Search

In this paper, Jordan-Kronecker invariants are calculated for all nilpotent 6- and 7-dimensional Lie algebras. We consider the Poisson bracket family, depending on the lambda parameter on a Lie coalgebra, i.e., on the linear space dual to a Lie algebra. For some space g proposed in the paper, two skew-symmetric matrices are defined for all points x on this linear space. To understand the behaviour of the matrix pencil (A - [lambda][BETA])([chi]), we consider Jordan-Kronecker invariants for this pencil and how they change with [chi] (the latter is done for 6-dimensional Lie algebras)., 1. Basic Definitions and Theorems Definition 1. A Poisson bracket is a bilinear skew-symmetric operation over functions f,g [right arrow] {f,g} satisfying the Jacobi identity and the Leibniz rule. The [...]

This paper is devoted to the generalization of the construction of a Jordan algebra of a Lie algebra and the known theorems on the local finite-dimensionality of Lie PI-algebras with an algebraic adjoint representation to Mal'tsev algebras., Introduction Jordan algebras of Lie algebras are defined in [9] by analogy with local algebras of associative algebras and triple Jordan systems. In the paper their generalized version for Mal'tsev [...]

Let A be an algebra over a field F generated by a set of matrices S. The paper considers algorithmic aspects of checking whether A coincides with the full matrix algebra. Laffey has shown that for F = C, under the assumption that S contains a Jordan matrix from a certain class, there is a fast method for checking whether A possesses nontrivial invariant subspaces and, consequently, coincides with the full algebra by Burnside's theorem. This paper extends the class to the largest subclass of Jordan matrices on which the algorithm works correctly. Examples demonstrating different types of behavior of other matrix systems are provided. Bibliography: 18 titles., 1. INTRODUCTION In studying a matrix subalgebra, it is necessary to determine whether it is a proper sub-algebra or coincides with the full matrix algebra. If a subalgebra is defined [...]

The paper discusses the questions of coincidence of the basic nil-radicals on classes of algebraic Lie algebras and proves the local finite-dimensionality of Lie algebras with an algebraic adjoint representation of bounded degree over fields of sufficiently large positive characteristic., 1. Introduction The present paper supplements the results of A. I. Kostrikin and E. I. Zel'manov obtained during the solution of the restricted Burnside problem for groups of single prime [...]

This paper belongs to our series of works on algebraic geometry over arbitrary algebraic structures. In this one, there will be investigated seven equivalences (namely: geometric, universal geometric, quasi-equational, universal, elementary, and combinations thereof) in specific classes of algebraic structures (equationally Noetherian, [q.sub.[omega]]-compact, [u.sub.[omega]]-compact, equational domains, equational co-domains, etc.). The main questions are the following: (1) Which equivalences coincide inside a given class K, which do not? (2) With respect to which equivalences a given class K is invariant, with respect to which it is not?, 1. Introduction In the series of papers on universal algebraic geometry [2-10], we consecutively develop the technique for investigations in algebraic geometry over an arbitrary algebraic structure [??] in a [...]

An embedded system of algebras of vertex subsets of weighted digraph is constructed. The properties of minimal spanning forests restricted to atoms of the corresponding algebras are investigated. Bibliography: 9 titles., 1. NOTATION AND DEFINITIONS The present paper is a continuation of paper [1]; the definitions and notation correspond to those adopted in it. The vertex set of a digraph G [...]

An embedded system of algebras of vertex subsets of weighted digraph is constructed. The properties of minimal spanning forests restricted to atoms of the corresponding algebras are investigated. Bibliography: 9 titles., UDC 519.17 1. NOTATION AND DEFINITIONS The present paper is a continuation of paper [1]; the definitions and notation correspond to those adopted in it. The vertex set of a [...]

In this paper, an analog of the conformable fractional derivative is defined in an arbitrary finite-dimensional commutative associative algebra. Functions taking values in the indicated algebras and having derivatives in the sense of a conformable fractional derivative are called [phi]-monogenic. A relation between the concepts of [phi]-monogenic and monogenic functions in such algebras has been established. Two new definitions have been proposed for the fractional derivative of the functions with values in finite-dimensional commutative associative algebras. Keywords. Conformable fractional derivative, fractional analytic functions, local fractional derivative, [alpha]-differentiable functions, [phi]-monogenic functions in commutative algebras., 1. Introduction The idea of fractional derivative was first raised by L'Hospital in 1 695. After introducing it, many new definitions have been formulated. The most well-known of them are [...]

