Periodic automorphisms of Takiff algebras, contractions, and $θ$-groups

Semi-direct products of Lie algebras and their invariants

The goal of this paper is to extend the standard invariant-theoretic design, well-developed in the reductive case, to the setting of representation of certain non-reductive groups. This concerns the following notions and results: the existence of generic stabilisers and generic isotropy groups for finite-dimensional representations; structure of the fields and algebras of invariants; quotient morphisms and structure of their fibres. One of the main tools for obtaining non-reductive Lie algebras is the semi-direct product construction. We observe that there are surprisingly many non-reductive L…

Automorphisms of finite order, periodic contractions, and Poisson-commutative subalgebras of $\mathcal S(\mathfrak g)$

Let $\mathfrak g$ be a semisimple Lie algebra, $\vartheta\in {\sf Aut}(\mathfrak g)$ a finite order automorphism, and $\mathfrak g_0$ the subalgebra of fixed points of $\vartheta$. Recently, we noticed that using $\vartheta$ one can construct a pencil of compatible Poisson brackets on $\mathcal S(\mathfrak g)$, and thereby a `large' Poisson-commutative subalgebra $\mathcal Z(\mathfrak g,\vartheta)$ of $\mathcal S(\mathfrak g)^{\mathfrak g_0}$. In this article, we study invariant-theoretic properties of $(\mathfrak g,\vartheta)$ that ensure good properties of $\mathcal Z(\mathfrak g,\vartheta)$…

关于局部紧压缩群酉表示理论的研究

On the unitary representation theory of locally compact contraction groups

本文研究局部紧压缩群及其半直积的酉表示，提出刻画type I或CCR群的问题，分类无挠群的不可约表示，并给出正则树自同构群的新例子。

The contraction category of graphs

We study the category whose objects are graphs of fixed genus and whose morphisms are contractions. We show that the corresponding contravariant module categories are Noetherian and we study two families of modules over these categories. The first takes a graph to a graded piece of the homology of its unordered configuration space and the second takes a graph to an intersection homology group whose dimension is given by a Kazhdan-Lusztig coefficient; in both cases we prove that the module is finitely generated. This allows us to draw conclusions about torsion in the homology groups of graph co…

Periodic automorphisms, compatible Poisson brackets, and Gaudin subalgebras

Let $\mathfrak g$ be a finite-dimensional Lie algebra. The symmetric algebra $\mathcal S(\mathfrak g)$ is equipped with the standard Lie-Poisson bracket. In this paper, we elaborate on a surprising observation that one naturally associates the second compatible Poisson bracket on $\mathcal S(\mathfrak g)$ to any finite order automorphism $θ$ of $\mathfrak g$. We study related Poisson-commutative subalgebras $\mathcal C$ of $\mathcal S(\mathfrak g)$ and associated Lie algebra contractions of $\mathfrak g$. To obtain substantial results, we have to assume that $\mathfrak g$ is semisimple. Then w…