Anti-commutative Gröbner-Shirshov basis of a free Lie algebra

Bokut, L. A.; Chen, Yuqun; Yu, Li

doi:10.1007/s11425-008-0168-y

Cited by 14 publications

(13 citation statements)

References 16 publications

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“…They created a theory of GS bases for every algebra of this class; thus, they found a linear basis of every quotient of R. [194], see also [15]. Two anticommutative GS bases of Lie K (X ) were found in [34,37], which yields the Hall and Lyndon-Shirshov linear bases respectively. -The Lie k-algebras presented by Chevalley generators and defining relations of types A n , B n , C n , D n , G 2 , F 4 , E 6 , E 7 , and E 8 .…”

Section: ((U)(v)) > (V)mentioning

confidence: 99%

Gröbner–Shirshov bases and their calculation

2014

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In this survey we give an exposition of the theory of Gröbner-Shirshov bases for associative algebras, Lie algebras, groups, semigroups, -algebras, operads, etc. We mention some new Composition-Diamond lemmas and applications.Keywords Gröbner basis · Gröbner-Shirshov basis · Composition-Diamond lemma · Congruence · Normal form · Braid group · Free semigroup · Chinese monoid · Plactic monoid · Associative algebra · Lie algebra · Lyndon-Shirshov basis · Lyndon-Shirshov word · PBW theorem · -algebra · Dialgebra · Semiring · Pre-Lie algebra · Rota-Baxter algebra · Category · ModuleSupported by the NNSF of China (11171118) The set of all associative Lyndon-Shirshov words in X NLSW(X)The set of all non-associative Lyndon-Shirshov words in X PBW theoremThe Poincare-Birkhoff-Witt theorem X * The free monoid generated byThe free commutative monoid generated by X X * *The set of all non-associative words (u) in X gp X |S The group generated by X with defining relations S sgp X |S The semigroup generated by X with defining relations S k A field K A commutative algebra over k with unity k X The free associative algebra over k generated by X k X |S The associative algebra over k with generators X and defining relations S S c A Gröbner-Shirshov completion of S I d(S)The ideal generated by a set S sThe maximal word of a polynomial s with respect to some ordering < Irr(S)The set of all monomials avoiding the subwords for all s ∈ S k [X ] The polynomial algebra over k generated by X Lie(X )The free Lie algebra over k generated by X Lie K (X )The free Lie algebra generated by X over a commutative algebra K

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Section: ((U)(v)) > (V)mentioning

confidence: 99%

Gröbner–Shirshov bases and their calculation

2014

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“…In this paper we give another approach to definition of LS basis and LS words following Shirshov's (Gröbner-Shirshov bases) theory for anti-commutative algebras [30]. In our previous paper [7] we gave the same kind of results for the Hall basis. To be more precise, in that paper, we had found an anti-commutative Gröbner-Shirshov basis of a free anticommutative (non-associative) algebra AC(X), such that the corresponding irreducible monomials (not containing the maximal monomials of the Gröbner-Shirshov basis) are exactly the Hall monomials.…”

Section: Introductionmentioning

confidence: 94%

Lyndon–Shirshov basis and anti-commutative algebras

Bokut¹,

Chen²,

Liu³

2013

Journal of Algebra

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Chen, Fox, Lyndon 1958 [10] and Shirshov 1958 [29] introduced non-associative Lyndon-Shirshov words and proved that they form a linear basis of a free Lie algebra, independently. In this paper we give another approach to definition of Lyndon-Shirshov basis, i.e., we find an anti-commutative Gröbner-Shirshov basis S of a free Lie algebra such that Irr(S) is the set of all non-associative Lyndon-Shirshov words, where Irr(S) is the set of all monomials of N(X), a basis of the free anti-commutative algebra on X, not containing maximal monomials of polynomials from S. Following from Shirshov's anti-commutative Gröbner-Shirshov bases theory [32], the set Irr(S) is a linear basis of a free Lie algebra.

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“…In the paper of L. A. Bokut, Y. Q. Chen and Y. Li [11], an application of Shirshov's CD-lemma for anti-commutative algebras (see [55]) is given. This application gives an anti-commutative Gröbner-Shirshov basis of a free Lie algebra.…”

Section: Anti-commutative Algebrasmentioning

confidence: 99%

“…By using this theorem, a Gröbner-Shirshov basis S in AC(X) is given in [11] which shows that the Hall words in X forms a basis for the free Lie algebra Lie(X), where …”

Section: S ∈ S and [Asb] Is A Normal S-word} Is A Basis Of The Algebrmentioning

confidence: 99%