Deformed preprojective algebras and symplectic reflection algebras for wreath products

Deformed Cartan matrices and generalized preprojective algebras I: Finite type

We give an interpretation of the $(q,t)$-deformed Cartan matrices of finite type and their inverses in terms of bigraded modules over the generalized preprojective algebras of Langlands dual type in the sense of Geiß-Leclerc-Schröer [Invent. math. 209 (2017)]. As an application, we compute the first extension groups between the generic kernels introduced by Hernandez-Leclerc [J. Eur. Math. Soc. 18 (2016)], and propose a conjecture that their dimensions coincide with the pole orders of the normalized $R$-matrices between the corresponding Kirillov-Reshetikhin modules.

变形Cartan矩阵与广义预投射代数II：一般型

Deformed Cartan matrices and generalized preprojective algebras II: General type

提出带多参数的变形对称化广义Cartan矩阵定义，借助广义预投射代数分次模给出范畴解释，推导逆矩阵的辫群组合公式，并在特定条件下与质量变形Cartan矩阵一致。

Harish-Chandra homomorphisms and symplectic reflection algebras for wreath-products

The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreath-product in terms of quantum hamiltonian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existence of such a construction has been conjectured in [EG]. We also present a new approach to reflection functors and shift functors for generalized preprojective algebras and symplectic reflection algebras associated with wreath-products.

Recollement of Deformed Preprojective Algebras and the Calogero-Moser Correspondence

The aim of this paper is to clarify the relation between the following objects: $ (a) $ rank 1 projective modules (ideals) over the first Weyl algebra $ A_1(\C)$; $ (b) $ simple modules over deformed preprojective algebras $ Π_λ(Q) $ introduced by Crawley-Boevey and Holland; and $ (c) $ simple modules over the rational Cherednik algebras $ H_{0,c}(S_n) $ associated to symmetric groups. The isomorphism classes of each type of these objects can be parametrized geometrically by the same space (namely, the Calogero-Moser algebraic varieties); however, no natural functors between the corresponding …

Preprojective algebras and MV polytopes

The purpose of this paper is to apply the theory of MV polytopes to the study of components of Lusztig's nilpotent varieties. Along the way, we introduce reflection functors for modules over the non-deformed preprojective algebra of a quiver.

Central extensions of preprojective algebras, the quantum Heisenberg algebra, and 2-dimensional complex reflection groups

We introduce a central extension of the preprojective algebra of a finite Dynkin quiver (depending on a regular weight for the corresponding root system), whose natural deformed version is flat (unlike that for the preprojective algebra). We calculate the Hilbert polynomial of the central extension, and show that it is a Frobenius algebra. As a corollary, we obtain the Hilbert series of the usual deformed preprojective algebra in which the deformation parameters are variables, and show that this algebra is Gorenstein (although it is not a flat module over the ring of parameters). The proofs ar…