In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. Representations play an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie algebras.
Finite-dimensional representations Representations
A complex representation of a group is an action by a group on a finite-dimensional vector space over the field . A representation of the Lie group G, acting on an n-dimensional vector space V over is then a smooth group homomorphism , where is the general linear group of all invertible linear transformations of under their composition. Since all n-dimensional spaces are isomorphic, the group can be identified with the group of the invertible, complex generally Smoothness of the map can be regarded as a technicality, in that any continuous homomorphism will automatically be smooth. Corollary 3.51
We can alternatively describe a representation of a Lie group as a linear action of on a vector space . Notationally, we would then write in place of for the way a group element acts on the vector .
A typical example in which representations arise in physics would be the study of a linear partial differential equation having symmetry group . Although the individual solutions of the equation may not be invariant under the action of , the space of all solutions is invariant under the action of . Thus, constitutes a representation of . See the example of SO(3), discussed below.
Basic definitions If the homomorphism is injective (i.e., a monomorphism), the representation is said to be faithful.
If a basis for the complex vector space V is chosen, the representation can be expressed as a homomorphism into general linear group . This is known as a matrix representation. Two representations of G on vector spaces V, W are equivalent if they have the same matrix representations with respect to some choices of bases for V and W.
Given a representation , we say that a subspace W of V is an invariant subspace if for all and . The representation is said to be irreducible if the only invariant subspaces of V are the zero space and V itself. For certain types of Lie groups, namely compact Theorem 4.28 and semisimple Section 10.3 groups, every finite-dimensional representation decomposes as a direct sum of irreducible representations, a property known as complete reducibility. For such groups, a typical goal of representation theory is to classify all finite-dimensional irreducible representations of the given group, up to isomorphism. (See the Classification section below.)
A unitary representation on a finite-dimensional inner product space is defined in the same way, except that is required to map into the group of unitary operators. If G is a compact Lie group, every finite-dimensional representation is equivalent to a unitary one.
Lie algebra representations Each representation of a Lie group G gives rise to a representation of its Lie algebra; this correspondence is discussed in detail in subsequent sections. See representation of Lie algebras for the Lie algebra theory.
An example: The rotation group SO(3)
In quantum mechanics, the time-independent Schrödinger equation, plays an important role. In the three-dimensional case, if has rotational symmetry, then the space of solutions to will be invariant under the action of SO(3). Thus, will—for each fixed value of —constitute a representation of SO(3), which is typically finite dimensional. In trying to solve , it helps to know what all possible finite-dimensional representations of SO(3) look like. 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. (The commutation relations among the angular momentum operators are just the relations for the Lie algebra of SO(3).) One subtlety of this analysis is that the representations of the group and the Lie algebra are not in one-to-one correspondence, a point that is critical in understanding the distinction between integer spin and half-integer spin.
Ordinary representations The rotation group SO(3) is a compact Lie group and thus every finite-dimensional representation of SO(3) decomposes as a direct sum of irreducible representations. The group SO(3) has one irreducible representation in each odd dimension. Section 4.7 For each non-negative integer , the irreducible representation of dimension can be realized as the space of homogeneous harmonic polynomials on of degree . Section 17.6 Here, SO(3) acts on in the usual way that rotations act on functions on :
The restriction to the unit sphere of the elements of are the spherical harmonics of degree .
If, say, , then all polynomials that are homogeneous of degree one are harmonic, and we obtain a three-dimensional space spanned by the linear polynomials , , and . If , the space is spanned by the polynomials , , , , and .
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. (See the role played by the spherical harmonics in the mathematical analysis of hydrogen.)
Projective representations If we look at the Lie algebra of SO(3), this Lie algebra is isomorphic to the Lie algebra of SU(2). By the representation theory of , there is then one irreducible representation of in every dimension. The even-dimensional representations, however, do not correspond to representations of the group SO(3). Proposition 4.35 These so-called "fractional spin" representations do, however, correspond to projective representations of SO(3). These representations arise in the quantum mechanics of particles with fractional spin, such as an electron.
