Representations up to homotopy of Lie algebroids.

*(English)*Zbl 1238.58010Representation theory is an important mathematical subject which has many applications, ranging from number theory and combinatorics to geometry, quantum mechanics and quantum field theory.

In this paper, the authors study a more general notion of representation in terms of Lie algebroids, representations up to homotopy. The main idea in the paper is to represent Lie algebroids in cochain complexes of vector bundles. After defining the notion of representations up to homotopy of Lie algebroids, the authors give some differential characterizations about it. Also it is introduced the Weil algebra of a Lie algebroid which is a generalization of the standard Weil algebra of a Lie algebra in the last section. And finally, it is presented the relationship between the Weil algebra associated to a Lie algebroid and Kalkman’s BRST algebra for equivariant cohomology.

In this paper, the authors study a more general notion of representation in terms of Lie algebroids, representations up to homotopy. The main idea in the paper is to represent Lie algebroids in cochain complexes of vector bundles. After defining the notion of representations up to homotopy of Lie algebroids, the authors give some differential characterizations about it. Also it is introduced the Weil algebra of a Lie algebroid which is a generalization of the standard Weil algebra of a Lie algebra in the last section. And finally, it is presented the relationship between the Weil algebra associated to a Lie algebroid and Kalkman’s BRST algebra for equivariant cohomology.

Reviewer: M. Habil Gürsoy (Malatya)