Yvette Kosmann-Schwarzbach

Exact Gerstenhaber algebras and Lie bialgebroids

We show that to any Poisson manifold and, more generally, to any triangular Lie bialgebroid in the sense of Mackenzie and Xu, there correspond two differential Gerstenhaber algebras in duality, one of which is canonically equipped with an operator generating the graded Lie algebra bracket, i.e. with the structure of a Batalin-Vilkovisky algebra.

Quasi-Poisson manifolds

Abstract. A quasi-Poisson manifold is a G-manifold equipped with an invariant bivector field whose Schouten bracket is the trivector field generated by the invariant element in ∧3g associated to an invariant inner product. We introduce the concept of the fusion of such manifolds, and we relate the quasi-Poisson manifolds to the previously introduced quasi-Hamiltonian manifolds with groupvalued moment maps.

Quasi, twisted, and all that... in Poisson geometry and Lie algebroid theory

Motivated by questions from quantum group and field theories, we review structures on manifolds that are weaker versions of Poisson structures, and variants of the notion of Lie algebroid. We give a simple definition of the Courant algebroids and introduce the notion of a deriving operator for the Courant bracket of the double of a proto-bialgebroid. We then describe and relate the various quasi-Poisson structures, which have appeared in the literature since 1991, and the twisted Poisson structures studied by Severa and Weinstein.

The lie bialgebroid of a Poisson-Nijenhuis manifold

We describe a new class of Lie bialgebroids associated with Poisson-Nijenhuis structures.

The Noether Theorems

This chapter will deal briefly with the results stated and proved by Noether in the Invariante Variationsprobleme1 [1918c]. Her originality in this article consisted in dealing with problems that arose either in classical mechanics (the first theorem) or in general relativity (the second theorem). We emphasize what has been ignored by most authors who have cited this article, that in it Noether treated a problem of very great generality, since she dealt with a Lagrangian of arbitrary order with an arbitrary number of independent variables,2 as well as an arbitrary number of dependent variables, and considered the invariance of such Lagrangians under the action of “groups of infinitesimal transformations.”

Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey ?

After a brief summary of the main properties of Poisson manifolds and Lie algebroids in general, we survey recent work on the modular classes of Poisson and twisted Poisson manifolds, of Lie algebroids with a Poisson or twisted Poisson structure, and of Poisson-Nijenhuis manifolds. A review of the spinor approach to the modular class concludes the paper.

The double of a Jacobian quasi-bialgebra

We show that the construction of the double of a Lie bialgebra can be extended to the case where a vector space is only equipped with the structure of a Jacobian quasi-bialgebra (also called a Lie quasi-bialgebra). In this case, the double is itself a Jacobian quasi-bialgebra and it is quasi-triangular. The more general case of the double of a proto-Lie bialgebra is also discussed. In the first section, the notions of exact, strictly exact, quasi-triangular and triangular Jacobian quasi-bialgebras are defined and their equivalence classes under twisting are studied.

Poisson and Symplectic Functions in Lie Algebroid Theory

Emphasizing the role of Gerstenhaber algebras and of higher derived brackets in the theory of Lie algebroids, we show that the several Lie algebroid brackets which have been introduced in the recent literature can all be defined in terms of Poisson and pre-symplectic functions in the sense of Roytenberg and Terashima. We prove that in this very general framework there exists a one-to-one correspondence between nondegenerate Poisson functions and symplectic functions. We also determine the differential associated to a Lie algebroid structure obtained by twisting a structure with background by both a Lie bialgebra action and a Poisson bivector.

Modular Classes of Regular Twisted Poisson Structures on Lie Algebroids

We derive a formula for the modular class of a Lie algebroid with a regular twisted Poisson structure in terms of a canonical Lie algebroid representation of the image of the Poisson map. We use this formula to compute the modular classes of Lie algebras with a twisted triangular r-matrix. The special case of r-matrices associated to Frobenius Lie algebras is also studied.

Quantum homogeneous spaces and quasi-Hopf algebras

We propose a formulation of the quantization problem of Manin quadruples, and show that a solution to this problem yields a quantization of the corresponding Poisson homogeneous spaces. We then solve both quantization problems in an example related to quantum spheres.