The second part of the paper describes in terms of generators and relations the Hochschild cohomology ring of a self-injective algebra of tree class [E.sub.6] with finite representation type. Bibliography: 15 titles., 1. INTRODUCTION The present paper continues a series of papers devoted to studying Hochschild cohomologies of self-injective algebra of finite representation type over an algebraically closed field. According to Riedtmann's [...]

In this paper, we study local and 2-local derivations of p-filiform Leibniz algebras, describe all local derivations of p-filiform non-Lie Leibniz algebras, and also show that there exist two-local derivations on these algebras that are not derivations. Keywords and phrases: Leibniz algebra, p-filiform Leibniz algebra, derivation, local derivation, 2local derivation. AMS Subject Classification: 17A32, 17B10, 17B20, UDC 512.554.38 1. Introduction. Local derivations were first considered by Kadison in [8] in 1990 and independently by Larson and Sourour in [10]. In these papers, certain conditions under which [...]

In this paper, we study questions related to the theory of stochastic processes on Lie nilpotent groups. In particular, we consider the stochastic process on the Heisenberg group [H.sub.3](R) whose trajectories satisfy the horizontal conditions in the stochastic sense relative to the standard contact structure on [H.sub.3](R). It is shown that this process is a homogeneous Markov process relative to the Heisenberg group operation. There was found a representation in the form of a Wiener integral for a one-parameter linear semigroup of operators for which the Heisenberg sublaplacian generated by basis vector fields of the corresponding Lie algebra L([H.sub.3]) is producing. The main method of solving the problem in this paper is using the path integrals technique, which indicates the common direction of further development of the results., UDC 512.813.52+517.955.4+517.983.37+517.987.4+519.216.22 Introduction In [12,17], for the first time there was calculated the heat kernel corresponding to the sublaplacian L on the Heisenberg group [H.sub.2n+1] (R), using the technique of [...]

The paper interrelates the simultaneous triangularization problem for matrix pairs with the Paz problem and known results on the length of the matrix algebra. The length function is applied to the Al'pin--Koreshkov algorithm, and it is demonstrated how its multiplicative complexity can be reduced. An asymptotically superior procedure for verifying the simultaneous triangularizability of a pair of complex matrices is provided. The procedure is based on results on the lengths of upper triangular matrix algebras. Also the definition of the hereditary length of an algebra is introduced, and the problem of computing the hereditary lengths of matrix algebras is discussed. Bibliography: 22 titles., 1

The Hochschild cohomology ring for self-injective algebras of tree class [E.sub.8] with finite representation type is described in terms of generators and relations. Bibliography: 17 titles., 1. INTRODUCTION The present paper continues (and completes) the series devoted to the study of the Hochschild cohomology of self-injective basic algebras over algebraically closed field of finite representation type. [...]

Hochschild cohomology algebra is described in term of generators and relations for a family of local algebras of semidihedral type. This family appears in famous K. Erdmann's classification only if the characteristic of the base field is equal to 2. Bibliography: 13 titles., INTRODUCTION The present paper is devoted to calculation of the Hochschild cohomology algebra HH*(R) for the so called 'exceptional' family of local algebras of semidihedral type. Recall that the algebras [...]

The present paper is devoted to the study of the dimension of the space of all derivations on homogeneous commutative regular algebras. We shall show that if [Formula omitted] is a Maharam homogeneous measure space with a finite countable-additive measure [Formula omitted] and [Formula omitted] is a homogeneous regular unital subalgebra in [Formula omitted] with the homogeneous Boolean algebra of idempotents [Formula omitted] then [Formula omitted] where [Formula omitted] is the weight of the Boolean algebra [Formula omitted] and [Formula omitted] is the transcendence degree of [Formula omitted]., Author(s): Shavkat Ayupov [sup.1] [sup.2], Karimbergen Kudaybergenov [sup.1] [sup.3], Khakimbek Karimov [sup.1] [sup.4] Author Affiliations: (1) grid.419209.7, 0000 0001 2110 259X, V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, , [...]

