JournalsjncgVol. 15, No. 2pp. 643–711

Polyvector fields and polydifferential operators associated with Lie pairs

  • Ruggero Bandiera

    Sapienza Università di Roma, Italy
  • Mathieu Stiénon

    Pennsylvania State University, University Park, USA
  • Ping Xu

    Pennsylvania State University, University Park, USA
Polyvector fields and polydifferential operators associated with Lie pairs cover
Download PDF

This article is published open access under our Subscribe to Open model.

Abstract

We prove that the spaces tot(Γ(ΛA)RTpoly)\operatorname{tot} \big(\Gamma ({\Lambda^\bullet A^\vee}) \otimes_R{{\mathcal{T}_{poly}^{\bullet}}}\big) and tot(Γ(ΛA)RDpoly)\operatorname{tot} \big(\Gamma ({\Lambda^\bullet A^\vee}) \otimes_R{{\mathcal{D}_{poly}^{\bullet}}}\big) associated with a Lie pair (L,A)(L,A) each carry an LL_\infty algebra structure canonical up to an LL_\infty isomorphism with the identity map as linear part. These two spaces serve, respectively, as replacements for the spaces of formal polyvector fields and formal polydifferential operators on the Lie pair (L,A)(L,A). Consequently, both HCE(A,Tpoly)\mathbb{H}^{\bullet}_{\operatorname{CE}}(A,{\mathcal{T}_{poly}^{\bullet}}) and HCE(A,Dpoly)\mathbb{H}^{\bullet}_{\operatorname{CE}}(A,{\mathcal{D}_{poly}^{\bullet}}) admit unique Gerstenhaber algebra structures. Our approach is based on homotopy transfer and the construction of a Fedosov dg Lie algebroid (i.e. a dg foliation on a Fedosov dg manifold).

Cite this article

Ruggero Bandiera, Mathieu Stiénon, Ping Xu, Polyvector fields and polydifferential operators associated with Lie pairs. J. Noncommut. Geom. 15 (2021), no. 2, pp. 643–711

DOI 10.4171/JNCG/416