Algebraic Structures and Applications: SPAS 2017, Västerås and Stockholm, Sweden, October 4-6

976 Pages·2020·9.395 MB·English

by  Sergei Silvestrov, Anatoliy Malyarenko, Milica Rančić

Springer Proceedings in Mathematics & Statistics
Sergei Silvestrov
Anatoliy Malyarenko
Milica Rančić   Editors
Algebraic
Structures and
Applications
SPAS 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 from
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More information about this series at http://www.springer.com/series/10533
Sergei Silvestrov Anatoliy Malyarenko
(cid:129) (cid:129)
č ć
Milica Ran i
Editors
Algebraic Structures
and Applications
ä å
SPAS 2017, V ster s and Stockholm,
–
Sweden, October 4 6
123
Editors
SergeiSilvestrov Anatoliy Malyarenko
Division of AppliedMathematics Division of AppliedMathematics
Schoolof Education, Culture Schoolof Education, Culture
andCommunication andCommunication
Mälardalen University Mälardalen University
Västerås,Sweden Västerås,Sweden
Milica Rančić
Division of AppliedMathematics
Schoolof Education, Culture
andCommunication
Mälardalen University
Västerås,Sweden
ISSN 2194-1009 ISSN 2194-1017 (electronic)
SpringerProceedings in Mathematics& Statistics
ISBN978-3-030-41849-6 ISBN978-3-030-41850-2 (eBook)
https://doi.org/10.1007/978-3-030-41850-2
MathematicsSubjectClassification(2010): 08-XX,16-XX,17-XX,00A69,60-XX
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Dedicated to
Professor Dmitrii S. Silvestrov’s
70th birthday
Preface
This book highlights the latest advances in algebraic structures and applications
focused on mathematical notions, methods, structures, concepts, problems, algo-
rithms, and computational methods important in natural sciences, engineering, and
modern technology. In particular, the book features mathematical methods
and models from rapidly expanding theory of Hom-algebra structures, from
noncommutative and non-associative algebras and rings associated to generaliza-
tions of differential calculus, quantum deformations of algebras, Lie algebras and
their generalizations, semi-groups and groups actions, constructive algebra with
interplay 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 large
research communities working on modern algebraic structures and their applica-
tions. The chapters cover both theory and applications, and are illustrated with a
wealth of ideas, theorems, notions, proofs, examples, open problems, and findings
on interplay of algebraic structures with other parts of Mathematics and with
applications to help readers grasp the material, and to encourage them to develop
new mathematical methods and concepts in their future research. Presenting new
methods and results, reviews of cutting-edge research, and open problems and
directions for future research, they will serve as a source of inspiration for a broad
range of researchers and research students in algebra, noncommutative geometry,
noncommutative analysis, applied algebraic structures, algebraic structures and
methods in computational and engineering mathematics, algebraic methods in
mathematical and theoretical physics, and other relevant areas of natural science
and engineering.
This work is one of the long-term outcomes of the International Conference
“Stochastic Processes and Algebraic Structures—From Theory Towards
Applications” (SPAS2017) and of the follow-up research efforts, seminars and
activities on algebraic structures and applications developed following the ideas,
vii
viii Preface
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 actively
contribute to the theory and applications of stochastic processes and algebraic
structures, methods, and models. SPAS2017 conference was co-organised by the
Mathematics and Applied Mathematics research environment MAM, Division of
Applied Mathematics, Mälardalen University, Västerås and the Department of
Mathematics, Stockholm University, Stockholm and was held in Västerås and
Stockholm,SwedenonOctober4–6,2017.TheSPAS2017conferencewasheldin
honour of Professor Dmitrii Silvestrov’s 70th birthday and his 50 years offruitful
service to mathematics, education, and international cooperation.
Representing the second of two volumes, the book consists of 40 contributed
chapters.
Chapter 1 considers a method of construction of the 3-Lie algebra from a Lie
algebraequippedwithananalogueofthenotionoftraceandcombinesitwithbased
on a Lie algebra construction of the Weil algebra, which is a universal model for
connection and curvature. To this end, in addition to universal connection and
curvature, 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 structure
constants of 3-Lie algebra. The chapter also considers one of the most important
applications 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-Lie
algebra.
Chapter 2 investigates the possibility of combining the usual Grassmann alge-
braswiththeirternaryZ -gradedcounterparts,thuscreatingamoregeneralalgebra
3
with quadratic and cubic constitutive relations coexisting together. The classifica-
tion of ternary and cubic algebras according to the symmetry properties of ternary
products under the action of the S permutation group is recalled. Instead of only
3
two kinds of binary algebras, symmetric or antisymmetric, one gets four different
generalizations. A particular case of algebras generated by two types of variables,
na andhA,satisfyingquadratic andcubicrelations respectively, nanb ¼(cid:2)nbna and
hAhBhC ¼jhBhChA, j¼e23pi, is considered. Differential calculus of the first order is
defined on these algebras, and its fundamental properties are investigated. The
invariance properties of the generalized algebras are also considered.
Chapter 3 gives a survey of methods for constructing ternary Lie algebras and
ternary Lie superalgebras, proposes a generalization of the Nambu-Hamilton
equation to a superspace, and shows that this generalization induces a family of
ternaryNambu-Poissonbracketsofevendegreefunctionsonasuperspace.Itisalso
