About this Title
Alexander Polishchuk, University of Oregon, Eugene, OR and Leonid Positselski, Independent University of Moscow, Moscow, Russia
Publication: University Lecture Series
Publication Year: 2005; Volume 37
ISBNs: 978-0-8218-3834-1 (print); 978-1-4704-2182-3 (online)
MathSciNet review: MR2177131
MSC: Primary 16S37; Secondary 16E30
Quadratic algebras, i.e., algebras defined by quadratic relations, often occur in various areas of mathematics. One of the main problems in the study of these (and similarly defined) algebras is how to control their size. A central notion in solving this problem is the notion of a Koszul algebra, which was introduced in 1970 by S. Priddy and then appeared in many areas of mathematics, such as algebraic geometry, representation theory, noncommutative geometry, $K$-theory, number theory, and noncommutative linear algebra.
The book offers a coherent exposition of the theory of quadratic and Koszul algebras, including various definitions of Koszulness, duality theory, Poincaré-Birkhoff-Witt-type theorems for Koszul algebras, and the Koszul deformation principle. In the concluding chapter of the book, they explain a surprising connection between Koszul algebras and one-dependent discrete-time stochastic processes.
Graduate students and research mathematicians interested in algebra.
Table of Contents
- Chapter 1. Preliminaries
- Chapter 2. Koszul algebras and modules
- Chapter 3. Operations on graded algebras and modules
- Chapter 4. Poincaré-Birkhoff-Witt bases
- Chapter 5. Nonhomogeneous quadratic algebras
- Chapter 6. Families of quadratic algebras and Hilbert series
- Chapter 7. Hilbert series of Koszul algebras and one-dependent processes
- Appendix A. DG-algebras and Massey products