Composition-Diamond Lemma for associative conformal algebras

Abstract

In this paper, we study the concept of associative n-conformal algebra over a field of characteristic 0 and establish Composition-Diamond lemma for a free associative n-conformal algebra. As an application, we construct Gröbner-Shirshov bases for Lie n-conformal algebras presented by generators and defining relations.

Gröbner–Shirshov bases and their calculation

“…-Associative algebras, Shirshov [207], Bokut [22], Bergman [11]; -Associative algebras over a commutative algebra, Mikhalev and Zolotykh [170]; -Associative -algebras, where is a group, Bokut and Shum [59]; -Lie algebras, Shirshov [207]; -Lie algebras over a commutative algebra, Bokut et al [31]; -Lie p-algebras over k with char k = p, Mikhalev [166]; -Lie superalgebras, Mikhalev [165,167]; -Metabelian Lie algebras, Chen and Chen [75]; -Quiver (path) algebras, Farkas et al [101]; -Tensor products of associative algebras, Bokut et al [30]; -Associative differential algebras, Chen et al [76]; -Associative (n−)conformal algebras over k with char k = 0, Bokut et al [45], Bokut et al [43]; -Dialgebras, Bokut et al [38]; -Pre-Lie (Vinberg-Koszul-Gerstenhaber, right (left) symmetric) algebras, Bokut et al. [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 al...…”

“…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].…”

Universal enveloping Poisson conformal algebras

“…In the next section, we present a general approach to the study of conformal algebras given by generators and relations, a sort of Composition-Diamond Lemma (CD-Lemma) for conformal algebras. Previous versions of the CD-Lemma for associative conformal algebras [4,5,15] work for bounded functions N. Our approach does not depend on N and, which is more important, we reduce the problem to modules over ordinary associative algebras. Therefore, one may apply available computer algebra packages for computations in conformal algebras within this approach.…”

Gröbner-Shirshov Bases: Some New Results

“…-Conformal associative algebras (C, (n), n ≥ 0, D) ( [12]). There are 6 types of compositions including inclusion, intersection, D-inclusion, Dintersection, left (right) multiplication by a generator.…”