DIRAC STRUCTURES OF OMNI-LIE ALGEBROIDS

Omni-Lie algebroids are generalizations of Alan Weinstein's omni-Lie algebras. A Dirac structure in an omni-Lie algebroid DE circle plus JE is necessarily a Lie algebroid together with a representation on E. We study the geometry underlying these Dirac structures in the light of reduction theory. In particular, we prove that there is a one-to-one correspondence between reducible Dirac structures and projective Lie algebroids in T = TM circle plus E; we establish the relation between the normalizer N(L) of a reducible Dirac structure L and the derivation algebra Der(b(L)) of the projective Lie algebroid b(L); we study the cohomology group H(center dot)(L, rho(L)) and the relation between N(L) and H(1)(L, rho(L)); we describe Lie bialgebroids using the adjoint representation; we study the deformation of a Dirac structure L, which is related with H(2)(L, rho(L)).MathematicsSCI(E)1ARTICLE81163-11852