Schedule of Talks.

The talks of the summer school will be given by the students. Therefore, participants are encouraged to choose one of the listed talks that they would be happy to give. Please choose also one or two possible alternatives. Email your choices to either Gwyn Bellamy or Olivier Dudas.

The deadline for submitting your choice of talk is the 27th of February, 2014. After the deadline, we will distribute the talks as fairly as possible. The organizer in charge of your talk will contact you to give more detailed information about what should be covered in the talk. You should feel free to email that organizer with any questions you might have.


Timetable

Tuesday Wednesday Thursday Friday
9-10 Intro L5 L10 L15A L15B
10-11 L1 L6 L11 L16
11.30-12.30 L2A L2B L7 L12 R
Lunch break
2.30-4.30 E1 E2 E3 E4
5-6 L3 L8 L13
6-7 L4 L9A L9B L14

Tuesday

Morning:
L1 Representations of quivers. Definition of quivers. Representations of quivers - equivalence with modules over the path algebra. The hereditary property. Description of simple, projective and injective modules. The Euler characteristic formula. The case of finite Dynkin diagrams and Gabriel's theorem, [4].

L2 Kac-Moody algebras and their quantum groups. Review of semi-simple Lie algebras, root systems and Weyl groups. Definition of Kac-Moody Lie algebras associated with generalized Cartan matrices, [20]. Quantum groups and the statement of the PBW theorem. The example of \(U_q(\mathfrak{sl}_2)\), [10].

Afternoon:
L3 Representations of quantum groups for semi-simple Lie algebras. Finite dimensional representations, semi-simplicity for generic \(q\). The definition of category \(\mathcal{O}\). Verma modules and their simple quotients (highest weight theory). The case of \(U_q(\mathfrak{sl}_2)\), [9].

L4 Affine Nil-Hecke algebras. Definition of the Hecke algebra. Description via the polynomial representation and Demazure operators. The center of the algebra. Description as a ring of matrices. See [13, Example 2.2 (3)] and [21, Section 2].

Wednesday

Morning:
L5 Quiver Hecke Algebras. Definition and first properties. Specializations and gradings; see [5], [13] or [21, Section 3]. The cyclotomic quotients and relation to cyclotomic Hecke algebras; see [6] and [18].

L6 Khovanov and Lauda's approach: diagrammatics. Description of the quiver Hecke algebra in terms of diagrams. Categorification of \(U_q(\mathfrak{n}_-)\). Statement of the Khovanov-Lauda conjecture, [13,14].

L7 Relation with canonical basis. Definition of the canonical/crystal basis for the negative part \(U_q(\mathfrak{n}_-)\) of a quantum group. The examples \(U_q(\mathfrak{sl}_2)\) and \(U_q(\mathfrak{sl}_3)\). Canonical basis for highest weight modules, [12,17].

Afternoon:
L8 Sheaves and their cohomology. Constructible sheaves and operations on sheaves. The cohomology of constructible sheaves, [1,8].

L9 The derived category of constructible sheaves. Derived functors. Derived and triangulated categories. The case of constructible sheaves, [1,8].

Thursday

Morning:
L10 Perverse sheaves. Perverse sheaves. Intersection cohomology complexes on stratified spaces. The decomposition theorem,[1,19]. See also Achar's videos from the Newton institute.

L11 Equivariant perverse sheaves in representation theory. A brief account of equivariant perverse sheaves and the equivariant decomposition theorem; see [7, Section 8]. The example of equivariant perverse sheaves on the nilpotent cone of \(\mathfrak{gl}(V)\) and the Springer correspondence [2]. See also [16] for an overview.

L12 Introduction to equivariant cohomology. Classifying spaces - example of \(B \mathbb{C}^{\times} = \mathbb{P}^{\infty}\). Definition of equivariant cohomolgy and first properties. The Kunneth formula. See [3] or [11] for an introduction to equivariant cohomology.

Afternoon:
L13 Computations in the equivariant world. The example of \(H_G^*(G / B) \simeq H_B^*(G/ G) = H^*_T(pt)\). The relation of \(H_G^*(G / B)\) to \(H^*(G / B)\). Kozsul duality between \(H_T(pt)\) and \(H^*(T)\). The example of \(H^*_G(G / B \times G / B)\) [3].

L14 The example of \(\mathfrak{sl}_2\). Identification of the algebra \(\mathrm{Ext}^*_G(\pi_! \mathbb{C}, \pi_! \mathbb{C})\) with \(H^*_G(G / B \times G / B)\), [7, Section 8.6]. Construction of the affine Nil-Hecke algebra as the ext-algebra \(\mathrm{Ext}^*_G(\pi_! \mathbb{C}, \pi_!\mathbb{C})\) [21, Section 5.3.2].

