nLab double affine Hecke algebra

Idea

Double Hecke algebras or double affine Hecke algebras (DAHA) or Cherednik algebras are a particular class of multiparametric families of associative algebras; one parameter is $q$ and then there are 1 or more parameters $t_\alpha$ where labels $\alpha$ depend on a root system. While Iwahori-Hecke algebra is a deformation of the group algebra of a Weyl group, DAHA are flat deformations of certain crossed product algebras (or twisted group algebras) involving Coxeter groups. Affine denotes the relation to affine Weyl group in DAHA case. Ivan Cherednik introduced DAHA in proving Macdonald conjecture?s about orthogonal polynomials attached to reduced root systems.

Notice that rational Cherednik algebras defined by Etingof and Ginzburg are not literally DAHAs, but certain degenerations of those and are a special case of symplectic reflection algebras.

Related concepts

• Hecke algebra

• Calogero-Moser system, Dunkl operator

• symplectic reflection algebra

• rational Cherednik algebra

References

• Wikipedia, Double affine Hecke algebra

Monograph:

• Ivan Cherednik, Double affine Hecke algebras, London Math. Soc. Lecture Note Series 319, Cambridge Univ. Press 2005

It has been introduced in a series of works in early 1990s:

• Ivan Cherednik, Double affine Hecke algebras. Knizhnik-Zamolodchikov equations, and Macdonald operators, IMRN (Duke Math. J.) 9 (1992), 171–180 MR1185831 (94b:17040), A unification of Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras, Invent. math. 106 (1991) 411–431 MR1128220 (93b:17040); Double affine Hecke algebras and Macdonald’s conjectures, Ann. of Math.(2) 141:1 (1995) 191–216.

An introduction into the monograph above is the preprin

• Ivan Cherednik, Introduction to double Hecke algebras, math.QA/0404307

• Ivan Cherednik, Y. Markov, Hankel transform via double Hecke algebra, math.QA/0004116

• Mark Haiman, Cherednik algebras, Macdonald polynomials and combinatorics, Proc. ICM, Madrid 2006, Vol. III, 843–872, djvu scan; author’s pdf

• Pavel Etingof, Lectures on Calogero-Moser systems, pdf

• Wee Liang Gan, Victor Ginzburg, Almost-commuting variety, $\mathcal{D}$-modules, and Cherednik algebras, With an appendix by Ginzburg. IMRP Int. Math. Res. Pap. 2006, 26439, 1–54, MR2008c:14062

A construction via factorization homology of quantum group representation categories is indicated in

• David Ben-Zvi, Adrien Brochier, David Jordan, Integrating quantum groups over surfaces: quantum character varieties and topological field theory (arXiv:1501.04652)

category: algebra