## 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

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 $g$ we have a linear form coming from the trace of the adjoint action of $g$ on itself

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

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 $E$ is a section $\xi$ of $E^*$ which satisfies, for all sections $x$ of $E$ the equation

(2)$\langle \xi, x\rangle \omega \otimes \lambda = [x,\omega] \otimes \lambda + \omega \otimes \mathcal{L}_x \lambda \,,$

where $\omega$ and $\lambda$ are nowhere-vanishing sections of $\wedge^\mathrm{top} E$ and $\wedge^\mathrm{top} T^* M$, 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 $g_k$ #.)

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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