In this paper, we study an irreducible decomposition structure of the [Formula omitted]-module direct image [Formula omitted] for the finite map [Formula omitted] We explicitly construct the simple components of [Formula omitted] by providing their generators and their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a [Formula omitted]-module decomposition of the polynomial ring localized at the discriminant of [Formula omitted]. Furthermore, we study the action of invariant differential operators on the higher Specht polynomials., Author(s): Ibrahim Nonkané [sup.1], Léonard Todjihounde [sup.2] Author Affiliations: (1) Departement d'économie et de mathématiques appliquées, IUFIC, Université Thomas Sankara, , 12 BP 417, Ouagadougou 12, Burkina Faso (2) grid.412037.3, [...]

The word maps [mathematical expression not reproducible], where D is a division algebra over a field K, are considered. It is proved that if [mathematical expression not reproducible] is the subgroup of G[L.sub.n](D), generated by transvections, and Z([E.sub.n](D)) is its center. Furthermore if, in addition, n > 2, then [??]([E.sub.n](D)) [contains] [E.sub.n](D) \ Z([E.sub.n](D)). The proof of the result is based on an analog of the 'Gauss decomposition with prescribed semisimple part' (introduced and studied in two papers of the second author with collaborators) in the case of the group G[L.sub.n](D), which is also considered in the present paper. Bibliography: 18 titles., UDC 512.7, 512.64, 512.81 Introduction Let [member of] be a group, and let w = w([x.sub.1], ..., [x.sub.m]) be a nontrivial word in m variables. Then one can define the [...]

In this paper, we aim at highlighting the significance of the A- and B-properties introduced by Finch (Bull Aust Math Soc 2:57-62, 1970b). These conditions turn out to capture interesting structural features of lattices of closed subspaces of complete inner vector spaces. Moreover, we generalise them to the context of effect algebras, establishing a novel connection between quantum structures (orthomodular posets, orthoalgebras, effect algebras) arising from the logico-algebraic approach to quantum mechanics., Author(s): D. Fazio [sup.1], A. Ledda [sup.1], F. Paoli [sup.1] Author Affiliations: (1) grid.7763.5, 0000 0004 1755 3242, Dipartimento di Pedagogia, Psicologia, Filosofia, Università di Cagliari, , via Is Mirrionis [...]

In this paper, we consider the T-space structure of the relatively free Grassmann algebra [F.sup.(3)] without unity over an infinite field of prime and zero characteristic. Our work is focused on T-spaces [W.sub.n] generated by all so-called n-words. A question about connections between [W.sub.r] and [W.sub.n] for different natural numbers r and n is investigated. The proved theorem on these connections allows us to construct the diagrams of inclusions, which, to some extent, clarify the structure of the algebra: the basic T-spaces produce infinite strictly descending chains of inclusions in the algebra [F.sup.(3)]., Introduction The structure of unitary closed T-spaces in the relatively free Grassmann algebra [F.sup.(3)] = k/[T.sup.(3)] was studied rather well in [6, 7]. Here F = k is a free [...]

This paper collects in a unified form well-known versions of the Anderson--Divinsky--Sulinski lemma for algebras that are nearly associative, and gives a number of its analogues for Lie algebras., 1. Introduction If the values of the radical in the Kurosh--Amitsur sense T on ideals of algebras from its class of definition are their ideals, then the class of T-semisimple [...]

In this paper, we consider the T-space structure of the relatively free Grassmann algebra [F.sup.(3)] without unity over an infinite field of prime and zero characteristic. Our work is focused on T-spaces [W.sub.n] generated by all n-words. A question about connections between [W.sub.r] and [W.sub.n] for different natural numbers r and n is investigated. The proved theorem on these connections allows us to construct the diagrams of inclusions that, to some extent, clarify the structure of the algebra: the basic T-spaces produce infinite strictly descending chains of inclusions in the algebra [F.sup.(3)]., Introduction The structure of unitary closed T-spaces in the relatively Grassmann algebra [F.sup.(3)] = k(1,[x.sub.1],...,[x.sub.i],...)/[T.sup.(3)] was studied rather well in [5,6]. Here F = k(1,[x.sub.1],...,[x.sub.i],...) is a free countably generated [...]

