## Journal of Differential Geometry

### Picard groups of Poisson manifolds

#### Abstract

For a Poisson manifold $M$ we develop systematic methods to compute its Picard group $\mathrm{Pic}(M)$, i.e., its group of self Morita equivalences. We establish a precise relationship between $\mathrm{Pic}(M)$, and the group of gauge transformations up to Poisson diffeomorphisms showing, in particular, that their connected components of the identity coincide; this allows us to introduce the Picard Lie algebra of $M$ and to study its basic properties. Our methods lead to the proof of a conjecture from “Poisson sigma models and symplectic groupoids, in Quantization of Singular Symplectic Quotients” [A.S. Cattaneo, G. Felder, Progress in Mathematics 198 (2001), 41] stating that $\mathrm{Pic}(\mathfrak{g}^*)$, for any compact simple Lie algebra agrees with the group of outer automorphisms of $\mathfrak{g}$.

#### Note

HB has had the support of Faperj, and RLF is partially supported by NSF grants DMS 1308472 and DMS 1405671. Both authors acknowledge the support of a Capes/Brazil–FCT/Portugal cooperation grant and the Ciências Sem Fronteiras program sponsored by CNPq.

#### Article information

Source
J. Differential Geom., Volume 109, Number 1 (2018), 1-38.

Dates
Received: 22 October 2015
First available in Project Euclid: 4 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1525399215

Digital Object Identifier
doi:10.4310/jdg/1525399215

Mathematical Reviews number (MathSciNet)
MR3798714

Zentralblatt MATH identifier
06868029

#### Citation

Bursztyn, Henrique; Fernandes, Rui Loja. Picard groups of Poisson manifolds. J. Differential Geom. 109 (2018), no. 1, 1--38. doi:10.4310/jdg/1525399215. https://projecteuclid.org/euclid.jdg/1525399215