Multiplicative dirac structures

Abstract

We introduce multiplicative Dirac structures on Lie groupoids, providing a unified framework to study both multiplicative Poisson bivectors (Poisson groupoids) and multiplicative closed 2-forms such as symplectic groupoids. We prove that for every source simply connected Lie groupoid G with Lie algebroid AG, there exists a one-to-one correspondence between multiplicative Dirac structures on G and Dirac structures on AG that are compatible with both the linear and algebroid structures of AG. We explain in what sense this extends the integration of Lie bialgebroids to Poisson groupoids and the integration of Dirac manifolds. We explain the connection between multiplicative Dirac structures and higher geometric structures such as LA- groupoids and lA-groupoids. © 2013 Mathematical Sciences Publishers.