Van Est integration in higher Lie theory

• Applicants: Professorin Dr. Madeleine Jotz; Professorin Chenchang Zhu, Ph.D.
• Subject Area: Mathematics
• Term: from 2020 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 446784784

• Final Report Year: 2024

Final Report Abstract

Lie algebroids and their higher versions known as Lie n-algebroids are differential graded manifolds that describe infinitesimal symmetries. The integration of these structures to structures encoding global symmetries – that is, to Lie groupoids and their higher versions – is a problem that has interested many mathematicians in the last decades. In the early 2000’s, Crainic and Fernandes proposed in a seminal paper a path-integration of Lie algebroids. This beautiful method has a lot of analytical difficulties. Now that the adjoint representation (up to homotopy) of a Lie algebroid is well-understood, we proposed in our project the study the integration problem from a new and more algebraic point of view, extending van Est’s method for integrating Lie algebras. Van Est’s theory is the differentiation of Lie group cocycles to Lie algebra cocycles. This is generalised to the setting of Lie groupoids and Lie algebroids. After the work on differentiation of a Lie n-groupoid to its tangent complex, we saw a possibility to generalise the Van Est theory to higher Lie groupoids. We found out that the Eilenberg-Zilber-type formula presented by E. Getzler links shifted differential 2-forms on higher Lie groupoids to certain pairings on their respective tangent complexes, in such a way as to give a Van Est theory for simplicial vector spaces. That is, it differentiates a pairing of simplicial vector spaces to a pairing of their corresponding tangent complexes, which are simply vector spaces complexes. Although it is easy and clear to define the dual for a vector space complex, it is not clear how to define a meaningful dual for a simplicial vector space. Lead by the correspondence of pairings, we have however found some nice concept of dual for simplicial vector spaces. This further leads to a version of 2-dual for a VB-Lie 2-groupoid.

Publications

• Differentiating L8 groupoids – Part I: Du Li, Leonid Ryvkin, Arne Wessel, Chenchang Zhu
  Du Li, Leonid Ryvkin, Arne Wessel & Chenchang Zhu
  
• The Controlling L8-Algebra, Cohomology and Homotopy of Embedding Tensors and Lie–Leibniz Triples. Communications in Mathematical Physics, 386(1), 269-304.
  Sheng, Yunhe; Tang, Rong & Zhu, Chenchang
  
• Hamiltonian Lie algebroids over Poisson manifolds. Journal of Symplectic Geometry, 22(4), 695-733.
  Blohmann, Christian; Ronchi, Stefano & Weinstein, Alan
  
• Lie theory and cohomology of relative Rota–Baxter operators. Journal of the London Mathematical Society, 109(2).
  Jiang, Jun; Sheng, Yunhe & Zhu, Chenchang