Dirac generating operators and Manin triples

Details Dirac generating operators and Manin triples Dirac generating operators and Manin triples

2008-03-17

arxiv.org/abs/0803.2376v3

J. Lond. Math. Soc. (2) 79 (2009), 399-421

Mathematics

Differential Geometry

26 pages, introduction rewritten, minor corrections

Scientific paper

10.1112/jlms/jdn084

Given a pair of (real or complex) Lie algebroid structures on a vector bundle $A$ (over $M$) and its dual $A^*$, and a line bundle $\module$ such that $\module\otimes\module=(\wedge^{\TOP} A^*\otimes\wedge^{\TOP} T^*M)$, there exist two canonically defined differential operators $\bdees$ and $\bdel$ on $\sections{\wedge A\otimes\module}$. We prove that the pair $(A,A^*)$ constitutes a Lie bialgebroid if, and only if, the square of $\bdirac =\bdees+\bdel$ is the multiplication by a function on $M$. As a consequence, we obtain that the pair $(A,A^*)$ is a Lie bialgebroid if, and only if, $\bdirac$ is a Dirac generating operator as defined by Alekseev & Xu \cite{AlekseevXu}. Our approach is to establish a list of new identities relating the Lie algebroid structures on $A$ and $A^*$ (Theorem \ref{Thm:C}).

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