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Journal of the London Mathematical Society Advance Access originally published online on February 2, 2009
Journal of the London Mathematical Society 2009 79(2):399-421; doi:10.1112/jlms/jdn084
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© 2009 London Mathematical Society

Dirac generating operators and Manin triples

Zhuo Chen

Department of Mathematics
Pennsylvania State University
109 McAllister Building
University Park, PA 16802
USA
chen_z@math.psu.edu

Mathieu Stiénon

Department of Mathematics
Pennsylvania State University
109 McAllister Building
University Park, PA 16802
USA

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 L such that L {otimes} L = ({wedge}top A* {otimes} {wedge}top T*M)1/2 exists, there exist two canonically defined differential operators d* and Formula on gcy({wedge}A {otimes} L). We prove that the pair (A, A*) constitutes a Lie bialgebroid if and only if the square of D = d* + Formula 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 D is a Dirac generating operator as defined by Alekseev and Xu. Our approach is to establish a list of new identities relating the Lie algebroid structures on A and A*.

2000 Mathematics Subject Classification 15A66, 17B63, 17B66, 53C15, 53D17, 55R65, 58A10.

Received May 1, 2008; published online February 2, 2009.


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