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