Omni-Lie Algebras

Abstract

We show that the space R^n x gl(n,R) with a certain antisymmetric bracket operation contains all n-dimensional Lie algebras. The bracket does not satisfy the Jacobi identity, but it does satisfy it for subalgebras which are isotropic under a certain symmetric bilinear form with values in R^n. We ask what the corresponding "group-like" object should be. The bracket may be obtained by linearizing at a point the bracket on TM + T*M introduced by T. Courant for the definition of Dirac structures, a notion which encompasses Poisson structures, closed 2-forms, and foliations.