Nonformal deformations of algebras of holomorphic functions

Abstract: Formal deformations of associative algebras are by now classical and relatively well-studied objects. They were introduced by Gerstenhaber in 1964, and they are interesting especially because of their relation to deformation quantization. By contrast, the theory of nonformal deformations is now at a much earlier stage of development. Roughly, a general feature of all existing approaches to nonformal deformations, which distinguishes them from formal deformations, is that the role of the "base" ring is now played by a certain algebra of functions (continuous, or smooth, or holomorphic...) rather than by the algebra of formal power series. This makes nonformal deformations quite attractive from the physical point of view, because they allow evaluating the deformed star product at concrete nonzero values of the deformation parameter (Planck's constant). In this talk, our main objects will be nonformal (or, more exactly, holomorphic) deformations of the algebras of holomorphic functions on the polydisc and on the ball in $\mathbb{C}^n$. We will discuss some properties of such deformations and their relation to formal deformations. If time permits, we will compare our approach to holomorphic deformations with S. Waldmann's approach, which is better adapted to deformation quantization, but which applies only to some proper subalgebras of the algebras of holomorphic functions.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( slides | video )

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Noncommutative geometry in NYC

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