Program
Representation theory seminar: Hall algebras and Cherednik algebras
Student No.:50
Time:15:10-17:00 (Wed.), 09-18~12-18
Instructor:Bangming Deng,Xiaoguang Ma,Fan Xu  
Place:Conference Room 3, Floor 2, Jin Chun Yuan West Building
Starting Date:2013-9-18
Ending Date:2013-12-18

 

 

Abstract:

 

Recently, Schiffmann and Vasserot showed that there exists a strong relation between Hall algebras of elliptic curves and double affine Hecke algebras (also called Cherednik algebras) of GL(n) (see [5]). Etingof, Oblomkov and Rains defined some generalization of Cherednik algebras and related it to weighted projective lines and multiplicative preprojective algebras (see [6]). The aim of this seminar is to understand these results.

 
The sketchy plan is as follows:
 
 

(1) Categories of coherent sheaves of weighted projective lines;

(2) Categories of coherent sheaves of elliptic curves;

(3) Hall algebras of weighted projective lines and elliptic curves;

(4) Double affine Hecke algebras;

If time permits,

(5) The theorem of Etingof, Oblomkov and Rains;

(6) The relation between Hall algebras of elliptic curves and Cherednik algebras.

 

 
 

References:

 

[1] K. Bruning, I. Burban: Coherent sheaves of elliptic curves. Lecture note, see homepage of Igor Burban.

[2] W. Geigle and H. Lenzing, A class of weighted projective curves arising in representation theory of finite-dimensional algebras, in Singularities, representation of algebras, and vector bundles (Lambrecht, 1985), 265–297, Lecture Notes in Math., 1273, Springer, Berlin, 1987.

[3] Igor Burban, Olivier Schiffmann, On the Hall algebra of an elliptic curve, I, arXiv:math/0505148.

[4] P. Etingof, Xiaoguang Ma, Lecture notes on Cherednik algebras, arXiv: 1001.0432.

[5] O. Schiffmann, E. Vasserot, The elliptic Hall algebra, Cherednick Hecke algebras and Macdonald polynomials, arXiv:0802.4001.

[6] Pavel Etingof, Alexei Oblomkov, Eric Rains, Generalized double affine Hecke algebras of rank 1 and quantized Del Pezzo surfaces, arXiv:math/0406480.