Related papers: Low dimensional bihamiltonian structures of topolo…

In analogy to the KP theory, the second Poisson structure for the dispersionless KP hierarchy can be defined on the space of commutative pseudodifferential operators $L=p^n+\sum_{j=-\infty}^{n-1}u_j p^j$. The reduction of the Poisson…

We study symplectic forms on hypersurface algebroids. These are a broad generalization of the $b^{k}$-Poisson structures studied extensively by Miranda, Scott, and collaborators, and their geometry is intimately related to the group of…

The purpose of this work is to study Lie superalgebroid structures on the space of superdifferential $1$-forms over the supermanifolds whose superfunctions are the differential forms on its underlying manifold. These superalgbroids are…

In this paper, we study compatible Leibniz algebras. We characterize compatible Leibniz algebras in terms of Maurer-Cartan elements of a suitable differential graded Lie algebra. We define a cohomology theory of compatible Leibniz algebras…

We make a study of Poisson structures of T*M which are graded structures when restricted to the fiberwise polynomial algebra, and give examples. A class of more general graded bivector fields which induce a given Poisson structure w on the…

This paper is devoted to a systematic study and classification of invariant affine or metric connections on certain classes of naturally reductive spaces. For any non-symmetric, effective, strongly isotropy irreducible homogeneous…

We formulate and prove examples of a conjecture which describes the W-algebras in type A as successive quantum Hamiltonian reductions of affine vertex algebras associated with several hook-type nilpotent orbits. This implies that the affine…

Berwick-Evens and Lerman recently showed that the category of vector fields on a geometric stack has the structure of a Lie $2$-algebra. Motivated by this work, we present a construction of graded weak Lie $2$-algebras associated with…

We develop the theory of Poisson and Dirac manifolds of compact types, a broad generalization in Poisson and Dirac geometry of compact Lie algebras and Lie groups. We establish key structural results, including local normal forms, canonical…

We consider the WDVV associativity equations in the four dimensional case. These nonlinear equations of third order can be written as a pair of six component commuting two-dimensional non-diagonalizable hydrodynamic type systems. We prove…

We show that a bi-flat F-structure $(\nabla,\circ,e,\nabla^*,*,E)$ on a manifold $M$ defines a differential bicomplex $(d_{\nabla},d_{E\circ\nabla^*})$ on forms with value on the tangent sheaf of the manifold. Moreover, the sequence of…

Using spectral sequences techniques we compute the bihamiltonian cohomology groups of the pencil of Poisson brackets of dispersionless KdV hierarchy. In particular this proves a conjecture of Liu and Zhang about the vanishing of such…

We construct lattices on six dimensional not completely solvable almost abelian Lie groups, for which the Mostow condition does not hold. For the corresponding compact quotients, we compute the de Rham cohomology (which does not agree in…

One of the four well-known series of simple Lie algebras of Cartan type is the series of Lie algebras of Special type, which are divergence-free Lie algebras associated with polynomial algebras and the operators of taking partial…

We generalize to the homotopy case a result of K. Mackenzie and P. Xu on relation between Lie bialgebroids and Poisson geometry. For a homotopy Poisson structure on a supermanifold $M$, we show that $(TM, T^*M)$ has a canonical structure of…

A family of naturally reductive pseudo-Riemannian spaces is constructed out of the representations of Lie algebras with ad-invariant metrics. We exhibit peculiar examples, study their geometry and characterize the corresponding naturally…

We introduce a class of potential submanifolds in pseudo-Euclidean spaces (each N-dimensional potential submanifold is a special flat torsionless submanifold in a 2N-dimensional pseudo-Euclidean space) and prove that each N-dimensional…

We consider the Hamiltonian structure of reduced fluid models obtained from a kinetic description of collisionless plasmas by Vlasov-Maxwell equations. We investigate the possibility of finding Poisson subalgebras associated with fluid…

Results about the following classes of finite-dimensional Lie algebras over a field of characteristic zero are presented: anisotropic (i.e., Lie algebras for which each adjoint operator is semisimple), regular (i.e., Lie algebras in which…

We introduce a Lie bialgebra structure on the central extension of the Lie algebra of differential operators on the line and the circle (with scalar or matrix coefficients). This defines a Poisson--Lie structure on the dual group of…