Springer Proceedings in Mathematics & Statistics

Sergei SilvestrovAnatoliy MalyarenkoMilica Rančić   Editors

Algebraic Structures and ApplicationsSPAS 2017, Västerås and Stockholm, Sweden, October 4–6

Springer Proceedings in Mathematics &Statistics

Volume 317

Springer Proceedings in Mathematics & Statistics

This book series features volumes composed of selected contributions fromworkshops and conferences in all areas of current research in mathematics andstatistics, including operation research and optimization. In addition to an overallevaluation of the interest, scientific quality, and timeliness of each proposal at thehands of the publisher, individual contributions are all refereed to the high qualitystandards of leading journals in the field. Thus, this series provides the researchcommunity with well-edited, authoritative reports on developments in the mostexciting areas of mathematical and statistical research today.

More information about this series at http://www.springer.com/series/10533

Sergei Silvestrov • Anatoliy Malyarenko •

Milica RančićEditors

Algebraic Structuresand ApplicationsSPAS 2017, Västerås and Stockholm,Sweden, October 4–6

123

EditorsSergei SilvestrovDivision of Applied MathematicsSchool of Education, Cultureand CommunicationMälardalen UniversityVästerås, Sweden

Anatoliy MalyarenkoDivision of Applied MathematicsSchool of Education, Cultureand CommunicationMälardalen UniversityVästerås, Sweden

Milica RančićDivision of Applied MathematicsSchool of Education, Cultureand CommunicationMälardalen UniversityVästerås, Sweden

ISSN 2194-1009 ISSN 2194-1017 (electronic)Springer Proceedings in Mathematics & StatisticsISBN 978-3-030-41849-6 ISBN 978-3-030-41850-2 (eBook)https://doi.org/10.1007/978-3-030-41850-2

Mathematics Subject Classification (2010): 08-XX, 16-XX, 17-XX, 00A69, 60-XX

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This Springer imprint is published by the registered company Springer Nature Switzerland AGThe registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated toProfessor Dmitrii S. Silvestrov’s70th birthday

Preface

This book highlights the latest advances in algebraic structures and applicationsfocused on mathematical notions, methods, structures, concepts, problems, algo-rithms, and computational methods important in natural sciences, engineering, andmodern technology. In particular, the book features mathematical methodsand models from rapidly expanding theory of Hom-algebra structures, fromnoncommutative and non-associative algebras and rings associated to generaliza-tions of differential calculus, quantum deformations of algebras, Lie algebras andtheir generalizations, semi-groups and groups actions, constructive algebra withinterplay with topology, knot theory, dynamical systems, functional analysis, per-turbation analysis of Markov chains and applications in financial mathematics,engineering mathematics, and networks analysis.

The book gathers selected, high-quality contributed chapters from several largeresearch communities working on modern algebraic structures and their applica-tions. The chapters cover both theory and applications, and are illustrated with awealth of ideas, theorems, notions, proofs, examples, open problems, and findingson interplay of algebraic structures with other parts of Mathematics and withapplications to help readers grasp the material, and to encourage them to developnew mathematical methods and concepts in their future research. Presenting newmethods and results, reviews of cutting-edge research, and open problems anddirections for future research, they will serve as a source of inspiration for a broadrange of researchers and research students in algebra, noncommutative geometry,noncommutative analysis, applied algebraic structures, algebraic structures andmethods in computational and engineering mathematics, algebraic methods inmathematical and theoretical physics, and other relevant areas of natural scienceand engineering.

This work is one of the long-term outcomes of the International Conference“Stochastic Processes and Algebraic Structures—From Theory TowardsApplications” (SPAS2017) and of the follow-up research efforts, seminars andactivities on algebraic structures and applications developed following the ideas,

vii

research and cooperations initiated at SPAS2017. This top quality focused inter-national conference brought together a selected group of mathematicians,researchers from related subjects and practitioners from industry who activelycontribute to the theory and applications of stochastic processes and algebraicstructures, methods, and models. SPAS2017 conference was co-organised by theMathematics and Applied Mathematics research environment MAM, Division ofApplied Mathematics, Mälardalen University, Västerås and the Department ofMathematics, Stockholm University, Stockholm and was held in Västerås andStockholm, Sweden on October 4–6, 2017. The SPAS2017 conference was held inhonour of Professor Dmitrii Silvestrov’s 70th birthday and his 50 years of fruitfulservice to mathematics, education, and international cooperation.

Representing the second of two volumes, the book consists of 40 contributedchapters.

Chapter 1 considers a method of construction of the 3-Lie algebra from a Liealgebra equipped with an analogue of the notion of trace and combines it with basedon a Lie algebra construction of the Weil algebra, which is a universal model forconnection and curvature. To this end, in addition to universal connection andcurvature, the chapter introduces new elements, and extends the action of the dif-ferential of Weil algebra to these new elements with the help of the structureconstants of 3-Lie algebra. The chapter also considers one of the most importantapplications of the Weil algebra in a field theory, the construction of the B.R.S.algebra, and an analogue of the B.R.S. algebra is constructed by means of a 3-Liealgebra.

Chapter 2 investigates the possibility of combining the usual Grassmann alge-bras with their ternary Z3-graded counterparts, thus creating a more general algebrawith quadratic and cubic constitutive relations coexisting together. The classifica-tion of ternary and cubic algebras according to the symmetry properties of ternaryproducts under the action of the S3 permutation group is recalled. Instead of onlytwo kinds of binary algebras, symmetric or antisymmetric, one gets four differentgeneralizations. A particular case of algebras generated by two types of variables,na and hA, satisfying quadratic and cubic relations respectively, nanb ¼ �nbna andhAhBhC ¼ jhBhChA, j ¼ e

2pi3 , is considered. Differential calculus of the first order is

defined on these algebras, and its fundamental properties are investigated. Theinvariance properties of the generalized algebras are also considered.

Chapter 3 gives a survey of methods for constructing ternary Lie algebras andternary Lie superalgebras, proposes a generalization of the Nambu-Hamiltonequation to a superspace, and shows that this generalization induces a family ofternary Nambu-Poisson brackets of even degree functions on a superspace. It is alsoshown that the construction of ternary quantum Nambu-Poisson bracket, based onthe trace of a matrix, can be extended to matrix Lie superalgebra glðm; nÞ by meansof the supertrace of a matrix. The method of constructing ternary Lie algebras withthe help of a derivation and an involution of a commutative, associative algebra isextended to commutative superalgebra with superinvolution and even degreederivation. A generalization of the Nambu-Hamilton equation in superspace is

viii Preface

proposed, and a family of ternary Nambu-Poisson brackets, defined with the help ofBerezinian, is introduced.

