北京雁栖湖应用数学研究院 北京雁栖湖应用数学研究院

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清华大学 "求真书院"
清华大学丘成桐数学科学中心
清华三亚国际数学论坛
上海数学与交叉学科研究院
BIMSA > Cherednik Algebras and DAHAs \(ICBS\)
Cherednik Algebras and DAHAs
The course is devoted to two families of algebras actively studied over the past two decades: the double affine Hecke algebras (DAHAs) introduced by Ivan Cherednik, and their rational version, the so-called Cherednik algebras introduced by Pavel Etingof and Victor Ginzburg. We will describe the foundations of their theory and discuss some applications and links - mostly, to integrable systems but also touching upon orthogonal polynomials, combinatorics, and geometry. To some extent, similar constructions exist at the elliptic level as well; these will be discussed towards the end of the course.
Professor Lars Aake Andersson
讲师
Oleg Chalykh
日期
2025年03月03日 至 04月28日
位置
Weekday Time Venue Online ID Password
周一,周三 13:30 - 15:05 A3-4-101 ZOOM 08 787 662 9899 BIMSA
修课要求
Basic notions of linear algebra and representation theory. Some knowledge of Coxeter groups, simple complex Lie algebras and root systems is desirable but not essential as all the necessary concepts will be introduced in the course. Some knowledge of integrable systems would be useful but is not required.
课程大纲
1. Background material: Coxeter groups and Hecke algebras, root systems and Dynkin diagrams, classical and quantum integrable systems.
2. Rational Cherednik algebras and Dunkl operators. Category O and KZ equations. Applications: Calogero-Moser systems; diagonal harmonics.
3. Affine Weyl groups and affine Hecke algebras. Double affine Hecke algebras and Cherednik operators.
4. Macdonald-Ruijsenaars systems, Macdonald polynomials, Macdonald symmetric functions.
5. Star-shaped affine Dynkin quivers, generalised DAHAs, and character varieties.
6. Global Cherednik algebras. Elliptic Cherednik algebras and elliptic Dunkl operators. Applications: elliptic Calogero-Moser system, Inozemtsev spin chain.
7. Elliptic Cherednik operators and Ruijsenaars systems.
参考资料
1. Etingof, P.: Calogero-Moser systems and representation theory. Vol. 4. European Mathematical Society, 2007.
2. Chalykh, O.: Dunkl and Cherednik Operators. Encyclopedia of Mathematical Physics (Second edition), vol. 3, 309-327. Academic Press, 2025.
3. Etingof, P., Ginzburg, V.: Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism. Invent. Math. 147, 243-348, 2002.
4. Kirillov Jr, A.: Lectures on affine Hecke algebras and Macdonald’s conjectures. Bull. AMS 34, no. 3, 251-292, 1997.
5. Macdonald, I. G.: Affine Hecke algebras and orthogonal polynomials. No. 157. Cambridge University Press, 2003.
6. Cherednik, I.: Double affine Hecke algebras. Vol. 319. Cambridge University Press, 2005.

Background references on Coxeter groups and root systems:
7. Humphreys, J.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics. Vol. 29. Cambridge University Press, 1990.
8. Bourbaki, N.: Elements of mathematics. Lie groups and Lie algebras. Chapters 4-6 (English translation). Springer, 2002.
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北京雁栖湖应用数学研究院
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No. 544, Hefangkou Village Huaibei Town, Huairou District Beijing 101408

北京市怀柔区 河防口村544号
北京雁栖湖应用数学研究院 101408

Tel. 010-60661855
Email. administration@bimsa.cn

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