Related papers: Dirac brackets and reduction of invariant bi-Poiss…

The main aim of this paper is the description of a large class of lattices in some nilpotent Lie groups, sometimes filiformes, carrying a flat left invariant linear connection anf often a left invariant symplectic form. As a consequence we…

Let $G$ be a Lie group, with an invariant non-degenerate symmetric bilinear form on its Lie algebra, let $\pi$ be the fundamental group of an orientable (real) surface $M$ with a finite number of punctures, and let $\bold C$ be a family of…

We prove that for any known Lie algebra $\frak{g}$ having none invariants for the coadjoint representation, the absence of invariants is equivalent to the existence of a left invariant exact symplectic structure on the corresponding Lie…

We develop a set of sufficient conditions for guaranteeing that an integrable system with a symmetry group $K$ on a manifold $M$ descends to an integrable system on a dense open subset of the quotient Poisson space $M/K$. The higher…

Let $X$ be a smooth, compact, oriented $4$-manifold. Building upon work of Li-Liu, Ruberman, Nakamura and Konno, we consider a families version of Seiberg-Witten theory and obtain obstructions to the existence of certain group actions on…

We introduce a gauge-theoretic integer lift of the Rohlin invariant of a smooth 4-manifold X with the homology of $S^1 \times S^3$. The invariant has two terms; one is a count of solutions to the Seiberg-Witten equations on X, and the other…

The purpose of this paper is to investigate shifted $(+1)$ Poisson structures in context of differential geometry. The relevant notion is shifted $(+1)$ Poisson structures on differentiable stacks. More precisely, we develop the notion of…

We construct explicitly a class of coboundary Poisson-Lie structures on the group of formal diffeomorphisms of ${\Bbb R}^n$. Equivalently, these give rise to a class of coboundary triangular Lie bialgebra structures on the Lie algebra $W_n$…

Let $X=G/\Gamma$ be the quotient of a semisimple Lie group $G$ by its non-cocompact arithmetic lattice. Let $H$ be a reductive algebraic subgroup of $G$ acting on $X$. We give several equivalent algebraic conditions on $H$ for the existence…

A study is made of left-invariant $\mathrm{G}_2$-structures with an exact 3-form on a Lie group $G$ whose Lie algebra $\mathfrak{g}$ admits a codimension-one nilpotent ideal $\mathfrak{h}$. It is shown that such a Lie group $G$ cannot admit…

Let $\mathcal{G}$ be a bundle gerbe with connection on a smooth manifold $M$, and let $\rho: G \rightarrow \operatorname{Diff}(M)$ be a smooth action of a Fr\'echet--Lie group $G$ on $M$ that preserves the isomorphism class of…

A Theorem due to Guillemin and Sternberg about geometric quantization of Hamiltonian actions of compact Lie groups $G$ on compact Kaehler manifolds says that the dimension of the $G$-invariant subspace is equal to the Riemann-Roch number of…

Given a Lie-Poisson completely integrable bi-Hamiltonian system on $\mathbb{R}^n$, we present a method which allows us to construct, under certain conditions, a completely integrable bi-Hamiltonian deformation of the initial Lie-Poisson…

The special structures that arise in symplectic topology (particularly Gromov--Witten invariants and quantum homology) place as yet rather poorly understood restrictions on the topological properties of symplectomorphism groups. This…

We presented a Hilbert-Mumford criterion for polystablility associated with an action of a real reductive Lie group $G$ on a real submanifold $X$ of a Kahler manifold $Z$. Suppose the action of a compact Lie group with Lie algebra…

The assumption in the main result of [Peter W. Michor: Basic Differential Forms for Actions of Lie Groups, Proc. AMS 124, 5 (1996) 1633-1642] is removed. Thus: A section of a Riemannian $G$-manifold $M$ is a closed submanifold $\Si$ which…

We show that generic symplectic quotients of a Hamiltonian $G$-space $M$ by the action of a compact connected Lie group $G$ are also symplectic quotients of the same manifold $M$ by a compact torus. The torus action in question arises from…

For a connected abelian Lie group T acting on a Poisson manifold (Y,{\pi}) by Poisson isomorphisms, the T-leaves of {\pi} in Y are, by definition, the orbits of the symplectic leaves of {\pi} under T, and the leaf stabilizer of a T-leaf is…

Given a Poisson structure (or, equivalently, a Hamiltonian operator) $P$, we show that its Lie derivative $L_{\tau}(P)$ along a vector field $\tau$ defines another Poisson structure, which is automatically compatible with $P$, if and only…

Let $\Sigma$ be a closed surface, $G$ a compact Lie group, with Lie algebra $g$, and $\xi \colon P \to \Sigma$ a principal $G$-bundle. In earlier work we have shown that the moduli space $N(\xi)$ of central Yang- Mills connections, for…