Non-commutative and Non-associative Algebra and Analysis Structures: SPAS 2019, Västerås, Sweden, September 30–October 2 (Springer Proceedings in Mathematics & Statistics, 426)

The goal of the 2019 conference on Stochastic Processes and Algebraic Structures held in SPAS2019, Västerås, Sweden, from September 30th to October 2nd 2019 was to showcase the frontiers of research in several important topics of mathematics, mathematical statistics, and its applications. The conference has been organized along the following tracks: 1. Stochastic processes and modern statistical methods in theory and practice, 2. Engineering Mathematics, 3. Algebraic Structures and applications. This book highlights the latest advances in algebraic structures and applications focused on mathematical notions, methods, structures, concepts, problems, algorithms, and computational methods for the natural sciences, engineering, and modern technology. In particular, the book features mathematical methods and models from non-commutative and non-associative algebras and rings associated to generalizations of differential calculus, quantumdeformations of algebras, Lie algebras, Lie superalgebras, color Lie algebras, Hom-algebras and their n-ary generalizations, semi-groups and group algebras, non-commutative and non-associative algebras and computational algebra interplay with q-special functions and q-analysis, topology, dynamical systems, representation theory, operator theory and functional analysis, applications of algebraic structures in coding theory, information analysis, geometry and probability theory. The book gathers selected, high-quality contributed chapters from several large research communities working on modern algebraic structures and their applications. The chapters cover both theory and applications, and are illustrated with a wealth of ideas, theorems, notions, proofs, examples, open problems, and results on the interplay of algebraic structures with other parts of Mathematics. The applications help readers grasp the material, and encourage them to develop new mathematical methods and concepts in their future research. Presenting new methods and results, reviews of cutting-edge research, open problems, and directions for future research, will serve as a source of inspiration for a broad range of researchers and students. Preface Contents Contributors 1 Index of Hom-Lie Algebras 1.1 Introduction 1.2 Preliminary 1.2.1 Hom-Lie Algebras 1.2.2 Representations of Hom-Lie Algebras 1.3 Index of Hom-Lie Algebras 1.3.1 For a Coadjoint Representation 1.3.2 For an Arbitrary Representation 1.3.3 Index of Twisted Lie Algebras 1.3.4 Index of Multiplicative Simple Hom-Lie Algebras 1.4 Index of Semidirect Products of Hom-Lie Algebras 1.4.1 Coadjoint Representations 1.4.2 The Stabilizer of an Arbitrary Point of mathfrakqast References 2 On Ternary (Hom-)Nambu-Poisson Algebras 2.1 Introduction 2.2 Ternary Nambu-Poisson Algebras Induced by Poisson Algebras 2.2.1 Examples 2.2.2 Constructing Poisson and Ternary Nambu-Poisson Algebras from Solvable Lie Algebras 2.3 Ternary Hom-Nambu-Poisson Algebras Induced by Hom-Poisson Algebras 2.3.1 Examples References 3 Classification, Centroids and Derivations of Two-Dimensional Hom-Leibniz Algebras 3.1 Introduction 3.2 Hom-Leibniz Algebras and Superalgebras 3.3 Symmetric (Two-Sided) Hom-Leibniz Superalgebras 3.4 Centroids and Derivations of Hom-Leibniz Superalgebras 3.5 Classification of Multiplicative 2-Dimensional Hom-Leibniz Algebras 3.6 Centroids and Derivations of 2-Dimensional Multiplicative Hom-Leibniz Algebras References 4 Color Hom-Lie Algebras, Color Hom-Leibniz Algebras and Color Omni-Hom-Lie Algebras 4.1 Introduction 4.2 Hom-Associative Algebras, Hom-Modules and Color