Quantum Groupoids

We introduce a general notion of quantum universal enveloping algebroids (QUE algebroids), or quantum groupoids, as a unification of quantum groups and star-products. Some basic properties are studied including the twist construction and the classical limits. In particular, we show that a quantum groupoid naturally gives rise to a Lie bialgebroid as a classical limit. Conversely, we formulate a conjecture on the existence of a quantization for any Lie bialgebroid, and prove this conjecture for the special case of regular triangular Lie bialgebroids. As an application of this theory, we study the dynamical quantum groupoid , which gives an interpretation of the quantum dynamical Yang–Baxter equation in terms of Hopf algebroids. Received: 6 April 2000 / Accepted: 15 August 2000     

A characterization of strict Jacobi-Nijenhuis manifolds through the theory of Lie algebroids

We obtain a characterization of strict Jacobi-Nijenhuis structures using the equivalent notions of generalized Lie bialgebroid and Jacobi bialgebroid.     

On Quasi-Jacobi and Jacobi-Quasi Bialgebroids

We study quasi-Jacobi and Jacobi-quasi bialgebroids and their relationships with twisted Jacobi and quasi Jacobi manifolds. We show that we can construct quasi-Lie bialgebroids from quasi-Jacobi bialgebroids, and conversely, and also that the structures induced on their base manifolds are related via a “quasi Poissonization”.      

On Jacobi quasi-Nijenhuis algebroids and Courant–Jacobi algebroid morphisms

We propose a definition of a Jacobi quasi-Nijenhuis algebroid and show that any such Jacobi algebroid has an associated quasi-Jacobi bialgebroid. Therefore, an associated Courant–Jacobi algebroid is also obtained. We introduce the notions of quasi-Jacobi bialgebroid morphism and Courant–Jacobi algebroid morphism, also providing some examples of Courant–Jacobi algebroid morphisms.     

The Local Structure of Lie Bialgebroids

We study the local structure of Lie bialgebroids at regular points. In particular, we classify all transitive Lie bialgebroids. In special cases, they are connected to classical dynamical r-matrices and matched pairs induced by Poisson group actions.     

Poisson Quasi-Nijenhuis Structures with Background

We define Poisson quasi-Nijenhuis structures with background on Lie algebroids and we prove that any generalized complex structure on a Courant algebroid which is the double of a Lie algebroid has an associated Poisson quasi-Nijenhuis structure with background. We prove that any Lie algebroid with a Poisson quasi-Nijenhuis structure with background constitutes, with its dual, a quasi-Lie bialgebroid. We also prove that any pair (π,ω) of a Poisson bivector and a 2-form induces a Poisson quasi-Nijenhuis structure with background and we observe that particular cases correspond to already known compatibilities between π and ω. This paper was presented as a poster in the conference “Poisson 2008”, EPFL, Lausanne, in July 2008.     

Formality for Lie Algebroids

Using Dolgushev’s generalization of Fedosov’s method for deformation quantization, we give a positive answer to a question of P. Xu: can one prove a formality theorem for Lie algebroids ? As a direct application of this result, we obtain that any triangular Lie bialgebroid is quantizable.     

Graded Lie Algebra Generating of Parastatistical Algebraic Relations

A new kind of graded Lie algebra (We call it Z2,2 graded Lie algebra) is introduced as a framework for formulating parasupersymmetric theories. By choosing suitable Bose subspace of the Z2,2 graded Lie algebra and using relevant generalized Jacobi identities, we generate the whole algebraic structure of parastatistics.     

Vertex Operators Arising from Jacobi–Trudi Identities

We give an interpretation of the boson-fermion correspondence as a direct consequence of the Jacobi–Trudi identity. This viewpoint enables us to construct from a generalized version of the Jacobi–Trudi identity the action of a Clifford algebra on the polynomial algebras that arrive as analogues of the algebra of symmetric functions. A generalized Giambelli identity is also proved to follow from that identity. As applications, we obtain explicit formulas for vertex operators corresponding to characters of the classical Lie algebras, shifted Schur functions, and generalized Schur symmetric functions associated to linear recurrence relations.     

Nambu–Dirac Structures for Lie Algebroids

The theory of Nambu–Poisson structures on manifolds is extended to the context of Lie algebroids in a natural way based on the derived bracket associated with the Lie algebroid differential. A new way of combining Nambu–Poisson structures and triangular Lie bialgebroids is described in this work. Also, we introduce the concept of a higher order Dirac structure on a Lie algebroid. This allows to describe both Nambu–Poisson structures and Dirac structures on manifolds in the same setting.     

