmeier (Boca Raton)
- Jan Schröer (Bonn)
- Sibylle Schroll (Leicester)
- Andrzej Skowroński (Toruń)
- Pu Zhang (Shanghai)
Workshop Participants
History and Achievements
Here are some of our favourite achievements (in a random order, highlighting the role of present and former members of the Bielefeld group where relevant, and with apologies for inaccuracies and omissions):
- Auslander-Reiten sequences [AR75, AS81], and the Auslander-Reiten quiver of an algebra [Riedtmann80]. Homological description of Auslander algebras [Auslander71], and higher versions [Iyama07a, Iyama07b].
- Brauer-Thrall conjectures [Roiter68, Bautista85, Ringel08], Multiplicative Basis Theorem [BGRS85], Indecomposables live in all smaller lengths [Bongartz13] and the Tame and Wild Theorem [Drozd79, Crawley-Boevey88, GNRSV93]. The list of minimal representation-infinite algebras [HV83, Bongartz84].
- Classification of families of algebras of finite or tame representation type [Gabriel72, Nazarova73, BD77, Riedtmann83, Skowronski89, Erdmann90, Skowronski97, Brustle04]. Classification of representations for major classes of tame algebras [GP68, GP71, DR76, Ringel75, Ringel84, BR87, Brustle01].
- Properties of representations of quivers: Connection with Coxeter groups [BGP73], Kac's Theorem [Kac80, Kac83] and conjectures [CV04, Hausel10, HLR13], properties of general representations [Schofield92] and moduli spaces [Reineke03], the Kerner bijection [Kerner91, CK94] and braid group action [Crawley-Boevey92b].
- Tilting theory and more recently tau-tilting theory [BB80, HR82, AHK07, AIR14]. Connection with derived categories, and their structure [Happel88, Rickard89, Krause00, BIK08, BIK11].
- Construction and classification of hereditary abelian categories, especially coherent sheaves on a weighted projective line [GL87, HRS96, Happel01].
- Hall algebras and applications to quantum groups [Ringel90, Green95, PX00, GLS06, GLS17].
- Relationship with cluster algebras, especially cluster tilting theory [MRZ03, BMRRT06, CC06, KR07, IY08].
- Gorenstein algebras and their stable categories [Buchweitz86, AB89, Happel91, Krause05].
- Quasi-hereditary algebras, including the existence of a canonical tilting module [Scott86, Ringel91, Donkin93, Iyama03].
- The study of pure-injective modules, especially the classification for tame hereditary algebras [Ringel79, JL89, Crawley-Boevey91, Crawley-Boevey92a, Prest98, Ringel98, KR00, Krause01].
Bibliography
| [AIR14] | , τ-tilting theory, Compos. Math. 150 (2014), 415–452, doi: |
| [AHK07] | L. Angeleri Hügel, D. Happel and H. Krause (eds.), Handbook of tilting theory, London Mathematical Society Lecture Note Series 332, Cambridge University Press Cambridge 2007, doi: |
| [Auslander71] | , Representation dimension of Artin algebras, Lecture Notes, Queen Mary College, London, 1971 |
| [AB89] | , The homological theory of maximal Cohen-Macaulay approximations, Colloque en l'honneur de Pierre Samuel (Orsay, 1987), Mém. Soc. Math. France (N.S.) 38 (1989), 5–37, doi: |
| [AR75] | , Representation theory of Artin algebras. III. Almost split sequences, Comm. Algebra 3 (1975), 239–294, doi: |
| [AS81] | , Almost split sequences in subcategories, J. Algebra 69 (1981), 426–454, doi: |
| [Bautista85] | , On algebras of strongly unbounded representation type, Comment. Math. Helv. 60 (1985), 392–399, doi: |
| [BGRS85] | , Representation-finite algebras and multiplicative bases, Invent. Math. 81 (1985), 217–285, doi: |
| [BIK08] | , Local cohomology and support for triangulated categories, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), 573–619, doi: |
| [BIK11] | , Stratifying modular representations of finite groups, Ann. of Math. (2) 174 (2011), 1643–1684, doi: |
| [BGP73] | , Coxeter functors and Gabriel's theorem, Uspekhi Mat. Nauk 28 (1973), 19–33;
translation: , Coxeter functors and Gabriel's theorem, Russ. Math. Surv. 28 (1973), 17–32, doi: |
| [BD77] | , The representation type of finite groups (Russian), Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 71 (1977), 24–41;
translation: , Representation type of finite groups, J. Sov. Math. 20 (1982), 2515–2528, doi: |
| [Bongartz84] | , Critical simply connected algebras, Manuscripta Math. 46 (1984), 117–136, doi: |
| [Bongartz13] | , Indecomposables live in all smaller lengths, Represent. Theory 17 (2013), 199–225, doi: |
