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The Atiyah algebroid of the path fibration over a Lie group | | A. Alekseev
; E. Meinrenken
; | | Date: |
24 Oct 2008 | | Abstract: | 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. | | Source: | arXiv, 0810.4402 | | Services: | Forum | Review | PDF | Favorites | |
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