Algebra/topology seminar
Speaker: Sinan Yalin
Title: Moduli spaces of bialgebras, higher Hochschild cohomology and formality
Abstract:
Algebras over props provide a good formalism to parametrize various structures of bialgebras as well as their homotopy version, which naturally appears in problems related to topology, geometry and mathematical physics. A relevant idea to understand their deformation theory is to gather them in a moduli space of algebraic structures in the setting of Toen-Vezzosi's derived algebraic geometry. Infinitesimal deformations of such structures are then controlled by tangent Lie (or L-infinity) algebras naturally providing the corresponding cohomology theories and obstruction theories.
In a work in collaboration with Gregory Ginot, we apply these results to several open problems relating the deformation theory of E_2-algebras with the deformation theory of associative and coassociative bialgebras. In particular, relying on the higher Deligne conjecture (now a theorem), we solve two longstanding conjectures of Gerstenhaber-Schack and Kontsevich: the existence of an E_3-structure refining the deformation complex of a dg bialgebra, and the E_3-formality of this deformation complex in the case of a symmetric bialgebra. This E_3-formality theorem provides, in turn, a new proof of Etingof-Kazdhan deformation quantization independant from the choice of a Drinfeld associator.