 
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 Klimc`ik and Strobl [7], and were studied by 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
