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MODULAR CLASSES OF REGULAR TWISTED POISSON STRUCTURES ON LIE ALGEBROIDS
 

Summary: MODULAR CLASSES OF REGULAR TWISTED POISSON
STRUCTURES ON LIE ALGEBROIDS
YVETTE KOSMANN­SCHWARZBACH AND MILEN YAKIMOV
Abstract. We derive a formula for the the modular class of a Lie algebroid
with a regular twisted Poisson structure in terms of a canonical Lie algebroid
representation of the image of the Poisson map. We use this formula to com­
pute the modular classes of Lie algebras with a twisted triangular r­matrix.
The special case of r­matrices associated to Frobenius Lie algebras is also
studied.
1. Introduction
Twisted Poisson structures on manifolds, whose definition we recall in Section
2.1, first appeared in the mathematical physics literature. They were introduced
in geometry by KlimŸc‘k and Strobl [7], and were studied by Ÿ Severa and Wein­
stein [13] who proved that they can be described as Dirac structures in Courant
algebroids. Roytenberg then showed in [12] that, more generally, twisted Poisson
structures on Lie algebroids appear in a natural way in his general theory of
twisting of Lie bialgebroids.
While the modular vector fields of Poisson manifolds were first defined by
Koszul in 1985, the theory of the modular classes of Poisson manifolds was de­
veloped by Weinstein in his 1997 article [15] and that of the modular classes of Lie

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics