Note on pre-Courant algebroid structures for parabolic geometries

This note aims to demonstrate that every parabolic geometry has a naturally defined per-Courant algebröıd structure. If the geometry is regular, this structure is a Courant algebröıd if and only if the the curvature κ of the Cartan connection vanishes. This note assumes familiarity with both parabolic geometry, and Courant algebröıds. See [ČS00] and [ČG02] for a good introduction to the first case, and [KS05] and [Vai05] for the second. Some of the basic definitions will be recalled here: Definition 0.1 (Parabolic Geometry). Let G be a semi-simple Lie group with Lie algebra g, and P a parabolic subgroup with Lie algebra p. A parabolic geometry on a manifold M is given by a principal P bundle P , an inclusion P ⊂ G, and a principal connection −→ω on G. This connection is required to satisfy the condition that −→ω |P is a linear isomorphism TP → g. Let A = P ×P g = G ×G g and denote by −→ ∇ the affine connection on A coming from −→ω . By construction, A also inherits an algebräıc bracket {, } and the Killing form B. It moreover has a well defined subbundle A(0) = P ×P p, and the properties of −→ ∇ give an equivalence A/A(0) ∼= T , thus a projection π : A → T . The Killing form B then defines an inclusion T ∗ ⊂ A, with (T ) = A(0). This implies that for v a one form, x any section of A: