# Disclaimer These notes are written from a physicist's perspective. Thus, they are not particularly rigorous, and might disturb mathematicians. A mathematician friend called my proofs "sloppy ass shit". You've been warned. # Index [[2-sphere]] [[Abelian Group]] [[Active and Passive Transformations]] [[Ad Homomorphism]] [[Adjoint Action]] [[Adjoint of a Linear Map]] [[Analytic Functions]] [[Anti-Unitary Operator]] [[Atlas]] [[Automorphism]] [[Baker-Campbell-Hausdorff Formula]] [[Banach Space]] [[Basis Transformation]] [[Basis Vectors of a Representation]] [[Basis]] [[Bilinear Form]] [[Boundary of a Set]] [[Character Table]] [[Character of a Representation]] [[Closed Set]] [[Closure of a Set]] [[Commutator]] [[Compactness]] [[Complex Coordinate Space]] [[Components]] [[Composite Map]] [[Conjugacy Class]] [[Connected Group]] [[Connected Sum]] [[Connectedness]] [[Constant Map]] [[Continuous Group]] [[Continuous Map]] [[Contraction]] [[Coordinate Basis]] [[Coordinate Chart]] [[Coordinate Transformations on Manifolds]] [[Coordinate Transformations]] [[Coset]] [[Cotangent Space]] [[Covector and Contravector]] [[Covering of a Set]] [[Currying]] [[Curve]] [[Cyclic Group of Order 2]] [[Cyclic Subgroup]] [[Determinant]] [[Diagonalizable Matrix]] [[Diffeomorphism]] [[Differentiable Function]] [[Differentiable Functions on a Manifold]] [[Differentiable Manifold]] [[Differentiable One-Form Field]] [[Differentiable Tensor Field]] [[Differentiable Vector Field]] [[Differentials of a Map]] [[Dimension]] [[Discrete Group]] [[Discrete Topology]] [[Domain, Co-domain, Image]] [[Dual Basis]] [[Dual Space]] [[Eigenvalue]] [[Einstein Summation Convention]] [[Equivalence Class]] [[Equivalence Relation]] [[Euclidean Norm]] [[Euclidean Space]] [[Euler Characteristic]] [[Fiber Bundle]] [[Functions on Manifolds]] [[General Linear Group]] [[Gradient]] [[Group Element Order]] [[Group Element]] [[Group Homomorphism]] [[Group Isomorphism]] [[Group Multiplication Table]] [[Group Order]] [[Group Rearrangement Theorem]] [[Group Representation]] [[Group]] [[Harmonic Polynomial]] [[Hausdorff Space]] [[Heisenberg Algebra]] [[Hermitian Form]] [[Hermitian Operator]] [[Hilbert Space]] [[Homeomorphism]] [[Homomorphism]] [[Homotopy Type]] [[Idempotent Operator]] [[Identity Component]] [[Identity Map]] [[Inclusion Map]] [[Index Notation]] [[Induced Lie Algebra Homomorphism]] [[Induced Topology]] [[Inner Product Space]] [[Inner Product]] [[Integral Curves]] [[Interior of a Set]] [[Internal Symmetry]] [[Inverse Map]] [[Irreducible Representation]] [[Isometry Group]] [[Isomorphism]] [[Jacobi Identity]] [[Jacobian]] [[Kernel of a Homomorphism]] [[Kernel of a Map]] [[Killing Form]] [[Levi-Civita Tensor]] [[Lie Algebra Homomorphism]] [[Lie Algebra Isomorphism]] [[Lie Algebra of Vector Fields]] [[Lie Algebra]] [[Lie Group]] [[Lie Product Formula]] [[Lie Subalgebra]] [[Linear Independence]] [[Linear Map]] [[Linear Operator]] [[Loop]] [[Lorentz Group in 4 Dimensions]] [[Lorentz Group]] [[Manifold]] [[Map]] [[Matrix Lie Group]] [[Matrix]] [[Metric Dual]] [[Metric Space]] [[Metric Topology]] [[Metric]] [[Minkowski Metric]] [[Neighborhood]] [[Non-Degeneracy (Quadratic Forms)]] [[Non-coordinate Basis]] [[Non-singular Matrix]] [[Normal Subgroup]] [[Normed Vector Space]] [[One-Form]] [[One-Parameter Subgroup]] [[Open Set]] [[Operator]] [[Orientability]] [[Orthogonal Group in 3 Dimensions]] [[Orthogonal Group]] [[Orthogonal Operator]] [[Orthogonality Relations for Character]] [[Orthogonality]] [[Outer Product]] [[Pfaffian]] [[Physicists' Definition of Lie Algebras]] [[Point Group]] [[Polynomial]] [[Primitive Character]] [[Projection Operator]] [[Proper Subspace]] [[Pseudo-norm]] [[Pull-Back and Push-Forward]] [[Quadratic Forms on Vector Spaces]] [[Quotient Group]] [[Quotient Space]] [[Real Coordinate Space]] [[Regular Representation]] [[Relation]] [[Relationship Between SO(3) and SU(2)]] [[Relationship Between SO(3,1) and SL(2,C)]] [[Relationship Between the Lie Groups]] [[Relative Topology]] [[Restricted Lorentz Group in 4 Dimensions]] [[Restricted Lorentz Group]] [[Restriction of a Map]] [[Rotation]] [[Scalar]] [[Schur's Lemma (Group Theory)]] [[Sesquilinear Form]] [[Similarity Transformation]] [[Simple Conectedness]] [[Simple Group]] [[Span]] [[Special Linear Group in Two Complex Dimensions]] [[Special Linear Group]] [[Special Orthogonal Group in 2 Dimensions]] [[Special Orthogonal Group in 3 Dimensions]] [[Special Orthogonal Group]] [[Special Unitary Group in 2 Dimensions]] [[Special Unitary Group]] [[Spherical Harmonics]] [[Square Matrix]] [[Square-Integrable Functions]] [[Standard Basis]] [[Stokes' theorem]] [[Structure Constants of a Lie Algebra]] [[Subgroup Index]] [[Subgroup]] [[Symmetric Group]] [[Symplectic Form]] [[Symplectic Group]] [[Symplectic Matrix]] [[Symplectic Orthogonal Complement]] [[Symplectic Structure]] [[Symplectic Vector Space]] [[Symplectomorphism]] [[Tangent Bundle]] [[Tangent Space]] [[Tensor Field]] [[Tensor]] [[The Decomposition Theorem for Reducible Representations]] [[Theorem on the Unitarity of Representations]] [[Topological Space]] [[Topology]] [[Torus]] [[Toy Index Thoerem]] [[Trace and Determinant as Homomorphisms]] [[Trace of Generators]] [[Trace]] [[Transposition]] [[Trivial Group]] [[Trivial Representation]] [[Trivial Topology]] [[Types of Maps]] [[Unitary Group in 2 Dimensions]] [[Unitary Group]] [[Unitary Operator]] [[Unitary Representation]] [[Usual Topology]] [[Vector Bundle]] [[Vector Field]] [[Vector Space Isomorphism]] [[Vector Space]] [[Vector Subspace]] [[Vector]] [[Wonderful Orthogonality Theorem]] [[g-Handled Torus]] # Bibliography Schutz - Geometrical Methods of Mathematical Physics Jeevanjee - An Introduction to Tensors and Group Theory for Physicists Dresselhaus - Applications of Group Theory to the Physics of Solids