Chapter 4 is concerned with properties of derivations, ðas; brÞ-derivations,generalized derivations and quasiderivations of n-BiHom-Lie algebras, and gen-eralized derivations of ðnþ 1Þ-BiHom-Lie algebras induced by n-BiHom-Liealgebras.

Chapter 5 is devoted to n-ary BiHom-algebras, generalizing BiHom-algebras,introduces an alternative concept of BiHom-Lie algebra called BiHom-Lie-Leibnizalgebra and studies various types of n-ary BiHom-Lie algebras andBiHom-associative algebras. It is shown that n-ary BiHom-Lie-Leibniz algebra canbe represented by BiHom-Lie-Leibniz algebra through fundamental objects, andconstructions of n-ary BiHom-Lie algebras induced by ðn� 1Þ-ary BiHom-Liealgebras are considered.

Chapter 6 aims to generalize the concepts of k-solvability and k-nilpotency,initially defined for n-Lie algebras, to n-Hom-Lie algebras and to study theirproperties. In particular, k-derived series and k-central descending series are definedand their properties are studied. It is shown that k-solvability is a radical property,and these properties are considered for ðnþ 1Þ-Hom-Lie algebras induced by n-Hom-Lie algebras.

Chapter 8 introduces and studies nilpotent and filiform Hom-Lie algebras, and aclassification of filiform Hom-Lie algebras of dimension n� 7 is presented.

In Chap. 9, the variety defined by a system of polynomial equations, containingboth structure constants of the skew-symmetric bilinear map and constantsdescribing the twisting linear endomorphism for Hom-Lie algebras, is considered.The equations are linear in the constants representing the endomorphism andnon-linear in the structure constants. When the Hom-Lie algebra is 3 or4-dimensional, the space of possible endomorphisms with minimum dimension isdescribed. For the 3-dimensional case, families of 3-dimensional Hom-Lie algebrasarising from a general nilpotent linear endomorphism are described up to isomor-phism together with non-isomorphic canonical representatives for all families.Furthermore, a list of 4-dimensional Hom-Lie algebras arising from a generalnilpotent linear endomorphism is presented.

In Chap. 10, the universal enveloping algebra of color Hom-Lie algebras isstudied. A construction of the free Hom-associative color algebra on a Hom-moduleis described for a certain type of color Hom-Lie algebras and is applied to obtain theuniversal enveloping algebra of those Hom-Lie color algebras. Finally, this con-struction is applied to obtain the extension of the well-known Poincaré-Birkhoff-Witt theorem for Lie algebras to the enveloping algebra of the certaintypes of color Hom-Lie algebra.

Chapter 11 reviews the current progress on Hom-Gerstenhaber algebras andHom-Lie algebroids, representations and cohomology of Hom-Lie algebroids,connection to a differential calculus and dual description for Hom-Lie algebroids,and the relationship between Hom-Lie algebroids and Hom-Gerstenhaber algebras.

Preface ix

In Chap. 12, a modified concept of strong Hom-associativity is introduced. It isproved that the basic “Yau twist” construction of a Hom-associative algebra froman associative algebra does in fact produce strongly Hom-associative algebras. It isproved that the axioms for a strongly Hom-associative algebra yield a confluentrewrite system, and a basis for the free strongly Hom-associative algebra is given afinite presentation through a parsing expression grammar.

Chapter 13 concerns Hom-Hopf algebras, Hom-Yetter-Drinfeld category, brai-ded tensor categories and subcategories, quasitriangular (or cobraided) Hom-Hopfalgebras, category of left Hom-modules or Hom-comodules, Radford biproductHom-Hopf algebra and generalizations, and results relating these structures, pre-senting conditions for classes of R-smash product Hom-algebras and T-smashcoproduct Hom-coalgebras to be a Hom-Hopf algebras and also providing somenontrivial examples.

In Chap. 14, the structural aspects of the f -quandle theory are used to classify, upto isomorphisms, all f -quandles of order n. The classification is based on aneffective algorithm that generates and checks all f -quandles for a given order.

Chapter 15 is devoted to noncommutatively graded algebras and generalizationsof various classical graded results to the noncommutatively graded situation, suchas results concerning identity elements, inverses, existence of limits and colimits,and adjointness of certain functors. In the particular instance of noncommutativelygraded Lie algebras, the existence of universal graded enveloping algebras andgraded version of the Poincaré-Birkhoff-Witt theorem are established.

Chapter 16 pertains to some aspects of differential calculus on associativealgebras with focus on the notion of a “symmetry” of a generalized zero curvatureequation and Bäcklund and forward, backward, and binary Darboux transforma-tions. A matrix version of the binary Darboux transformation and application to aninfinite system of equations is considered. Finally, a recent work on a deformationof the matrix binary Darboux transformation in bidifferential calculus, leading to atreatment of integrable equations with sources is reviewed.

Chapter 17 presents interesting ideas and approaches regarding the DixmierConjecture, its generalizations, and analogues.

Chapter 18 considers commutants in crossed product algebras, for algebras ofpiece-wise constant functions on the real line acted on by the group of integers Z.The algebra of piece-wise constant functions does not separate points of the realline, and interplay of the action with separation properties of the points or subsetsof the real line by the function algebra become essential for many properties of thecrossed product algebras and their subalgebras. Properties of this class of crossedproduct algebras and interplay with dynamics of the actions are investigated, andthe commutants and changes in the commutants are studied in the crossed productsfor the canonical generating commutative function subalgebras of the algebra ofpiece-wise constant functions with common jump points when arbitrary number ofjump points are added or removed. In Chap. 19, the Ore extension algebra for thealgebra of functions with finite support on a countable set is considered. Explicitformulas are derived for twisted derivations and the centralizer, and the center of theOre extension algebra under specific conditions is described. Chapter 20 is devoted

