Affinity of Cherednik algebras on projective space

Lecture Notes on Cherednik Algebras

Rational Cherednik algebras

We survey a number of results about rational Cherednik algebra representation theory and its connection to symplectic singularities and their resolutions. Mathematics Subject Classification (2000). Primary 16G, 17B; Secondary 20C, 53D.

Cherednik algebras and Yangians

We construct a functor from the category of modules over the trigonometric (resp. rational) Cherednik algebra of type gll to the category of integrable modules of level l over a Yangian for the loop algebra sln (resp. over a subalgebra of this Yangian called the Yangian deformed double loop algebra) and we establish that it is an equivalence of categories if l + 2 < n.

Microlocalization of Rational Cherednik Algebras

We construct a microlocalization of the rational Cherednik algebras H of type Sn. This is achieved by a quantization of the Hilbert scheme Hilb C2 of n points in C2. We then prove the equivalence of the category of H -modules and that of modules over its microlocalization under certain conditions on the parameter.

Koszul Algebras and Sheaves over Projective Space

We are going to show that the sheafication of graded Koszul modules KΓ over Γn = K [x0, x1...xn] form an important subcategory ∧ KΓ of the coherents sheaves on projective space, Coh(P n). One reason is that any coherent sheave over P n belongs to ∧ KΓup to shift. More importantly, the category KΓ allows a concept of almost split sequence obtained by exploiting Koszul duality between graded Kosz...