Shirshov Bases Research Articles

Linear algebraic approach to Gröbner–Shirshov basis theory

We construct a new efficient algorithm for finding Gröbner–Shirshov bases for noncommutative algebras and their representations. This algorithm uses the Macaulay matrix [F.S. Macaulay, On some formula in elimination, Proc. London Math. Soc. 33 (1) (1902) 3–27], and can be viewed as a representation theoretic analogue of the F 4 algorithm developed by J.C. Faugère. We work out some examples of universal enveloping algebras of Lie algebras and of their representations to illustrate the algorithm.

Gröbner–Shirshov bases for some braid groups

Presenting 2-generator Artin groups A ( m ) and braid groups B 3 and B 4 as towers of HNN extensions of free groups, we obtain Gröbner–Shirshov bases, normal forms and rewriting systems for these groups.

Representations of Ariki–Koike Algebras and Gröbner–Shirshov Bases

In this paper, we investigate the structure of Ariki–Koike algebras and their Specht modules using Gröbner–Shirshov basis theory and combinatorics of Young tableaux. For a multipartition λ, we find a presentation of the Specht module Sλ given by generators and relations, and determine its Gröbner–Shirshov pair. As a consequence, we obtain a linear basis of Sλ consisting of standard monomials with respect to the Gröbner–Shirshov pair. We show that this monomial basis can be canonically identified with the set of cozy tableaux of shape λ. 2000 Mathematics Subject Classification 16Gxx, 05Exx.

Gröbner–Shirshov Bases for Universal Enveloping Conformal Algebras of Simple Conformal Lie Superalgebras of Type WN

For simple conformal Lie superalgebras of type WN, Grobner–Shirshov bases of their universal enveloping associative conformal algebras are found. The universal enveloping algebras considered correspond to a minimal locality function for which there is an injective embedding.

Gröbner–Shirshov Bases: From their Incipiency to the Present

In this paper, the history and the main results of the theory of Grobner–Shirshov bases are given for commutative, noncommutative, Lie, and conformal algebras from the beginning (1962) to the present time. The problem of constructing a base of a free Lie algebra is considered, as well as the problem of studying the structure of free products of Lie algebras, the word problem for Lie algebras, and the problem of embedding an arbitrary Lie algebra into an algebraically closed one. The modern form of the composition-diamond lemma (the CD lemma) is presented. The rewriting systems for groups are considered from the point of view of Grobner–Shirshov bases. The important role of conformal algebras is treated, the statement of the CD lemma for associative conformal algebras is given, and some examples are considered. An analog of the Hilbert basis theorem for commutative conformalalgebras is stated. Bibliography: 173 titles.

GRBNER–SHIRSHOV BASES FOR THE KAC–MOODY ALGEBRAS OF THE TYPE

ABSTRACT The Grbner–Shirshov bases for the Kac–Moody algebras of the type are constructed.

Hecke algebras, Specht modules and Gröbner–Shirshov bases

In this paper, we study the structure of Specht modules over Hecke algebras using the Gröbner–Shirshov basis theory for the representations of associative algebras. The Gröbner–Shirshov basis theory enables us to construct Specht modules in terms of generators and relations. Given a Specht module S q λ , we determine the Gröbner–Shirshov pair ( R q , R q λ ) and the monomial basis G ( λ ) consisting of standard monomials. We show that the monomials in G ( λ ) can be parameterized by the cozy tableaux. Using the division algorithm together with the monomial basis G ( λ ), we obtain a recursive algorithm of computing the Gram matrices. We discuss its applications to several interesting examples including Temperley–Lieb algebras.

Gröbner–Shirshov Bases for Irreducible sln + 1-Modules

We determine the Gröbner–Shirshov bases for finite-dimensional irreducible representations of the special linear Lie algebra sln+1 and construct explicit monomial bases for these representations. We also show that each of these monomial bases is in 1–1 correspondence with the set of semistandard Young tableaux of a given shape.

Gröbner–Shirshov Bases for Lie Superalgebras and Their Universal Enveloping Algebras

We show that a set of monic polynomials in a free Lie superalgebra is a Gröbner–Shirshov basis for a Lie superalgebra if and only if it is a Gröbner–Shirshov basis for its universal enveloping algebra. We investigate the structure of Gröbner–Shirshov bases for Kac–Moody superalgebras and give explicit constructions of Gröbner–Shirshov bases for classical Lie superalgebras.

ON A SHIRSHOV BASIS OF RELATIVELY FREE ALGEBRAS OF COMPLEXITY $ n$

A Shirshov basis is a set of elements of an algebra over which has bounded height in the sense of Shirshov.A description is given of Shirshov bases consisting of words for associative or alternative relatively free algebras over an arbitrary commutative associative ring with unity. It is proved that the set of monomials of degree at most is a Shirshov basis in a Jordan PI-algebra of degree . It is shown that under certain conditions on (satisfied by alternative and Jordan PI-algebras), if each factor of with nilpotent projections of all elements of is nilpotent, then is a Shirshov basis of if generates as an algebra.Bibliography: 12 titles.