A. Alekseev, H. Bursztyn, E. Meinrenken
Pure Spinors on Lie groups
Preprint series: ESI preprints

MSC:

15A66 Clifford algebras, spinors

58F05 Hamiltonian and Lagrangian systems; symplectic geometry, See also {70Hxx, 81S10}

Abstract: For any manifold $M$, the direct sum $\TM=TM\oplus T^*M$ carries a
natural inner product given by the pairing of vectors and
covectors. Differential forms on $M$ may be viewed as spinors for the
corresponding Clifford bundle, and in particular there is a notion
of \emph{pure spinor}.
In this paper, we study pure spinors and Dirac structures
in the case when $M=G$ is a Lie group with a bi-invariant
pseudo-Riemannian metric, e.g. $G$ semi-simple. The
applications of our theory include the construction of distinguished
volume forms on conjugacy classes in $G$, and a new approach to the
theory of quasi-Hamiltonian $G$-spaces.