Courant Algebroids in Parabolic Geometry

Details Courant Algebroids in Parabolic Geometry Courant Algebroids in Parabolic Geometry

2011-12-29

arxiv.org/abs/1112.6425v3

Physics

Mathematical Physics

24 pages, Section 4.3 added

Scientific paper

Let $g$ be a semisimple Lie algebra, $h$ be a Lie subalgebra of $g$, and $(G,H)$ denote the corresponding pair of connected Lie groups. To a smooth manifold $M$, a parabolic geometry associates a principal $P$-bundle, where $P$ is a parabolic subgroup of a semisimple Lie group $G$, and a Cartan connection. We show that the adjoint tractor bundle of a parabolic geometry, which is isomorphic to the Atiyah algebroid of the principal $P$-bundle, can be endowed with the structure of a Courant algebroid, modulo a topological obstruction to the Jacobi identity. As a result, we are able to show that if the Atiyah algebroid of a principal $H$-bundle admits a pre-Courant algebroid structure, then $h$ is a parabolic subalgebra of $g$.

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