Lie Groups
1
Introduction
1.1
Notation and Some definitions
2
Lie Groups
3
Lie Algebra
4
Matrix Exponential
4.1
Computing Exponential
5
The Lie Algebra of a Matrix Lie Group
6
Relationships Between Lie Groups and Lie Algebras
7
Code Implantation
7.1
Introduction
7.2
Matrix Exponential Algorithms
7.2.1
Method 1: Direct Power Series
7.2.2
Method 2: Eigenvalue Decomposition
7.2.3
Method 3: Padé Approximation (expm package)
7.2.4
Performance Comparison
7.3
Seven Fundamental Properties
7.3.1
Property 1: Identity Element
7.3.2
Property 2: Transpose
7.3.3
Property 3: Adjoint (Conjugate Transpose)
7.3.4
Property 4: Conjugation
7.3.5
Property 5: Determinant
7.3.6
Property 6: Commuting Matrices
7.3.7
Property 7: Lie Product Formula
7.3.8
Summary: Theorem @ref(thm:thm1)
7.4
Baker-Campbell-Hausdorff Formula
7.4.1
Background
7.4.2
Implimentation
7.4.3
Convergence Analysis
7.5
SO(3) Rotation Group
7.5.1
Mathematical Background
7.5.2
Basis for
\(\mathfrak{so}(3)\)
7.5.3
3D Visualization: Rotation Path
7.6
One-Parameter Subgroups
7.7
Lie Bracket and Cross Product
7.8
SU(2): Special Unitary Group
7.8.1
Definition and Structure
7.8.2
Pauli Matrices
7.8.3
Pauli Commutation Relations
7.8.4
SU(2) Basis Elements
7.8.5
Double Cover Property
7.8.6
Determinant Evolution Along Path
7.8.7
Covering Map Φ: SU(2) → SO(3)
7.9
Quantum Rotation Operators
7.9.1
Definition
7.9.2
Implementation
7.9.3
Properties Verification
7.9.4
Rotation Composition
7.9.5
Action on Spin States
7.10
Lie Bracket: General Properties
7.11
Conclusions
References
Ashan De Silva
Lie Groups and Lie Algbera
References