nLab Cartan-Dirac structure on a Lie group



\infty-Lie theory

∞-Lie theory (higher geometry)


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For GG a Lie group and ,\langle -,-\rangle a non-degenerate binary invariant polynomial on its Lie algebra 𝔤\mathfrak{g}, there is a canonical Dirac structure on GG, a subbundle of the generalized tangent bundle TGT *GT 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 GG.

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

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


The Cartan-dirac structure appears first as example 4.2 in

See also example 3.4 in

and example 2.13 in the lecture notes

More discussion is in section 7.2 of (BCWZ).

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

Last revised on March 20, 2013 at 14:06:02. See the history of this page for a list of all contributions to it.