Deformation quantization of Poisson structures associated to Lie algebroids.
Neumaier, Nikolai, Waldmann, Stefan (2009)
SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]
Marius Crainic, Ieke Moerdijk (2008)
Journal of the European Mathematical Society
We introduce a new cohomology for Lie algebroids, and prove that it provides a differential graded Lie algebra which “controls” deformations of the structure bracket of the algebroid.
Alexander B. Goncharov, Richard Kenyon (2013)
Annales scientifiques de l'École Normale Supérieure
We show that the dimer model on a bipartite graph on a torus gives rise to a quantum integrable system of special type, which we call acluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space of line bundles with connections on the graph . The sum of Hamiltonians is essentially the partition function of the dimer model. We say that two such graphs and areequivalentif the Newton polygons of the corresponding partition functions...
Zhang-Ju Liu, Ping Xu (2001)
Annales de l’institut Fourier
The purpose of this paper is to establish a connection between various objects such as dynamical -matrices, Lie bialgebroids, and Lagrangian subalgebras. Our method relies on the theory of Dirac structures and Courant algebroids. In particular, we give a new method of classifying dynamical -matrices of simple Lie algebras , and prove that dynamical -matrices are in one-one correspondence with certain Lagrangian subalgebras of .
Ping Xu (2003)
Annales scientifiques de l'École Normale Supérieure
Arnlind, Joakim, Hoppe, Jens (2010)
SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]
Yvette Kosmann-Schwarzbach, Juan Monterde (2002)
Annales de l’institut Fourier
We define the divergence operators on a graded algebra, and we show that, given an odd Poisson bracket on the algebra, the operator that maps an element to the divergence of the hamiltonian derivation that it defines is a generator of the bracket. This is the “odd laplacian”, , of Batalin-Vilkovisky quantization. We then study the generators of odd Poisson brackets on supermanifolds, where divergences of graded vector fields can be defined either in terms of berezinian volumes or of graded connections. Examples...
Henrique Bursztyn, Olga Radko (2003)
Annales de l’institut Fourier
We study gauge transformations of Dirac structures and the relationship between gauge and Morita equivalences of Poisson manifolds. We describe how the symplectic structure of a symplectic groupoid is affected by a gauge transformation of the Poisson structure on its identity section, and prove that gauge-equivalent integrable Poisson structures are Morita equivalent. As an example, we study certain generic sets of Poisson structures on Riemann surfaces: we find complete gauge-equivalence invariants...
D. Iglesias-Ponte, J. C. Marrero (2002)
Extracta Mathematicae
Fernand Pelletier, Patrick Cabau (2008)
Banach Center Publications
The notion of generalized PN manifold is a framework which allows one to get properties of first integrals of the associated bihamiltonian system: conditions of existence of a bi-abelian subalgebra obtained from the momentum map and characterization of such an algebra linked with the problem of separation of variables.
Azzouz Awane (2008)
Revista Matemática Complutense
Claude Roger (2009)
Archivum Mathematicum
We shall give a survey of classical examples, together with algebraic methods to deal with those structures: graded algebra, cohomologies, cohomology operations. The corresponding geometric structures will be described(e.g., Lie algebroids), with particular emphasis on supergeometry, odd supersymplectic structures and their classification. Finally, we shall explain how BV-structures appear in Quantum Field Theory, as a version of functional integral quantization.
Zhuo Chen, Daniele Grandini, Yat-Sun Poon (2015)
Complex Manifolds
Holomorphic Poisson structures arise naturally in the realm of generalized geometry. A holomorphic Poisson structure induces a deformation of the complex structure in a generalized sense, whose cohomology is obtained by twisting the Dolbeault @-operator by the holomorphic Poisson bivector field. Therefore, the cohomology space naturally appears as the limit of a spectral sequence of a double complex. The first sheet of this spectral sequence is simply the Dolbeault cohomology with coefficients in...
Pickrell, Doug (2008)
SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]
Marius Crainic, Chenchang Zhu (2007)
Annales de l’institut Fourier
We discuss the integrability of Jacobi manifolds by contact groupoids, and then look at what the Jacobi point of view brings new into Poisson geometry. In particular, using contact groupoids, we prove a Kostant-type theorem on the prequantization of symplectic groupoids, which answers a question posed by Weinstein and Xu. The methods used are those of Crainic-Fernandes on -paths and monodromy group(oid)s of algebroids. In particular, most of the results we obtain are valid also in the non-integrable...
Pantelis A. Damianou, Rui Loja Fernandes (2008)
Annales de l’institut Fourier
It is well-known that the Poisson-Nijenhuis manifolds, introduced by Kosmann-Schwarzbach and Magri form the appropriate setting for studying many classical integrable hierarchies. In order to define the hierarchy, one usually specifies in addition to the Poisson-Nijenhuis manifold a bi-hamiltonian vector field. In this paper we show that to every Poisson-Nijenhuis manifold one can associate a canonical vector field (no extra choices are involved!) which under an appropriate assumption defines an...
Eva Miranda (2014)
Open Mathematics
The main purpose of this paper is to present in a unified approach to different results concerning group actions and integrable systems in symplectic, Poisson and contact manifolds. Rigidity problems for integrable systems in these manifolds will be explored from this perspective.
Janusz Grabowski (2000)
Banach Center Publications
We present a general theorem describing the isomorphisms of the local Lie algebra structures on the spaces of smooth (real-analytic or holomorphic) functions on smooth (resp. real-analytic, Stein) manifolds, as, for example, those given by Poisson or contact structures. We admit degenerate structures as well, which seems to be new in the literature.
Ben Amar, Nabiha (2003)
Journal of Lie Theory
Vaisman, Izu (2003)
Balkan Journal of Geometry and its Applications (BJGA)