Generalized Lie bialgebroids and Jacobi structures

Abstract

The notion of a generalized Lie bialgebroid (a generalization of the notion of a Lie bialgebroid) is introduced in such a way that a Jacobi manifold has associated a canonical generalized Lie bialgebroid. As a kind of converse, we prove that a Jacobi structure can be defined on the base space of a generalized Lie bialgebroid. We also show that it is possible to construct a Lie bialgebroid from a generalized Lie bialgebroid and, as a consequence, we deduce a duality theorem. Finally, some special classes of generalized Lie bialgebroids are considered: triangular generalized Lie bialgebroids and generalized Lie bialgebras.

32 pages

References in corpus (3)

• Tangent Lifts of Poisson and Related Structures
• Generalized Lie bialgebroids and Jacobi structures
• Some linear Jacobi structures on vector bundles

Cited by in corpus (52)

• Generalized Lie bialgebroids and Jacobi structures
• Graded contact manifolds and contact Courant algebroids
• The graded Jacobi algebras and (co)homology
• Contact manifolds and generalized complex structures
• AV-differential geometry: Poisson and Jacobi structures
• Lie algebroid structures and Lagrangian systems on affine bundles
• Some linear Jacobi structures on vector bundles
• Remarks on Contact and Jacobi Geometry
• Differential calculus on a Lie algebroid and Poisson manifolds
• Integration of Dirac-Jacobi structures
• Differential graded contact geometry and Jacobi structures
• Twisted Jacobi manifolds, twisted Dirac-Jacobi structures and quasi-Jacobi bialgebroids
• Lie brackets on affine bundles
• Infinitesimal Automorphisms of VB-groupoids and algebroids
• Lie algebroid foliations and ${\mathcal{E}}^1(M)$-Dirac structures
• Poisson-Jacobi reduction of homogeneous tensors
• E-Courant algebroids
• Jacobi-Lie symmetry and Jacobi-Lie T-dual sigma models on group manifolds
• Jacobi bundles and the BFV-complex
• Gauged sigma-models with nonclosed 3-form and twisted Jacobi structures
• Jacobi-Lie T-plurality
• Classification of real low dimensional Jacobi(generalized)-Lie bialgebras
• Jacobi-Nijenhuis algebroids and their modular classes
• Classical r-matrices of real low-dimensional Jacobi-Lie bialgebras and their Jacobi-Lie groups
• On quasi-Jacobi and Jacobi-quasi bialgebroids
• Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review
• On Jacobi quasi-Nijenhuis algebroids and Courant-Jacobi algebroid morphisms
• Poisson-Nijenhuis groupoids
• Frame-independent mechanics:geometry on affine bundles
• Jacobi Structures of Evolutionary Partial Differential Equations
• Characteristic foliation of twisted Jacobi manifolds
• Generalized Lie bialgebras and Jacobi structures on Lie groups
• Odd Jacobi manifolds: general theory and applications to generalised Lie algebroids
• Jacobi quasi-Nijenhuis Algebroids
• The Schroedinger operator as a generalized Laplacian
• Jacobi-Lie symmetry in WZW model on the Heisenberg Lie group $H_4$
• Jacobi algebroids and quasi Q-manifolds
• Homology and modular classes of Lie algebroids
• Higher omni-Lie algebroids
• Jacobi Geometry and Hamiltonian Mechanics: the Unit-Free Approach
• Dirac structures for generalized Courant and Courant algebroids
• Gauge transformations of Jacobi structures and contact groupoids
• Jacobi groupoids and generalized Lie bialgebroids
• On characterization of Poisson and Jacobi structures
• On Generalized Lie Bialgebroids
• On a Lichnerowicz type cohomology attached to a function
• On twisted contact groupoids and on integration of twisted Jacobi manifolds
• Jacobi structures on affine bundles
• Reduction of Jacobi manifolds via Dirac structures theory
• The Deformation $L_{\infty }$ algebra of a Dirac--Jacobi structure
• Hom-Poisson-Nijenhuis structures on Hom-Lie algebroids and Hom-Dirac structures on Hom-Courant algebroids
• Twisted cohomology of Lie algebroids