Formality theorem with coefficients in a module

Details Formality theorem with coefficients in a module Formality theorem with coefficients in a module

2006-04-18

arxiv.org/abs/math/0604386v2

Mathematics

Quantum Algebra

44 pages. I removed the second formality theorem and improved the part about applications

Scientific paper

In this article, $X$ will denote a ${\cal C}^{\infty}$ manifold. In a very famous article, Kontsevich showed that the differential graded Lie algebra (DGLA) of polydifferential operators on $X$ is formal. Calaque extended this theorem to any Lie algebroid. More precisely, given any Lie algebroid $E$ over $X$, he defined the DGLA of $E$-polydifferential operators, $\Gamma (X, ^{E}D^{*}_{poly})$, and showed that it is formal. Denote by $\Gamma (X, ^{E}T^{*}_{poly})$ the DGLA of $E$-polyvector fields. Considering $M$, a module over $E$, we define $\Gamma (X, ^{E}T_{poly}^{*}(M))$ the $\Gamma (X, ^{E}T^{*}_{poly})$-module of $E$-polyvector fields with values in $M$. Similarly, we define the $\Gamma (X, ^{E}D^{*}_{poly})$-module of $E$-polydifferential operators with values in $M$, $\Gamma (X, ^{E}D^{*}_{poly}(M))$. We show that there is a quasi-isomorphism of $L_{\infty}$-modules over $\Gamma (X, ^{E}T^{*}_{poly})$ from $\Gamma (X, ^{E}T^{*}_{poly}(M))$ to $\Gamma (X, ^{E}D^{*}_{poly}(M))$. Our result extends Calaque 's (and Kontsevich's) result.

Affiliated with

Also associated with

No associations

Rating

Formality theorem with coefficients in a module does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Formality theorem with coefficients in a module, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Formality theorem with coefficients in a module will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-195633