Deformation Theory of Algebras and Their Diagrams
About this Title
Martin Markl, Academy of Sciences of the Czech Republic, Praha, Czech Republic
Publication: CBMS Regional Conference Series in Mathematics
Publication Year 2012: Volume 116
ISBNs: 978-0-8218-8979-4 (print); 978-0-8218-9192-6 (online)
MathSciNet review: MR2931635
MSC: Primary 16S80; Secondary 13D10, 16E40, 17B55
This book brings together both the classical and current aspects of deformation theory. The presentation is mostly self-contained, assuming only basic knowledge of commutative algebra, homological algebra and category theory. In the interest of readability, some technically complicated proofs have been omitted when a suitable reference was available. The relation between the uniform continuity of algebraic maps and topologized tensor products is explained in detail, however, as this subject does not seem to be commonly known and the literature is scarce.
The exposition begins by recalling Gerstenhaber's classical theory for associative algebras. The focus then shifts to a homotopy-invariant setup of Maurer-Cartan moduli spaces. As an application, Kontsevich's approach to deformation quantization of Poisson manifolds is reviewed. Then, after a brief introduction to operads, a strongly homotopy Lie algebra governing deformations of (diagrams of) algebras of a given type is described, followed by examples and generalizations.
Graduate students and research mathematicians interested in deformations of algebras, moduli spaces, algebraic geometry, and/or algebraic topology.
Table of Contents
- Chapter 1. Basic notions
- Chapter 2. Deformations and cohomology
- Chapter 3. Finer structures of cohomology
- Chapter 4. The gauge group
- Chapter 5. The simplicial Maurer-Cartan space
- Chapter 6. Strongly homotopy Lie algebras
- Chapter 7. Homotopy invariance and quantization
- Chapter 8. Brief introduction to operads
- Chapter 9. $L_\infty $-algebras governing deformations
- Chapter 10. Examples
- 11. Index