Related papers: On the free Lie algebra with multiple brackets
An $N$-free poset is a poset whose comparability graph does not embed an induced path with four vertices. We use the well-quasi-order property of the class of countable $N$-free posets and some labelled ordered trees to show that a…
It is shown, that each Lifting cocycle $\Psi_{2n+1},\Psi_{2n+3},\Psi_{2n+5},...$ ([Sh1], [Sh2]) on the Lie algebra $\Dif_n$ of polynomial differential operators on an $n$-dimensional complex vector space is the sum of two cocycles, its even…
We study holomorphic Poisson manifolds and holomorphic Lie algebroids from the viewpoint of real Poisson geometry. We give a characterization of holomorphic Poisson structures in terms of the Poisson Nijenhuis structures of Magri-Morosi and…
We recover some recent results by Dotsenko, Shadrin and Vallette on the Deligne groupoid of a pre-Lie algebra, showing that they follow naturally by a pre-Lie variant of the PBW Theorem. As an application, we show that Kapranov's $L_\infty$…
The space of Lie algebra cohomology is usually described by the dimensions of components of certain degree even for the adjoint module as coefficients when the spaces of cochains and cohomology can be endowed with a Lie superalgebra…
We look at the Poisson structure on the total space of the dual bundle to the Lie algebroid arising from a matched pair of Lie groups. This dual bundle, with the natural semidirect product group structure, becomes a Poisson-Lie group as…
This paper is about the relation of the geometry of Lie groupoids over a fixed compact manifold and the geometry of their (infinite-dimensional) bisection Lie groups. In the first part of the paper we investigate the relation of the…
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…
We study the classical and quantum $G$ extended superconformal algebras from the hamiltonian reduction of affine Lie superalgebras, with even subalgebras $G\oplus sl(2)$. At the classical level we obtain generic formulas for the Poisson…
We introduce a family of compatible Poisson brackets on the space of $2\times 2$ polynomial matrices, which contains the reflection equation algebra bracket. Then we use it to derive a multi-Hamiltonian structure for a set of integrable…
In the first part of our paper (Sections 1, 2 and 3) we reprove results of B. Plotkin, G. Zhitomirski. On automorphisms of categories of free algebras of some varieties, Journal of Algebra, 306:2, (2006), 344 -- 367 for the case of…
A graded poset structure is defined for the sets of Littlewood-Richardson (LR) tableaux that count the multiplicity of an irreducible GL(n)-module in the tensor product of irreducibles indexed by a sequence of rectangular partitions. This…
In this paper, we study invariant Poisson structures on homogeneous manifolds, which serve as a natural generalization of homogeneous symplectic manifolds previously explored in the literature. Our work begins by providing an algebraic…
It is shown that the non-trivial cocycles on simple Lie algebras may be used to introduce antisymmetric multibrackets which lead to higher-order Lie algebras, the definition of which is given. Their generalised Jacobi identities turn out to…
We point out, and draw some consequences of, the fact that the Poisson Lie group G* dual to G=GL_n(C) (with its standard complex Poisson structure) may be identified with a certain moduli space of meromorphic connections on the unit disc…
This is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable properties and which are expected to play a major role in Representation theory. The…
In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular…
Earlier an arbitrary poset $P$ was proved to be isomorphic to the collection of subsets of a space $M$ with two closures which are closed in the first closure and open in the other. As a space $M$ for this representation an algebraic dual…
N-Lie algebra structures on smooth function algebras given by means of multi-differential operators, are studied. Necessary and sufficient conditions for the sum and the wedge product of two $n$-Poisson sructures to be again a multi-Poisson…
The homology of free Lie algebras with coefficients in tensor products of the adjoint representation working over Q contains important information on the homological properties of polynomial outer functors on free groups. The latter…