Hom-Lie Algebras 4.3 (σ,τ)-Differential Graded Commutative Color Algebra 4.4 Representations of Color Hom-Lie Algebras References 5 On (σ,τ)-Derivations of Group Algebra as Category Characters 5.1 Introduction 5.2 General Definitions and Preliminaries 5.3 Groupoid and Characters 5.4 Quasi-inner (σ,τ)-Derivations 5.5 (σ,τ)-Nilpotent Groups 5.5.1 General Case of (σ,τ)-Nilpotent Groups 5.5.2 The Case of Inner Endomorphisms 5.5.3 Heisenberg Group 5.6 (σ,τ)-FC Groups References 6 Decomposition of Complete Color Hom-Lie Algebras 6.1 Introduction 6.2 Preliminaries on Color Hom-Lie Algebras and Their Representation and Derivations 6.3 Decomposition of Complete Hom-Lie Superalgebras References 7 Hom-Prealternative Superalgebras 7.1 Introduction 7.2 Hom-Prealternative Algebras and Bimodules 7.3 Hom-Prealternative and Hom-Alternative Superalgebras 7.3.1 Prealternative Superalgebras 7.3.2 Bimodules of Hom-Prealternative Superalgebras References 8 Spectral Analysis of Equations over Quaternions 8.1 Introduction 8.1.1 Motivation 8.1.2 Overview 8.2 Preliminaries 8.2.1 The Quaternions mathscrH 8.2.2 A Representation (mathscrC2, π) of mathscrH 8.3 Quaternion-Spectral Forms 8.3.1 Quaternion-Spectral Forms of mathscrH 8.3.2 Similarity on q-Spectral Forms in mathcalH2 8.3.3 Quaternion-Spectral Equivalence 8.3.4 Quaternion-Spectral Mapping Theorem 8.3.5 The Quaternion-Spectralization σ 8.4 Some Algebraic Structures on mathscrH 8.4.1 Classification of mathscrH 8.4.2 The Quaternions mathscrH and the Lie Group SU2(mathscrC) 8.5 Monomial Equations on mathscrH 8.6 Certain Quadratic Equations on mathscrH 8.7 Linear Equations on mathscrH References 9 Multiplication and Linear Integral Operators on Lp Spaces Representing Polynomial Covariant Type Commutation Relations 9.1 Introduction 9.2 Preliminaries and Notations 9.3 Operator Representations of Covariance Commutation Relations 9.3.1 Representations of Covariance Commutation Relations by Integral and Multiplication Operators on Lp Spaces References 10 Representations of Polynomial Covariance Type Commutation Relations by Piecewise Function Multiplication and Composition Operators 10.1 Introduction 10.2 Preliminaries and Notations 10.3 Representations by Operators Involving Piecewise Functions 10.4 Representations Involving Inner Superposition Operators 10.5 Representations Involving Weighted Composition Operators 10.5.1 Representations by Operators on C[α,β] References 11 Nearly Associative and Nearly Hom-Associative Algebras and Bialgebras 11.1 Introduction 11.2 Nearly Associative Algebras: Basic Definitions and Properties 11.3 Classification of the Two-Dimensional Nearly Associative Algebras 11.4 Bimodules and Matched Pairs Nearly Associative Algebras 11.5 Manin Triple and Bialgebra of Nearly Associative Algebras 11.6 Hom-Lie Admissible, G-Hom-Associative, Flexible and Anti-flexible Hom-Algebras 11.7 Nearly Hom-Associative Algebras, Bimodules and Matched Pairs References 12 On Generalized q-Hyperbolic Functions in the Spirit of Kapteyn, with Corresponding q-Lie Group 12.1 Introduction 12.2 q-Analogues of the Results by Muldoon, Ungar and Kapteyn 12.3 The Corresponding Matrix Formulas 12.4 Connections with the Triangle Operator 12.5 Appendix: Discussion References 13 Divisibility in Hom-Algebras, Single-Element Properties in Non-associative Algebras and Twisted Derivations 13.1 Introduction, Definitions and Notations 13.2 Twisted Derivations. Review of Some Well-Known Results 13.3 Divisibility in Hom-Associative Algebras 13.3.1 Divisibility in Unital Hom-Associative Algebras 13.3.2 Divisibility in Hom-Unital Hom-Associative Algebras 13.4 Sandwich Twisted Derivations and Pivot Commutation 13.4.1 Approach to Relations 13.4.2 Sandwich Twisted Derivatives in Unital