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October 10, 2006

Kosmann-Schwarzbach & Weinstein on Lie Algebroid Classes

Posted by Urs Schreiber

On October 24 Yvette Kosmann-Schwarzbach will give a talk in the Hamburg math colloquium on

Modular classes and relative modular classes in Lie algebroid theory

based on work with Alan Weinstein

Y. Kosmann-Schwarzbach & A. Weinstein
Relative modular classes of Lie algebroids
math.DG/0508515

and Camille Laurent-Gengoux

Y. Kosmann-Schwarzbach & C. Laurent-Gengoux
The modular class of a twisted Poisson structure
math.SG/0505663 .

A Lie algebroid # is to a Lie groupoid as a Lie algebra is to a Lie group #.

On each Lie algebra gg we have a linear form coming from the trace of the adjoint action of gg on itself

(1)χ g:xTr(ad x). \chi^g : x \mapsto \mathrm{Tr}(\mathrm{ad}_x) \,.

This is a 1-cocycle in Lie algebra cohomology.

This construction may be generalized to Lie algebroids. The corresponding 1-class in Lie algebroid cohomology is called the modular class of the Lie algebroid.

Technically, the modular class of a Lie algebroid EE is a section ξ\xi of E *E^* which satisfies, for all sections xx of EE the equation

(2)ξ,xωλ=[x,ω]λ+ω xλ, \langle \xi, x\rangle \omega \otimes \lambda = [x,\omega] \otimes \lambda + \omega \otimes \mathcal{L}_x \lambda \,,

where ω\omega and λ\lambda are nowhere-vanishing sections of topE\wedge^\mathrm{top} E and topT *M\wedge^\mathrm{top} T^* M , respectively. (See p. 4 of the first paper cited above for details.)

There are “twisted” versions of Lie algebroids, where the familiar equations only hold up to higher coherence. These should be nothing but 2- (and higher) algebroids, but are not usually addressed this way.

The most prominent example is the Courant algebroid #. (I once thought I convinced myself that the Courant algebroid over a point is nothing but the Baez-Crans Lie 2-algebra g kg_k #.)

There is something some people call a Dirac structure on a Courant algebroid. Kosmann-Schwarzbach et al. call this a twisted Poisson structure.

In the second of the above papers a notion of modular class for such a “Lie algebroid with twisted Poisson structure” is defined. One of the main points of these papers is to relate this definition to a generalization of the ordinary modular class of a Lie algebroid called the relative modular class.

Maybe I’ll discuss more of the details after I have heard the talk.

Posted at October 10, 2006 12:52 PM UTC

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