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Quadratic Algebras
Alexander Polishchuk, University of Oregon, Eugene, OR, and Leonid Positselski, Independent University of Moscow, Russia
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University Lecture Series
2005; 159 pp; softcover
Volume: 37
ISBN-10: 0-8218-3834-2
ISBN-13: 978-0-8218-3834-1
List Price: US$39
Member Price: US$31.20
Order Code: ULECT/37
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This book introduces recent developments in the study of algebras defined by quadratic relations. 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 authors give 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.

The book can be used by graduate students and researchers working in algebra and any of the above-mentioned areas of mathematics.

Readership

Graduate students and research mathematicians interested in algebra.

Reviews

"The authors are leading experts in the field, and the book is a rather complete statement of the art of these subjects. Many known results are unified and generalized. The book is recommended to anybody interested in these subjects."

-- Mathematical Reviews

Table of Contents

  • Preliminaries
  • Koszul algebras and modules
  • Operations on graded algebras and modules
  • Poincaré-Birkhoff-Witt bases
  • Nonhomogeneous quadratic algebras
  • Families of quadratic algebras and Hilbert series
  • Hilbert series of Koszul algebras and one-dependent processes
  • DG-algebras and Massey products
  • Bibliography
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