Algebras References 14 On Lie-Type Constructions over Twisted Derivations 14.1 Introduction 14.2 Definitions and Notations 14.3 Some Results on Twisted Derivations 14.4 Composition of (σ,τ)-Derivations 14.4.1 mathbbZ-Grading Property of Commutator on (σ,τ)-Derivations 14.4.2 mathbbZ2-Grading on (σ,τ)-Derivations 14.4.3 mathbbZ2-Grading on (σ,τ)-Derivations over n-Ary Algebras 14.5 Hom-Algebras of Generalized Jacobians for (σ,τ)-Derivations 14.5.1 Concerning (σ,σ)-Derivations 14.5.2 When σneqτ 14.5.3 With Equal Commutation Relations 14.5.4 More General Commutation Relations References 15 An Application of Twisted Group Rings in Secure Group Communications 15.1 Introduction 15.2 Algebraic Setting 15.3 Key Management over Twisted Group Rings 15.4 Secure Group Key Management References 16 Construction and Characterization of n-Ary Hom-Bialgebras and n-Ary Infinitesimal Hom-Bialgebras 16.1 Introduction 16.2 Basics and Notations 16.2.1 Hom-Associative Algebras 16.2.2 Hom-Associative Coalgebras 16.2.3 Hom-Bialgebras and Infinitesimal Hom-Bialgebras 16.3 n-Ary Bialgebras of Associative Type 16.4 n-Ary Bialgebras of Hom-Associative Type 16.5 From Infinitesimal (Hom)-Bialgebras to Ternary Infinitesimal (Hom)-Bialgebras 16.5.1 From Infinitesimal Bialgebras to Ternary Infinitesimal Bialgebras 16.5.2 From Infinitesimal Hom-Bialgebras to Ternary Infinitesimal Hom-Bialgebras References 17 Network Rewriting Utility Description 17.1 Introduction 17.2 Example: The Conference Problem 17.3 Program Composition 17.3.1 Development History 17.3.2 Implementation Language 17.4 The Network Datatype Library 17.4.1 Pure Networks 17.4.2 Networks with Feedback 17.4.3 Rich Networks 17.5 The Completion Utility 17.5.1 Algorithms 17.5.2 The Database 17.5.3 Inspecting Database Contents 17.5.4 Included Completion Problems 17.6 Availability References 18 Double Constructions of BiHom-Frobenius Algebras 18.1 Introduction 18.2 Bimodules and Matched Pairs of Hom-associative Algebras 18.2.1 Bimodules of Hom-associative Algebras 18.2.2 Matched Pairs of Hom-associative Algebras 18.3 Double Constructions of Involutive Hom-Frobenius Algebras and Antisymmetric Infinitesimal Hom-bialgebras 18.3.1 Double Constructions of Involutive Hom-Frobenius Algebras 18.3.2 Antisymmetric Infinitesimal Hom-bialgebras 18.4 Double Constructions of Involutive BiHom-Frobenius Algebras 18.4.1 Bimodule and Matched Pair of BiHom-Associative Algebras 18.4.2 Double Constructions of Involutive BiHom-Frobenius Algebras 18.5 Double Constructions of Involutive Symplectic Hom-associative Algebras 18.5.1 Hom-dendriform Algebras 18.5.2 mathcalO -operators and Hom-dendriform Algebras 18.5.3 Bimodules and Matched Pairs of Hom-dendriform Algebras 18.5.4 Double Constructions of Involutive Symplectic Hom-associative Algebras 18.5.5 Hom-dendriform D-bialgebras 18.6 Matched Pairs of BiHom-Associative Algebras 18.6.1 Bihom-dendriform Algebras 18.6.2 mathcalO -operators and BiHom-Dendriform Algebras 18.6.3 Bimodules and Matched Pairs of BiHom-Dendriform Algebras 18.7 Concluding Remarks References 19 On Classification of (n+1)-Dimensional n-Hom-Lie Algebras with Nilpotent Twisting Maps 19.1 Introduction 19.2 Preliminaries 19.3 Properties of (n+1)-Dimensional N-Hom-Lie Algebras 19.4 The Hom-Nambu-Filippov Identity 19.5 Lists of 4-Dimensional 3-Hom-Lie Algebras 19.5.1 Nilpotent α with Kernel of Dimension 1 19.5.2 Nilpotent α with Kernel of Dimension 2 19.5.3 Diagonalisable α with Kernel of Dimension 2 References 20 On Classification of (n+1)-Dimensional n-Hom-Lie Algebras for n =4,5,6 and Nilpotent Twisting Map with 2-Dimensional Kernel 20.1 Introduction 20.2 Definitions and Properties of n-Hom-Lie Algebras 20.3 Lists of (n+1)-Dimensional n-Hom-Lie Algebras in Various Arities