Dirac generating operators and Manin triples

Zhuo Chen, Mathieu Stiénon

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

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 ⊗ ℒ = (∧top A* ⊗ ∧top T*M)1/2 exists, there exist two canonically defined differential operators d* and ∂ on (∧A ⊗ ℒ). We prove that the pair (A, A*) constitutes a Lie bialgebroid if and only if the square of D = d* + ∂ 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*.

Original languageEnglish (US)
Pages (from-to)399-421
Number of pages23
JournalJournal of the London Mathematical Society
Volume79
Issue number2
DOIs
StatePublished - Apr 2009

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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