The lie bialgebroid of a Poisson-Nijenhuis manifold

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

Symmetry Groups and New Exact Solutions to （2＋1）-Dimensional Variable Coefficient Canonical Generalized KP Equation

In this paper, the modified CK＇s direct method to find symmetry groups of nonlinear partial differential equation is extended to （2＋1）-dimensional variable coeffficient canonical generalized KP （VCCGKP） equation. As a result, symmetry groups, Lie point symmetry group and Lie symmetry for the VCCGKP equation are obtained. In fact, the Lie point symmetry group coincides with that obtained by the standard Lie group approach. Applying the given Lie symmetry, we obtain five types of similarity reductions and a lot of new exact solutions, including hyperbolic function solutions, triangular periodic solutions, Jacobi elliptic function solutions and rational solutions, for the VCCGKP equation.     

Symmetry Groups and New Exact Solutions to (2+1)-Dimensional Variable Coefficient Canonical Generalized KP Equation

In this paper, the modified CK's direct method to find symmetry groups of nonlinear partial differential equation is extended to (2 1)-dimensional variable coefficient canonical generalized KP (VCCGKP) equation. As a result, symmetry groups, Lie point symmetry group and Lie symmetry for the VCCGKP equation are obtained. In fact, the Lie point symmetry group coincides with that obtained by the standard Lie group approach. Applying the given Lie symmetry, we obtain five types of similarity reductions and a lot of new exact solutions, including hyperbolic function solutions, triangular periodic solutions, Jacobi elliptic function solutions and rational solutions, for the VCCGKP equation.     

Non-Commutative Batalin-Vilkovisky Algebras,Homotopy Lie Algebras and the Courant Bracket

We consider two different constructions of higher brackets. First, based on a Grassmann-odd, nilpotent Δ operator, we define a non-commutative generalization of the higher Koszul brackets, which are used in a generalized Batalin-Vilkovisky algebra, and we show that they form a homotopy Lie algebra. Secondly, we investigate higher, so-called derived brackets built from symmetrized, nested Lie brackets with a fixed nilpotent Lie algebra element Q. We find the most general Jacobi-like identity that such a hierarchy satisfies. The numerical coefficients in front of each term in these generalized Jacobi identities are related to the Bernoulli numbers. We suggest that the definition of a homotopy Lie algebra should be enlarged to accommodate this important case. Finally, we consider the Courant bracket as an example of a derived bracket. We extend it to the “big bracket” of exterior forms and multi-vectors, and give closed formulas for the higher Courant brackets.     

Synthetic Hamiltonian mechanics

This paper deals with some infinitesimal aspects of Hamiltonian mechanics from the standpoint of synthetic differential geometry. Fundamental results concerning Hamiltonian vector fields, Poisson brackets, and momentum mappings are discussed. The significance of the Lie derivative in the synthetic context is also consistently stressed. In particular, the notion of an infinitesimally Euclidean space is introduced, and the Jacobi identity of vector fields with respect to Lie brackets is established naturally for microlinear, infinitesimally Euclidean spaces by using Lie derivatives instead of a highly combinatorial device such as P. Hall's 42-letter identity.     

BV Quantization of Topological Open Membranes

We study bulk-boundary correlators in topological open membranes. The basic example is the open membrane with a WZ coupling to a 3-form. We view the bulk interaction as a deformation of the boundary string theory. This boundary string has the structure of a homotopy Lie algebra, which can be viewed as a closed string field theory. We calculate the leading order perturbative expansion of this structure. For the 3-form field we find that the C-field induces a trilinear bracket, deforming the Lie algebra structure. This paper is the first step towards a formal universal quantization of general quasi-Lie bialgebroids.Dept. of Particle Physics, Weizmann Institute, Rehovot, IsraelMathematics Graduate Center, CUNY, New York, USA     

Courant Algebroids and Strongly and Strongly Homotopy Lie Algebras

Courant algebroids are structures which include as examples the doubles of Lie bialgebras and the bundle TM T^*M with the bracket introduced by T. Courant for the study of Dirac structures. Within the category of Courant algebroids one can construct the doubles of Lie bialgebroids, the infinitesimal objects for Poisson groupoids. We show that Courant algebroids can be considered as strongly homotopy Lie algebras.     

Jacobi Last Multiplier and Lie Symmetries: A Novel Application of an Old Relationship

Abstract

After giving a brief account of the Jacobi last multiplier for ordinary differential equations and its known relationship with Lie symmetries, we present a novel application which exploits the Jacobi last multiplier to the purpose of finding Lie symmetries of first-order systems. Several illustrative examples are given.