Matrix groups and continuous symmetry

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Requisitos previos obligatorios, hasta el fondo: 41

Aprenderás a

• Identify the classical matrix groups and decide which are compact and which are connected.
• Use unit quaternions to describe rotations of ℝ³ and construct the two-to-one homomorphism SU(2) → SO(3).
• Compute matrix exponentials and show that t ↦ exp(tX) is a one-parameter subgroup whose tangent vector is X.

Definición

• O(n) = {A ∈ ℝⁿˣⁿ : AᵀA = I} and SO(n) = {A ∈ O(n) : det A = 1}.
• U(n) = {A ∈ ℂⁿˣⁿ : A*A = I} and SU(n) = {A ∈ U(n) : det A = 1}.
• The matrix exponential is exp(X) = Σ_{k≥0} Xᵏ / k!, which converges for every square matrix X.
• The Lie algebra of a matrix group G is the set of matrices X with exp(tX) ∈ G for all real t.

Teorema

• Every element of SO(3) is a rotation about some axis (Euler's rotation theorem).
• SU(2) is homeomorphic to the 3-sphere, and there is a surjective homomorphism SU(2) → SO(3) with kernel {±I}.
• det(exp X) = e^{tr X}, so exp maps trace-zero matrices into SL(n).
• The Lie algebra of SO(n) is the space of skew-symmetric n × n matrices.

Errores frecuentes

• Thinking exp(X + Y) = exp(X) exp(Y) for all matrices; this holds when X and Y commute but fails in general.
• Believing rotations of ℝ³ commute like rotations of the plane; SO(3) is non-abelian.

Conduce a

Descubrimientos opcionales

• Mathematical foundations of quantum mechanics Spin-½ particles are described by representations of SU(2), the double cover of the rotation group SO(3); this is why rotating an electron's state by 2π multiplies it by −1.
• Functors Taking the tangent space at the identity turns a homomorphism of matrix groups into a linear map of Lie algebras, compatibly with composition: a functor from matrix groups to Lie algebras.
• ← Symmetry: slides, turns and flips

Itinerarios de aprendizaje

• Mathematical physics

Lecturas complementarias

• John Stillwell, Naive Lie Theory (2008)
• Andrew Baker, Matrix Groups: An Introduction to Lie Group Theory (2002)