Representation of a Lie group

A representation of the Lie group G, acting on an n-dimensional vector space V over

A typical example in which representations arise in physics would be the study of a linear partial differential equation having symmetry group

If a basis for the complex vector space V is chosen, the representation can be expressed as a homomorphism into general linear group

If G is a compact Lie group, every finite-dimensional representation is equivalent to a unitary one.

The representation theory of SO(3) plays a key role, for example, in the mathematical analysis of the hydrogen atom.

Every standard textbook on quantum mechanics contains an analysis which essentially classifies finite-dimensional irreducible representations of SO(3), by means of its Lie algebra.

, then all polynomials that are homogeneous of degree one are harmonic, and we obtain a three-dimensional space

As noted above, the finite-dimensional representations of SO(3) arise naturally when studying the time-independent Schrödinger equation for a radial potential, such as the hydrogen atom, as a reflection of the rotational symmetry of the problem.

These representations arise in the quantum mechanics of particles with fractional spin, such as an electron.

This problem goes under the name of "addition of angular momentum" or "Clebsch–Gordan theory" in the physics literature.

In the case of the group SU(3), for example, the irreducible representations are labeled by a pair

The Lie correspondence may be employed for obtaining group representations of the connected component of the G. Roughly speaking, this is effected by taking the matrix exponential of the matrices of the Lie algebra representation.

These form (representatives of) the zeroth homotopy group of G. For example, in the case of the four-component Lorentz group, representatives of space inversion and time reversal must be put in by hand.

If Π : G → GL(V) is a group representation for some vector space V, then its pushforward (differential) at the identity, or Lie map,

The main result of this section is the following:[13] From this we easily deduce the following: A projective representation is one in which each

In quantum physics, it is natural to allow projective representations in addition to ordinary ones, because states are really defined only up to a constant.

Note, however, that the representation of a given group element as a product of exponentials is very far from unique, so it is very far from clear that

Let T be a maximal torus in G. By Schur's lemma, the irreducible representations of T are one dimensional.

These representations can be classified easily and are labeled by certain "analytically integral elements" or "weights."

Not only does this formula gives a lot of useful information about the representation, but it plays a crucial role in the proof of the theorem of the highest weight.

on a finite-dimensional vector space V is "unitarizable," meaning that it is possible to choose an inner product on V so that each

by Here are some important examples in which unitary representations of a Lie group have been analyzed.

In quantum physics, one is often interested in projective unitary representations of a Lie group

The reason for this interest is that states of a quantum system are represented by vectors in a Hilbert space

are then required to satisfy the homomorphism property up to a constant: We have already discussed the irreducible projective unitary representations of the rotation group SO(3) above; considering projective representations allows for fractional spin in addition to integer spin.

are in one-to-one correspondence with ordinary unitary representations of the universal cover of

Important examples where Bargmann's theorem applies are SO(3) (as just mentioned) and the Poincaré group.

The latter case is important to Wigner's classification of the projective representations of the Poincaré group, with applications to quantum field theory.

In this case, to get an ordinary representation, one has to pass to the Heisenberg group, which is a one-dimensional central extension of

(This claim follows from Schur's lemma and holds even if the representations are not assumed ahead of time to be finite dimensional.)

Sophus Lie , the originator of Lie theory . The theory of manifolds was not discovered in Lie's time, so he worked locally with subsets of The structure would today be called a local group .
Here V is a finite-dimensional vector space, GL( V ) is the set of all invertible linear transformations on V and is its Lie algebra. The maps π and Π are Lie algebra and group representations respectively, and exp is the exponential mapping. The diagram commutes only up to a sign if Π is projective.