It is well known that Reed--Muller codes over a prime field are radical powers of a corresponding group algebra. The case of a nonprime field is less studied in terms of equalities and inclusions between Reed--Muller codes and radical powers. In this paper, we prove that Reed--Muller codes in the case of a nonprime field of arbitrary characteristic are distinct from radical powers and provide necessary and sufficient conditions for inclusions between these codes and the powers of the radical., 1. Introduction Let p be a prime number and q = [p.sup.l], l [greater than or equal to] 1. Consider the field Q = [F.sub.q] of characteristic p and order [...]

A universal algebra is called Hopfian if any of its surjective endomorphisms is an automorphism, and co-Hopfian if injective endomorphisms are automorphisms. In this paper, necessary and sufficient conditions are found for Hopfianity and co-Hopfianity of unitary acts over groups. It is proved that a coproduct of finitely many acts (not necessarily unitary) over a group is Hopfian if and only if every factor is Hopfian., 1. Introduction A universal algebra A is called Hopfian if any surjective endomorphism [phi]: A [right arrow] A is an automorphism; it is co-Hopfian if any injective endomorphism [psi]: A [...]

Our paper is devoted to the investigations of doubly alternative zero divisors of the real Cayley--Dickson split-algebras. We describe their annihilators and orthogonalizers and also establish the relationship between centralizers and orthogonalizers for such elements. Then we obtain an analogue of the real Jordan normal form in the case of the split-octonions. Finally, we describe commutativity, orthogonality, and zero divisor graphs of the split-complex numbers, the split-quaternions, and the split-octonions in terms of their diameters and cliques. To the memory of V. T. Markov, 1. Introduction The zero divisors of the Cayley--Dickson algebras are of special interest; however, the problem of their identification and description of their annihilators is rather difficult. Some attempts to [...]

In a previous paper (Funct. Anal. Appl., 3 (2015), 205-209), we defined quantum Markov states. Here we recall this definition and present a proof of the results from that paper (which are given there without proofs). We give a definition of a quantum hidden Markov state generated by a function of a quantum Markov process and show how it is related to other definitions of such states. Our definitions work for quantum Markov fields on [Z.sup.N] and on graphs. We consider an example with the Cayley tree. Bibliography: 5 titles., UDC 519.2 1. QUANTUM MARKOV STATES Let [M.sub.d] be the algebra of complex matrices of order d. Set [mathematical expression not reproducible] The algebra [mathematical expression not reproducible] is embedded [...]

In this paper, we construct anticliques for noncommutative operator graphs generated by generalized Pauli matrices. It is shown that application of entangled states for the construction of the code space K allows one to substantially increase the dimension of a noncommutative operator graph for which the projection on K is an anticlique. Bibliography: 10 titles., UDC 517.5 1. INTRODUCTION In the paper [1], the authors introduced operator systems which are subspaces V of the algebra of all bounded linear operators in a Hilbert space H [...]

A large class of solid solution models are formulated on the premise that exchange of chemical species takes place on a finite number of unique crystallographic sites, and that the thermodynamic properties of the solution are a function of the proportions of species occupying each of the sites. These models are broadly classified as being of Bragg-Williams-type. They form an excellent first order approximation of non-ideal mixing and long-range order. In this article we present the mathematical framework common to all Bragg-Williams models, introducing necessary concepts from geometry, set theory and linear algebra. We combine this with a set of mathematical tools which we have found useful in building and using such models. We include several worked examples to illustrate key concepts and provide general expressions which can be used for all models. This paper is split into two parts. In the first part, we show how the valences of the species occupying each site and the total charge of the species involved in site exchange are sufficient to define the space of valid site occupancies of a solid solution, and to compute the endmembers bounding that space. We show that this space can be visualised as a polytope, i.e, an n-dimensional polyhedron, and we describe the relationship between site-occupancy space and compositional space. In the second part of the paper, we present the linear algebra required to transform descriptions of modified van Laar and subregular solution models from one independent endmember basis to another. The same algebra can also be used to derive macroscopic endmember interactions from microscopic site interactions. This algebra is useful both in the initial design of solution models, and when performing thermodynamic calculations in restricted chemical subsystems. A polytope description of solid solutions is used in the thermodynamic software packages Perple_X and burnman. The algorithms described in this paper are made available as python code., Author(s): Robert Myhill [sup.1], James A. D. Connolly [sup.2] Author Affiliations: (1) grid.5337.2, 0000 0004 1936 7603, School of Earth Sciences, University of Bristol, , Wills Memorial Building, Queen's Road, [...]

