Formal exponential maps and Hochschild cohomology associated with dg manifolds
Open Access
- Author:
- Seol, Seokbong
- Graduate Program:
- Mathematics (PHD)
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- June 02, 2022
- Committee Members:
- Ping Xu, Co-Chair & Dissertation Advisor
Mathieu Stienon, Co-Chair & Dissertation Advisor
Nigel Higson, Major Field Member
Adrian Ocneanu, Major Field Member
Martin Bojowald, Outside Unit & Field Member
Alexei Novikov, Professor in Charge/Director of Graduate Studies - Keywords:
- differential graded manifolds
Hochschild cohomology
Atiyah class
homotopy Lie algebras
Duflo theorem - Abstract:
- In this dissertation, we study two different aspects related to dg manifolds. The first part is devoted to the study of the relation between ‘formal exponential maps,’ the Atiyah class, and Kapranov L-infinity algebras associated with dg manifolds in the smooth context. We prove that, for a dg manifold, a ‘formal exponential map’ exists if and only if the Atiyah class vanishes. Inspired by Kapranov’s construction of a homotopy Lie algebra associated with the holomorphic tangent bundle of a complex manifold, we prove that the space of vector fields on a dg manifold admits an L-infinity algebra structure, unique up to isomorphism, whose unary bracket is the Lie derivative w.r.t. the homological vector field, whose binary bracket is a 1-cocycle representative of the Atiyah class, and whose higher multibrackets can be computed by a recursive formula. For the dg manifold arising from a complex manifold X, we prove that this L-infinity algebra structure is quasi- isomorphic to the standard L-infinity algebra structure on the Dolbeault complex. The second part is devoted to the study of Hochschild cohomology of a dg manifold arising from a Lie algebra in terms of Keller admissible triples. We prove that a Keller admissible triple induces an isomorphism of Gerstenhaber algebras between Hochschild cohomologies of the direct-sum type for dg algebras. As an application, we show that the Hochschild cohomology of the dg algebra of smooth functions on a dg manifold arising from a Lie algebra g is isomorphic to the Hochschild cohomology of the universal enveloping algebra Ug. Furthermore, we give a new concrete proof of the Kontsevich–Duflo theorem for finite-dimensional Lie algebras.