x Preface

to description for the centralizer of the coefficient ring in the skew PBW extensionsand in PBW extensions of the algebra of functions with finite support on acountable set. In Chap. 21, simple explicit formulas for reordering elements in analgebra with three generators and Lie type relations are derived. Centralizers andcenters are computed as an example of an application of these formulas. Chapter 22is devoted to a general class of multi-parametric family of unital associativecomplex algebras defined by commutation relations associated with group orsemigroup actions of dynamical systems and iterated function systems.A generalization of commutation relations in three generators is also considered,modifying Lie algebra type commutation relations, typical for usual differential ordifference operators, to relations satisfied by more general twisted differenceoperators associated with general twisting maps. General reordering and nestedcommutator formulas for arbitrary elements in these algebras are presented, andsome special cases are considered, generalizing some well-known results inmathematics and physics. In Chap. 23, operator representations of deformed Lietype commutation relations, associated with group or semigroup actions ofdynamical systems and iterated function systems are considered, and it is shownthat some multi-parameter deformed symmetric difference and multiplicationoperators satisfy these commutation relations. The operator representations areconsidered also in the context of twisted derivations. Chapters 24 and 25 aredevoted to characterization of Lie polynomials in the associative algebras bygenerators and relations. Chapter 24 provides a review of some results about Liepolynomials in finitely-generated associative algebras with defining relations thatinvolve deformed commutation relations which appear in study of quantum groups,q-deformed oscillator algebras, Hom-algebra structures, quantum algebras,orthogonal polynomials and special functions, discrete and generalised differentialand integral calculus, extensions of homological algebra structures and algebraiccombinatorics. In Chap. 25, for the q-deformed Heisenberg algebra, that is theunital associative algebra with two generators A;B satisfying the q-deformedcommutation relation AB� qBA ¼ I, a characterization is obtained for all the Liepolynomials in A;B, and basis and graded structure and commutation relations forassociated Lie algebras are studied for torsion-type case, that is if q is a root ofunity.

Chapter 26 is a survey considering the fractional operators in q-calculus. Startingfrom the fractional versions of q-Pochhammer symbol, the notions of the fractionalq-integral and q-derivative are generalised by introducing variable lower bound ofintegration and their properties, examples and counterexamples are considered. InChap. 27, the q;x-special functions are considered. By introducing the new vari-able, a calculus consisting of two dual exponential, hyperbolic, and trigonometricfunctions is developed. The concept of even and odd functions is extended, theformulas for chain rule, Leibniz theorem, and generalised additions for the threeabove functions are considered, and to enable trigonometric formulas with halfargument and de Moivre theorem, Ward numbers and a generalization of rationalnumbers are introduced.

Preface xi

Chapters 28–31 study semigroups and semigroup actions. In Chap. 28, theoverview introduction to the basic constructive algebraic structures with apartnesswith special emphasize on a set and semigroup with apartness is given with thepurpose of better understanding of constructive algebra in Bishop’s style positionfor algebraists, as well as for the ones who apply algebraic knowledge who mightwonder what is constructive algebra all about. In the context of basic constructivealgebraic structures, constructive analogues of isomorphism theorems are given andtwo points of view, classical and constructive are considered. Chapter 29 is anoverview devoted to the semigroups with focus on decomposability of a certaintype of semigroups with finiteness conditions into a semilattice of archimedeansemigroups. Finiteness condition in terms of elements of the semigroup and itssubsemigroups, in terms of ideals or congruences of certain types, and characteri-zations using connections between their elements and their special subsets areconsidered. Some applications of presented classes of semigroups and their semi-lattice decompositions in certain types of ring constructions are presented. In Chap.30, the fixed and common fixed point problems of selfmappings that are satisfyingcertain type of generalized integral contractions in the setup of multiplicative metricspaces are investigated. Well-posedness results for fixed point problem of mapsunder restrictions of integral type contractions are obtained. Moreover, the periodicpoint results for generalized integral type contraction mappings are also obtained.Chapter 31 is concerned with the existence of common fixed points of cycliccontractive mappings satisfying generalized integral contractive conditions inmultiplicative metric spaces. The well-posedness of common fixed point results andperiodic point results of cyclic contractions are also established. These resultsestablish some of the general common fixed point theorems for self-mappings.

Vandermonde type matrices, determinants and determinant varieties, their sta-tionary and extreme points and their connections to differential equations, orthog-onal polynomials, and applications are studied in Chaps. 32–34. In Chap. 32, thevalues of the determinant of Vandermonde matrices with real elements are analyzedboth visually and analytically over the unit sphere in various dimensions. For threedimensions some generalized Vandermonde matrices are analyzed visually. Theextreme points of the ordinary Vandermonde determinant on finite-dimensional unitspheres are given as the roots of rescaled Hermite polynomials and a recursionrelation is provided for the coefficients. Analytical expressions for these roots arealso given for dimension three to seven. A transformation of the optimizationproblem is provided and some relations between the ordinary and generalizedVandermonde matrices involving limits are discussed. In Chap. 33, results onoptimising the Vandermonde determinant on different surfaces defined by uni-variate polynomials are discussed. The coordinates of the extreme points are givenas roots of polynomials. Applications in curve fitting and electrostatics are alsobriefly discussed. In Chap. 34, Wishart matrices and their eigenvalues and theLaguerre ensembles of the spectrum of Wishart matrix are considered. The chapterstudies how to express extreme points of the joint eigenvalue probability densitydistribution of a Wishart matrix using optimisation techniques for the Vandermondedeterminant over certain surfaces implicitly defined by univariate polynomials.

xii Preface

Chapter 35 is devoted to applications in financial engineering connected toapproximations of the price of a financial instruments using cubature formulae onWiener space in the infinite-dimensional setting. An algebraic method for calcu-lating cubature formulae on Wiener space is described. Chapter 36 deals withapplications to a market model with four correlated factors and two stochasticvolatilities, one of which is rapid-changing, while another one is slow-changing intime. An advanced Monte Carlo method based on the theory of cubature in Wienerspace, is used to find the no-arbitrage price of the European call option in the abovemodel. Chapter 37 concerns with applications of Lie symmetry analysis to pricingweather derivatives by partial differential equations. Weather derivatives are pricedconsidering weather variables such as rainfall, temperature, humidity, and wind asthe underlying asset. In this chapter, using the Feynman-Kac theorem and therainfall indexes, the partial differential equation (PDE) that governs the price of anoption is derived, the Lie analysis theory is applied to solve this PDE, the groupclassification is used to find the invariant solutions. Chapter 38 is concerned withperturbed Markov chains important for analysis and ranking in networks, and inparticular with the detailed perturbation analysis for stationary distributions,effective explicit series representations for the corresponding stationary distribu-tions and upper bounds for the deviation and asymptotic expansions with respect tothe perturbation parameter. Chapter 39 is about interaction of the nearest-neighborsand next nearest-neighbors for the mixed type p-adic k-Ising model with spin values�1; þ 1f g and p-adic Gibbs measures for the mixed type p-adic k-Ising model on

the Cayley tree of arbitrary order. Chapter 40 is about the similarity boundary layerequations governing magneto-fluid dynamic steady incompressible laminarboundary layer flow for a point sink with an applied magnetic field, heat, and masstransfer. The series solution method has been effectively implemented to the relatedintegro-differential equation. The condition at infinity is applied to related Padeapproximants of the obtained series solution. The features of the flow characteristic,heat, and mass transfer have been analyzed and discussed with respect to thepertinent parameters viz magnetic and suction/injection parameters. It has beenfound that the magnetic field increases the skin friction but it reduces the heattransfer. Comparison of the obtained results for some particular cases of the presentstudy has been done with the earlier results and they have been found to be in agood agreement.

