Yuqun Chen

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: 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.

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).

Grobner-Shirshov bases for Lie algebras over a commutative algebra

Abstract In this paper we establish a Grobner–Shirshov bases theory for Lie algebras over commutative rings. As applications we give some new examples of special Lie algebras (those embeddable in associative algebras over the same ring) and non-special Lie algebras (following a suggestion of P.M. Cohn (1963) [28] ). In particular, Cohnʼs Lie algebras over the characteristic p are non-special when p = 2 , 3 , 5 . We present an algorithm that one can check for any p, whether Cohnʼs Lie algebras are non-special. Also we prove that any finitely or countably generated Lie algebra is embeddable in a two-generated Lie algebra.

Gröbner–Shirshov bases for semirings

Abstract In the paper we derive a Grobner–Shirshov algorithm for semirings and commutative semirings. As applications, we obtain Grobner–Shirshov bases and A. Blassʼs (1995) and M. Fiore and T. Leinsterʼs (2004) normal forms of the semirings N [ x ] / ( x = 1 + x + x 2 ) and N [ x ] / ( x = 1 + x 2 ) , correspondingly.

Grobner-Shirshov bases for metabelian Lie algebras

Abstract In this paper, we establish the Grobner-Shirshov bases theory for metabelian Lie algebras. As applications, we find the Grobner-Shirshov bases for partial commutative metabelian Lie algebras related to circuits, trees and some cubes.

Gröbner–Shirshov Basis for HNN Extensions of Groups and for the Alternating Group

In this article, we generalize the Shirshovs Composition Lemma by replacing the monomial order for others. By using Gröbner–Shirshov bases, the normal forms of HNN extension of a group and the alternating group are obtained.

Gröbner–Shirshov Bases for Schreier Extensions of Groups

In this article, by using the Gröbner–Shirshov bases, we give characterizations of the Schreier extensions of groups when the group is presented by generators and relations. An algorithm to find the conditions of a group to be a Schreier extension is obtained. By introducing a special total order, we obtain the structure of the Schreier extension by an HNN group.

COMPOSITION-DIAMOND LEMMA FOR λ-DIFFERENTIAL ASSOCIATIVE ALGEBRAS WITH MULTIPLE OPERATORS

In this paper, we establish the Composition-Diamond lemma for λ-differential associative algebras over a field K with multiple operators. As applications, we obtain Grobner–Shirshov bases of free λ-differential Rota–Baxter algebras. In particular, linear bases of free λ-differential Rota–Baxter algebras are obtained and consequently, the free λ-differential Rota–Baxter algebras are constructed by words.