L. A. Bokut

Algorithmic and Combinatorial Algebra

1. Composition Method for Associative Algebras. 2. Free Lie Algebras. 3. The Composition Method in the Theory of Lie Algebras. 4. Amalgamated Products of Lie Algebras. 5. Decision Problems and Embedding Theorems in the Theory of Varieties of Lie Algebras. 6. The Word Problem and Embedding Theorems in the Theory of the Varieties of Groups. 7. The Problem of Endomorph Reducibility and Relatively Free Groups with the Word Problem Undecidable. 8. The Constructive Method in the Theory of HNN-Extensions. Groups with Standard Normal Form. 9. The Constructive Method for HNN-Extensions and the Conjugacy Problem for Novikov-Boone Groups. A1: Calculations in Free Groups. A2: Algorithmic Properties of Wreath Products of Groups. A3: Survey of the Theory of Absolutely Free Algebras. Bibliography. Index.

Gröbner–Shirshov bases and their calculation

In this survey we give an exposition of the theory of Gröbner–Shirshov bases for associative algebras, Lie algebras, groups, semigroups, Ω-algebras, operads, etc. We mention some new Composition-Diamond lemmas and applications.

Gröbner–Shirshov Bases: From their Incipiency to the Present

In this paper, the history and the main results of the theory of Gröbner–Shirshov bases are given for commutative, noncommutative, Lie, and conformal algebras from the beginning (1962) to the present time. The problem of constructing a base of a free Lie algebra is considered, as well as the problem of studying the structure of free products of Lie algebras, the word problem for Lie algebras, and the problem of embedding an arbitrary Lie algebra into an algebraically closed one. The modern form of the composition-diamond lemma (the CD lemma) is presented. The rewriting systems for groups are considered from the point of view of Gröbner–Shirshov bases. The important role of conformal algebras is treated, the statement of the CD lemma for associative conformal algebras is given, and some examples are considered. An analog of the Hilbert basis theorem for commutative conformalalgebras is stated. Bibliography: 173 titles.

GRÖBNER-SHIRSHOV BASES: SOME NEW RESULTS

In this survey article, we report some new results of Groebner-Shirshov bases, including new Composition-Diamond lemmas, applications of some known Composition-Diamond lemmas and content of some expository papers.

Composition–Diamond lemma for tensor product of free algebras☆

Abstract In this paper, we establish a Composition–Diamond lemma for the tensor product k 〈 X 〉 ⊗ k 〈 Y 〉 of two free algebras over a field. As an application, we construct a Grobner–Shirshov basis in k 〈 X 〉 ⊗ k 〈 Y 〉 by lifting a given Grobner–Shirshov basis in the tensor product k [ X ] ⊗ k 〈 Y 〉 in which k [ X ] is the polynomial algebra.

Markov and Artin Normal Form Theorem for Braid Groups

In this article, we will present the results of Artin–Markov on braid groups by using the Gröbner–Shirshov basis. As a consequence, we can reobtain the normal form of Artin–Markov–Ivanovsky as an easy corollary.

GRÖBNER–SHIRSHOV BASES AND EMBEDDINGS OF ALGEBRAS

In this paper, by using Grobner–Shirshov bases, we show that in the following classes, each (respectively, countably generated) algebra can be embedded into a simple (respectively, two-generated) algebra: associative differential algebras, associative Ω-algebras, associative λ-differential algebras. We show that in the following classes, each countably generated algebra over a countable field k can be embedded into a simple two-generated algebra: associative algebras, semigroups, Lie algebras, associative differential algebras, associative Ω-algebras, associative λ-differential algebras. We give another proofs of the well known theorems: each countably generated group (respectively, associative algebra, semigroup, Lie algebra) can be embedded into a two-generated group (respectively, associative algebra, semigroup, Lie algebra).

GRÖBNER-SHIRSHOV BASES FOR THE BRAID SEMIGROUP

We found Groebner-Shirshov basis for the braid semigroup

Grobner-Shirshov bases for Lie algebras over a commutative algebra

B^+_{n+1}

GRÖBNER–SHIRSHOV BASES FOR FREE INVERSE SEMIGROUPS

. It gives a new algorithm for the solution of the word problem for the braid semigroup and so for the braid group.