Transverse Poisson Structures to Adjoint orbits in semi-simple Lie algebras

Cluster X-varieties for dual Poisson-Lie groups I

We associate a family of cluster X-varieties to the dual Poisson-Lie group G* of a complex semi-simple Lie group G of adjoint type given with the standard Poisson structure. This family is described by the W-permutohedron associated to the Lie algebra g of G: vertices being labeled by cluster X-varieties and edges by new Poisson birational isomorphisms, on appropriate seed X-tori, called saltation. The underlying combinatorics is based on a factorization of the Fomin-Zelevinsky twist maps into mutations and other new Poisson birational isomorphisms on seed X-tori called tropicalmutations, asso…

Deformed Poisson W-algebras of Type A

For the algebraic group $SL_{l+1}(\mathbb{C})$ we describe a system of positive roots associated to conjugacy classes in its Weyl group. Using this we explicitly describe the algebra of regular functions on certain transverse slices to conjugacy classes in $SL_{l+1}(\mathbb{C})$ as a polynomial algebra of invariants. These may be viewed as an algebraic group analogue of certain graded parabolic invariants that generate the (graded) W-algebra in type A.

The Poisson geometry of the conjugation quotient map for simple algebraic groups and deformed Poisson W-algebras

We define Poisson structures on certain transversal slices to conjugacy classes in complex simple algebraic groups introduced in arXiv:0809.0205. These slices are associated to the elements of the Weyl group, and the Poisson structures on them are analogous to the Poisson structures introduced by J. de Boer, T. Tjin and A. Premet in papers arXiv:hep-th/9211109 and http://www.maths.man.ac.uk/DeptWeb/Homepages/aap/Reprints/Transverse.ps on the Slodowy slices in complex simple Lie algebras. The quantum deformations of these Poisson structures are known as W-algebras of finite type. As an applicat…

Dirac 结构的爆破

Blowups of Dirac structures

本文刻画了实扭曲 Dirac 结构沿余维数大于1的子流形爆破可提升的条件，即子流形横截或不变且横截李代数具有相同常数高度，并恢复了 Polishchuk 关于 Poisson 结构的定理。

On integrability of some bi-Hamiltonian two field systems of PDE

We continue the study of integrability of bi-Hamiltonian systems with a compatible pair of local Poisson structures (H_0,H_1), where H_0 is a strongly skew-adjoint operator. This is applied to the construction of some new two field integrable systems of PDE by taking the pair (H_0,H_1) in the family of compatible Poisson structures that arose in the study of cohomology of moduli spaces of curves.

泊松代数的通用构造及其应用

Universal constructions for Poisson algebras. Applications

本文定义了泊松代数的通用代数，证明其对有限维泊松代数存在，构造了相关伴随函子，刻画了其代数结构，并给出了自同构群描述与阿贝尔群分次分类两项应用。

Equivariant quantization of Poisson homogeneous spaces and Kostant's problem

Let $\mathfrak g$ be a finite dimensional split semisimple Lie algebra and $λ$ a weight of $\mathfrak g$. Let $F$ be the algebra of quantized regular functions on the connected simply connected group $G$ corresponding to $\mathfrak g$. In the present paper we introduce a certain subspace $F'$ of $F$ (which is not necessary a subalgebra of $F$) and endow it with an associative $\star$-product using the so-called reduced fusion element. We prove that the algebra $(F',\star)$ is isomorphic to $(L(λ))_{fin}$, where $L(λ)$ is the irreducible highest weight $\check{U}_q\mathfrak g$-module and "$fin$…

Poisson orders on large quantum groups

We develop a Poisson geometric framework for studying the representation theory of all contragredient quantum super groups at roots of unity. This is done in a uniform fashion by treating the larger class of quantum doubles of bozonizations of all distinguished pre-Nichols algebras arXiv:1405.6681 belonging to a one-parameter family; we call these algebras \emph{large} quantum groups. We prove that each of these quantum algebras has a central Hopf subalgebra giving rise to a Poisson order in the sense of arXiv:math/0201042.. We describe explicitly the underlying Poisson algebraic groups and Po…