BIREP — Representations of finite dimensional algebras at Bielefeld

• BIREP
• Faculty of Mathematics
• Bielefeld University

 

Seminar on Representation Theory ( = Seminar zur Darstellungstheorie (eKVV 241012)) (SoSe 2014)

Wednesday 10-12 h in V5-227 (2 SWS)
Organizer: Prof. Dr. Henning Krause, Dr. Julia Sauter
Content: Selected Topics from Representation Theory of finite dimensional algebras (and beyond)
This is a seminar with varying topics, suggestions are always welcome. In particular, phd students are encouraged to formulate their research needs and interests and we will try to incorporate this in the seminar. Usually we speak English and in exceptional cases German.
At the moment, we plan the following three topics (with the person in charge of planning the talks):

• Quantum cluster algebras (Philipp Lampe)
• Quasiheredity of Schur algebras (Henning Krause)
• Auslander-Reiten theory revisited (Julia Sauter)

Schedule of talks

Please be aware that the schedule might change. On the 30th of April the seminar is from 14:15h-16h. Please, observe that on the 14th of May we are in V2-210 from 10:15h-12h.

• 9th of April: Lusztig's canonical basis for universal enveloping algebras, (Philipp Lampe)
• 16th of April: Quantum cluster algebras, (Florian Gellert)
• 23rd of April: Cluster structures on quantized universal enveloping algebras, (Philipp Lampe)
• 30th of April, 14-16h: Quasiheredity of the Schur algebra I, [ABW] (Henning Krause)
• 7th of May: Quasiheredity of the Schur algebra II, [ABW] (Henning Krause)
• 14th of May, 10:15h in V2-210: Higher Representation-Infinite Algebras from Geometric Tilting Objects (Ragnar Buchweitz)
• 21th of May: Introduction to AR-theory, [ASS] (Apolonia Gottwald)
• 28th of May: The derived category of a finite dimensional algebra, [H] (Rebecca Reischuk)
• 4th of June: Noncrossing partitions parametrizing clusters - preparations for the workshop on noncrossing partitions (Florian Gellert)
• 11th of June: (no seminar)
• 18th of June: Auslander-Reiten duality for exact categories [LZ], (Andrew Hubery)
• 25th of June: Examples of categories with almost split sequences [A], (Julia Sauter)
• 2nd of July: Auslander-Reiten theory in triangulated categories, [K], [KL] (Fajar Yuliawan)
• 9th of July: An introduction to Higher Auslander-Reiten theory, [HIO] (Yu Zhou)
• 16th of July: (no seminar)

Literature

• [A] Auslander: The what, where and why of almost split sequences
• [ABW] Akin, Buchsbaum, Weymann: Schur functors and Schur complexes
• [AIR] Amiot, Iyama, Reiten: Stable categories of Cohen-Macaulay modules and cluster categories (arxiv)
• [ASS] Assem, Simson, Skowronski: Elements of the representation theory of associative algebras. Vol. 1.
• [H] Happel: Triangulated categories in the representation theory of finite-dimensional algebras
• [HIO] Herschend, Iyama, Oppermann: n-representation infinite algebras (arxiv)
• [IR] Iyama, Reiten: Introduction to tau-tilting (arxiv)
• [IO] Iyama , Oppermann: Stable categories of Cohen-Macaulay modules and cluster categories (arxiv)
• [K] Krause: Auslander-Reiten theory via Brown Representability
• [KL] Krause, Jue Le: Auslander-Reiten formula for complexes of modules
• [LZ] Lenzing, Zuazua: Auslander-Reiten duality for abelian categories

Literature for Quantum cluster algebras

• [BZ] Berenstein, A., Zelevinsky, A.: Quantum cluster algebras. Adv. Math. 195 (2), 405--455 (2005)
• [GL] Gellert, F., Lampe, P.: Quantisation Spaces of Cluster Algebras (2014). preprint: http://arxiv.org/abs/1402.1094
• [GLS] Geiss, C., Leclerc, B., Schröer, J.: Cluster structures on quantum coordinate rings. Selecta Math. (N.S.) 19(2), 337--397 (2013)
• [GY] Goodearl, K.R., Yakimov, M.T.: Quantum cluster algebra structures on quantum nilpotent algebras (2013). preprint: arXiv 1309.7869
• [Gr] Grabowski, J.E.: Graded cluster algebras. Preprint: arXiv 1309.6170 (2013)
• [GL] Grabowski, J.E., Launois, S.: Graded quantum cluster algebras and an application to quantum grassmannians. Preprint: arXiv 1301.2133 (2013)
• [HL] Hernandez, D., Leclerc, B.: Cluster algebras and quantum affine algebras. Duke Math. J. 154(2), 265--341 (2010)
• [La1] Lampe, P.: A quantum cluster algebra of Kronecker type and the dual canonical basis. Int. Math. Res. Not. IMRN (13), 2970--3005 (2011)
• [La2] Lampe, P.: Quantum cluster algebras of type A and the dual canonical basis. Proc. London Math. Soc. (108), 1--43 (2014)