nLab Cartan-Dirac structure on a Lie group

Idea

For $G$ a Lie group and $\langle -,-\rangle$ a non-degenerate binary invariant polynomial on its Lie algebra $\mathfrak{g}$, there is a canonical Dirac structure on $G$, a subbundle of the generalized tangent bundle $T G \oplus T^* G$ which is maximal isotropic with respect to the canonical pairing and preserved by the Courant bracket twisted by the canonical differential 3-form $\langle -,[-,-]\rangle$ on $G$.

The leaves of this structure are the conjugacy classes of $G$. There is a differential 2-forms $\theta_g$ on each conjugacy class $\iota \colon \mathcal{C} \hookrightarrow G$, such that $d \theta_g = \iota^* \langle -,[-,-]\rangle$.

This Dirac structure was first observed in (Ševera-Weinstein). The name “Cartan-Dirac structure” was introduced in (BCWZ).

References

The Cartan-dirac structure appears first as example 4.2 in

• Pavol Ševera, Alan Weinstein, Poisson geometry with a 3-form background (arXiv:math/0107133)

See also example 3.4 in

• Henrique Bursztyn, Marius Crainic, Alan Weinstein, Chenchang Zhu, Integration of twisted Dirac brackets (arXiv:math/0303180)

and example 2.13 in the lecture notes

• Henrique Bursztyn, Alan Weinstein, Poisson geometry and Morita equivalence (arXiv:math/0402347)

More discussion is in section 7.2 of (BCWZ).

Another class of Dirac structures on Lie groups – multiplicative Dirac structuress – is discussed in

• Cristian Ortiz, Multiplicative Dirac structures on Lie groups, Comptes Rendus Academie des Sciences Paris, Ser. I 346 (2008) (arXiv:0906.2373)