In connection with the problem of describing holomorphically homogeneous real hypersurfaces in the space [??], we study five-dimensional real Lie algebras realized as algebras of holomorphic vector fields on such manifolds. We prove the following assertion: If on a holomorphically homogeneous real hypersurface M of the space [??], there is a decomposable, solvable, five-dimensional Lie algebra of holomorphic vector fields having a full rank near some point P [member of] M, then this surface is either degenerate near P in the sense of Levy or is a holomorphic image of an affine-homogeneous surface. Keywords and phrases: homogeneous manifold, holomorphic transformation, decomposable Lie algebra, vector field, real hypersurface in [??]. AMS Subject Classification: 32V40, 53B25, 17B66, 1. Introduction. In this paper, we discuss the problem of local description of homogeneous real hypersurface in the three-dimensional complex space. The study of such surfaces is based on the [...]

In this paper, we discuss methods and approaches for definition of multiplication of distributions, which is not defined in general in the classical theory. We show that this problem is related to the fact that the operator of multiplication by a smooth function is nonclosable in the space of distributions. We give the general method of construction of new objects called new distributions, or mnemofunctions, that preserve essential properties of usual distributions and produce algebras as well. We describe various methods of embedding of distribution spaces into algebras of mnemofunctions. All ideas and considerations are illustrated by the simplest example of the distribution space on a circle. Some effects arising in study of equations with distributions as coefficients are demonstrated by example of a linear first-order differential equation., CONTENTS 1. Introduction 147 2. Line Distribution Spaces 148 3. Multiplication Problem for Distributions 149 4. Closures of Nonclosable Operators 151 5. Algebras of Mnemofunctions 157 6. Space of Periodic [...]

In this paper we study the class of laterally complete commutative unital regular algebras A over arbitrary fields. We introduce a notion of passport [GAMMA](X) for a faithful regular laterally complete A-modules X, which consist of uniquely defined partition of unity in the Boolean algebra of all idempotents in A and of the set of pairwise different cardinal numbers. We prove that A-modules X and Y are isomorphic if and only if [GAMMA](X) = [GAMMA](Y). Further we study Banach A-modules in the case A = [C.sub.[infinity]](Q) or A = [C.sub.[infinity]](Q)+i*[C.sub.[infinity]](Q). We establish the equivalence of all norms in a finite-dimensional (respectively, [sigma]-finite-dimensional) A-module and prove an Aversion of Riesz Theorem, which gives the criterion of a finite-dimensionality (respectively, [sigma]-finite-dimensionality) of a Banach A-module., CONTENTS 1. Introduction 129 2. Preliminaries 130 3. Classification of Faithful l-Complete A-modules 135 4. Banach [C.sub.[infinify]](Q)-modules 138 References 144 1. Introduction The development of the theory of Baer *-algebras [...]

The paper presents new results on varieties of commutative metabelian algebras over a field of zero characteristic. We study the structure of the multilinear part of the variety of all commutative metabelian algebras as a module of the symmetric group. Two almost nilpotent varieties are introduced and studied in this class of algebras. We prove the nonexistence of other almost nilpotent commutative metabelian varieties of subexponential growth., UDC 512.5 Throughout the paper, we shall use the term algebra to denote a vector space with one binary bilinear operation. According to the definition given by A. G. Kurosh, [...]

