Related papers: Linear Poisson structures on R^4

We develop here a simple quantisation formalism that make use of Lie algebra properties of the Poisson bracket. When the brackets $\{H,\phi_i\}$ and $\{\phi_i,\phi_j\}$, where $H$ is the Hamiltonian and $\phi_i$ are primary and secondary…

Let L be a finite dimensional Lie algebra over an algebraically closed field k of characteristic zero. We provide necessary and also some sufficient conditions in order for its Poisson center and semi-center to be polynomial algebras over…

We compute $\frac{1}{2}$-derivations on the deformed generalized Heisenberg-Virasoro algebras and on not-finitely graded Heisenberg-Virasoro algebras $\widehat{W}_n(G)$, $\widetilde{W}_n(G)$, and $\widetilde{HW}_n(G)$. We classify all…

All results concern characteristic 2. Two procedures that to every simple Lie algebra assign simple Lie superalgebras, most of the latter new, are offered. We prove that every simple finite-dimensional Lie superalgebra is obtained as the…

In this paper we introduce poly-Poisson structures as a higher-order extension of Poisson structures. It is shown that any poly-Poisson structure is endowed with a polysymplectic foliation. It is also proved that if a Lie group acts…

We describe the automorphism groups of elliptic Poisson algebras on polynomial algebras in three variables and give an explicit set of generators and defining relations for this group.

We study higher-degree generalizations of symplectic groupoids, referred to as {\em multisymplectic groupoids}. Recalling that Poisson structures may be viewed as infinitesimal counterparts of symplectic groupoids, we describe "higher''…

We show that the quantisation of a connected simply-connected Poisson-Lie group admits a left-covariant noncommutative differential structure at lowest deformation order if and only if the dual of its Lie algebra admits a pre-Lie algebra…

A study is made of real Lie algebras admitting a hypersymplectic structure, and we provide a method to construct such hypersymplectic Lie algebras. We use this method in order to obtain the classification of all hypersymplectic structures…

We show that every closed oriented smooth 4-manifold admits a complete singular Poisson structure in each homotopy class of maps to the 2-sphere. The rank of this structure is 2 outside a small singularity set, which consists of finitely…

In this work we study a particular class of Lie bialgebras arising from Hermitian structures on Lie algebras such that the metric is ad-invariant. We will refer to them as Lie bialgebras of complex type. These give rise to Poisson Lie…

We develop a theory of noncommutative Poisson extensions. For an augmented dg algebra \(A\), we show that any shifted double Poisson bracket on \(A\) induces a graded Lie algebra structure on the reduced cyclic homology. Under the…

We compute $\frac{1}{2}$-derivations on the deformative Schr\"{o}dinger-Witt algebra, on not-finitely graded Witt algebras $W_n(G)$, and on not-finitely graded Heisenberg-Witt algebra $HW_n(G)$. We classify all transposed Poisson structures…

We study the shifted Poisson structure on the cochain complex C*(g) of a graded Lie algebra arising from shifted Lie bialgebra structure on g. We apply this to construct a 1-shifted Poisson structures on an infinitesimal quotient of the…

A family of Poisson structures, parametrised by an arbitrary odd periodic function $\phi$, is defined on the space $\cW$ of twisted polygons in $\RR^\nu$. Poisson reductions with respect to two Poisson group actions on $\cW$ are described.…

Object of investigation are almost hypercomplex manifolds with Hermitian-Norden metrics of the lowest dimension. The considered manifolds are constructed on 4-dimensional Lie groups. It is established a relation between the classes of a…

We introduce Poisson triple systems, which are vector spaces with 3 trilinear operations satisfying 9 polynomial identities of degree 5. We show that every Poisson triple system has a universal enveloping Poisson algebra. Finally, we…

We classify real Poisson structures on complex toric manifolds of type $(1,1)$ and initiate an investigation of their Poisson cohomology. For smooth toric varieties, such structures are necessarily algebraic and are homogeneous quadratic in…

We develop a new approach to deformation quantizations of Lie bialgebras and Poisson structures which goes in two steps. In the first step one associates to any Poisson (resp. Lie bialgebra) structure a so called quantizable Poisson (resp.…

The automorphisms of all 4-dimensional, real Lie Algebras are presented in a comprehensive way. Their action on the space of $4\times 4$, real, symmetric and positive definite, matrices, defines equivalence classes which are used for the…