Västerås, Sweden Sergei SilvestrovNovember 2019 Anatoliy Malyarenko

Milica Rančić

Preface xiii

Acknowledgements This collective book project has been realized thanks to the strategic supportoffered by Mälardalen University for the research and research education in Mathematics con-ducted by the research environment Mathematics and Applied Mathematics (MAM) in theestablished research specialization of Educational Sciences and Mathematics at the School ofEducation, Culture and Communication at Mälardalen University. We are grateful to both theDepartment of Mathematics, Stockholm University, Stockholm and to the Mathematics andApplied Mathematics research environment MAM, Division of Applied Mathematics, The Schoolof Education, Culture and Communication at Mälardalen University, Västerås for support andcooperation in joint organization of the successful international SPAS2017 conference that lead tothis book. We also wish to extend our thanks to the Swedish International DevelopmentCooperation Agency (Sida) and International Science Programme in Mathematical Sciences (ISP),Nordplus program of Nordic Council of Ministers, International Mathematical Union, as well asmany other national and international funding organisations, and the research and educationenvironments and institutions of the individual researchers and research teams who contributed tosuccess of SPAS2017 and to this collective book. Finally, we especially thank all the authors fortheir excellent research contributions to this book. We also thank the staff of the publisher Springerfor their excellent efforts and cooperation in publication of this collective book. All contributedchapters have been reviewed and we are grateful to the reviewers for their work.

xiv Preface

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII

Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXVII

1 Weil Algebra, 3-Lie Algebra and B.R.S. Algebra . . . . . . . . . . . . 1Viktor Abramov1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 3-Lie Algebra Induced by a Lie Algebra . . . . . . . . . . . . . . . 31.3 Weil Algebra of Induced 3-Lie Algebra . . . . . . . . . . . . . . . 61.4 B.R.S. Algebra and 3-Lie Algebra . . . . . . . . . . . . . . . . . . . 10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Algebras with Ternary Composition Law CombiningZ2 and Z3 Gradings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Viktor Abramov, Richard Kerner and Olga Liivapuu2.1 Classification of Ternary and Cubic Algebras . . . . . . . . . . . 132.2 Examples of Z3-Graded Ternary Algebras . . . . . . . . . . . . . . 182.3 Generalized Z2 � Z3-Graded Ternary Algebra . . . . . . . . . . . 212.4 Two Distinct Gradings: Z3 � Z2 Versus Z6 . . . . . . . . . . . . . 242.5 First Order Differential Calculus Over Z2 and Z3

Skew-Symmetric Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 272.6 Graded Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 332.7 Invariance Groups of Z2 and Z3 Skew-Symmetric

Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3 Ternary Lie Superalgebras and Nambu-Hamilton Equationin Superspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Viktor Abramov and Priit Lätt3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 n-Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

xv

3.3 n-Lie Superalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4 Extension of Nambu Approach to Superspace . . . . . . . . . . . 71References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4 Generalized Derivations of n-BiHom-Lie Algebras . . . . . . . . . . . 81Amine Ben Abdeljelil, Mohamed Elhamdadi, Ivan Kaygorodovand Abdenacer Makhlouf4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.2 Basic Review of n-BiHom-Lie Algebras . . . . . . . . . . . . . . . 824.3 Derivations, ðas; brÞ-Derivations and Generalized

Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.4 Quasiderivations of n-BiHom-Lie Algebras . . . . . . . . . . . . . 904.5 Generalized Derivations of ðnþ 1Þ-BiHom-Lie Algebras

Induced by n-BiHom-Lie Algebras . . . . . . . . . . . . . . . . . . . 93References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5 On n-ary Generalization of BiHom-Lie Algebrasand BiHom-Associative Algebras . . . . . . . . . . . . . . . . . . . . . . . . 99Abdennour Kitouni, Abdenacer Makhlouf and Sergei Silvestrov5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.2 BiHom-Lie Algebras and n-BiHom-Lie Algebras . . . . . . . . . 1015.3 Associative Type n-ary BiHom-Algebras . . . . . . . . . . . . . . . 1105.4 ðnþ 1Þ-BiHom-Lie Algebras Induced by n-BiHom-Lie

Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6 On Solvability and Nilpotency for n-Hom-Lie Algebras andðnþ 1Þ-Hom-Lie Algebras Induced by n-Hom-Lie Algebras . . . 127Abdennour Kitouni, Abdenacer Makhlouf and Sergei Silvestrov6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.3 Solvability and Nilpotency of n-Hom-Lie Algebras . . . . . . . 1376.4 Solvability and Nilpotency of ðnþ 1Þ-Hom-Lie Algebras

Induced by n-Hom-Lie Algebras . . . . . . . . . . . . . . . . . . . . . 1456.5 Iterated Constructions and ðnþ kÞ-Hom-Lie Algebras

Induced by n-Hom-Lie Algebras . . . . . . . . . . . . . . . . . . . . . 1496.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7 Multiplicative n-Hom-Lie Color Algebras . . . . . . . . . . . . . . . . . 159Ibrahima Bakayoko and Sergei Silvestrov7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1597.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1617.3 Constructions of n-Hom-Lie Color Algebras . . . . . . . . . . . . 1657.4 Hom-Modules over n-Hom-Lie Color Algebras . . . . . . . . . . 172

xvi Contents

7.5 Generalized Derivation of Color Hom-Algebrasand Their Color Hom-Subalgebras . . . . . . . . . . . . . . . . . . . 174

