Gröbner-Shirshov bases for quantum enveloping algebras

Bokut, L. A.; Malcolmson, Peter

doi:10.1007/bf02785535

Cited by 39 publications

(29 citation statements)

References 11 publications

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“…Different approaches were developed by Ringel [191,192], Green [110], and Kharchenko [131][132][133][134][135]. GS bases of quantum enveloping algebras are unknown except for the case A n , see [55,86,195,220] [151]). New approaches to plactic monoids via GS bases in the alphabets of row and column generators are found in [29].…”

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

confidence: 99%

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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%

“…Consider the set S + consisting of Jimbo's relations: [86] shows that S + is a GS basis for k X | S + = U + q (A N ) [55]. The proof is different from the argument of Bokut and Malcolmson [55].…”

Section: Corollary 6 (Kac [117]) For Every Symmetrizable Cartan Matrimentioning

confidence: 99%

Gröbner–Shirshov bases and their calculation

2014

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“…The original method appeared in studies of simple finite dimensional Lie algebras. Then it has been extended into the field of quantum algebra in a lot of publications (see, for example [9], [31], [42]). By means of this method the investigation of the Drinfeld-Jimbo enveloping algebra amounts to a consideration of its positive and negative homogeneous components, quantum Borel sub-algebras.…”

Section: Introductionmentioning

confidence: 99%

A Combinatorial approach to the quantification of Lie algebras

Kharchenko¹

2002

Pacific J. Math.

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We propose a notion of a quantum universal enveloping algebra for any Lie algebra defined by generators and relations which is based on the quantum Lie operation concept. This enveloping algebra has a PBW basis that admits a monomial crystallization by means of the Kashiwara idea. We describe all skew primitive elements of the quantum universal enveloping algebras for the classical nilpotent algebras of the infinite series defined by the Serre relations and prove that the above set of PBW-generators for each of these enveloping algebras coincides with the Lalonde-Ram basis of the ground Lie algebra with a skew commutator in place of the Lie operation. The similar statement is valid for Hall-Shirshov basis of any Lie algebra defined by one relation, but it is not so in the general case.

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“…By [8], we can get a Gröbner-Shirshov basis T for U(G(A)), where T consists of the following relations:…”

Section: Verma Module Over Kac-moody Algebrasmentioning

confidence: 99%

Composition-diamond lemma for modules

Chen

Zhong

2010

Czech Math J

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In this paper we give some relationships among the Gröbner-Shirshov bases in free associative algebras, free left modules and "double-free" left modules (free modules over a free algebra). We give the Chibrikov's Composition-Diamond lemma for modules and show that Kang-Lee's Composition-Diamond lemma follows from this lemma. As applications, we also deal with highest weight module over the Lie algebra sl 2 , Verma module over a Kac-Moody algebra, Verma module over Lie algebra of coefficients of a free conformal algebra and the universal enveloping module for a Sabinin algebra.

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Gröbner-Shirshov bases for quantum enveloping algebras

Cited by 39 publications

References 11 publications

Gröbner–Shirshov bases and their calculation

Gröbner–Shirshov bases and their calculation

A Combinatorial approach to the quantification of Lie algebras

Composition-diamond lemma for modules

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