عنوان

Algorithmic and Combinatorial Algebra

9401120021

9789401120029

b594972

Algorithmic and Combinatorial Algebra

[Book]

by L.A. Bokut', G.P. Kukin.

Dordrecht

Springer Netherlands

1994

(xvi, 386 pages)

Mathematics and Its Applications, 255.

1 Composition Method for Associative Algebras --; 1.1 Introduction --; 1.2 Free Semigroups and Free Groups --; 1.3 The Composition Lemma --; 1.4 The Composition Lemma for Semigroup Algebras --; 1.5 The Generalised Clifford Algebra and Some Other Examples --; 1.6 Finite-Dimensional Representation of a Generalised Clifford Algebra --; 1.7 More on Embeddings into Simple Algebras --; 2 Free Lie Algebras --; 2.1 Introduction --; 2.2 The Definition of Free Lie Algebras --; 2.3 Projective Algebras --; 2.4 Elementary Transformations and Automorphisms of Free Algebras --; 2.5 Lie Algebra Derivations --; 2.6 The Ideal of Codimension 1 --; 2.7 Constructing Generators for an Arbitrary Subalgebra in a Free Lie Algebra --; 2.8 The Shirshov Theorem on Free Lie Algebra Subalgebras --; 2.9 Automorphisms of Free Lie Algebras of Finite Rank --; 2.10 A Criterion for a Lie Algebra to be Free --; 2.11 Bases of a Free Lie Algebra --; 2.12 Construction of Free Lie Algebras --; 2.13 Universal Enveloping Associative Algebra --; 2.14 On Subrings of Free Rings with Operators --; 2.15 Embedding Lie Rings into Associative Rings with Operators --; 2.16 Restricted Lie Algebras --; 2.17 Relatively Free Lie Algebras --; 2.18 Embedding Countable-Dimensional Lie Algebras into Lie Algebras with Two Generators --; 2.19 The Residual Finiteness of Associative and Lie Algebras --; 2.20 Residual Finiteness of Free Rings and Algebras --; 3 The Composition Method in the Theory of Lie Algebras --; 3.1 Introduction --; 3.2 The Composition Lemma --; 3.3 Formulation of Decision Problems. One-Relator Lie Algebras --; 3.4 Embedding Lie Algebras into Simple Lie Algebras --; 3.5 The Main Algorithmic Problems for Lie Algebras are Unsolvable --; 3.6 Unrecognizable Markov Properties for Finitely Presented Lie Algebras --; 3.7 Defining Relations of a Subalgebra --; 3.8 Residual Finiteness and Decision Problems --; 3.9 On Residual Finiteness of One-Relator Lie Algebras --; 3.10 Constructing Free Resolutions --; 3.11 Cohomological Dimension of Almost Free Lie Algebras --; 4 Amalgamated Products of Lie Algebras --; 4.1 Introduction --; 4.2 Definition of Amalgamated Products --; 4.3 Constructing the Free Product of Associative Algebras without 1 --; 4.4 Constructing the Amalgamated Product of Lie Algebras --; 4.5 Subalgebras of the Free Product of Lie Algebras --; 4.6 Generators of a Subalgebra of the Free Product of Lie Algebras --; 4.7 Decomposition of a Free Product into the Sum of Two Subalgebras, One Free --; 4.8 Decomposition of a Subalgebra of a Free Product into the Sum of Two Subalgebras, One Free --; 4.9 The Theorem on Subalgebras of an Amalgamated Product of Lie Algebras --; 4.10 Free Subalgebras in a Free Product of Lie Algebras --; 4.11 The Case in Which the Kurosh Formula Almost Holds --; 4.12 Supplementary Facts on Free Products --; 4.13 Residual Finiteness of Free Products of Associative and Lie Algebras --; 4.14 Residual Finiteness of Free Soluble Lie Algebras with respect to Inclusion into Finitely Generated Subalgebras --; 4.15 On Residual Finiteness of Free Soluble Groups with respect to Inclusion into Finitely Generated Subgroups --; 4.16 On Residual Properties of Free Products of Lie Algebras. Central Systems in Free Products --; 5 The Problem of Endomorph Reducibility and Relatively Free Groups with the Word Problem Unsolvable --; 5.1 Introduction --; 5.2 When the Problem of Endomorph Reducibility for Relatively Free Rings is Unsolvable --; 5.3 When the Problem of Endomorph Reducibility is Solvable --; 5.4 The Problem of Endomorph Reducibility for Relatively Free Groups --; 5.5 The Variety R Included in NN --; 5.6 The Free Group T of the Variety

This major volume presents an authoritative overview of developments in algorithmic and combinatorial algebra which have been achieved during the last forty years or so, with an emphasis on the results obtained by the Novosibirsk school of A.I. Mal'tchev and A.I. Shirshov and followers. The book has nine chapters. These deal with Applications of the Composition (or Diamond) Lemma to associative and Lie algebras (Chapters 1 and 3), to subalgebras of free Lie algebras and free products of Lie algebras (Chapters 2 and 4), to word problems and embedding theorems in varieties of Lie algebras and groups (Chapters 5--7) and to the constructive theory of HNN-extensions and its use in analysing the word and conjugacy problems in the Novikov--Boone groups (Chapters 8 and 9). Many results described here appear for the first time in a monograph. The volume concludes with a discussion of three applications. For graduate students and researchers whose work involves algorithmic and combinatorial algebra and its applications.

Algebra.

Algorithms.

Mathematics.

by L.A. Bokut', G.P. Kukin.

G P Kukin

L A Bokut'

[Book]

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