References 21 Deforming Algebras with Anti-involution via Twisted Associativity 21.1 Introduction 21.2 Algebras with Involution 21.3 Hom-Structures 21.3.1 Review of Hom-Associative Results 21.3.2 Hom-Associativity of Type II 21.3.3 Example: Temperley-Lieb Algebras 21.4 Discussion References 22 Admissible Hom-Novikov-Poisson and Hom-Gelfand-Dorfman Color Hom-Algebras 22.1 Introduction 22.2 Preliminaries and Some Results 22.2.1 -Commutative Hom-associative color Hom-algebras 22.2.2 On Hom-Novikov Color Hom-algebras 22.2.3 On Hom-Lie color Hom-algebras 22.3 Admissible Hom-Novikov-Poisson Color Hom-algebras 22.3.1 Constructions and Bimodules of (Admissible) Hom-Novikov-Poisson Color Hom-Algebras 22.3.2 Tensor Products of Admissible Hom-Novikov-Poisson Color Hom-Algebras 22.4 Hom-Gelfand-Dorfman Color Hom-algebras References 23 The Wishart Distribution on Symmetric Cones 23.1 Introduction 23.1.1 Symmetric Cones and Jordan Algebra 23.1.2 Trace, Determinant and Minimal Polynomials 23.1.3 The Gamma Function of a Cone 23.2 The Wishart Ensembles on Symmetric Cones 23.2.1 Lassalle Measure on Symmetric Cones and Probability Distribution 23.2.2 Degenerate Wishart Ensembles on Symmetric Cones 23.3 Conclusion References 24 Induced Ternary Hom-Nambu-Lie Algebras 24.1 Introduction 24.2 Preliminaries 24.3 Induced Ternary Hom-Nambu Lie Algebras from Hom-Lie Algebras with Nilpotent Linear Endomorphism 24.4 Canonical Forms of Induced Ternary Hom-Nambu-Lie Algebras 24.5 Some examples References 25 Commutants in Crossed Products for Piecewise Constant Function Algebras Related to Multiresolution Analysis 25.1 Introduction 25.2 Definitions and Preliminary Results 25.2.1 Algebraic Crossed Products 25.2.2 Automorphisms Induced by Bijections 25.3 Crossed Products for Piecewise Constant Function Algebras and Multiresolution Analysis 25.3.1 Crossed Products for Piecewise Constant Function Algebras 25.3.2 Piecewise Constant Function Algebras Generated by the Haar Scaling Function 25.4 A Comparison of Commutants in Nested Spaces 25.4.1 Comparison of C(mathcalA0) and C(mathcalAj) for Some jinmathbbZ>0 References 26 Constacyclic and Skew Constacyclic Codes Over a Finite Commutative Non-chain Ring 26.1 Introduction 26.2 Basic Concepts and Results 26.3 Constacyclic Codes Over R 26.4 Skew Constacyclic Codes Over R 26.5 Gray Images of Constacyclic Codes Over R 26.6 Conclusion References 27 Wallis Type Formula and a Few Versions of the Number π in q-Calculus 27.1 Introduction 27.2 On q-numbers 27.3 On the q-Wallis Formula 27.4 The Geometrical Interpretation 27.5 The Number πq from the q-gamma Function 27.6 Other Versions of πq 27.7 Conclusion References 28 On (λ,μ,γ)-Derivations of BiHom-Lie Algebras 28.1 Introduction 28.2 Definitions and Preliminary Results 28.3 Generalized Derivations of BiHom-Lie Algebras 28.4 Classification of Multiplicative 2-Dimensional BiHom-Lie Algebras 28.5 Centroids and Derivations of 2-Dimensional Multiplicative BiHom-Lie Algebras References 29 HNN-Extension of Involutive Multiplicative Hom-Lie Algebras 29.1 Introduction 29.2 Involutive Hom-Algebras 29.3 HNN-Extension of Involutive Hom-Associative Algebras 29.4 HNN-Extension of Involutive Hom-Lie Algebras References 30 Two-Sided Noncommutative Gröbner Basis on Quiver Algebras 30.1 Introduction 30.2 Noncommutative Gröbner Basis in Polynomial Ring 30.2.1 Mora's Algorithm 30.3 Path Algebra 30.3.1 Basics to Noncommutative Gröbner Basis in a Path Algebra 30.4 One-Side Gröbner Bases in Path Algebra 30.4.1 Left Gröbner Bases in Path Algebra 30.4.2 Right Gröbner Basis in a Path Algebra 30.5 Two-Sided Gröbner Bases 30.5.1 Division Algorithms 30.5.2 Two-Sided S-Polynomial 30.5.3 The Main Theorem References Appendix Author Index Author Index Subject Index Index