W-algebras are a class of non-commutative algebras related to the classical universal enveloping algebras. They can be defined as a subquotient of U(g) related to a choice of nilpotent element e and compatible nilpotent subalgebra m. The definition is a quantum analogue of the classical construction of Hamiltonian reduction.
We define a quantum version of Hamiltonian reduction by stages and use it to construct intermediate reductions between different W-algebras U(g,e) in type A.This allows us to express the W-algebra U(g,e') as a subquotient of U(g,e) for nilpotent elements e' covering e. It also produces a collection of (U(g,e),U(g,e'))-bimodules analogous to the generalised Gel'fand-Graev modules used in the classical definition of the W-algebra; these can be used to obtain adjoint functors between the corresponding module categories.
The category of modules over a W-algebra has a full subcategory defined in a parallel fashion to that of the Bernstein-Gel'fand-Gel'fand (BGG) category O; this version of category O(e) for W-algebras is equivalent to an infinitesimal block of O by an argument of Miličić and Soergel. We therefore construct analogues of the translation functors between the different blocks of O, in this case being functors between the categories O(e) for different W-algebras U(g,e). This follows an argument of Losev, and realises the category O(e') as equivalent to a full subcategory of the category O(e) where e' is greater than e in the refinement ordering.
Generalized affine Grassmannian slices provide geometric realizations for weight spaces of representations of semisimple Lie algebras. They are also Coulomb branches, symplectic dual to Nakajima quiver varieties. In this paper, we prove that neighbouring generalized affine Grassmannian slices are related by Hamiltonian reduction by the action of the additive group. We also prove a weaker version of the same result for their quantizations, algebras known as truncated shifted Yangians.
Presilting and silting subcategories in extriangulated categories were introduced by Adachi and Tsukamoto recently. In this paper, we prove that the Gabriel-Zisman localization $\mathcal B/({\rm thick}\mathcal W)$ of an extriangulated category $\mathcal B$ with respect to a presilting subcategory $\mathcal W$ satisfying certain condition can be realized as a subfactor category of $\mathcal B$. This generalizes the result by Iyama-Yang for silting reduction on triangulated categories. Then we discuss the relation between silting subcategories and tilting subcategories in extriangulated categori…
Examples of exact categories in representation theory are given by the category of Delta-filtered modules over quasi-hereditary algebras, but also by various categories related to matrix problems, such as poset representations or representations of bocses. Motivated by the matrix problem background, we study in this article the reduction of exact structures, and consider the poset Ex(A) of all exact structures on a fixed additive category A. This poset turns out to be a complete lattice, and under suitable conditions results of Enomoto's imply that it is boolean. We initiate in this article a …
A connection between minimal nilpotent orbits of types A and D via Hamiltonian reduction
State Key Laboratory of Mathematical Sciences, Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
