Gröbner–Shirshov Bases for Replicated Algebras

Shirshov published six papers on Lie algebras in which he found the following results (in order of publication. 1953–1962):

In this paper, we construct free commutative integro-differential algebras by applying the method of Gröbner–Shirshov bases. We establish the Composition-Diamond Lemma for free commutative differential Rota–Baxter (DRB) algebras of order n. We also obtain a weakly monomial order on these algebras, allowing us to obtain Gröbner–Shirshov bases for free commutative integro-differential algebras on a set. We finally generalize the concept of functional monomials to free differential algebras with arbitrary weight and generating sets from which to construct a canonical linear basis for free commutative integro-differential algebras.

The additive primitive length of an element [Formula: see text] of a relatively free algebra [Formula: see text] in a variety of algebras [Formula: see text] is equal to the minimal number [Formula: see text] such that [Formula: see text] can be presented as a sum of [Formula: see text] primitive elements. We give an upper bound for the additive primitive length of the elements in the [Formula: see text]-generated polynomial algebra over a field of characteristic 0, [Formula: see text]. The bound depends on [Formula: see text] and on the degree of the element. We show that if the field has more than two elements, then the additive primitive length in free [Formula: see text]-generated nilpotent-by-abelian Lie algebras is bounded by 5 for [Formula: see text] and by 6 for [Formula: see text]. If the field has two elements only, then our bounds are 6 for [Formula: see text] and 7 for [Formula: see text]. This generalizes a recent result of Ela Aydın for two-generated free metabelian Lie algebras. In all cases considered in the paper, the presentation of the elements as sums of primitive elements can be found effectively in polynomial time.

In 2008, Bokut obtained a Gröbner–Shirshov basis [Formula: see text] of the braid group [Formula: see text] in the Artin–Garside generators and showed that [Formula: see text]-irreducible words of the [Formula: see text] coincided with the Garside normal forms of words. Using this basis [Formula: see text], we obtain the concrete expression of the [Formula: see text]-irreducible words, i.e. normal forms, of the [Formula: see text], and hence give a new understanding of the word problem and Garside normal forms of braid groups.

This article gives a local answer to the coquecigrue problem for Leibniz algebras, that is, the problem of finding a generalization of the (Lie) group structure such that Leibniz algebras are the corresponding tangent algebra structure. Using links between Leibniz algebra cohomology and Lie rack cohomology, we generalize the integration of a Lie algebra into a Lie group by proving that every Leibniz algebra is isomorphic to the tangent Leibniz algebra of a local Lie rack. This article ends with an example of a Leibniz algebra integration in dimension 5.

The composition method was developed by A. I. Shirshov. To begin with, in [372], he proved the theorem on embedding a Lie algebra of finite or countable dimension into a two-generated Lie algebra, applying this method. In [278], the notion of a composition was introduced and a lemma, the first variant of a composition lemma, was proven (see [278], Lemma 4). The composition lemma in its modern form appeared in the work of L. A. Bokut’ [53]. Therein A. I. Shirshov also presented applications of the method in deciding the word problem for Lie algebras given by generators and defining relations and to studying one-relator Lie algebras. Next, in [373] he employed the composition method in his constructing a basis of the amalgamated free product of Lie algebras.

Gröbner bases in universal enveloping algebras of Leibniz algebras

Deformations and rigidity in varieties of Lie algebras

In this paper the main result is the rigidity of varietally free Lie algebras within the varieties where they are free. In fact, these algebras are usually free in some larger varieties, such as various types of the commutator varieties, and this is demonstrated in this paper as well. A related paper is [2] where, among some other results, we claimed the rigidity of free metabelian algebras within the respective variety of Lie algebras.

Let F(x,y) be a relatively free algebra of rank 2 in some variety of algebras over a field K of characteristic 0. In this paper we consider the problem whether p(x,y) ∈ F(x,y) is a primitive element (i.e. an automorphic image of x): (i) If F(x,y)/(p(x,y)) ≅ F(z), the relatively free algebra of rank 1 (ii) If p(f,g) is primitive for some injective endomorphism (f,g) of F(x,y) (iii) If p(x,y) is primitive in a relatively free algebra of larger rank. These problems have positive solutions for polynomial algebras in two variables. We give the complete answer for the free metabelian associative and Lie algebras and some partial results for free associative algebras.

Algebra word problems inculcate in learners the skills and ability to think critically and look for answers to problems in society and the world. Using an explanatory sequential mixed design, the study investigated the errors made by pre-tertiary students in algebraic word problems in a selected school in the Asutifi North District of the Ahafo Region of Ghana. Whereas the Newman Error Analysis Model was used to find the types of errors students make when translating and solving algebraic word problems, protocols from topics relating to the study were carefully selected and used as interview guides alongside inspiration from the Newman Error Analysis Model. Quota and simple random sampling techniques were used to select ninety-eight (98) respondents. The Algebraic Word Problem Achievement Test (AWPAT) was given to the ninety-eight students who had studied several topics on word problems, followed by a structured interview to elicit more information from the respondents. The scores gained from the marked test items illustrated that students made more transformation errors, which happened indirectly from a lack of understanding of the concept. A careful analysis of the written responses of the five (5) students interviewed also revealed that, lack of comprehension directly translated into students’ inability to transform algebraic word problems into mathematics equations. In the same vein, the study showed that the main cause of students’ failure to translate and solve algebraic word problems was students’ lack of understanding of the concept of “word problem” among others. The study recommends that students be motivated by encouragement and praise to arouse their interest in translating and solving algebraic word problems.

Bases, filtrations and module decompositions of free Lie algebras

Quadratic algebras are generalizations of Lie algebras which include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by B\^ocher contractions of the conformal Lie algebra $\mathfrak{so}(4,\mathbb {C})$ to itself. In 2 dimensions there are two kinds of quadratic algebras, nondegenerate and degenerate. In the geometric case these correspond to 3 parameter and 1 parameter potentials, respectively. In a previous paper we classified all abstract parameter-free nondegenerate quadratic algebras in terms of canonical forms and determined which of these can be realized as quadratic algebras of 2D nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces, and studied the relationship between B\^ocher contractions of these systems and abstract contractions of the free quadratic algebras. Here we carry out an analogous study of abstract parameter-free degenerate quadratic algebras and their possible geometric realizations. We show that the only free degenerate quadratic algebras that can be constructed in phase space are those that arise from superintegrability. We classify all B\^ocher contractions relating degenerate superintegrable systems and, separately, all abstract contractions relating free degenerate quadratic algebras. We point out the few exceptions where abstract contractions cannot be realized by the geometric B\^ocher contractions.

In this study, we use the graph painting problem in Lie algebra to develop an entertaining puzzle mobile game, known as water pipe saving combat. There are two modes in this game. One is single-player mode, which is designed based on the algorithm of a linear graph painting. The other is online mode, which adds the idea of Tetris for entertainment. In the future, we hope that this game not only entertaining, but also providing a useful tool for the study of graphing painting problem in Lie algebra.

In the present paper, methods and algorithms for numerical solution of spectral problems and some problems in algebra related to them for one- and two-parameter polynomial and rational matrices are considered. A survey of known methods of solving spectral problems for polynomial matrices that are based on the rank factorization of constant matrices, i.e., that apply the singular value decomposition (SVD) and the normalized decomposition (the QR factorization), is given. The approach to the construction of methods that makes use of rank factorization is extended to one- and two-parameter polynomial and rational matrices. Methods and algorithms for solving some parametric problems in algebra based on ideas of rank factorization are presented. Bibliography: 326titles.