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

8 On Classification of Filiform Hom-Lie Algebras . . . . . . . . . . . . 189Abdenacer Makhlouf and Mourad Mehidi8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1898.2 Generalities on Hom-Lie Algebras . . . . . . . . . . . . . . . . . . . 1908.3 Solvable, Nilpotent and Filiform Hom-Lie Algebras . . . . . . . 1958.4 Constructions by Twist of Hom-Lie Algebras . . . . . . . . . . . 1978.5 Filiform Hom-Lie Algebras Varieties . . . . . . . . . . . . . . . . . 1998.6 Filiform Hom-Lie Algebras of Dimension � 5 . . . . . . . . . . 2068.7 Filiform Hom-Lie Algebras of Dimension 6 . . . . . . . . . . . . 2068.8 Filiform Hom-Lie Algebras of Dimension 7 . . . . . . . . . . . . 2108.9 Multiplicative Filiform Hom-Lie Algebras . . . . . . . . . . . . . . 216References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

9 Classification of Low-Dimensional Hom-Lie Algebras . . . . . . . . 223Elvice Ongong’a, Johan Richter and Sergei Silvestrov9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2249.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2259.3 Nilpotent Linear Endomorphism Case . . . . . . . . . . . . . . . . . 237References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

10 Enveloping Algebras of Certain Types of Color Hom-LieAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257Abdoreza Armakan and Sergei Silvestrov10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25710.2 Basic Concepts on Hom-Lie Algebras and Color

Quasi-Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26010.3 The Universal Enveloping Algebra . . . . . . . . . . . . . . . . . . . 26510.4 The Poincare–Birkhoff–Witt Theorem . . . . . . . . . . . . . . . . . 26810.5 Hom-Lie Superalgebras Case . . . . . . . . . . . . . . . . . . . . . . . 279References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

11 A Review on Hom-Gerstenhaber Algebras and Hom-LieAlgebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285Satyendra Kumar Mishra and Sergei Silvestrov11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28511.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28811.3 Hom-Lie Algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29311.4 Hom-Gerstenhaber Algebras and Hom-Lie-Rinehart

Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

Contents xvii

11.5 Special Classes of Hom-Gerstenhaber Algebrasand Hom-Lie Algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . 306

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

12 Strong Hom-Associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317Lars Hellström12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31712.2 Canyons and Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31912.3 Applying the Diamond Lemma . . . . . . . . . . . . . . . . . . . . . . 32312.4 A Combinatorial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 330References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

13 On Hom-Yetter-Drinfeld Category . . . . . . . . . . . . . . . . . . . . . . . 339Tianshui Ma, Sergei Silvestrov and Huihui Zheng13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33913.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34013.3 A Class of Braided Tensor Category . . . . . . . . . . . . . . . . . . 34513.4 Twisted Tensor Biproduct Hom-Hopf Algebra . . . . . . . . . . . 350References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

14 On the Classification of f -Quandles . . . . . . . . . . . . . . . . . . . . . . 359Indu Rasika Churchill, Mohamed Elhamdadiand Nicolas Van Kempen14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35914.2 Basics of Quandles and f -Quandles . . . . . . . . . . . . . . . . . . 36014.3 Classification of f -Quandles up to Isomorphism . . . . . . . . . . 364References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

15 Noncommutatively Graded Algebras . . . . . . . . . . . . . . . . . . . . . 371Patrik Nystedt15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37115.2 Graded Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37315.3 The Graded Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . 38015.4 Noncommutatively Graded Lie Algebras . . . . . . . . . . . . . . . 381References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

16 Differential Calculi on Associative Algebras and IntegrableSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385Aristophanes Dimakis and Folkert Müller-Hoissen16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38516.2 Bidifferential Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38816.3 Symmetries in Bidifferential Calculus . . . . . . . . . . . . . . . . . 39516.4 A Matrix Version of the Binary Darboux Transformation . . . 40116.5 An Infinite System of Equations in Bidifferential Calculus

with Solutions Generated by the Binary DarbouxTransformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

xviii Contents

16.6 Deformation of Binary Darboux Transformationsand Integrable Systems with Sources . . . . . . . . . . . . . . . . . . 405

16.7 Conclusions and Further Remarks . . . . . . . . . . . . . . . . . . . . 407References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

17 The Jacobian Conjecture2n Implies the Dixmier Problemn . . . . 411Vladimir V. Bavula17.1 The Jacobian Conjecture2n implies the Dixmier Problemn . . . 41117.2 The Inversion Formula for the Weyl Algebras,

the Polynomial Algebras and Their Tensor Products . . . . . . 41917.3 Every Endomorphism of the Algebra I1

Is an Automorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

18 Commutants in Crossed Product Algebras for PiecewiseConstant Functions on the Real Line . . . . . . . . . . . . . . . . . . . . . 427Alex Behakanira Tumwesigye, Johan Richter and Sergei Silvestrov18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42818.2 Definitions and a Preliminary Result . . . . . . . . . . . . . . . . . . 42918.3 Commutants in Crossed Product Algebras for Piecewise

Constant Functions on the Real Line . . . . . . . . . . . . . . . . . . 43018.4 Jump Points Added Arbitrarily . . . . . . . . . . . . . . . . . . . . . . 43218.5 Finitely Many Jump Points Added . . . . . . . . . . . . . . . . . . . 43318.6 An Example with Two Jump Points Added . . . . . . . . . . . . . 43618.7 Comparison of Commutants for General Sets . . . . . . . . . . . 440References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

19 Ore Extensions of Function Algebras . . . . . . . . . . . . . . . . . . . . . 445Alex Behakanira Tumwesigye, Johan Richter and Sergei Silvestrov19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44519.2 Ore Extensions. Basic Preliminaries . . . . . . . . . . . . . . . . . . 44619.3 Derivations on Algebras of Functions on a Finite Set . . . . . . 44719.4 Centralizers in Ore Extensions for Functional Algebras . . . . 45219.5 Infinite Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . 45819.6 The Skew Power Series Ring . . . . . . . . . . . . . . . . . . . . . . . 46219.7 The Skew-Laurent Ring A½x; x�1; ~r� . . . . . . . . . . . . . . . . . . 464References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

20 Centralizers in PBW Extensions . . . . . . . . . . . . . . . . . . . . . . . . 469Alex Behakanira Tumwesigye, Johan Richter and Sergei Silvestrov20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46920.2 Definitions and Preliminary Notions . . . . . . . . . . . . . . . . . . 47020.3 Centralizers in Skew PBW Extensions . . . . . . . . . . . . . . . . 472

Contents xix

20.4 Skew PBW Extensions of Function Algebras . . . . . . . . . . . 474References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