Historically, dense vegetation cover near buildings has caused power disruptions during weather phenomena. These types of severe storms impact the coast of Florida each year. However, challenges exist for obtaining both power outage data and calculating the impact of tree cover. NASA's Nighttime Lights, Black Marble, VNP46 product is utilized to analyze the natural and built environments. One aspect of the built environment that can be mapped with the Black Marble data is the megawatts of electricity used by the electrical power grid based on the magnitude of emitted nighttime light energy. This paper discusses using Black Marble data and other landscape variables within a probabilistic model to examine spatial patterns and map electricity outages with Bayesian networks. The research results indicate a high probability of a significant power outage when dense vegetation is present, but nuances in our naura and built environments like electric substations and land cover type alter the chance of reducing energy emissions. KEYWORDS: Nighttime lights, Map algebra, Probabilistic algebra, Bayes' theorem southeastern geographer, 62(2) 2022: pp. 128-146, INTRODUCTION A weather phenomenon with high winds and precipitation, such as a seasonal storm, can damage the electrical utility system. Along with being economically and socially disruptive, power outages may [...]

Character measure is a probability measure on irreducible representations of a semisimple Lie algebra. It appears from the decomposition into irreducibles of tensor power of a fundamental representation. In this paper we calculate the asymptotics of character measure on representations of s[o.sub.2n+1] in the regime near the boundary of weight diagram. We find out that it converges to a Poisson-type distribution. Bibliography: 8 titles., 1. INTRODUCTION The probability measures that appear from a decomposition of representation into irreducibles are actively studied for many decades. One of the most famous asymptotic results is the Vershik-Kerov-Logan-Shepp [...]

Let R be an associative ring with 1 and G = GL(n, R) the general linear group of degree n [greater than or equal to] 3 over R. A goal of the paper is to calculate the relative centralizers of the relative elementary subgroups or the principal congruence subgroups, corresponding to an ideal A[??]R modulo the relative elementary subgroups or the principal congruence subgroups, corresponding to another ideal B [??] R. Modulo congruence subgroups, the results are essentially easy exercises in linear algebra. But modulo the elementary subgroups, they turned out to be quite tricky, and definitive answers are obtained only over commutative rings or, in some cases, only over Dedekind rings/Dedekind rings of arithmetic type. Bibliography: 43 titles., 1. Introduction Let F, H [less than or equal to] G be two subgroups of G. We consider the centralizer of F modulo H, [Please download the PDF to view [...]

The famous 'algebraic Mayer-Vietoris theorem' is usually stated for complexes over an Abelian category. In the present paper, this theorem is generalized for complexes over a preabelian category. The proofs are based on the technique and results of the relative homological algebra developed by the author earlier. Bibliography: 4 titles., 1. Introduction We extend to preabelian categories the so called 'Mayer-Vietoris algebraic sequence' which is usually stated for the category of chain complexes over an arbitrary Abelian category (cf., for [...]

In this paper, we propose an abstract scheme for the study of special operators and apply this scheme to examining elliptic pseudo-differential operators and related boundary-value problems on manifolds with nonsmooth boundaries. In particular, we consider cases where boundaries may contain conical points, edges of various dimensions, and even peak points. Using the constructions proposed, we present well-posed formulations of boundary-value problems for elliptic pseudo-differential equations on manifolds discussed in Sobolev-Slobodecky spaces. Keywords and phrases: local-type operator, operator symbol, ellipticity, Fredholm property, index, pseudo-differential operator. AMS Subject Classification: 35S05, 47B37, 1. Introduction. In the theory of pseudo-differential operators and the corresponding equations and boundary-value problem, a key role is played by the notion of the symbol of a pseudo-differential operator. [...]

In this paper, we give a general insight into the ideas that make ground for the developing of universal algebraic geometry and logical geometry. We specify the role of algebraic logic as one of the major instruments of the whole theory. The problem of the sameness of geometries of algebraic and definable sets for different algebras is considered as ans illuminating example how algebra, geometry, model theory, and algebraic logic work together., 1. Informal Introduction of B. Plotkin The whole story began for me in the 80s of the last century. Some practical discussions have led to the problem of constructing an [...]

In this paper, the classical theorem on the image of the solvable radical of a finite-dimensional Lie algebra over a field of characteristic zero under the action of its derivation is generalized to locally generalized special Lie algebras., 1. Introduction The image of the greatest solvable ideal of a finite-dimensional Lie algebra over a field of characteristic zero under the action of any of its derivations is included [...]

