Nicolas Guay

On the category O for rational Cherednik algebras

Abstract We study the category 𝒪 of representations of the rational Cherednik algebra AW attached to a complex reflection group W. We construct an exact functor, called Knizhnik-Zamolodchikov functor: 𝒪→ℋW-mod, where ℋW is the (finite) Iwahori-Hecke algebra associated to W. We prove that the Knizhnik-Zamolodchikov functor induces an equivalence between 𝒪/𝒪tor, the quotient of 𝒪 by the subcategory of AW-modules supported on the discriminant, and the category of finite-dimensional ℋW-modules. The standard AW-modules go, under this equivalence, to certain modules arising in Kazhdan-Lusztig theory of “cells”, provided W is a Weyl group and the Hecke algebra ℋW has equal parameters. We prove that the category 𝒪 is equivalent to the module category over a finite dimensional algebra, a generalized “q-Schur algebra” associated to W.

Projective modules in the category O for the Cherednik algebra

Abstract We study projective objects in the category O c of the rational Cherednik algebra introduced recently by Berest, Etingof and Ginzburg. We prove that it has enough projectives and that it is a highest weight category in the sense of Cline, Parshall and Scott, and therefore satisfies an analog of the BGG-reciprocity formula for a semisimple Lie algebra.

From quantum loop algebras to Yangians

The main purpose of this note is to give a proof of a statement of V. Drinfeld in [Dr1] regarding Yangians and quantum loop algebras, namely how the former can be constructed as limit forms of the latter. We also apply the same ideas to twisted quantum loop algebras to recover the (non-twisted) Yangians.

On the category for rational Cherednik algebras

Abstract We study the category 𝒪 of representations of the rational Cherednik algebra AW attached to a complex reflection group W. We construct an exact functor, called Knizhnik-Zamolodchikov functor: 𝒪→ℋW-mod, where ℋW is the (finite) Iwahori-Hecke algebra associated to W. We prove that the Knizhnik-Zamolodchikov functor induces an equivalence between 𝒪/𝒪tor, the quotient of 𝒪 by the subcategory of AW-modules supported on the discriminant, and the category of finite-dimensional ℋW-modules. The standard AW-modules go, under this equivalence, to certain modules arising in Kazhdan-Lusztig theory of “cells”, provided W is a Weyl group and the Hecke algebra ℋW has equal parameters. We prove that the category 𝒪 is equivalent to the module category over a finite dimensional algebra, a generalized “q-Schur algebra” associated to W.

Twisted Yangians for symmetric pairs of types B, C, D

We study a class of quantized enveloping algebras, called twisted Yangians, associated with the symmetric pairs of types B, C, D in Cartan’s classification. These algebras can be regarded as coideal subalgebras of the Yangian for orthogonal or symplectic Lie algebras. They can also be presented as quotients of a reflection algebra by additional symmetry relations. We prove an analogue of the Poincaré–Birkoff–Witt Theorem, determine their centres and study also extended reflection algebras.

Twisted Yangians of small rank

We study quantized enveloping algebras called twisted Yangians associated with the symmetric pairs of types CI, BDI and DIII (in Cartans classification) when the rank is small. We establish isomorphisms between these twisted Yangians and the well known Olshanskiis twisted Yangians of types AI and AII, and also with the Molev-Ragoucy reflection algebras associated with symmetric pairs of type AIII. We also construct isomorphisms with twisted Yangians in Drinfelds original presentation.

Representations of twisted Yangians of types B, C, D: I

We initiate a theory of highest weight representations for twisted Yangians of types B, C, D and we classify the finite-dimensional irreducible representations of twisted Yangians associated to symmetric pairs of types CI, DIII and BCD0.

Twisted Yangians, twisted quantum loop algebras and affine Hecke algebras of type BC

We study twisted Yangians of type AIII which have appeared in the literature under the name of reflection algebras. They admit q-versions which are new twisted quantum loop algebras. We explain how these can be defined equivalently either via the reflection equation or as coideal subalgebras of Yangians of gln (resp. of quantum loop algebras of gln). The connection with affine Hecke algebras of type BC comes from a functor of Schur-Weyl type between their module categories.

Equivalences between three presentations of orthogonal and symplectic Yangians

We prove the equivalence of two presentations of the Yangian