### Deformation Theory Seminar

Wednesday, February 14, 2018 - 2:00pm

#### Murray Gerstenhaber

U Penn

Location

University of Pennsylvania

DRL 4C2

Universal Deformation Formulas (UDFs) for the deformation of associative algebras  play a key role in deformation quantization. Here we present examples  for certain classes of infinitesimals. A basic separable 2-cocycle $F$ of an associative algebra $A$ is one for which there exist commuting derivations $D_1,\dots, D_n$ of $A$ such that $F = \sum_{ij}a_{ij}D_i \smile D_j$, where the $a_{ij}$ are central elements of $A$. When $A$ is defined over the rationals,  there is a natural definition of the exponential of such a cocycle. With this $\exp \hbar F$ defines a formal one-parameter family of deformations of $A$, where $\hbar$ is a deformation parameter. The rational quantization of smooth functions on a smooth manifold using a bivector field as an infinitesimal deformation is a special case.