The Atiyah algebroid of the path fibration over a Lie group

Details The Atiyah algebroid of the path fibration over a Lie group The Atiyah algebroid of the path fibration over a Lie group

2008-10-24

arxiv.org/abs/0810.4402v1

Lett. Math. Phys. 90 (2009), no. 1-3, 23-58

Mathematics

Differential Geometry

31 pages

Scientific paper

Let G be a connected Lie group, LG its loop group, and PG->G the principal LG-bundle defined by quasi-periodic paths in G. This paper is devoted to differential geometry of the Atiyah algebroid A=T(PG)/LG of this bundle. Given a symmetric bilinear form on the Lie algebra g and the corresponding central extension of Lg, we consider the lifting problem for A, and show how the cohomology class of the Cartan 3-form on G arises as an obstruction. This involves the construction of a 2-form on PG with differential the pull-back of the Cartan form. In the second part of this paper we obtain similar LG-invariant primitives for the higher degree analogues of the Cartan form, and for their G-equivariant extensions.

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