Dirac Structures of Omni-Lie Algebroids
Omni-Lie algebroids are generalizations of Alan Weinstein’s omni-Lie algebras. A Dirac structure in an omni-Lie algebroid is necessarily a Lie algebroid together with a representation on . We study the geometry underlying these Dirac structures in the light of reduction theory. In particular, we prove that there is a one-to-one correspondence between reducible Dirac structures and projective Lie algebroids in ; we establish the relation between the normalizer of a reducible Dirac structure and the derivation algebra of the projective Lie algebroid ; we study the cohomology group and the relation between and ; we describe Lie bialgebroids using the adjoint representation; we study the deformation of a Dirac structure , which is related with .
Keywords: omni-Lie algebroid, Dirac structures, local Lie algebras, reduction, normalizer, deformation
Mathematics Subject Classification 2000: 17B66, 58H05
Lie algebroids (and local Lie algebras in the sense of Kirillov ) are generalizations of Lie algebras that naturally appear in Poisson geometry (and its variations, e.g., Jacobi manifolds in the sense of Lichnerowicz )(see for a detailed description of this subject). Courant algebroids are combinations of Lie algebroids and quadratic Lie algebras. It was originally introduced in by T. Courant where he first called them Dirac manifolds, and then were re-named after him in (see also an alternate definition ) by Liu, Weinstein and Xu to describe the double of a Lie bialgebroid. Recently, several applications of Courant algebroids and Dirac structures have been found in different fields, e.g., Manin pairs and moment maps ; generalized complex structures ; -algebras and symplectic supermanifolds ; gerbes as well as BV algebras and topological field theories .
Motivated by an integrability problem of the Courant bracket, A. Weinstein gives a linearization of the Courant bracket at a point , which had been studied from several aspects recently . A. Weinstein has shown that an omni-Lie algebra structure can encode all Lie algebra structures on a vector space, the next step is, logically, to find out candidates that could encode all Lie algebroid structures on a vector bundle. In a recent work , we have given a definitive answer to this question. Over there, a generalized Courant algebroid structure is defined on the direct sum bundle , where and are the gauge Lie algebroid and the jet bundle of a vector bundle respectively. Such a structure is called an omni-Lie algebroid since it reduces to the omni-Lie algebra if the base manifold is a point . Furthermore, an omni-Lie algebroid is the first example of -Courant algebroids .
It is well known that the theory of Dirac structures has wide and deep applications in both mathematics and physics (e.g., ). In , only some special Dirac structures were studied. The authors proved that there is a one-to-one correspondence between Dirac structures coming from bundle maps and Lie algebroid (local Lie algebra) structures on when ( is a line bundle). In other words, Dirac structures that are graphs of maps actually underlines the geometric objects of Lie algebroids, or local Lie algebras.
As a continuation of , the present paper explores what a general Dirac structure of the omni-Lie algebroid can encode. For a vector space , Weinstein proved that Dirac structures in the omni-Lie algebra correspond to Lie algebra structures on subspaces of in . For a vector bundle over , Dirac structures in the omni-Lie algebroid turn out to be more complicated than that of omni-Lie algebras. The key concept that we need is a projective Lie algebroid, which is a subbundle , equipped with a Lie algebroid structure such that the anchor is the projection from to . A Dirac structure is called reducible if is a regular subbundle of . We will see that any Dirac structure is reducible if (Lemma LABEL:Lem:dimention). The main result is Theorem LABEL:Thm:Main, which claims a one-to-one correspondence between reducible Dirac structures in and projective Lie algebroids in . In fact, the projection of a reducible Dirac structure to yields a projective Lie algebroid and, conversely, a projective Lie algebroid can be uniquely lifted to a Dirac structure using a connection in .
Furthermore, using the falling operator , we establish a connection between the derivation algebra of a projective Lie algebroid and the normalizer of the corresponding lifted Dirac structure . We prove that, for any , . Conversely, any can be lifted to an element in . Another observation is that, to any Dirac structure , there associates a representation of on , namely (Proposition LABEL:pro:representation). So there is an associated cohomology group . We will see that the normalizer of is related with and the deformation of is related with .
This paper is organized as follows. In Section 2 we recall the basic properties of omni-Lie algebroids. In Section LABEL:DiracReduct, we state the main result of this paper — the correspondence between reducible Dirac structures and projective Lie algebroids. In Section 4, several interesting examples are discussed. In Section LABEL:sec:1cohomology, we study the relation between the normalizer of a reducible Dirac structure and Lie derivations. In Section LABEL:sec:2cohomology, we give some applications of the related cohomologies of Dirac structures.
2 Omni-Lie Algebroids
We use the following convention throughout the paper: denotes a vector bundle over a smooth manifold (we assume that is not a zero bundle), the usual deRham differential of forms and an arbitrary point in . By we denote the direct sum and use , , respectively, to denote the projection from to and .
First, we briefly review the notion of omni-Lie algebroids defined in , which generalizes omni-Lie algebras defined by A. Weinstein in . Given a vector bundle , let be the first jet bundle of , and the gauge Lie algebroid of . These two vector bundles associate, respectively, with the jet sequence: