Abstract
First, we prove a decomposition formula for any multiplicative differential form on a Lie groupoid \(\cal{G}\). Next, we prove that if \(\cal{G}\) is a Poisson Lie groupoid, then the space \(\Omega_{\text{mult}}^{\bullet}(\cal{G})\) of multiplicative forms on \(\cal{G}\) has a differential graded Lie algebra (DGLA) structure. Furthermore, when combined with Ω•(M), which is the space of forms on the base manifold M of \(\cal{G},\Omega_{\text{mult}}^{\bullet}(\cal{G})\) forms a canonical DGLA crossed module. This supplements a previously known fact that multiplicative multi-vector fields on \(\cal{G}\) form a DGLA crossed module with the Schouten algebra Γ(∧•A) stemming from the Lie algebroid A of \(\cal{G}\).
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 12071241) and the National Key Research and Development Program of China (Grant No. 2021YFA1002000). We gratefully acknowledge Ping Xu from Penn State University for helpful discussions and helpful comments. Special thanks go to Cristian Ortiz from University of Sao Paulo, who brought to our attention the fact that Theorem 5.5 we presented in the initial version of this manuscript had already been discovered. We have clarified this in the current version.
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Chen, Z., Lang, H. & Liu, Z. Multiplicative forms on Poisson groupoids. Sci. China Math. (2024). https://doi.org/10.1007/s11425-023-2231-1
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DOI: https://doi.org/10.1007/s11425-023-2231-1