| [BB80] | , Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors, in: Representation theory II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Mathematics 832, Springer Berlin-New York 1980, 103–169, doi: |
| [Brustle01] | , Kit algebras, J. Algebra 240 (2001), 1–24, doi: |
| [Brustle04] | , Tame tree algebras, J. Reine Angew. Math. 567 (2004), 51–98, doi: |
| [BMRRT06] | , Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), 572–618, doi: |
| [Buchweitz86] | , Maximal Cohen–Macaulay modules and Tate-cohomology over Gorenstein rings, Universität Hannover (1986), 155 pp |
| [BR87] | , Auslander-Reiten sequences with few middle terms and applications to string algebras, Comm. Algebra 15 (1987), 145–179, doi: |
| [CC06] | , Cluster algebras as Hall algebras of quiver representations, Comment. Math. Helv. 81 (2006), 595–616, doi: |
| [Crawley-Boevey88] | , On tame algebras and bocses, Proc. London Math. Soc. (3) 56 (1988), 451–483, doi: |
| [Crawley-Boevey91] | , Tame algebras and generic modules, Proc. London Math. Soc. (3) 63 (1991), 241–265, doi: |
| [Crawley-Boevey92a] | , Modules of finite length over their endomorphism rings, in: Representations of algebras and related topics (Kyoto, 1990), London Math. Soc. Lecture Note Ser. 168, Cambridge Univ. Press Cambridge 1992, 127–184, doi: |
| [Crawley-Boevey92b] | , Exceptional sequences of representations of quivers, in: Representations of algebras (Ottawa, ON, 1992), CMS Conf. Proc. 14, Amer. Math. Soc. Providence RI 1993, 117–124 |
| [CK94] | , A functor between categories of regular modules for wild hereditary algebras, Math. Ann. 298 (1994), 481–487, doi: |
| [CV04] | , Absolutely indecomposable representations and Kac-Moody Lie algebras (with an appendix by Hiraku Nakajima), Invent. Math. 155 (2004), 537–559, doi: |
| [DR76] | |
| [Donkin93] | , On tilting modules for algebraic groups, Math. Z. 212 (1993), 39–60, doi: |
| [Drozd79] | , Tame and wild matrix problems (Russian), in: Representations and quadratic forms (Russian), Akad. Nauk Ukrain. SSR Inst. Mat. 154, Kiev 1979, 39–74;
translation: , Tame and Wild Matrix Problems, in: Amer. Math. Soc. Transl. (2) Vol. 128 (1986), 31–55, doi: |
| [Erdmann90] | |
| [Gabriel72] | , Unzerlegbare Darstellungen I, Manuscripta Math. 6 (1972), 71–103, doi: |
| [GNRSV93] | , Tame and wild subspace problems, Ukraïn. Mat. Zh. 45 (1993), 313–352;
translation: , Tame and wild subspace problems, Ukrainian Math. J. 45 (1993), 335–372, doi: |
| [GL87] | , A class of weighted projective curves arising in representation theory of finite-dimensional algebras, in: Singularities, representation of algebras, and vector bundles (Lambrecht, 1985), Lecture Notes in Math. 1273, Springer Berlin 1987, 265–297, doi: |
| [GLS06] | , Rigid modules over preprojective algebras, Invent. Math. 165 (2006), 589–632, doi: |
| [GLS17] | , Quivers with relations for symmetrizable Cartan matrices I: Foundations, Invent. Math. 209 (2017), 61–158, doi: |
| [GP68] | , Indecomposable representations of the Lorentz group, Uspekhi Mat. Nauk 23 (1968), 3–60;
translation: , Indecomposable representations of the Lorentz group, Russ. Math. Surv. 23 (168), 1–58, doi: |
| [GP71] | , Quadruples of subspaces of a finite-dimensional vector space, Dokl. Akad. Nauk SSSR 197 (1971), 762–765; translation: , Quadruples of subspaces of a finite-dimensional vector space, Sov. Math. Dokl. 12 (1971), 535–539 |
| [Green95] | , Hall algebras, hereditary algebras and quantum groups, Invent. Math. 120 (1995), 361–377, doi: |
| [Happel88] | |
| [Happel91] | , On Gorenstein algebras, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991), in: Representation Theory of Finite Groups and Finite-Dimensional Algebras (Proceedings of the Conference at the University of Bielefeld from May 15–17, 1991), Progr. Math. 95, Birkhäuser Basel 1991, 389–404, doi: |
| [Happel01] | , A characterization of hereditary categories with tilting object, Invent. Math. 144 (2001), 381–398, doi: |
| [HRS96] | |
| [HR82] | , Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), 399–443, doi: |
| [HV83] | , Minimal algebras of infinite representation type with preprojective component, Manuscripta Math. 42 (1983), 221–243, doi: |
| [Hausel10] | , Kac's conjecture from Nakajima quiver varieties, Invent. Math. 181 (2010), 21–37, doi: |