21 Reordering, Centralizers and Centers in an Algebrawith Three Generators and Lie Type Relations . . . . . . . . . . . . . 491John Musonda, Sten Kaijser and Sergei Silvestrov21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49121.2 Some Commutator Formulas in AR;J;Q . . . . . . . . . . . . . . . . 49521.3 Centralizers and the Center in AR;J;Q . . . . . . . . . . . . . . . . . 50121.4 Linear Transformation of Generators . . . . . . . . . . . . . . . . . . 50421.5 An Expression for ½Qk; SmTn� . . . . . . . . . . . . . . . . . . . . . . . 50521.6 Centralizers of S; T and Q, and the Center . . . . . . . . . . . . . . 506References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

22 Reordering in Noncommutative Algebras Associatedwith Iterated Function Systems . . . . . . . . . . . . . . . . . . . . . . . . . 509John Musonda, Johan Richter and Sergei Silvestrov22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51022.2 Commutation Relations and Reordering . . . . . . . . . . . . . . . 51222.3 Reordering Formulas for Sj;Q-Elements . . . . . . . . . . . . . . . 51522.4 Commutator Formulas for Sj;Q-Elements . . . . . . . . . . . . . . 51922.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52522.6 Linear Transformation of the Sj-Generators . . . . . . . . . . . . . 54122.7 Reordering Formulas for Rj;Q-Elements . . . . . . . . . . . . . . . 54722.8 Some Operator Representations . . . . . . . . . . . . . . . . . . . . . 549References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550

23 Twisted Difference Operator Representations of Deformed LieType Commutation Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 553John Musonda, Johan Richter and Sergei Silvestrov23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55323.2 Operator Representations and Derivations . . . . . . . . . . . . . . 55523.3 First Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55823.4 Generalization of the First Representation . . . . . . . . . . . . . . 56023.5 Another Generalization of the First Representation . . . . . . . . 566References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571

24 Torsion-Type q-Deformed Heisenberg Algebraand Its Lie Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575Rafael Reno Cantuba and Sergei Silvestrov24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57524.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57724.3 Consequences of q Being a Root of Unity on Structure

Constants and Commutators . . . . . . . . . . . . . . . . . . . . . . . . 582

xx Contents

24.4 The Lie Algebra Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592

25 Lie Polynomial Characterization Problems . . . . . . . . . . . . . . . . 593Rafael Reno Cantuba and Sergei Silvestrov25.1 Introduction to a Lie Polynomial Characterization

Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59325.2 Lie Polynomials in the Free Algebra . . . . . . . . . . . . . . . . . . 59425.3 The Linearly Twisted Commutation Relation

AB ¼ rðBAÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59525.4 The Macfarlane-Biedenharn q-Oscillator Algebra . . . . . . . . . 59625.5 The Fairlie-Odesskii Algebra . . . . . . . . . . . . . . . . . . . . . . . 59825.6 The Universal Askey–Wilson Algebra . . . . . . . . . . . . . . . . . 599References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600

26 Relations Between the Fractional Operators in q-Calculus . . . . . 603Sergei Silvestrov, Predrag M. Rajković, Sladjana D. Marinkovićand Miomir S. Stanković26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60326.2 Basic Definitions and Statements . . . . . . . . . . . . . . . . . . . . 60426.3 The Fractional q-Integral . . . . . . . . . . . . . . . . . . . . . . . . . . 61026.4 The Fractional q-Derivative of Riemann–Liouville Type . . . 61526.5 The Fractional q-Derivative of Caputo Type . . . . . . . . . . . . 61726.6 The Relations Between Fractional q-Operators . . . . . . . . . . . 620References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624

27 On the Exponential and Trigonometric q;x-SpecialFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625Thomas Ernst27.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62527.2 Preliminary Definitions and Theorems . . . . . . . . . . . . . . . . . 62627.3 On the q;x-Addition with Applications to q;x-Special

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63127.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651

28 Isomorphism Theorems for Basic Constructive AlgebraicStructures with Special Emphasize On ConstructiveSemigroups with Apartness—An Overview . . . . . . . . . . . . . . . . 653Melanija Mitrović and Sergei Silvestrov28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65428.2 Algebraic Structures within CLASS . . . . . . . . . . . . . . . . . . 65628.3 Algebraic Structures within BISH . . . . . . . . . . . . . . . . . . . . 66428.4 Conclusion Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685

Contents xxi

29 Semilatice Decompositions of Semigroups. Hereditarinessand Periodicity—An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 687Melanija Mitrović and Sergei Silvestrov29.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68829.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69029.3 Certain Decomposition of a Semigroup . . . . . . . . . . . . . . . . 69629.4 Completely p-Regular Semigroups . . . . . . . . . . . . . . . . . . . 70529.5 Periodic Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71129.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719

30 Common Fixed Point for Integral Type Contractive Mappingsin Multiplicative Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 723Talat Nazir and Sergei Silvestrov30.1 Introduction. Fixed and Common Fixed Points

of Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72330.2 Multiplicative Metric Space . . . . . . . . . . . . . . . . . . . . . . . . 72530.3 Common Fixed Points of Mappings in Multiplicative

Metric Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72730.4 Common Fixed Point Results for Integral Type

Contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72830.5 Well-Posedness Results for Common Fixed Points . . . . . . . . 73530.6 Periodic Points of Contractive Mappings . . . . . . . . . . . . . . . 73730.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740

31 Cyclic Contractions and Common Fixed Point Results ofIntegral Type Contractions in Multiplicative Metric Spaces . . . 743Talat Nazir and Sergei Silvestrov31.1 Introduction. Cyclic Contraction Mappings with

Restrictions of Integral Type . . . . . . . . . . . . . . . . . . . . . . . . 74331.2 Common Fixed Points of Cyclic Contraction Mappings . . . . 74431.3 Well-Posedness Results for Cyclic Contraction Mappings . . . 75231.4 Periodic Points of Cyclic Contractions . . . . . . . . . . . . . . . . 755References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757

32 Extreme Points of the Vandermonde Determinant on theSphere and Some Limits Involving the GeneralizedVandermonde Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761Karl Lundengård, Jonas Österberg and Sergei Silvestrov32.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76132.2 Optimizing the Vandermonde Determinant over

the Unit Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76732.3 Some Limit Theorems Involving the Generalized

Vandermonde Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785

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32.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789

33 Extreme Points of the Vandermonde Determinant on SurfacesImplicitly Determined by a Univariate Polynomial . . . . . . . . . . 791Asaph Keikara Muhumuza, Karl Lundengård, Jonas Österberg,Sergei Silvestrov, John Magero Mango and Godwin Kakuba33.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79133.2 Some Applications of the Vandermonde Determinant

and Its Extreme Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79333.3 Extreme Points of the Vandermonde Determinant on

