**University Lecture Series**

Volume: 37;
2005;
159 pp;

MSC: Primary 16; 13; 60;

**Electronic ISBN: 978-1-4704-2182-3
Product Code: ULECT/37.E**

List Price: $41.00

AMS Member Price: $32.80

MAA Member Price: $36.90

#### Supplemental Materials

# Quadratic Algebras

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*Alexander Polishchuk; Leonid Positselski*

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.

#### Reviews & Endorsements

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

# Table of Contents

## Quadratic Algebras

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Introduction vii8 free
- Chapter 1. Preliminaries 114 free
- Chapter 2. Koszul algebras and modules 1932
- 1. Koszulness 1932
- 2. Hilbert series 2134
- 3. Koszul complexes 2538
- 4. Distributivity and n-Koszulness 2942
- 5. Homomorphisms of algebras and Koszulness. I 3245
- 6. Homomorphisms of algebras and Koszulness. II 3750
- 7. Koszul algebras in algebraic geometry 4053
- 8. Infinitesimal Hopf algebra associated with a Koszul algebra 4558
- 9. Koszul algebras and monoidal functors 4962
- 10. Relative Koszulness of modules 5366

- Chapter 3. Operations on graded algebras and modules 5568
- Chapter 4. Poincaré–Birkhoff–Witt Bases 8194
- 2. PBW-theorem 8295
- 3. PBW-bases and Koszulness 8497
- 4. PBW-bases and operations on quadratic algebras 8598
- 5. PBW-bases and distributing bases 8699
- 6. Hilbert series of PBW-algebras 87100
- 7. Filtrations on quadratic algebras 88101
- 8. Commutative PBW-bases 91104
- 9. Z-algebras 95108
- 10. Z-PBW-bases 96109
- 11. Three-dimensional Sklyanin algebras 98111
- 1. PBW-bases 8194

- Chapter 5. Nonhomogeneous Quadratic Algebras 101114
- 1. Jacobi identity 101114
- 2. Nonhomogeneous PBW-theorem 103116
- 3. Nonhomogeneous quadratic modules 104117
- 4. Nonhomogeneous quadratic duality 105118
- 5. Examples 108121
- 6. Nonhomogeneous duality and cohomology 111124
- 7. Bar construction for CDG- lgebras and modules 112125
- 8. Homology of completed cobar-complexes 117130

- Chapter 6. Families of quadratic algebras and Hilbert series 119132
- 1. Openness of distributivity 119132
- 2. Deformations of Koszul algebras 120133
- 3. Upper bound for the number of Koszul Hilbert series 122135
- 4. Generic quadratic algebras 123136
- 5. Examples with small dim A[sub(1)] and dim A[sub(2)] 125138
- 6. Koszulness is not constructible 127140
- 7. Families of quadratic algebras over schemes 128141

- Chapter 7. Hilbert series of Koszul algebras and one-dependent processes 133146
- 1. Conjectures on Hilbert series of Koszul algebras 133146
- 2. Koszul inequalities 135148
- 3. Koszul duality and inequalities 138151
- 4. One-dependent processes 139152
- 5. PBW-algebras and two-block-factor processes 142155
- 6. Operations on one-dependent processes 143156
- 7. Hilbert space representations of one-dependent processes 146159
- 8. Hilbert series of one-dependent processes 147160
- 9. Hermitian construction of one- ependent processes 149162
- 10. Modules over one-dependent processes 151164

- Appendix A. DG-algebras and Massey products 153166
- Bibliography 155168
- Back Cover Back Cover1176