Related papers: Integrable systems associated to the filtrations o…
The following integrability theorem for vertex operator algebras V satisfying some finiteness conditions(C_2-cofinite and CFT-type) is proved: the vertex operator subalgebra generated by a simple Lie subalgebra {\frak g} of the weight one…
A method to construct both classical and quantum completely integrable systems from (Jordan-Lie) comodule algebras is introduced. Several integrable models based on a so(2,1) comodule algebra, two non-standard Schrodinger comodule algebras,…
A new highly symmetrical model of the compact Lie algebra $\mathfrak{g}^c_2$ is provided as a twisted ring group for the group $\mathbb{Z}_2^3$ and the ring $\mathbb{R}\oplus\mathbb{R}$. The model is self-contained and can be used without…
Lagrangian multiforms provide a variational framework to describe integrable hierarchies. The case of Lagrangian $1$-forms covers finite-dimensional integrable systems. We use the theory of Lie dialgebras introduced by Semenov-Tian-Shansky…
Let G be a reductive group over an algebraically closed field k of separably good characteristic p>0 for G. Under these assumptions a Springer isomorphism from the reduced nilpotent scheme of the Lie algebra of G to the reduced unipotent…
We review some basic theorems on integrability of Hamiltonian systems, namely the Liouville-Arnold theorem on complete integrability, the Nekhoroshev theorem on partial integrability and the Mishchenko-Fomenko theorem on noncommutative…
We associate to a semisimple complex Lie algebra $\mathfrak{g}$ a sequence of polynomials $P_{\ell,\mathfrak{g}}(x)\in\mathbb{Q}[x]$ in $r$ variables, where $r$ is the rank of $\mathfrak{g}$ and $\ell=0,1,2,\ldots $. The polynomials…
We first present a filtration on the ring L of Laurent polynomials such that the direct sum decomposition of its associated graded ring gr L agrees with the direct sum decomposition of gr L, as a module over the complex general linear Lie…
A new class of integrable mappings and chains is introduced. Corresponding $(1+2)$ integrable systems invariant with respect to such discrete transformations are presented in an explicit form. Their soliton-type solutions are constructed in…
We study (two-sided) ideals $I$ in the enveloping algebra $\U(\frak g_\infty)$ of an infinite-dimensional Lie algebra $\frak g_\infty$ obtained as the union (equivalently, direct limit) of an arbitrary chain of embeddings of simple…
It is pointed out that affine Lie algebras appear to be the natural mathematical structure underlying the notion of integrability for two-dimensional systems. Their role in the construction and classification of 2D integrable systems is…
A method to construct integrable deformations of Hamiltonian systems of ODEs endowed with Lie-Poisson symmetries is proposed by considering Poisson-Lie groups as deformations of Lie-Poisson (co)algebras. Moreover, the underlying Lie-Poisson…
The purpose of this article is to investigate relations between W-superalgebras and integrable super-Hamiltonian systems. To this end, we introduce the generalized Drinfel'd-Sokolov (D-S) reduction associated to a Lie superalgebra $g$ and…
We study a novel $n(n+1)/2$-dimensional non-semisimple Lie algebra $\mathfrak{g}_n$, a generalisation of both $\mathfrak{sl}_2(\mathbb{K})$ and the two-photon Lie algebra $\mathfrak{h}_6$. We investigate its properties, including its…
In recent work Bertram Kostant and Nolan Wallach ([KW1], [KW2]) have defined an interesting action of a simply connected Lie group $A$ isomorphic to \mathbb{C}^{{n\choose 2}} on gl(n) using a completely integrable system derived from…
We consider nonholonomic geodesic flows of left-invariant metrics and left-invariant nonintegrable distributions on compact connected Lie groups. The equations of geodesic flows are reduced to the Euler-Poincare-Suslov equations on the…
A class of simple filtered Lie algebras of polynomial growth with increasing filtration is distinguished and presentations of these algebras are explicitely described for the simplest examples. Lie (super)algebras of this class appear in…
The connections between Euler's equations on central extensions of Lie algebras and Euler's equations on the original, extended algebras are described. A special infinite sequence of central extensions of nilpotent Lie algebras constructed…
We generalize a result of Kostant and Wallach concerning the algebraic integrability of the Gelfand-Zeitlin vector fields to the full set of strongly regular elements in $gl(n,\mathbb{C})$. We use decomposition classes to stratify the…
Recently, E.Feigin introduced a very interesting contraction $\mathfrak q$ of a semisimple Lie algebra $\mathfrak g$ (see arXiv:1007.0646 and arXiv:1101.1898). We prove that these non-reductive Lie algebras retain good invariant-theoretic…