Gröbner-Shirshov Bases: Normal Forms, Combinatorial and Decision Problems in Algebra

✍ Scribed by Leonid Bokut; Yuqun Chen; Kyriakos Kalorkoti

• Publisher: WSPC
• Year: 2020
• Tongue: English
• Leaves: 307
• Category: Library

✦ Synopsis

The book is about (associative, Lie and other) algebras, groups, semigroups presented by generators and defining relations. They play a great role in modern mathematics. It is enough to mention the quantum groups and Hopf algebra theory, the Kac-Moody and Borcherds algebra theory, the braid groups and Hecke algebra theory, the Coxeter groups and semisimple Lie algebra theory, the plactic monoid theory. One of the main problems for such presentations is the problem of normal forms of their elements. Classical examples of such normal forms give the Poincaré-Birkhoff-Witt theorem for universal enveloping algebras and Artin-Markov normal form theorem for braid groups in Burau generators. What is now called Gröbner-Shirshov bases theory is a general approach to the problem. It was created by a Russian mathematician A I Shirshov (1921-1981) for Lie algebras (explicitly) and associative algebras (implicitly) in 1962. A few years later, H Hironaka created a theory of standard bases for topological commutative algebra and B Buchberger initiated this kind of theory for commutative algebras, the Gröbner basis theory. The Shirshov paper was largely unknown outside Russia. The book covers this gap in the modern mathematical literature. Now Gröbner-Shirshov bases method has many applications both for classical algebraic structures (associative, Lie algebra, groups, semigroups) and new structures (dialgebra, pre-Lie algebra, Rota-Baxter algebra, operads). This is a general and powerful method in algebra.

✦ Table of Contents

Contents
Preface
Chapter 1. Introduction
1.1. The Euclidean algorithm
1.2. The Gaussian elimination algorithm
1.3. Systems of polynomial equations
Chapter 2. Free algebras
2.1. Semigroups and groups
2.2. Noncommutative polynomials
2.3. Free commutative algebras
2.4. Free Lie algebras
Chapter 3. Composition-Diamond Lemma
3.1. Grobner bases for commutative algebras
3.2. Grobner–Shirshov bases for associative algebras
3.3. Grobner–Shirshov bases for tensor product of free algebras
3.4. Grobner–Shirshov bases for Lie algebras
Chapter 4. Applications of Grobner–Shirshov bases
4.1. Normal form for semigroups and groups
4.2. Free product of algebras and groups
4.3. Modules over associative algebras
4.4. Replicated algebras
4.5. Associative conformal algebras
4.6. Lifting of Grobner bases to Grobner–Shirshov bases
4.7. Lie algebra with unsolvable word problem
4.8. Grobner–Shirshov basis for the Drinfeld–Kohno Lie algebra
Chapter 5. Grobner-Shirshov bases for Lie algebras over a commutative algebra
5.1. Preliminaries
5.2. Composition-Diamond lemma for LiekY
5.3. Examples of Shirshov and Cartier
5.4. Cohn conjecture
5.5. Other applications
Chapter 6. Decision problems for groups
6.1. Decision problems
6.2. Computability
6.3. Group theoretic tools
6.4. Unsolvability of the word and conjugacy problems for finitely presented groups
6.5. The word problem and r.e. Turing degrees
6.6. The Higman embedding theorem
6.7. The conjugacy problem and r.e. Turing degrees
Chapter 7. (Co)Homology and Gr¨obner–Shirshov basis
7.1. Preliminaries
7.2. Algebraic discrete Morse theory
7.3. Group extensions
7.4. (Singular) Extensions of Algebras
Bibliography
Index