Operations on representations In this section, we describe three basic operations on representations., Section 4.3 See also the corresponding constructions for representations of a Lie algebra.
Direct sums If we have two representations of a group , and , then the direct sum would have as the underlying vector space, with the action of the group given by
for all , and .
Certain types of Lie groups—notably, compact Lie groups—have the property that every finite-dimensional representation is isomorphic to a direct sum of irreducible representations. In such cases, the classification of representations reduces to the classification of irreducible representations. See Weyl's theorem on complete reducibility.
Tensor products of representations
If we have two representations of a group , and , then the tensor product of the representations would have the tensor product vector space as the underlying vector space, with the action of uniquely determined by the assumption that
for all and . That is to say, .
The Lie algebra representation associated to the tensor product representation is given by the formula:, Proposition 4.18
The tensor product of two irreducible representations is usually not irreducible; a basic problem in representation theory is then to decompose tensor products of irreducible representations as a direct sum of irreducible subspaces. This problem goes under the name of "addition of angular momentum" or "Clebsch–Gordan theory" in the physics literature.
Dual representations
Let be a Lie group and be a representation of G. Let be the dual space, that is, the space of linear functionals on . Then we can define a representation by the formula
where for any operator , the transpose operator is defined as the "composition with " operator:
(If we work in a basis, then is just the usual matrix transpose of .) The inverse in the definition of is needed to ensure that is actually a representation of , in light of the identity .
The dual of an irreducible representation is always irreducible, Proposition 4.22 but may or may not be isomorphic to the original representation. In the case of the group SU(3), for example, the irreducible representations are labeled by a pair of non-negative integers. The dual of the representation associated to is the representation associated to . Chapter 6, Exercise 3. See also Chapter 10, Exercise 10
Lie group versus Lie algebra representations
Overview In many cases, it is convenient to study representations of a Lie group by studying representations of the associated Lie algebra. In general, however, not every representation of the Lie algebra comes from a representation of the group. For example, this underlies the distinction between integer spin and half-integer spin in quantum mechanics. On the other hand, if G is a simply connected group, then a theorem Theorem 5.6 says that we do, in fact, get a one-to-one correspondence between the group and Lie algebra representations.
Let be a Lie group with Lie algebra , and assume that a representation of is at hand. The Lie correspondence may be employed for obtaining group representations of the connected component of . Roughly speaking, this is effected by taking the matrix exponential of the matrices of the Lie algebra representation. A subtlety arises if is not simply connected. This may result in projective representations or, in physics parlance, multi-valued representations of . These are actually representations of the universal covering group of .
These results will be explained more fully below.
The Lie correspondence gives results only for the connected component of the groups, and thus the other components of the full group are treated separately by giving representatives for matrices representing these components, one for each component. These form (representatives of) the zeroth homotopy group of . For example, in the case of the four-component Lorentz group, representatives of space inversion and time reversal must be put in by hand. Further illustrations will be drawn from the representation theory of the Lorentz group below.
The exponential mapping
If is a Lie group with Lie algebra , then we have the exponential map from to , written as
If is a matrix Lie group, the expression can be computed by the usual power series for the exponential. In any Lie group, there exist neighborhoods of the identity in and of the origin in with the property that every in can be written uniquely as with . That is, the exponential map has a local inverse. In most groups, this is only local; that is, the exponential map is typically neither one-to-one nor onto.
Lie algebra representations from group representations It is always possible to pass from a representation of a Lie group to a representation of its Lie algebra If is a group representation for some vector space , then its pushforward (differential) at the identity, or Lie map, is a Lie algebra representation. It is explicitly computed using, Theorem 3.28
A basic property relating and involves the exponential map:
The question we wish to investigate is whether every representation of arises in this way from representations of the group . As we shall see, this is the case when is simply connected.
Group representations from Lie algebra representations The main result of this section is the following:, Theorem 5.6 Theorem: If is simply connected, then every representation of the Lie algebra of comes from a representation of itself. From this we easily deduce the following: Corollary: If is connected but not simply connected, every representation of comes from a representation of , the universal cover of . If is irreducible, then descends to a projective representation of .