shown that the construction of ternary quantum Nambu-Poisson bracket, based on
thetraceofamatrix,canbeextendedtomatrixLiesuperalgebraglðm;nÞbymeans
ofthesupertraceofamatrix.ThemethodofconstructingternaryLiealgebraswith
the help of a derivation and an involution of a commutative, associative algebra is
extended to commutative superalgebra with superinvolution and even degree
derivation. A generalization of the Nambu-Hamilton equation in superspace is
Preface ix
proposed,andafamilyofternaryNambu-Poissonbrackets,definedwiththehelpof
Berezinian, 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 ðnþ1Þ-BiHom-Lie algebras induced by n-BiHom-Lie
algebras.
Chapter 5 is devoted to n-ary BiHom-algebras, generalizing BiHom-algebras,
introduces analternative concept ofBiHom-LiealgebracalledBiHom-Lie-Leibniz
algebra and studies various types of n-ary BiHom-Lie algebras and
BiHom-associativealgebras.Itisshownthatn-aryBiHom-Lie-Leibnizalgebracan
be represented by BiHom-Lie-Leibniz algebra through fundamental objects, and
constructions of n-ary BiHom-Lie algebras induced by ðn(cid:2)1Þ-ary BiHom-Lie
algebras 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 their
properties.Inparticular,k-derivedseriesandk-centraldescendingseriesaredefined
and their properties are studied. It is shown that k-solvability is a radical property,
and these properties are considered for ðnþ1Þ-Hom-Lie algebras induced by n-
Hom-Lie algebras.
Chapter8introducesandstudiesnilpotentandfiliformHom-Liealgebras,anda
classification offiliform Hom-Lie algebras of dimension n(cid:3)7 is presented.
InChap.9,thevarietydefinedbyasystemofpolynomialequations,containing
both structure constants of the skew-symmetric bilinear map and constants
describing the twisting linear endomorphism for Hom-Lie algebras, is considered.
The equations are linear in the constants representing the endomorphism and
non-linear in the structure constants. When the Hom-Lie algebra is 3 or
4-dimensional, the space of possible endomorphisms with minimum dimension is
described.Forthe3-dimensionalcase,familiesof3-dimensionalHom-Liealgebras
arising 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 general
nilpotent linear endomorphism is presented.
In Chap. 10, the universal enveloping algebra of color Hom-Lie algebras is
studied.AconstructionofthefreeHom-associativecoloralgebraonaHom-module
isdescribedforacertaintypeofcolorHom-Liealgebrasandisappliedtoobtainthe
universal 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 certain
types of color Hom-Lie algebra.
Chapter 11 reviews the current progress on Hom-Gerstenhaber algebras and
Hom-Lie algebroids, representations and cohomology of Hom-Lie algebroids,
connection to a differential calculus and dual description for Hom-Lie algebroids,
andtherelationshipbetweenHom-LiealgebroidsandHom-Gerstenhaberalgebras.
x Preface
In Chap. 12, a modified conceptof strong Hom-associativity is introduced. It is
proved that the basic “Yau twist” construction of a Hom-associative algebra from
anassociativealgebradoesinfactproducestronglyHom-associativealgebras.Itis
proved that the axioms for a strongly Hom-associative algebra yield a confluent
rewritesystem,andabasisforthefreestronglyHom-associativealgebraisgivena
finite 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-Hopf
algebras, category of left Hom-modules or Hom-comodules, Radford biproduct
Hom-Hopf algebra and generalizations, and results relating these structures, pre-
senting conditions for classes of R-smash product Hom-algebras and T-smash
coproduct Hom-coalgebras to be a Hom-Hopf algebras and also providing some
nontrivial examples.
InChap.14,thestructuralaspectsofthef-quandletheoryareusedtoclassify,up
to isomorphisms, all f-quandles of order n. The classification is based on an
effective algorithm that generates and checks all f-quandles for a given order.
Chapter15isdevotedtononcommutativelygradedalgebrasandgeneralizations
of various classical graded results to the noncommutatively graded situation, such
as results concerning identity elements, inverses, existence of limits and colimits,
and adjointness of certain functors. In the particular instance of noncommutatively
graded Lie algebras, the existence of universal graded enveloping algebras and
graded version of the Poincaré-Birkhoff-Witt theorem are established.
Chapter 16 pertains to some aspects of differential calculus on associative
algebras with focus on the notion of a “symmetry” of a generalized zero curvature
equation and Bäcklund and forward, backward, and binary Darboux transforma-
tions.AmatrixversionofthebinaryDarbouxtransformationandapplicationtoan
infinite system of equations is considered. Finally, a recent work on a deformation
of the matrix binary Darboux transformation in bidifferential calculus, leading to a
treatment of integrable equations with sources is reviewed.
Chapter 17 presents interesting ideas and approaches regarding the Dixmier
Conjecture, its generalizations, and analogues.
Chapter 18 considers commutants in crossed product algebras, for algebras of
piece-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 real
line, and interplay of the action with separation properties of the points or subsets
ofthereallinebythefunctionalgebrabecomeessentialformanypropertiesofthe
crossed product algebras and their subalgebras. Properties of this class of crossed
product algebras and interplay with dynamics of the actions are investigated, and
thecommutantsandchangesinthecommutantsarestudiedinthecrossedproducts
for the canonical generating commutative function subalgebras of the algebra of
piece-wiseconstantfunctionswithcommonjumppointswhenarbitrary numberof
jump points are added or removed. In Chap. 19, the Ore extension algebra for the
algebra of functions with finite support on a countable set is considered. Explicit
formulasarederivedfortwistedderivationsandthecentralizer,andthecenterofthe
Oreextensionalgebraunderspecificconditionsisdescribed.Chapter20isdevoted