Quadratic Algebras ( University Lecture Series )

Publication series ：University Lecture Series

Author： Alexander Polishchuk;Leonid Positselski  

Publisher： American Mathematical Society‎

Publication year： 2005

E-ISBN: 9781470421823

P-ISBN(Paperback): 9780821838341

Subject： O153.3 Ring theory

Keyword： Algebra and Algebraic Geometry

Language： ENG

Access to resources Favorite

Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.

Quadratic Algebras

Description

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.

Chapter

Cover

Title

Copyright

Contents

Introduction

Chapter 1. Preliminaries

0. Conventions and notation

1. Bar constructions

2. Quadratic algebras and modules

3. Diagonal cohomology

4. Minimal resolutions

5. Low-dimensional cohomology

6. Lattices and distributivity

7. Lattices of vector spaces

Chapter 2. Koszul algebras and modules

1. Koszulness

2. Hilbert series

3. Koszul complexes

4. Distributivity and n-Koszulness

5. Homomorphisms of algebras and Koszulness. I

6. Homomorphisms of algebras and Koszulness. II

7. Koszul algebras in algebraic geometry

8. Infinitesimal Hopf algebra associated with a Koszul algebra

9. Koszul algebras and monoidal functors

10. Relative Koszulness of modules

Chapter 3. Operations on graded algebras and modules

1. Direct sums, free products and tensor products

2. Segre products and Veronese powers. I

3. Segre products and Veronese powers. II

4. Internal cohomomorphism

5. Koszulness cannot be checked using Hilbert series

Chapter 4. Poincaré–Birkhoff–Witt Bases

2. PBW-theorem

3. PBW-bases and Koszulness

4. PBW-bases and operations on quadratic algebras

5. PBW-bases and distributing bases

6. Hilbert series of PBW-algebras

7. Filtrations on quadratic algebras

8. Commutative PBW-bases

9. Z-algebras

10. Z-PBW-bases

11. Three-dimensional Sklyanin algebras

1. PBW-bases

Chapter 5. Nonhomogeneous Quadratic Algebras

1. Jacobi identity

2. Nonhomogeneous PBW-theorem

3. Nonhomogeneous quadratic modules

4. Nonhomogeneous quadratic duality

5. Examples

6. Nonhomogeneous duality and cohomology

7. Bar construction for CDG- lgebras and modules

8. Homology of completed cobar-complexes

Chapter 6. Families of quadratic algebras and Hilbert series

1. Openness of distributivity

2. Deformations of Koszul algebras

3. Upper bound for the number of Koszul Hilbert series

4. Generic quadratic algebras

5. Examples with small dim A[sub(1)] and dim A[sub(2)]

6. Koszulness is not constructible

7. Families of quadratic algebras over schemes

Chapter 7. Hilbert series of Koszul algebras and one-dependent processes

1. Conjectures on Hilbert series of Koszul algebras

2. Koszul inequalities

3. Koszul duality and inequalities

4. One-dependent processes

5. PBW-algebras and two-block-factor processes

6. Operations on one-dependent processes

7. Hilbert space representations of one-dependent processes

8. Hilbert series of one-dependent processes

9. Hermitian construction of one- ependent processes

10. Modules over one-dependent processes

Appendix A. DG-algebras and Massey products

Bibliography

Back Cover

The users who browse this book also browse


No browse record.