Surfaces Defined a Low Degree Univariate Polynomial . . . . 79633.4 Critical Points on the Sphere Defined by a p-norm . . . . . . . 80233.5 Some Results for Cubes and Intersections of Planes . . . . . . . 81433.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817

34 Optimization of the Wishart Joint Eigenvalue ProbabilityDensity Distribution Based on the VandermondeDeterminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819Asaph Keikara Muhumuza, Karl Lundengård, Jonas Österberg,Sergei Silvestrov, John Magero Mango and Godwin Kakuba34.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82034.2 Overview of Random Matrix Theory . . . . . . . . . . . . . . . . . 82434.3 Classical Random Matrix Ensembles . . . . . . . . . . . . . . . . . . 82534.4 The Vandermonde Determinant and Joint Eigenvalue

Probability Densities for Random Matrices . . . . . . . . . . . . . 82734.5 Optimising the Joint Eigenvalue Probability

Density Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83134.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836

35 An Algebraic Method for Pricing Financial Instrumentson Post-crisis Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839Anatoliy Malyarenko, Hossein Nohrouzian and Sergei Silvestrov35.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83935.2 Some Financial Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84035.3 The Cuchiero–Fontana–Gnoatto Model . . . . . . . . . . . . . . . . 84135.4 The Kusuoka–Bayer–Teichmann Framework . . . . . . . . . . . . 84335.5 An Algebraic Method for Finding Cubature Formulae . . . . . 84635.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855

Contents xxiii

36 Advanced Monte Carlo Pricing of European Options in aMarket Model with Two Stochastic Volatilities . . . . . . . . . . . . . 857Betuel Canhanga, Anatoliy Malyarenko, Jean-Paul Murara, Ying Niand Sergei Silvestrov36.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85736.2 Stochastic Cubature Formulae . . . . . . . . . . . . . . . . . . . . . . . 85836.3 The Simulation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 86336.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86536.5 Conclusions and Further Remarks . . . . . . . . . . . . . . . . . . . . 87136.6 Cubature on a Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . 871References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873

37 Lie Symmetry Analysis on Pricing Weather Derivativesby Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 875Clarinda Nhangumbe, Ebrahim Fredericks and Betuel Canhanga37.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87537.2 Outline of Lie Symetries Method for PDEs . . . . . . . . . . . . . 88237.3 The PDE of Ornstein-Uhlenbeck Process . . . . . . . . . . . . . . . 89037.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 899References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 900

38 Perturbation Analysis for Stationary Distributions of MarkovChains with Damping Component . . . . . . . . . . . . . . . . . . . . . . . 903Dmitrii Silvestrov, Sergei Silvestrov, Benard Abola,Pitos Seleka Biganda, Christopher Engström, John Magero Mangoand Godwin Kakuba38.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90438.2 Markov Chains with Damping Component (MCDC) . . . . . . 90538.3 Stationary Distributions of MCDC . . . . . . . . . . . . . . . . . . . 90938.4 Perturbation Model for MCDC . . . . . . . . . . . . . . . . . . . . . . 91338.5 Rate of Convergence for Stationary Distributions

of Perturbed MCDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91538.6 Asymptotic Expansions for Stationary Distributions

of Perturbed MCDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 920References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 930

39 Uniqueness of p-Adic Gibbs Measures for p-Adic k-IsingModel on Cayley Tree of Arbitrary Order . . . . . . . . . . . . . . . . 935Mutlay Dogan39.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93539.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93639.3 Uniqueness of p-adic Gibbs Measures . . . . . . . . . . . . . . . . . 93839.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943

xxiv Contents

40 On Pade Approximants Series Solutions of MHD FlowEquations with Heat and Mass Transfer Due to a Point Sink . . . 945Imran Chandarki and Brijbhan Singh40.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94540.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94640.3 Mathematical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94940.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95340.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 961

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967

Contents xxv

Contributors

Benard Abola Division of Applied Mathematics, School of Education, Cultureand Communication, Mälardalen University, Västerås, Sweden

Viktor Abramov Institute of Mathematics and Statistics, University of Tartu,Tartu, Estonia

Abdoreza Armakan Department of Pure Mathematics, Faculty of Mathematicsand Computer Science, Shahid Bahonar University of Kerman, Kerman, Iran

Ibrahima Bakayoko Département de Mathématiques, Université de N’Zérékoré,Nzerekore, Guinea

Vladimir V. Bavula Department of Pure Mathematics, University of Sheffield,Hicks Building, Sheffield, UK

Amine Ben Abdeljelil Department of Mathematics, University of South Florida,Tampa, FL, USA

Pitos Seleka Biganda Division of Applied Mathematics, School of Education,Culture and Communication, Mälardalen University, Västerås, Sweden

Betuel Canhanga Faculty of Sciences, Department of Mathematics, Informaticand Computer Sciences, Eduardo Mondlane University, Maputo, Mozambique

Rafael Reno Cantuba Mathematics and Statistics Department, De La SalleUniversity, Metro Manila, Philippines

Imran Chandarki Department of General Science and Engineering, N. B. NavaleSinhgad College of Engineering, Solapur, Maharashtra, India

Indu Rasika Churchill Department of Mathematics, State University of NewYork at Oswego, Oswego, NY, USA

Aristophanes Dimakis Department of Financial and Management Engineering,University of the Aegean, Chios, Greece

xxvii

Mutlay Dogan Faculty of Pure and Applied Sciences, School of Mathematics,Physics and Technology, University of Bahamas, Nassau, Bahamas

Mohamed Elhamdadi Department of Mathematics, University of South Florida,Tampa, FL, USA

Christopher Engström Division of Applied Mathematics, School of Education,Culture and Communication, Mälardalen University, Västerås, Sweden

Thomas Ernst Department of Mathematics, Uppsala University, Uppsala, Sweden

Ebrahim Fredericks Faculty of Science, Department of Mathematic and AppliedMathematics, University of Cape Town, Welgelegen, Cape Town, South Africa

Lars Hellström Division of Applied Mathematics, School of Education, Cultureand Communication, Mälardalen University, Västerås, Sweden

Sten Kaijser Department of Mathematics, Uppsala University, Uppsala, Sweden

Godwin Kakuba Department of Mathematics, School of Physical Sciences,Makerere University, Kampala, Uganda

