Lie Groups & Lie Algebras
Continuous symmetry groups and the tangent algebras that generate them.
A Lie group is a group that is also a smooth manifold, with smooth multiplication and inversion. Examples: $\mathbb R^n$ under $+$, $S^1 = U(1)$, $SO(n)$ (rotations), $SU(n)$ (special unitary), $\mathrm{GL}_n(\mathbb R)$.
The Lie algebra $\mathfrak g = T_e G$ is the tangent space at the identity, with bracket $[X, Y] = XY - YX$ (for matrix groups). The exponential map sends Lie-algebra elements to group elements:
$$\exp: \mathfrak g \to G, \quad \exp(t X) = e^{tX} = I + tX + \tfrac{t^2}{2}X^2 + \cdots$$Examples:
- $SO(2)$: $R(\theta) = e^{\theta J}$ with $J = \begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}$.
- $SU(2)$: $U = e^{-i\theta\, \hat n \cdot \boldsymbol\sigma/2}$, double cover of $SO(3)$.
- Heisenberg group: $[X, Y] = Z$, $[X, Z] = [Y, Z] = 0$.
Lie's third theorem: every (real, finite-dimensional) Lie algebra is the algebra of a simply connected Lie group, unique up to isomorphism. Symmetries of physical systems are organized by their Lie groups; their reps tell you the spectrum (e.g., angular-momentum quantum numbers from $\mathrm{SU}(2)$ irreps).