Friday

Morning:
L15 Lusztig's geometric construction of canonical bases. Flag varieties on quivers. Convolution of perverse sheaves. The example of the Dynkin quiver \(\bullet \longrightarrow \bullet\), [15,17,22].

L16 Quiver Hecke algebras as Ext-algebras. Based on [21, Section 5]. Description of the PIM for the quiver Hecke algebra in terms of perverse sheaves.

R Parity sheaves and modular representation theory of Quiver Hecke algebras. Research talk given by Geordie Williamson.

Exercise sessions

In addition to the talks, there will be four exercise sessions: the first on quivers and semi-simple Lie algebras, the second on affine Nil-Hecke algebras and quiver Hecke algebras, the third on sheaves and the last one on further topics.

References

[1] A. A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981).

[2] W. Borho and R. MacPherson, Représentations des groupes de Weyl et homologie d'intersection pour les variétés nilpotentes, C. R. Acad. Sci. Paris Sér. I Math, vol. 292 (1981), no. 15, 707-710.

[3] M. Brion, Equivariant cohomology and equivariant intersection theory, notes from the summer school "Theories des representations et geometrie algebrique" (Montreal), available at http://www-fourier.ujf-grenoble.fr/~mbrion/notesmontreal.pdf, 1997.

[4] M. Brion, Representations of quivers, notes from the summer school "Geometric Methods in Representation Theory" (Grenoble), available at http://www-fourier.ujf-grenoble.fr/~mbrion/notes_quivers_rev.pdf, 2008.

[5] J. Brundan, Quiver Hecke algebras and categorification, preprint available at http://arxiv.org/pdf/1301.5868v2.pdf, 2013.

[6] J. Brundan and A. Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, Invent. Math. 178 (2009), no. 3, 451-484.

[7] N. Chriss and V. Ginzburg, Representation theory and complex geometry, Birkhauser Boston Inc., Boston, MA, 1997.

[8] A. Dimca, Sheaves in topology, Universitext, Springer-Verlag, Berlin, 2004.

[9] J. Humphreys, Representations of semisimple Lie algebras in the BGG category \(\mathscr{O}\), Graduate Studies in Mathematics, vol. 94, American Mathematical Society, Providence, RI, 2008.

[10] J. C. Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics, vol. 6, American Mathematical Society, Providence, RI, 1996.

[11] J. C. Jantzen, Moment graphs and representations, notes from the summer school "Geometric Methods in Representation Theory" (Grenoble), available at http://home.imf.au.dk/jantzen/grnbl.pdf, 2008.

[12] M. Kashiwara, Global crystal bases of quantum groups, Duke Math. J., vol. 69(2) (1993), 455--485.

[13] M. Khovanov and A. D. Lauda, A diagrammatic approach to categorification of quantum groups. I, Represent. Theory, vol. 13 (2009), 309-347.

[14] M. Khovanov and A. D. Lauda, A diagrammatic approach to categorification of quantum groups II, Trans. Amer. Math. Soc., vol. 363 (2011), no. 5, 2685-2700.

[15] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447-498.

[16] G. Lusztig, Intersection cohomology methods in representation theory, Proceedings of the International Congress of Mathematicians, Vol. I,II (Kyoto, 1990), pp. 155-174.

[17] G. Lusztig, Introduction to quantum groups, Progress in Mathematics, Vol. 110, Birkhäuser Boston Inc., 1993.

[18] A. Mathas, Cyclotomic quiver Hecke algebras of type \(A\), preprint available at http://arxiv.org/pdf/1310.2142v1.pdf, 2013.

[19] K. McGerty, Constructible sheaves, perverse sheaves and character sheaves, lecture notes available at http://www.mathematik.uni-kl.de/charsheaves/PDF/mcgerty.pdf, 2013.

[20] N. Perrin, Introduction to Kac-Moody Lie algebras, lecture notes available at http://relaunch.hcm.uni-bonn.de/fileadmin/perrin/km-chapter3.pdf, 2012.

[21] R. Rouquier, Quiver Hecke algebras and 2-Lie algebras, Algebra Colloq. 19 (2012), no. 2, 359-410.

[22] O. Schiffmann, Lectures on canonical and crystal bases of Hall algebras, notes available at http://arxiv.org/pdf/0910.4460v2, 2009.






Support

The research school is supported by the Deutsche Forschungsgemeinschaft, CNRS and EPSRC. 

CNRSCorsica
DFG