Vol. 266, No. 2, 2013

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Multiplicative Dirac structures

Cristián Ortiz

Vol. 266 (2013), No. 2, 329–365

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 𝒞𝒜-groupoids.

Lie groupoids, Lie algebroids, multiplicative Dirac structures
Mathematical Subject Classification 2010
Primary: 53D17, 53D18
Received: 12 December 2012
Revised: 14 May 2013
Accepted: 19 May 2013
Published: 12 November 2013
Cristián Ortiz
Departamento de Matemática
Universidade Federal do Paraná
Setor de Ciências Exatas - Centro Politécnico
81531-990 Curitiba