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Search: MSC category 58H05 ( Pseudogroups and differentiable groupoids [See also 22A22, 22E65] )

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1. CMB Online first

Lang, Honglei; Sheng, Yunhe; Wade, Aissa
$\mathsf{VB}$-Courant algebroids, $\mathsf{E}$-Courant algebroids and generalized geometry
In this paper, we first discuss the relation between $\mathsf{VB}$-Courant algebroids and $\mathsf{E}$-Courant algebroids and construct some examples of $\mathsf{E}$-Courant algebroids. Then we introduce the notion of a generalized complex structure on an $\mathsf{E}$-Courant algebroid, unifying the usual generalized complex structures on even-dimensional manifolds and generalized contact structures on odd-dimensional manifolds. Moreover, we study generalized complex structures on an omni-Lie algebroid in detail. In particular, we show that generalized complex structures on an omni-Lie algebra $\operatorname{gl}(V)\oplus V$ correspond to complex Lie algebra structures on $V$.

Keywords:$\mathsf{VB}$-Courant algebroid, $\mathsf{E}$-Courant algebroid, omni-Lie algebroid, generalized complex structure, algebroid-Nijenhuis structure
Categories:53D17, 18B40, 58H05

2. CMB 2015 (vol 58 pp. 575)

Martinez-Torres, David
The Diffeomorphism Type of Canonical Integrations Of Poisson Tensors on Surfaces
A surface $\Sigma$ endowed with a Poisson tensor $\pi$ is known to admit canonical integration, $\mathcal{G}(\pi)$, which is a 4-dimensional manifold with a (symplectic) Lie groupoid structure. In this short note we show that if $\pi$ is not an area form on the 2-sphere, then $\mathcal{G}(\pi)$ is diffeomorphic to the cotangent bundle $T^*\Sigma$. This extends results by the author and by Bonechi, Ciccoli, Staffolani, and Tarlini.

Keywords:Poisson tensor, Lie groupoid, cotangent bundle
Categories:58H05, 55R10, 53D17

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