Part I Non-Associative and Non-Commutative Structures for Physics

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1 Part I Non-Associative and Non-Commutative Structures for Physics 1 Moufang Transformations and Noether Currents... 3 Eugen Paal 1.1 Introduction Moufang Loops and Mal tsev Algebras Birepresentations Moufang Noether Currents and ETC... 6 References Weakly Nonassociative Algebras, Riccati and KP Hierarchies... 9 Aristophanes Dimakis and Folkert Müller-Hoissen 2.1 Introduction NonassociativityandKP A Class of WNA Algebras and a Matrix Riccati Hierarchy WNA Algebras and Solutions of the Discrete KP Hierarchy From WNA to Gelfand Dickey Sato Conclusions References Applications of Transvectants Chris Athorne 3.1 Introduction Transvectants Hirota Padé Hyperelliptic References ix

2 x 4 Automorphisms of Finite Orthoalgebras, Exceptional Root Systems and Quantum Mechanics Artur E. Ruuge and Fred Van Oystaeyen 4.1 Introduction Saturated Configurations Non-Colourable Configurations The E 6 Case Orthoalgebras Generated by E Conclusions References A Rewriting Approach to Graph Invariants Lars Hellström 5.1 Background Graph Theory TheProblem Semigraphs ApplyingtheDiamondLemma ClassificationofInvariants References Part II Non-Commutative Deformations, Quantization, Homological Methods, and Representations 6 Graded q-differential Algebra Approach to q-connection Viktor Abramov 6.1 Introduction Graded q-differential Algebra q-connection and Its Curvature Matrix of a q-connection References On Generalized N-Complexes Coming from Twisted Derivations Daniel Larsson and Sergei D. Silvestrov 7.1 Introduction General Framework of (σ,τ)-derivations Generalized N-ComplexesandanExample References Remarks on Quantizations, Words and R-Matrices Hilja L. Huru 8.1 Introduction Multiplicative Cohomologies of Monoids Graded Modules LettersandWords... 94

3 xi 8.5 Quantizations of R-Matrices References Connections on Modules over Singularities of Finite and Tame CM Representation Type Eivind Eriksen and Trond Stølen Gustavsen 9.1 Introduction Preliminaries Obstruction Theory ResultsandExamples References Computing Noncommutative Global Deformations of D-Modules Eivind Eriksen 10.1 Introduction Noncommutative Global Deformations of D-Modules Computing Noncommutative Global Deformations Calculations for D-Modules on Elliptic Curves References Comparing Small Orthogonal Classes Gabriella D Este 11.1 Introduction Preliminaries ProofsandExamples References Part III Groups and Actions 12 How to Compose Lagrangian? Eugen Paal and Jüri Virkepu 12.1 Introduction General Method for Constructing Lagrangians Lagrangian for SO(2) PhysicalInterpretation Lagrangian for the Affine Transformations of the Line References Semidirect Products of Generalized Quaternion Groups by a Cyclic Group Peeter Puusemp 13.1 Introduction Semidirect Products of Q n by C A Description of G 1, G 2 and G 3 by Their Endomorphisms References

4 xii 14 A Characterization of a Class of 2-Groups by Their Endomorphism Semigroups Tatjana Gramushnjak and Peeter Puusemp 14.1 Introduction The Group G The Group G The Group G References Adjoint Representations and Movements Maido Rahula and Vitali Retšnoi 15.1 Introduction Generalized Leibnitz Rule Tangent Group Linear Group GL(2,R) TheOperatorofCenter Discriminant Parabola Relations to Moments in Probability Theory Conclusion References Applications of Hypocontinuous Bilinear Maps in Infinite-Dimensional Differential Calculus Helge Glöckner 16.1 Introduction PreliminariesandBasicFacts Differentiability Properties of Compositions with Hypocontinuous Bilinear Mappings Holomorphic Families of Operators Locally Convex Poisson Vector Spaces References Part IV Quasi-Lie, Super-Lie, Hom-Hopf and Super-Hopf Structures and Extensions, Deformations and Generalizations of Infinite-Dimensional Lie Algebras 17 Hom-Lie Admissible Hom-Coalgebras and Hom-Hopf Algebras Abdenacer Makhlouf and Sergei Silvestrov 17.1 Introduction Hom-Algebra and Hom-Coalgebra Structures Hom-Lie Admissible Hom-Coalgebras Hom-Hopf Algebras References

5 xiii 18 Bosonisation and Parastatistics K. Kanakoglou and C. Daskaloyannis 18.1 Introduction and Definitions (Super-)Lie and (Super-)Hopf Algebraic Structure of the Parabosonic P (n) B and Parafermionic P(n) F Algebras Bosonisation as a Technique of Reducing Supersymmetry Discussion References Deformations of the Witt, Virasoro, and Current Algebra Martin Schlichenmaier 19.1 Introduction Deformations of Lie Algebras Krichever Novikov Algebras The Geometric Families The Geometric Background Examples for the Degenerated Situations References Conformal Algebras in the Context of Linear Algebraic Groups Pavel Kolesnikov 20.1 Introduction Categories of Conformal Algebras Associative (G)-Conformal Algebras Conformal Endomorphism Algebra over a Linear Algebraic Group References Lie Color and Hom-Lie Algebras of Witt Type and Their Central Extensions Gunnar Sigurdsson and Sergei Silvestrov 21.1 Introduction Central Extensions of Witt-Type Lie Color Algebras Central Extensions of Γ -Graded Hom-Lie Algebras of Witt Type. 252 References A Note on Quasi-Lie and Hom-Lie Structures of σ-derivations of C[z ±1 1,...,z±1 n ] Lionel Richard and Sergei Silvestrov 22.1 Introduction Framework Sufficient Condition Laurent Polynomials References

6 xiv Part V Commutative Subalgebras in Noncommutative Algebras 23 Algebraic Dependence of Commuting Elements in Algebras Sergei Silvestrov, Christian Svensson, and Marcel de Jeu 23.1 Introduction Description of the Problem: Commuting Elements in an Algebra AreGiven,ThenFindCurvesTheyLieon Burchnall Chaundy Construction for Differential Operators Burchnall Chaundy Theory for the q-deformed Heisenberg Algebra References Crossed Product-Like and Pre-Crystalline Graded Rings Johan Öinert and Sergei D. Silvestrov 24.1 Introduction PreliminariesandDefinitions The Commutant of A 0 in a Crossed Product-Like Ring The Center of a Crossed Product-Like Ring A 0 σ α M Intersection Theorems Examples of Crossed Product-Like and Crystalline Graded Rings References Decomposition of the Enveloping Algebra so(5) Čestmír Burdík and Ondřej Navrátil 25.1 Introduction The Lie Algebra so(5) The Highest Weight Vectors Conclusion References Index...303

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