Quadratic Algebras

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Brand new. We distribute directly for the publisher. 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, Poincar??-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. Read more Show Less

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Overview

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.

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Product Details

  • ISBN-13: 9780821838341
  • Publisher: American Mathematical Society
  • Publication date: 12/28/2005
  • Series: University Lecture Series , #37
  • Pages: 159
  • Product dimensions: 6.80 (w) x 9.90 (h) x 0.40 (d)

Table of Contents

Ch. 1 Preliminaries 1
Ch. 2 Koszul algebras and modules 19
Ch. 3 Operations on graded algebras and modules 55
Ch. 4 Poincare-Birkhoff-Witt bases 81
Ch. 5 Nonhomogeneous quadratic algebras 101
Ch. 6 Families of quadratic algebras and Hilbert series 119
Ch. 7 Hilbert series of Koszul algebras and one-dependent processes 133
App. A DG-algebras and Massey products 153
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