Constructing the classical limit for quantum systems on compact semisimple Lie algebras

Integrable G-Strands on semisimple Lie groups

The present paper derives systems of partial differential equations that admit a quadratic zero curvature representation for an arbitrary real semisimple Lie algebra. It also determines the general form of Hamilton's principles and Hamiltonians for these systems and analyzes the linear stability of their equilibrium solutions in the examples of $\mathfrak{so}(3)$ and $\mathfrak{sl}(2,\mathbb{R})$.

Virtual copies of semisimple Lie algebras in enveloping algebras of semidirect products and Casimir operators

Given a semidirect product $\frak{g}=\frak{s}\uplus\frak{r}$ of semisimple Lie algebras $\frak{s}$ and solvable algebras $\frak{r}$, we construct polynomial operators in the enveloping algebra $\mathcal{U}(\frak{g})$ of $\frak{g}$ that commute with $\frak{r}$ and transform like the generators of $\frak{s}$, up to a functional factor that turns out to be a Casimir operator of $\frak{r}$. Such operators are said to generate a virtual copy of $\frak{s}$ in $\mathcal{U}(\frak{g})$, and allow to compute the Casimir operators of $\frak{g}$ in closed form, using the classical formulae for the invaria…

On exceptional nilpotents in semisimple Lie algebras

We classify all pairs (m,e), where m is a positive integer and e is a nilpotent element of a semisimple Lie algebra, which arise in the classification of simple rational W-algebras.

Non-semisimple Lie algebras with Levi factor \frak{so}(3), \frak{sl}(2,R) and their invariants

We analyze the number N of functionally independent generalized Casimir invariants for non-semisimple Lie algebras \frak{s}\overrightarrow{% oplus}_{R}\frak{r} with Levi factors isomorphic to \frak{so}(3) and \frak{sl}(2,R) in dependence of the pair (R,\frak{r}) formed by a representation R of \frak{s} and a solvable Lie algebra \frak{r}. We show that for any dimension n >= 6 there exist Lie algebras \frak{s}\overrightarrow{\oplus}_{R}\frak{r} with non-trivial Levi decomposition such that N(\frak{s}% \overrightarrow{oplus}_{R}\frak{r}) = 0.

On linear-quadratic Poisson pencils on trivial central extensions of semisimple Lie algebras

The paper is devoted to quadratic Poisson structures compatible with the canonical linear Poisson structures on trivial 1-dimensional central extensions of semisimple Lie algebras. In particular, we develop the general theory of such structures and study related families of functions in involution. We also show that there exists a 10-parametric family of quadratic Poisson structures on $\gl(3)^*$ compatible with the canonical linear Poisson structure and containing the 3-parametric family of quadratic bivectors recently introduced by Vladimir Sokolov. The involutive family of polynomial functi…

New approach to representation theory of semisimple Lie algebras and quantum algebras

A method to construct in explicit form the generators of the simple roots of an arbitrary finite-dimensional representation of a quantum or standard semisimple algebra is found. The method is based on general results from the global theory of representations of semisimple groups. The rank two algebras $A_2$, $B_2=C_2$, $D_2$ and $G_2$ are considered as examples. The generators of the simple roots are presented as solutions of a system of finite difference equations and given in the form of $N_l\times N_l$ matrices, where $N_l$ is the dimension of the representation.

紧致连通半单李群的Wehrl型熵问题

Wehrl-type entropy problem for compact connected semisimple Lie groups

本文通过二阶变分法证明紧致连通半单李群中相干投影子唯一极小化Wehrl熵并最大化Husimi幂矩，将问题归结为最高权向量的极矩性质。

Geometry and Topology of Coadjoint Orbits of Semisimple Lie Groups

Orbits of coadjoint representations of classical compact Lie groups have a lot of applications. They appear in representation theory, geometrical quantization, theory of magnetism, quantum optics etc. As geometric objects the orbits were the subject of much study. However, they remain hard for calculation and application. We propose simple solutions for the following problems: an explicit parameterization of the orbit by means of a generalized stereographic projection, obtaining a Kählerian structure on the orbit, introducing basis two-forms for the cohomology group of the orbit.

预李结构与半单李代数

Pre-Lie Structures for Semisimple Lie Algebras

本文研究了半单李代数与预李结构的可接受性，证明S3-结合代数是任何复数域上李代数的通用预李结构。

Integrable triples in semisimple Lie algebras

We classify all integrable triples in simple Lie algebras, up to equivalence. The importance of this problem stems from the fact that for each such equivalence class one can construct the corresponding integrable hierarchy of bi-Hamiltonian PDE. The simplest integrable triple $(f,0,e)$ in $\mathfrak{sl}_2$ corresponds to the KdV hierarchy, and the triple $(f,0,e_θ)$, where $f$ is the sum of negative simple root vectors and $e_θ$ is the highest root vector of a simple Lie algebra, corresponds to the Drinfeld-Sokolov hierarchy.

Airy structures for semisimple Lie algebras

We give a complete classification of Airy structures for finite-dimensional simple Lie algebras over $\mathbb C$, and to some extent also over $\mathbb R$, up to isomorphisms and gauge transformations. The result is that the only algebras of this type which admit any Airy structures are $\mathfrak{sl}_2$, $\mathfrak{sp}_4$ and $\mathfrak{sp}_{10}$. Among these, each admits exactly two non-equivalent Airy structures. Our methods apply directly also to semisimple Lie algebras. In this case it turns out that the number of non-equivalent Airy structures is countably infinite. We have derived a num…