Ivan Kaygorodov Universidade Federal do ABC, CMCC, Santo André, Brazil

Richard Kerner LPTMC, Paris, France

Abdennour Kitouni Division of Applied Mathematics, School of Education,Culture and Communication, Mälardalen University, Västerås, Sweden

Priit Lätt Institute of Mathematics and Statistics, University of Tartu, Tartu,Estonia

Olga Liivapuu Institute of Technology, Estonian University of Life Sciences,Tartu, Estonia

Karl Lundengård Division of Applied Mathematics, School of Education,Culture and Communication, Mälardalen University, Västerås, Sweden

Tianshui Ma Henan Engineering Laboratory for Big Data Statistical Analysis andOptimal Control, Henan Normal University, Xinxiang, China

Abdenacer Makhlouf IRIMAS-département de Mathématiques, Université deHaute Alsace, Mulhouse, France

Anatoliy Malyarenko Division of Applied Mathematics, School of Education,Culture and Communication, Mälardalen University, Västerås, Sweden

John Magero Mango Department of Mathematics, School of Physical Sciences,Makerere University, Kampala, Uganda

Sladjana D. Marinković Faculty of Electronic Engineering, Department ofMathematics, University of Niš, Niš, Serbia

Mourad Mehidi Université de Haute-Alsace, Mulhouse, France

xxviii Contributors

Satyendra Kumar Mishra Theoretical Statistics and Mathematics Unit, IndianStatistical Institute, Bangalore Centre, Bangalore, India

Melanija Mitrović Department of Mathematics and Informatics, Faculty ofMechanical Engineering, University of Niš, Niš, Serbia

Asaph Keikara Muhumuza Department of Mathematics, Busitema University,Tororo, Uganda;Division of Applied Mathematics, School of Education, Culture andCommunication, Mälardalen University, Västerås, Sweden

FolkertMüller-Hoissen Max Planck Institute for Dynamics and Self-Organization,Göttingen, Germany;Institute for Theoretical Physics, University of Göttingen, Göttingen, Germany

Jean-Paul Murara Division of Applied Mathematics, Mälardalen University,Västerås, Sweden

John Musonda Department of Mathematics and Statistics, School of NaturalSciences, The University of Zambia, Lusaka, Zambia

Talat Nazir Department of Mathematics, COMSATS Institute of InformationTechnology, Abbottabad, Pakistan;Division of Applied Mathematics, School of Education, Culture andCommunication, Mälardalen University, Västerås, Sweden

Clarinda Nhangumbe Faculty of Science, Department of Mathematic andInformatic, Eduardo Mondlane University, Maputo, Mozambique;Faculty of Science, Department of Mathematic and Applied Mathematics,University of Cape Town, Welgelegen, Cape Town, South Africa

Ying Ni Division of Applied Mathematics, Mälardalen University, Västerås,Sweden

Hossein Nohrouzian Division of Applied Mathematics, School of Education,Culture and Communication, Mälardalen University, Västerås, Sweden

Patrik Nystedt Department of Engineering Science, University West, Trollhättan,Sweden

Elvice Ongong’a School of Mathematics, University of Nairobi, Nairobi, Kenya;Division of Applied Mathematics, School of Education, Culture andCommunication, Mälardalen University, Västerås, Sweden

Jonas Österberg Division of Applied Mathematics, School of Education, Cultureand Communication, Mälardalen University, Västerås, Sweden

Predrag M. Rajković Faculty of Mechanical Engineering, Department ofMathematics, University of Niš, Niš, Serbia

Contributors xxix

Johan Richter Department of Mathematics and Natural Sciences, BlekingeInstitute of Technology, Karlskrona, Sweden

Dmitrii Silvestrov Department of Mathematics, Stockholm University, Stockholm,Sweden

Sergei Silvestrov Division of Applied Mathematics, School of Education, Cultureand Communication, Mälardalen University, Västerås, Sweden

Brijbhan Singh Department of Mathematics, Dr. Babasaheb AmbedkarTechnological University, Lonere, Raigad, Maharashtra, India

Miomir S. Stanković Faculty of Occupation Safety, Department of Mathematics,University of Niš, Niš, Serbia

Alex Behakanira Tumwesigye Department of Mathematics, College of NaturalSciences, Makerere University, Kampala, Uganda

Nicolas Van Kempen Department of Mathematics, State University of New Yorkat Oswego, Oswego, NY, USA

Huihui Zheng Henan Engineering Laboratory for Big Data Statistical Analysisand Optimal Control, Henan Normal University, Xinxiang, China

xxx Contributors

Chapter 1Weil Algebra, 3-Lie Algebra and B.R.S.Algebra

Viktor Abramov

Abstract We consider a method that allows to construct the 3-Lie algebra if wehave a Lie algebra equipped with an analogue of the notion of trace. At the sametime, it is well known that, based on a Lie algebra, we can construct theWeil algebra,which is a universal model for connection and curvature. In this paper, we propose ananswer to the question of how one could extend the construction of the Weil algebrafrom a Lie algebra to the induced 3-Lie algebra. To this end, in addition to universalconnection and curvature, we introduce new elements and extend the action of thedifferential ofWeil algebra to these new elements with the help of structure constantsof 3-Lie algebra. Since one of the most important applications of Weil algebra in afield theory is the construction of B.R.S. algebra, we propose an analogue of B.R.S.algebra constructed by means of a 3-Lie algebra.

Keywords n-Lie algebra · 3-Lie algebra · Weil algebra · Connection ·Curvature · Equivariant differential forms

MSC 2010 Classification Primary 17B60 · Secondary 17B56

1.1 Introduction

n-Lie algebra, where n ≥ 2, is a generalization of the notion of Lie algebra, which isthe particular case of n-Lie algebra, when n = 2. Here, an integer n is the number ofelements of algebra, which is required to compose a product. Thus, a multiplicationlaw of n-Lie algebra g is an n-arymultilinear mapping g×g×. . .×g (n times) → g,which is totally skew-symmetric and satisfies the Filippov–Jacobi identity, whichis also called fundamental identity in applications of n-Lie algebras in Nambu’sgeneralization of Hamiltonian mechanics and field theories.

V. Abramov (B)Institute of Mathematics and Statistics, University of Tartu, J. Liivi 2, 50409 Tartu, Estoniae-mail: viktor.abramov@ut.ee

© Springer Nature Switzerland AG 2020S. Silvestrov et al. (eds.), Algebraic Structures and Applications,Springer Proceedings in Mathematics & Statistics 317,https://doi.org/10.1007/978-3-030-41850-2_1

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