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# $\mathsf{VB}$-Courant algebroids, $\mathsf{E}$-Courant algebroids and generalized geometry

Published:2018-03-08
Printed: Sep 2018
• Honglei Lang,
Max Planck Institute for Mathematics, Bonn D-53111, Germany
• Yunhe Sheng,
Department of Mathematics, Jilin University, Chanchung, 130012, Jilin, China
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