II - Deformations of algebras in noncommutative geometry

Summary

These are significantly expanded lecture notes for the author’s minicourse at MSRI in June 2012. In these notes, following, e.g., [98, 156, 101], we give an example-motivated review of the deformation theory of associative algebras in terms of the Hochschild cochain complex as well as quantization of Poisson structures, and Kontsevich’s formality theorem in the smooth setting. We then discuss quantization and deformation via Calabi–Yau algebras and potentials. Examples discussed include Weyl algebras, enveloping algebras of Lie algebras, symplectic reflection algebras, quasihomogeneous isolated hypersurface singularities (including du Val singularities), and Calabi–Yau algebras.

The exercises are a great place to learn the material more detail. There are detailed solutions provided, which the reader is encouraged to consult if stuck. There are some starred (parts of) exercises which are quite difficult, so the reader can feel free to skip these (or just glance at them).

There are a lot of remarks, not all of which are essential; so many of them can be skipped on a first reading.

We will work throughout over a field k. A lot of the time we will need it to have characteristic zero; feel free to assume this always.

These notes are based on my lectures for MSRI’s 2012 summer graduate workshop on noncommutative algebraic geometry. I am grateful to MSRI and the organizers of the Spring 2013 MSRI program on noncommutative algebraic geometry and representation theory for the opportunity to give these lectures; to my fellow instructors and scientific organizers Gwyn Bellamy, Dan Rogalski, and Michael Wemyss for their help and support; to the excellent graduate students who attended the workshop for their interest, excellent questions, and corrections; and to Chris Marshall and the MSRI staff for organizing the workshop. I am grateful to Daniel Kaplan and Michael Wong for carefully studying these notes and providing many corrections.