The purpose of this paper is to generalize results of R. M. W. Wood on monomial bases for the mod 2 Steenrod algebra to the mod p Steenrod algebra, p > 2., UDC 515.14 Dedicated to Anatoly Timofeevich Fomenko on the occasion of his 70th anniversary 1. Introduction and Basic Definitions This paper is concerned with some special bases of the mod [...]

A characterization of the prime radical of loops as the set of strongly Engel elements was given in our earlier paper. In this paper, some properties of the prime radical of loops are considered. Also a connection between the prime radical of the loop of units of an alternative ring and the prime radical of this ring is given., UDC 512.548.77+512.554.5 1. Prime Radicals of Some Algebraic Structures 1.1. The Prime Radical of a Loop. The notion of a radical in group theory was suggested by A. Kurosh [6]. [...]

In this paper, a structure theorem for the symmetries of a graded flat cosymbol algebra of differential operators is proved. Together with a lemma on equivariant polynomials also proved in the paper, this theorem gives an upper bound on the dimension of the graded Lie algebra associated with the symmetries of geodesic flow on a smooth variety. Bibliography: 14 titles, UDC 512 1. Basic notions Let A be a unitary commutative algebra over a ring K with identity. If P and Q are A-modules, then for every a [member of] [...]

The Hochschild cohomology groups for algebras of semidihedral type that lie in the family SD[(2B).sub.2] (in the famous K. Erdmann's classification) over an algebraically closed field with characteristic different from two are computed. The calculation, relies upon the minimal projective bimodule resolution for algebras from the above family that was constructed in the previous author's paper. Bibliography: 33 titles., UDC 512.5 1. INTRODUCTION In the present paper, we compute the Hochschild cohomology groups for algebras of the family SD[(2B).sub.2] presented in the famous K. Erdmann's classification [1] over an [...]

Earlier [1], a new class of quaternionic so-called G-monogenic (differentiable in the meaning of Gateaux) mappings was considered. In the present paper, we introduce quaternionic H-monogenic (differentiable in the sense of Hausdorff) mappings and establish a relation between G- and H-monogenic mappings. The equivalence of different definitions of a G-monogenic mapping is proved. Keywords. Algebra of complex quaternions, G-monogenic mappings, Morera's theorem, H-monogenic mappings., 1. Introduction Many papers (see, e.g., [1-23]) are devoted to the problem of definition of an analytic function in associative (commutative or non-commutative) algebras. In particular, this problem is considered [...]

The paper represents a series of comments to the K. A. Zhevlakov and I. P. Shestakov theorem on the existence of a locally finite in the sense of Shirshov over an ideal of the ground ring radical on the class of algebras that are algebraic over this ideal and belong to some sufficiently good homogeneous variety. It is shown in detail how the given theorem includes Plotkin's and Kuz'min's theorems on the existence of a locally finite radical on the classes of algebraic Lie and Mal'tsev algebras. There is adduced its generalization to locally finite extensions of ideally algebraic Lie and alternative algebras., UDC 512.552.12+512.554.36 1. Introduction Two main objectives of the present paper are the demonstration of deductibility of results of B. I. Plotkin [18] and E. N. Kuz'min [12] on the [...]

The paper considers the problem of computing the lengths of matrix incidence algebras over a field whose cardinality is strictly less than the matrix order n. For n = 3, 4, the lengths of all such algebras over the field of two elements are determined. In the case where the ground field and the number n are arbitrary but the Jacobson radical of the algebra has nilpotency index 2, an upper bound for the length is provided. In addition, the incidence algebras isomorphic to a direct sum of triangular matrix algebras of order 2 and an algebra of diagonal matrices are considered. It is shown that the lengths of these algebras over the field of two elements can take only two distinct values, which can be determined exactly. Moreover, the diagonal number of a matrix incidence algebra is introduced and bounded above. Bibliography: 24 titles., 1. Introduction Investigations of generating systems of matrix algebras have a long history and are continued till nowadays. One of the first results on this topic is Burnside's theorem (the [...]