Abstract: The Erlangen programme of F. Klein (influenced by S. Lie) defines geometry as a study of invariants under a certain transitive group action. This approach proved to be fruitful much beyond the traditional geometry. For example, special relativity is the study of invariants of Minkowski space-time under the Lorentz group action. Another example is complex analysis as study of objects invariant under the conformal maps.

These notes systematically apply the Erlangen approach to various areas of mathematics. In the first instance we consider the group SL₂(ℝ) in details as well as the corresponding geometrical and analytical invariants with their interrelations. Consequently the course has a multi-subject nature touching algebra, geometry and analysis.

Key words and phrases. Erlangen program, SL(2,R), special linear group, Heisenberg group, symplectic group, Hardy space, Segal-Bargmann space, Clifford algebra, dual numbers, double numbers, Cauchy-Riemann-Dirac operator, Möbius transformations, covariant functional calculus, Weyl calculus (quantization), quantum mechanics, Schrödinger representation, metaplectic representation

2000 Mathematics Subject Classification. Primary [; Secondary 2.

010]43A8530G30, 42C40, 46H30, 47A13, 81R30, 81R60

• Contents
• Preface

• Erlangen Programme: Preview
  • Make a Guess in Three Attempts
  • Covariance of FSCc
  • Invariants: Algebraic and Geometric
  • Joint Invariants: Orthogonality
  • Higher-order Joint Invariants: Focal Orthogonality
  • Distance, Length and Perpendicularity
  • The Erlangen Programme at Large
• Groups and Homogeneous Spaces
  • Groups and Transformations
  • Subgroups and Homogeneous Spaces
  • Differentiation on Lie Groups and Lie Algebras
  • Integration on Groups
• Homogeneous Spaces from the Group SL₂(ℝ)
  • The Affine Group and the Real Line
  • One-dimensional Subgroups of SL₂(ℝ)
  • Two-dimensional Homogeneous Spaces
  • Elliptic, Parabolic and Hyperbolic Cases
  • Orbits of the Subgroup Actions
  • Unifying EPH Cases: The First Attempt
  • Isotropy Subgroups
• The Extended Fillmore–Springer–Cnops Construction
  • Invariance of Cycles
  • Projective Spaces of Cycles
  • Covariance of FSCc
  • Origins of FSCc
  • Projective Cross Ratio
• Indefinite Product Space of Cycles
  • Cycles: an Appearance and the Essence
  • Cycles as Vectors
  • Invariant Cycle Product
  • Zero-radius Cycles
  • Cauchy–Schwarz Inequality and Tangent Cycles
• Joint Invariants of Cycles: Orthogonality
  • Orthogonality of Cycles
  • Orthogonality Miscellanea
  • Ghost Cycles and Orthogonality
  • Actions of FSCc Matrices
  • Inversions and Reflections in Cycles
  • Higher-order Joint Invariants: Focal Orthogonality
• Metric Invariants in Upper Half-Planes
  • Distances
  • Lengths
  • Conformal Properties of Möbius Maps
  • Perpendicularity and Orthogonality
  • Infinitesimal-radius Cycles
  • Infinitesimal Conformality
• Global Geometry of Upper Half-Planes
  • Compactification of the Point Space
  • (Non)-Invariance of The Upper Half-Plane
  • Optics and Mechanics
  • Relativity of Space-Time
• Invariant Metric and Geodesics
  • Metrics, Curves’ Lengths and Extrema
  • Invariant Metric
  • Geodesics: Additivity of Metric
  • Geometric Invariants
  • Invariant Metric and Cross-Ratio
• Conformal Unit Disk
  • Elliptic Cayley Transforms
  • Hyperbolic Cayley Transform
  • Parabolic Cayley Transforms
  • Cayley Transforms of Cycles
• Unitary Rotations
  • Unitary Rotations—an Algebraic Approach
  • Unitary Rotations—a Geometrical Viewpoint
  • Rebuilding Algebraic Structures from Geometry
  • Invariant Linear Algebra
  • Linearisation of the Exotic Form
  • Conformality and Geodesics
• Cycles Cross Ratio: an Invitation
  • Preliminaries: Projective space of cycles
  • Fractional linear transformations and the invariant product
  • Cycles cross ratio
  • Discussion and generalisations
• Extension of Lie Geometry: Ensembles and their Implementation
  • Introduction
  • Möbius–Lie Geometry and the cycle Library
  • Ensembles of Interrelated Cycles and the figure Library
  • Mathematical Usage of the Library
  • To Do List
• Poincaré Extension of Möbius Transformations
  • Introduction
  • Geometric construction
  • Möbius transformations and Cycles
  • Extending cycles
  • Homogeneous spaces of cycles
  • Triples of intervals
  • Geometrisation of cycles
  • Concluding remarks
• Conformal Parametrisation of Loxodromes
  • Introduction
  • Preliminaries: Fractional Linear Transformations and Cycles
  • Fractional Linear Transformations and Loxodromes
  • Three-cycle Parametrisation of Loxodromes
  • Applications of Three-Cycle Parametrisation
  • Discussion and Open Questions
• Continued Fractions, Möbius Transformations and Cycles
  • Introduction
  • Continued Fractions
  • Möbius Transformations and Cycles
  • Continued Fractions and Cycles
  • Multi-dimensional Möbius maps and cycles
  • Continued fractions from Clifford algebras and horocycles

