Noncommutative Deformation Theory is aimed at mathematicians and physicists studying the local structure of moduli spaces in algebraic geometry. This book introduces a general theory of noncommutative deformations, with applications to the study of moduli spaces of representations of associative algebras and to quantum theory in physics. An essential part of this theory is the study of obstructions of liftings of representations using generalised (matric) Massey products. Suitable for researchers in algebraic geometry and mathematical physics interested in the workings of noncommutative algebraic geometry, it may also be useful for advanced graduate students in these fields.
Classical Deformation Theory. Noncommutative Algebras and Simple Modules. Noncommutative Deformation Theory. The Noncommutative Phase Space. A Cosmological Toy Model. Moduli of Endomorphisms of Rank Three.
[Noncommutative Deformation Theory] gives both an introduction to noncommutative deformation theory, and an application of the machinery in mathematical physics. […] The material on noncommutative deformation theory and Massey products is a generalisation and combination of several papers of the authors, culminating in a clean development of a deformation theory of multiple objects in abelian categories. It is nicely explained which kinds of properties one needs to impose on the objects and the categories to mimic the results in commutative deformation theory, such as the existence of a pro-representable hull. The material on mathematical physics is based on work of the second author, and is more recent.
- Pieter Belmans, Springer Nature Review, October 2017
The book is well written, with many detailed examples, making it ideal for graduate students and mathematicians interested in the subject matter. The reviewer looks forward to future applications and developments in this important subject.
-Adam Nyman, Mathematical Reviews, May 2018