Higher Lie Theory

• Applicant: Dr. Leonid Ryvkin
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 539126009

Lie theory concerns itself with the relation between global symmetries, like rotations of space, and their infinitesimal counterparts, indicating in which direction each point of space moves. Classically the global symmetries are encoded by Lie groups and the infinitesimal ones by Lie algebras. These two types of objects correspond to each other: A Lie group can be differentiated into a Lie algebra and each (finite-dimensional) Lie algebra can be integrated into a Lie group. This project concerns itself with higher theory which aims to establish a higher analogue of the above correspondence. Therein, Lie groups are replaced by higher Lie groupoids (Lie n-groupoids) which can be modeled by kan simplicial manifolds and Lie algebras are substituted by Lie n-algebroids, or differential graded manifolds. The aim of this project is to advance the correspondence between higher Lie groupoids and higher Lie algebroids. On the differentiation side, we intend to establish general equations for the bracket structure of the Lie n-algebroid associated to a Lie n-groupoid. On the side of integration, our objective is to provide a procedure by applying techniques from the theory of singular foliations. Furthermore we will investigate, when this procedure yields finite-dimensional and smooth objects.

• DFG Programme: Research Grants