Reflection Groups. A Contribution to the Handbook of Algebra

TL;DR: In this paper, it was shown that for an arbitrary finite root system, the periodicity conjecture of Al.B.Zamolodchikov concerning Y-systems always holds.

TL;DR: In this paper, the authors define a class of vertex operator algebras labelled by complex reflection groups, which are extensions of the super Virasoro algebra obtained by introducing additional generators, in correspondence with the invariants of the complex reflection group.

TL;DR: In a course on Cherednik algebras given by the first author at MIT in the Fall of 2009 as mentioned in this paper, the goal was to give an introduction to cherednik algebra, and to review the web of connections between them and other mathematical objects.

TL;DR: In this paper, the authors define and study a class of vertex operator algebras labelled by complex reflection groups, which are extensions of the super Virasoro algebra obtained by introducing additional generators, in correspondence with the invariants of the complex reflection group G.

TL;DR: In this paper, Semisimple Lie Algebras and root systems are used for representation theory, isomorphism and conjugacy theorem, and existence theorem for representation.

TL;DR: The invariant bilinear form and the generalized casimir operator are integral representations of Kac-Moody algebras and the weyl group as mentioned in this paper, as well as a classification of generalized cartan matrices.

TL;DR: In this article, the authors consider a class of Lie algebras in which every subalgebra is a subideal, and they show that it is possible to construct a locally coalescent class of these classes.

TL;DR: In this article, a classification of finite and affine reflection groups is presented, including Coxeter groups, Hecke algebras and Kazhdan-Lusztig polynomials.