• Representation Theory
  • Representations
  • Decomposition of Representations
  • Schur’s Lemma
  • Induced Representations
• Wavelets on Groups
  • Wavelet Transform on Groups
  • Square Integrable Representations
  • Fundamentals of Wavelets on Homogeneous Spaces
• Hypercomplex Linear Representations
  • Hypercomplex Characters
  • Induced Representations
  • Similarity and Correspondence: Ladder Operators
• Covariant Transform
  • Extending Wavelet Transform
  • Examples of Covariant Transform
  • Symbolic Calculi
  • Contravariant Transform
  • Composing the Co- and Contravariant Transforms
• Analytic Functions
  • Induced Covariant Transform
  • Induced Wavelet Transform and Cauchy Integral
  • The Cauchy-Riemann (Dirac) and Laplace Operators
  • The Taylor Expansion
  • Wavelet Transform in the Unit Disk and Other Domains
• Affine Group: the Real and Complex Techniques in Harmonic Analysis
  • Introduction
  • Two approaches to harmonic analysis
  • Affine group and its representations
  • Covariant transform
  • The contravariant transform
  • Intertwining properties of covariant transforms
  • Composing the covariant and the contravariant transforms
  • Transported norms

• Covariant and Contravariant Calculi
  • Functional Calculus as an Algebraic Homomorphism
  • Intertwining Group Actions on Functions and Operators
  • Jet Bundles and Prolongations
  • Spectrum and Spectral Mapping Theorem
  • Functional Model and Spectral Distance
  • Covariant Pencils of Operators

• Preview: Quantum and Classical Mechanics
  • Axioms of Mechanics
  • “Algebra” of Observables
  • Non-Essential Noncommutativity
  • Quantum Mechanics from the Heisenberg Group
  • Classical Noncommutativity
  • Summary
• The Heisenberg Group
  • The Symplectic Form and the Heisenberg group
  • Lie algebra of the Heisenberg group
  • Automorphisms of the Heisenberg group
  • Subgroups of ℍⁿ and Homogeneous Spaces
• Representations of the Heisenberg Group
  • Left Regular Representations and Its Subrepresentations
  • Induced Representations on Homogeneous Spaces
  • Co-adjoint Representation and Method of Orbits
  • Stone–von Neumann Theorem
  • Shale–Weil Representation
• Harmonic Oscillator and Ladder Operators
  • p-Mechanic Formalism
  • Elliptic characters and Quantum Dynamics
  • Ladder Operators and Harmonic Oscillator
  • Hyperbolic Quantum Mechanics
  • Parabolic (Classical) Representations on the Phase Space
• Wavelet Transform, Uncertainty Relation and Analyticity
  • Induced Wavelet (Covariant) Transform
  • The Uncertainty Relation
  • The Gaussian
  • Right Shifts and Analyticity
  • Uncertainty and Analyticity
  • Hardy Space on the Real Line
  • Contravariant Transform and Relative Convolutions
  • Norm Estimations of Relative Convolutions
• Open Problems
  • Geometry
  • Analytic Functions
  • Functional Calculus
  • Quantum Mechanics
• Supplementary Material
  • Dual and Double Numbers
  • Classical Properties of Conic Sections
  • Comparison with Yaglom’s Book
  • Other Approaches and Results
  • FSCc with Clifford Algebras
• How to Use the Software
  • Viewing Colour Graphics
  • Installation of CAS
  • Using the CAS and Computer Exercises
  • Library for Cycles
  • Predefined Objects at Initialisation
• References
• Index

This document was translated from L^AT_EX by H^EV^EA.