| [HLR13] | , Positivity for Kac polynomials and DT-invariants of quivers, Ann. of Math. (2) 177 (2013), 1147–1168, doi: |
| [Iyama03] | , Finiteness of representation dimension, Proc. Amer. Math. Soc. 131 (2003), 1011–1014, doi: |
| [Iyama07a] | , Auslander correspondence, Adv. Math. 210 (2007), 51–82, doi: |
| [Iyama07b] | , Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories, Adv. Math. 210 (2007), 22–50, doi: |
| [IY08] | , Mutation in triangulated categories and rigid Cohen-Macaulay modules, Invent. Math. 172 (2008), 117–168, doi: |
| [JL89] | |
| [Kac80] | , Infinite root systems, representations of graphs and invariant theory, Invent. Math. 56 (1980), 57–92, doi: |
| [Kac83] | , Root systems, representations of quivers and invariant theory, in: Invariant theory (Montecatini, 1982), Lecture Notes in Math. 996, Springer Berlin 1983, 74–108, doi: |
| [KR07] | , Cluster-tilted algebras are Gorenstein and stably Calabi-Yau, Adv. Math. 211 (2007), 123–151, doi: |
| [Kerner91] | , Stable components of wild tilted algebras, J. Algebra 142 (1991), 37–57, doi: |
| [Krause00] | , Smashing subcategories and the telescope conjecture—an algebraic approach, Invent. Math. 139 (2000), 99–133, doi: |
| [Krause01] | |
| [Krause05] | , The stable derived category of a Noetherian scheme, Compos. Math. 141 (2005), 1128–1162, doi: |
| [KR00] | H. Krause and C.M. Ringel (eds.), Infinite length modules. Invited lectures from the conference held in Bielefeld, September 7–11, 1998, Trends in Mathematics, Birkhäuser Verlag Basel 2000, x+439 pp, doi: |
| [MRZ03] | , Generalized associahedra via quiver representations, Trans. Amer. Math. Soc. 355 (2003), 4171–4186, doi: |
| [Nazarova73] | , Representations of quivers of infinite type (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 752–791;
translation: , Representations of quivers of infinite type, Math. USSR Izv. 7 (1973), 749–792, doi: |
| [PX00] | , Triangulated categories and Kac-Moody algebras, Invent. Math. 140 (2000), 563–603, doi: |
| [Prest98] | , Ziegler spectra of tame hereditary algebras, J. Algebra 207 (1998), 146–164, doi: |
| [Reineke03] | , The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli, Invent. Math. 152 (2003), 349–368, doi: |
| [Riedtmann80] | , Algebren, Darstellungsköcher, Überlagerungen und zurück, Comment. Math. Helv. 55 (1980), 199–224, doi: |
| [Riedtmann83] | , Representation-finite self-injective algebras of class $D_n$, Compositio Math. 49 (1983), 231–282 |
| [Rickard89] | , Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), 436–456, doi: |
| [Ringel75] | , The indecomposable representations of the dihedral 2-groups, Math. Ann. 214 (1975), 19–34, doi: |
| [Ringel79] | , Infinite-dimensional representations of finite-dimensional hereditary algebras, in: Symposia Mathematica, Vol. XXIII (Conf. Abelian Groups and their Relationship to the Theory of Modules, INDAM, Rome, 1977), Academic Press London-New York 1979, 321–412 |
| [Ringel84] | |
| [Ringel90] | , Hall algebras and quantum groups, Invent. Math. 101 (1990), 583–591, doi: |
| [Ringel91] | , The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z. 208 (1991), 209–223, doi: |
| [Ringel98] | , The Ziegler spectrum of a tame hereditary algebra, Colloq. Math. 76 (1998), 105–115, doi: |
| [Ringel08] | , The first Brauer-Thrall conjecture, in: Models, modules and abelian groups, Walter de Gruyter Berlin 2008, 369–374, doi: |
| [Roiter68] | , Unboundedness of the dimensions of the indecomposable representations of an algebra which has infinitely many indecomposable representations (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 1275–1282 |
| [Schofield92] | , General representations of quivers, Proc. London Math. Soc. (3) 65 (1992), 46–64, doi: |
| [Scott86] | , Simulating algebraic geometry with algebra. I. The algebraic theory of derived categories, in: The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), Proc. Sympos. Pure Math. 47, Part 2, Amer. Math. Soc. Providence RI 1987, 271–281, doi: |
| [Skowronski89] | , Selfinjective algebras of polynomial growth, Math. Ann. 285 (1989), 177–199, doi: |
| [Skowronski97] | , Simply connected algebras of polynomial growth, Compositio Math. 109 (1997), 99–133, doi: |
Contact
If you have any questions about the workshop, please contact the organisers at birep.