Shirshov composition techniques in Lie superalgebras (noncommutative Gröbner bases)

Abstract

The paper contains a development o[ A. I. Shirshov's composition techniques for colored Lie superalgebras: the concepts o/a composition and of a stable set are introduced for elements of those algebras, and different modifications of the composition lemma are proved. Bibliography: 36 titles.The investigation of canonical forms of elements has deep roots in algebra. The theory of canonical bases in algebraic systems goes back to the theory of groups and graphs ([1-3]). The theory of GrSbner bases of ideals in polynomial rings is now a well-developed branch of computer algebra. It gives the possibility of obtaining a series of effective algorithms for symbolic calculations (see, for example, [4]). At the present time, one can see increasing interest in the computer aspects of noncommutative (and even nonassociative) rings (see [5]).The composition method in Lie algebras was introduced by A. I. Shirshov in [6] (see also [7]). This method became an important tool in solving algorithmic problems in the theory of Lie algebras (see, for example, [8][9][10]). Expositions of the junction lemma in the associative case and relations, with computer algebra can be found in the following texts: in [11] by L. A. Bokut'; in [12] by G. Bergman; in [13] by V. N. Latyshev; and in the review [14] by V. A. Ufnarovsky.A general approach to standard bases from the homological point of view was suggested by E. S. Golod in [15]. We also mention [16] by D. Anick, in which special resolutions were used to construct standard bases in the homogeneous case. A. B. Verevkin and A. V. Kondratiev [17] created the package GRAAL of applied programs for calculations in associative algebras on the basis of GrSbner bases in ideals of the free associative algebra.In this article, compositions of elements of the free color Lie (p-)superalgebra are introduced. The main result is a version of the composition lemma for Lie superalgebras. It gives us the possibility of introducing the notion of a Gr6bner basis for ideals in free Lie superalgebras. We also discuss some connections with associative GrSbner bases. As an illustration, we consider some bases of the free product of color Lie (p-)superalgebras with the amalgamated superalgebra. Possible applications of the presented techniques in Lie superalgebras are connected with different algorithmic problems (note that the role of Shirshov's composition lemma is well known in solutions of a series of algorithmic problems in the theory of Lie algebras) and with packages of programs for calculations in Lie superalgebras given by generators and defining relations.Some particular cases of the results of this paper were obtained earlier in [18,19].

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].…”

Monomial algebras

“…The notion of the canonical form and many results about the combinatorial analysis in Lie algebras can be generalized to the "supercase" also (see, for example, [37,83,38]). …”

Gröbner–Shirshov bases for Lie algebras over a commutative algebra

“…Up to now, different versions of Composition-Diamond lemma are known for the following classes of algebras apart those mentioned above: (color) Lie super-algebras [38][39][40], Lie p-algebras [39], associative conformal algebras [14], modules [34,26] (see also [24]), right-symmetric algebras [11], dialgebras [9], associative algebras with multiple operators [13], Rota-Baxter algebras [10], and so on.…”