Gröbner-Shirshov bases for Rota-Baxter algebras

Bokut, L. A.; Chen, Yu; Deng, Xiaoyuan

doi:10.1007/s11202-010-0097-1

Cited by 38 publications

(41 citation statements)

References 56 publications

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“…[35], -Associative Rota-Baxter algebras over k with char k = 0, Bokut et al [32]; -L-algebras, Bokut et al [33]; -Associative -algebras, Bokut et al [41]; -Associative differential -algebras, Qiu and Chen [185]; --algebras, Bokut et al [33]; -Differential Rota-Baxter commutative associative algebras, Guo et al [111]; -Semirings, Bokut et al [40]; -Modules over an associative algebra, Golod [108], Green [109], Kang and Lee [123,124], Chibrikov [90]; -Small categories, Bokut et al [36]; -Non-associative algebras, Shirshov [206]; -Non-associative algebras over a commutative algebra, Chen et al [81]; -Commutative non-associative algebras, Shirshov [206]; -Anti-commutative non-associative algebras, Shirshov [206]; -Symmetric operads, Dotsenko and Khoroshkin [98].…”

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

confidence: 99%

“…Some new CD-lemmas for -algebras have appeared: for associative conformal algebras [45] and n-conformal algebras [43], for the tensor product of free algebras [30], for metabelian Lie algebras [75], for associative -algebras [41], for color Lie superalgebras and Lie p-superalgebras [165,166], for Lie superalgebras [167], for associative differential algebras [76], for associative Rota-Baxter algebras [32], for L-algebras [33], for dialgebras [38], for pre-Lie algebras [35], for semirings [40], for commutative integro-differential algebras [102], for difference-differential modules and differencedifferential dimension polynomials [225], for λ-differential associative -algebras [185], for commutative associative Rota-Baxter algebras [186], for algebras with differential type operators [111].…”

Section: Cd-lemmas For -Algebrasmentioning

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%

Section: Cd-lemmas For -Algebrasmentioning

confidence: 99%

Gröbner–Shirshov bases and their calculation

2014

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“…The study of Rota-Baxter type operators, however, is more challenging than their differential counterpart as can be expected already by comparing integral calculus with differential calculus. Nevertheless the method of Gröbner-Shirshov bases has been successfully applied to the study of Rota-Baxter algebras, differential Rota-Baxter algebras and integro-differential algebras [11,13,22]. We show in this paper that the methods of rewriting systems and Gröbner-Shirshov bases apply more generally to Rota-Baxter type algebras as well.…”

Section: P(x)p(y) = P(xp(y) + P(x)y − P(xy)) Leroux's Td Operator P(mentioning

confidence: 85%

“…Thus, we have achieved a uniform construction of the free objects for all the 14 categories of operated algebras whose defining identities are listed in the conjecture. Our construction generalizes and includes as special cases the known constructions for various operated algebras [11,13,14,18,31].…”

Section: P(x)p(y) = P(xp(y) + P(x)y − P(xy)) Leroux's Td Operator P(mentioning

confidence: 99%

Rota-Baxter type operators, rewriting systems and Gröbner-Shirshov bases

Guo

Sit

Gao

et al. 2015

ACM Commun. Comput. Algebra

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Abstract. In this paper we apply the methods of rewriting systems and Gröbner-Shirshov bases to give a unified approach to a class of linear operators on associative algebras. These operators resemble the classic Rota-Baxter operator, and they are called Rota-Baxter type operators. We characterize a Rota-Baxter type operator by the convergency of a rewriting system associated to the operator. By associating such an operator to a Gröbner-Shirshov basis, we obtain a canonical basis for the free algebras in the category of associative algebras with that operator. This construction include as special cases several previous ones for free objects in similar categories, such as those of Rota-Baxter algebras and Nijenhuis algebras.

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“…The classification of relative locations can be quite subtle in some structures, especially when a non-identity operator is present, such as in differential algebras, Rota-Baxter algebras [12] and, more generally, operated algebras [5,7,13,17]. Further by [11] free operated semigroups have natural combinatorial presentation as rooted trees which serve as the baby models for Feynman graphs [8,14].…”

Section: Introductionmentioning

confidence: 99%

Relative locations of subwords in free operated semigroups and Motzkin words

Zheng

Guo

2014

Front. Math. China

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Abstract. Bracketed words are basic structures both in mathematics (such as Rota-Baxter algebras) and mathematical physics (such as rooted trees) where the locations of the substructures are important. In this paper we give the classification of the relative locations of two bracketed subwords of a bracketed word in an operated semigroup into the separated, nested and intersecting cases. We achieve this by establishing a correspondence between relative locations of bracketed words and those of words by applying the concept of Motzkin words which are the algebraic forms of Motzkin paths.

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Gröbner-Shirshov bases for Rota-Baxter algebras

Cited by 38 publications

References 56 publications

Gröbner–Shirshov bases and their calculation

Gröbner–Shirshov bases and their calculation

Rota-Baxter type operators, rewriting systems and Gröbner-Shirshov bases

Relative locations of subwords in free operated semigroups and Motzkin words

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