arXiv research

New Lie groups found for Poisson diffeomorphisms.

The paper explores the connection between Poisson-Lie structures and invariant volume forms in Hamiltonian dynamics.

We show that for any coboundary Poisson Lie group G, the Poisson structure on G^* is linearizable at the group unit. This strengthens a result of Enriquez-Etingof-Marshall, who had established formal linearizability of G^* for quasi-triangular Poisson Lie groups G. We also prove linearizability properties for the group…

We introduce and study some mixed product Poisson structures on product manifolds associated to Poisson Lie groups and Lie bialgebras. For quasitriangular Lie bialgebras, our construction is equivalent to that of fusion products of quasi-Poisson G-manifolds introduced by Alekseev, Kosmann- Schwarzbach, and Meinrenken. …

Poisson sigma models represent an interesting use of Poisson manifolds for the construction of a classical field theory. Their definition in the language of fibre bundles is shown and the corresponding field equations are derived using a coordinate independent variational principle. The elegant form of equations of mot…

The notions of \emph{Poisson Lie group} and \emph{Poisson homogeneous space} are extended to the Dirac category. The theorem of Drinfel$'$d (\cite{Drinfeld93}) on the one-to-one correspondence between Poisson homogeneous spaces of a Poisson Lie group and a special class of Lagrangian subalgebras of the Lie bialgebra as…

A Riemann-Lie algebra is a Lie algebra $\cal G$ such that its dual ${\cal G}^*$ carries a Riemannian metric compatible (in the sense introduced by th author in C. R. Acad. Paris, t. 333, Série I, (2001) 763-768) with the canonical linear Poisson sructure of ${\cal G}^*$. The notion of Riemann-Lie algebra has its origin…

New structure on unitary group of Hilbert space.

A Poisson-Lie group acting by the coadjoint action on the dual of its Lie algebra induces on it a non-trivial class of quadratic Poisson structures extending the linear Poisson bracket on the coadjoint orbits.

We propose a Poisson-Lie analog of the symplectic induction procedure, using an appropriate Poisson generalization of the reduction of symplectic manifolds with symmetry. Having as basic tools the equivariant momentum maps of Poisson actions, the double group of a Poisson-Lie group and the reduction of Poisson manifold…

The paper defines projective structures for Lie bialgebras and Poisson-Lie groups.

New cohomology groups generalize Euler number for Lie superalgebras.

Let $(G,Π_{G},\tilde{g})$ be a Poisson-Lie group equipped with a left invariant pseudo-Riemannian metric $\tilde{g}$ and let $(TG,Π_{TG},\tilde{g}^{c})$ be the Sanchez de Alvarez tangent Poisson-Lie group of $G$ equipped with the left invariant pseudo-Riemannian metric $\tilde{g}^{c}$, complete lift of $\tilde{g}$. In …

A Lie group G in a group pair (D,G), integrating a Lie algebra g in a Manin pair (d,g) has a quasi-Poisson structure. We define the quasi-Poisson actions of such Lie groups G, that generalize the Poisson actions of Poisson Lie groups. We define and study the moment maps for those quasi-Poisson actions which are quasi-h…

Symplectic manifolds which are homogeneous spaces of Poisson-Lie groups are studied in this paper. We show that these spaces are, under certain assumptions, covering spaces of dressing orbits of the Poisson-Lie groups which act on them. The effect of the Poisson induction procedure on such spaces is also examined, thus…

It is shown that the cotangent bundle of a matched pair Lie group is itself a matched pair Lie group. The trivialization of the cotangent bundle of a matched pair Lie group are presented. On the trivialized space, the canonical symplectic two-form and canonical Poisson bracket are explicitly written. Various symplectic…

A Riemann-Poisson Lie group is a Lie group endowed with a left invariant Riemannian metric and a left invariant Poisson tensor which are compatible in the sense introduced in C.R. Acad. Sci. Paris sér. {\bf I 333} (2001) 763-768. We study these Lie groups and we give a characterization of their Lie algebras. We give al…

We establish a 1:1 correspondence between Poisson-Lie group actions on integrable Poisson manifolds and twisted multiplicative hamiltonian actions on source 1-connected symplectic groupoids. For an action of a Poisson-Lie group $G$ on a Poisson manifold $M$, we find an explicit description of the lifted hamiltonian act…

Describes reconstructing Poisson structures from Lie group actions.

We investigate some infinite dimensional Lie algebras and their associated Poisson structures which arise from a Lie group action on a manifold. If $G$ is a Lie group, $\g$ its Lie algebra and $M$ is a manifold on which $G$ acts, then the set of smooth maps from $M$ to $\g$ has at least two Lie algebra structures, both…

We prove a 2-categorical analogue of a classical result of Drinfeld: there is a one-to-one correspondence between connected, simply-connected Poisson Lie 2-groups and Lie 2-bialgebras. In fact, we also prove that there is a one-to-one correspondence between connected, simply connected quasi-Poisson 2-groups and quasi-L…

We discuss relations between linear Nambu-Poisson structures and Filippov algebras and define Filippov algebroids which are n-ary generalizations of Lie algebroids. We also prove results describing multiplicative Nambu- Poisson structures on Lie groups. In particular, we show that simple Lie groups do not admit multipl…

Poisson actions of Poisson Lie groups have an interesting and rich geometric structure. We will generalize some of this structure to Dirac actions of Dirac Lie groups. Among other things, we extend a result of Jiang-Hua-Lu, which states that the cotangent Lie algebroid and the action algebroid for a Poisson action form…

In this paper we develope a theory of reduction for classical systems with Poisson Lie groups symmetries using the notion of momentum map introduced by Lu. The local description of Poisson manifolds and Poisson Lie groups and the properties of Lu's momentum map allow us to define a Poisson reduced space.

