Representations up to homotopy of Lie algebroids
ثبت نشده
چکیده
We introduce and study the notion of representation up to homotopy of a Lie algebroid, paying special attention to examples. We use representations up to homotopy to define the adjoint representation of a Lie algebroid and show that the resulting cohomology controls the deformations of the structure. The Weil algebra of a Lie algebroid is defined and shown to coincide with Kalkman’s BRST model for equivariant cohomology in the case of group actions. The relation of this algebra with the integration of Poisson and Dirac structures is explained in [3].
منابع مشابه
Double Lie algebroids and representations up to homotopy
Weshow that a double Lie algebroid, togetherwith a chosen decomposition, is equivalent to a pair of 2-term representations up tohomotopy satisfying compatibility conditions which extend the notion of matched pair of Lie algebroids. We discuss in detail the double Lie algebroids arising from the tangent bundle of a Lie algebroid and the cotangent bundle of a Lie bialgebroid.
متن کامل0 Connections up to homotopy and characteristic classes ∗
The aim of this note is to clarify the relevance of “connections up to homotopy” [4, 5] to the theory of characteristic classes, and to present an application to the characteristic classes of Lie algebroids [3, 5, 7] (and of Poisson manifolds in particular [8, 13]). We have already remarked [4] that such connections up to homotopy can be used to compute the classical Chern characters. Here we p...
متن کامل0 Chern characters via connections up to homotopy ∗
1 Introduction: The aim of this note is to point out that Chern characters can be computed using curvatures of " connections up to homotopy " , and to present an application to the vanishing theorem for Lie algebroids. Classically, Chern characters are computed with the help of a connection and its curvature. However, one often has to relax the notion of connection so that one gains more freedo...
متن کاملDifferential Operators and Actions of Lie Algebroids
We demonstrate that the notions of derivative representation of a Lie algebra on a vector bundle, of semi-linear representation of a Lie group on a vector bundle, and related concepts, may be understood in terms of representations of Lie algebroids and Lie groupoids, and we indicate how these notions extend to derivative representations of Lie algebroids and semi-linear representations of Lie g...
متن کاملHorizontal Subbundle on Lie Algebroids
Providing an appropriate definition of a horizontal subbundle of a Lie algebroid will lead to construction of a better framework on Lie algebriods. In this paper, we give a new and natural definition of a horizontal subbundle using the prolongation of a Lie algebroid and then we show that any linear connection on a Lie algebroid generates a horizontal subbundle and vice versa. The same correspo...
متن کامل