Related papers: On integrability of transverse Lie-Poisson structu…

We study invariant Nijenhuis $(1,1)$-tensors on a homogeneous space $G/K$ of a reductive Lie group $G$ from the point of view of integrability of a Hamiltonian system of differential equations with the $G$-invariant Hamiltonian function on…

We introduce a combinatorial method to construct indefinite Ricci-flat metrics on nice nilpotent Lie groups. We prove that every nilpotent Lie group of dimension $\leq6$, every nice nilpotent Lie group of dimension $\leq7$ and every…

The link between (super)-affine Lie algebras as Poisson brackets structures and integrable hierarchies provides both a classification and a tool for obtaining superintegrable hierarchies. The lack of a fully systematic procedure for…

The structure of a solvable Lie groups admitting an Einstein left-invariant metric is, in a sense, completely determined by the nilradical of its Lie algebra. We give an easy-to-check necessary and sufficient condition for a nilpotent…

Let \theta be an involution of the semisimple Lie algebra g and g=k+p be the associated Cartan decomposition. The nilpotent commuting variety of (g,\theta) consists in pairs of nilpotent elements (x,y) of p such that [x,y]=0. It is…

A wide class of Hamiltonian systems with N degrees of freedom and endowed with, at least, (N-2) functionally independent integrals of motion in involution is constructed by making use of the two-photon Lie-Poisson coalgebra. The set of…

This paper does not contain any new results, it is just an attempt to present, in a systematic way, one construction which establishes an interesting relationship between some ideas and notions well-known in the theory of integrable systems…

We establish combinatorial formulas for the index of a class of matrix Lie algebras whose matrix forms are encoded by strict partial orderings.

Simple deformations, with a parameter $\epsilon$, of classical $R$-matrices which follow from decomposition of appropriate Lie algebras, are considered. As a result nonstandard Lax representations for some well known integrable systems are…

In this paper, we shall use a method based on the theory of extensions of left-symmetric algebras to classify complete left-invariant affine real structures on solvable non-unimodular three-dimensional Lie groups.

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…

Let $P$ be a Poisson algebra with a Lie bracket $\{, \}$ over a field $\F$ of characteristic $p\geq 0$. In this paper, the Lie structure of $P$ is investigated. In particular, if $P$ is solvable with respect to its Lie bracket, then we…

An equivalent condition for an element of a Lie algebra acting nilpotently in all its representations is obtained. Namely, it should belong to the derived algebra and go via factoring over the radical to a nilpotent element of the…

This article will discussing on $\frac{1}{2}$-derivations of quasi-filiform Lie algebras of maximum length. The non-trivial transposed Poisson algebras with the quasi-filiform Lie algebras of maximum length are constructed by using…

Given a first order dynamical system possessing a commutative algebra of dynamical symmetries, we show that, under certain conditions, there exists a Poisson structure on an open neighbourhood of its regular (not necessarily compact)…

I show that simple finite vertex algebras are commutative, and that the Lie conformal algebra structure underlying a reduced (i.e., without nilpotent elements) finite vertex algebra is nilpotent.

Premet has conjectured that the nilpotent variety of any finite-dimensional restricted Lie algebra is an irreducible variety. In this paper, we prove this conjecture in the case of Hamiltonian Lie algebra. and show that its nilpotent…

We prove nilpotency results for Lie algebras over an arbitrary field admitting a derivation, which satisfies a given polynomial identity $r(t)=0$. For the polynomial $r=t^n-1$ we obtain results on the nilpotency of Lie algebras admitting a…

We consider a class of map, recently derived in the context of cluster mutation. In this paper we start with a brief review of the quiver context, but then move onto a discussion of a related Poisson bracket, along with the Poisson algebra…

In this paper we analyze the structure of some subalgebras of quantized enveloping algebras corresponding to unipotent and solvable subgroups of a simple Lie group G. These algebras have the non--commutative structure of iterated algebras…