Paper connects Stokes phenomena to quantum groups and Poisson-Lie groups.

We provide an alternative method for obtaining of compatible Poisson structures on Lie groups by means of the adjoint representations of Lie algebras. In this way, we calculate some compatible Poisson structures on four dimensional and nilpotent six dimensional symplectic real Lie groups. Then using Magri-Morosi's theo…

We identify the cotangent bundle Lie algebroid of a Poisson homogeneous space G/H of a Poisson Lie group G as a quotient of a transformation Lie algebroid over G. As applications, we describe the modular vector fields of G/H, and we identify the Poisson cohomology of G/H with coefficients in powers of its canonical lin…

Let $G_¶$ be a compact simple Poisson-Lie group equipped with a Poisson structure $¶$ and $(M, ø)$ be a symplectic manifold. Assume that $M$ carries a Poisson action of $G_¶$ and there is an equivariant moment map in the sense of Lu and Weinstein which acts to the dual Poisson-Lie group $G^*_¶$, $\m: M\rightarrow G^*_¶…

The paper explores geometric and algebraic structures on Lie groups.

In this paper we study left invariant CR structures on Lie groups which are compatible with geometric properties as Poisson and kahler properties.

The study investigates linearizability of Poisson structures on groupoids.

Using the basic Lie symmetry method, we find the most general Lie point symmetries group of the $\nabla u=f(u)$ Poisson's equation, which has a subalgebra isomorphic to the $3-$dimensional special Euclidean group ${\rm SE}(3)$ or group of rigid motions of ${\Bbb R}^3$. Looking the adjoint representation of ${\rm SE}(3)…

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 having an irregular singularity at the origin. The Riemann-Hilbert map for such co…

This work is devoted to the study of a class of Poisson-Lie groups endowed with left invariant metrics. The triples $(G,π,<,>)$ are considered, where $G$ is a simply connected Lie group, ?$π$ is a multiplicative Poisson tensor and $<,>$ is a left invariant riemannian metric such that Hawkins conditions are satisfied. H…

In this paper we prove rigidity theorems for Poisson Lie group actions on Poisson manifolds. In particular, we prove that close infinitesimal momentum maps associated to Poisson Lie group actions are equivalent using a normal form theorem for SCI spaces. When the Poisson structure of the acted manifold is integrable, t…

Let $G$ be a complex reductive connected algebraic group equipped with the Sklyanin bracket. A classification of Poisson homogeneous $G$-spaces with connected isotropy subgroups is given. This result is based on Drinfeld's correspondence between Poisson homogeneous $G$-spaces and Lagrangian subalgebras in the double $D…

A $n$-dimensional Lie group $G$ equipped with a left invariant symplectic form $\om^+$ is called a symplectic Lie group. It is well-known that $\om^+$ induces a left invariant affine structure on $G$. Relatively to this affine structure we show that the left invariant Poisson tensor $π^+$ corresponding to $\om^+$ is po…

Lie bialgebra structures are reviewed and investigated in terms of the double Lie algebra, of Manin- and Gauß-decompositions. The standard R-matrix in a Manin decomposition then gives rise to several Poisson structures on the correponding double group, which is investigated in great detail.

A well known result of Drinfeld classifies Poisson Lie groups $(H,Π)$ in terms of Lie algebraic data in the form of Manin triples $(\mathfrak{d},\mathfrak{g},\mathfrak{h})$; he also classified compatible Poisson structures on $H$-homogeneous spaces $H/K$ in terms of Lagrangian subalgebras $\mathfrak{l}\subset\mathfrak{…

This work is motivated by a result of Drinfeld on Poisson homogeneous spaces. For each Poisson manifold $P$ with a Poisson action by a Poisson Lie group $G$, we describe a Lie algebroid structure on the direct sum vector bundle $P \times {\frak g} \oplus T^*P$, where ${\frak g}$ is the Lie algebra of $G$. It is built o…

We provide a quasi-Poisson version of the Drinfeld's correspondence between Poisson homogeneous spaces and Lagrangian subalgebras.

Groupoids are mathematical structures able to describe symmetry properties more general than those described by groups. They were introduced (and named) by H. Brandt in 1926. Around 1950, Charles Ehresmann used groupoids with additional structures (topological and differentiable) as essential tools in topology and diff…

A group, defined as set with associative multiplication and inverse, is a natural structure describing the symmetry of a space. The concept of group generalizes to group objects internal to other categories than sets. But there are yet more general objects that can still be thought of as groups in many ways, such as qu…

Study exact Lie bialgebras from flat Lie groups, classifying them.

For a particular class of backgrounds, equations of motion for string sigma models targeted in mutually dual Poisson-Lie groups are equivalent. This phenomenon is called the Poisson-Lie T-duality. On the level of the corresponding string effective actions, the situation becomes more complicated due to the presence of t…

Introduces $b$-Lie groups and studies their symplectic structures and reductions.

The paper defines a new Lie groupoid concept for infinite dimensions.

New star-product defined on Poisson manifolds using Toeplitz operators.