Lie Groups

Prerequisites: Group Theory. Full path (2 pages)

Why is it interesting?
Intuitive
Concrete
Abstract

Why is it interesting?

Lie groups describe continuous symmetry and lie at the heart of modern physics. For example, the symmetry group of the standard model of particle physics and the best spacetime symmetry group that we know (the Poincare group) are Lie groups.

Intuitive

A Lie group is a continuous set of transformations that satisfy the group axioms. A good example for a Lie group is the symmetry group of the circle. A rotation by $5^\circ$ is a symmetry of the circle and a rotation by $0.00001^\circ$ is a symmetry, too. In contrast, the symmetry group of a square is not continuous. A rotation by $90^\circ$ is a symmetry, whereas a rotation by $5^\circ$ is not a symmetry.

Concrete

The diagram below shows some low-dimensional, but important, Lie groups and their relationships.

For a more detailed explanation of this diagram and much more, see Fun with Symmetry.

For book recommendations, see here.

Abstract

A Lie group is a group $G$ that is also a smooth manifold, such that multiplication $G\times G\to G$ and inversion $G\to G$ are smooth maps. By Cartan's closed subgroup theorem, every closed subgroup of a Lie group is an embedded Lie subgroup. Hence all matrix groups defined as closed subsets of $GL(n,\mathbb{R})$ or $GL(n,\mathbb{C})$ , such as $O(n)$ , $SU(n)$ , $Sp(2n,\mathbb{R})$ and $SO(3,1)$ , are Lie groups.

Lie algebra. Left translation $L_g(h) = gh$ is a diffeomorphism. A vector field $V$ with $(L_g)_*V = V$ for all $g$ is fixed by its value at the identity, so the left-invariant vector fields form a vector space isomorphic to $T_eG$ . It is closed under the commutator of vector fields, and this makes $T_eG$ into the Lie algebra $\mathfrak{g}$ of $G$ . Dually, the Maurer-Cartan form $\theta$ is the $\mathfrak{g}$ -valued one-form that maps a tangent vector $v\in T_gG$ to $(L_{g^{-1}})_*v\in T_eG$ . For matrix groups $\theta = g^{-1}dg$ , and it satisfies $d\theta + \theta\wedge\theta = 0 .$

Global structure. The Lie algebra only sees a neighbourhood of the identity. The identity component $G_0$ is a normal subgroup and $G/G_0$ is discrete; for example $O(3,1)$ has four components, related by parity and time reversal. Connected groups with the same Lie algebra are quotients of the same simply connected group by discrete central subgroups, as in $SO(3) = SU(2)/\mathbb{Z}_2$ . Physical statements that depend on this global information include the existence of half-integer spin, the allowed magnetic charges in a gauge theory, and the periodicity of $\theta$ -angles.

Compact and non-compact groups. Every Lie group carries a left-invariant measure, the Haar measure, unique up to normalisation. For compact $G$ the total volume is finite, and averaging over the group shows that every finite-dimensional representation is equivalent to a unitary one and is a direct sum of irreducible ones. The Peter-Weyl theorem states that the matrix elements of the irreducible representations of a compact group form an orthogonal basis of $L^2(G)$ . A connected non-compact simple Lie group, such as the Lorentz group, has no non-trivial finite-dimensional unitary representations. The gauge group of the Standard Model is compact and its matter fields sit in finite-dimensional unitary representations. The Lorentz and Poincaré groups are non-compact, so the unitary representations that describe particles are infinite dimensional.

Path to this page

Leads to: Lie Algebras, SO(2), SO(3), Sp(1), SU(2), U(1).