Lie Groups

Source lecture(s): PHY622 Lec10-11

Intuition

Continuous groups whose elements are parameterized smoothly—like rotations forming SO(3).

Formal Definition

A group that is also a smooth manifold, with multiplication and inversion smooth maps.

Mathematical Formulation

Near identity: \(g(\theta)\approx I + i\theta^a T_a + O(\theta^2)\). Generators \(T_a\) satisfy \([T_a,T_b]=if_{abc}T_c\).

Derivation

Take an infinitesimal parameter \(\epsilon\); expand \(g(\epsilon)=I+\epsilon X+O(\epsilon^2)\). The generator \(X\) is the tangent vector at identity.

Worked Example

SO(2) is the circle group \(e^{i\theta}\); generator \(J=i\) with \([J,J]=0\).

Common Mistakes

• Forgetting that Lie brackets measure non-commutativity.
• Treating global and infinitesimal parameters interchangeably.

Related Concepts

• Symmetry Groups
• Group Representations
• SU(2) and SO(3)

Quiz

Q1. What is the Lie algebra of a Lie group?

Answer

Tangent space at identity with bracket.