Group representation
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 formula_1. A representation of the Lie group "G", acting on an "n"-dimensional vector space "V" over formula_1 is then a smooth group homomorphism 
formula_3,
where formula_4 is the general linear group of all invertible linear transformations of formula_5 under their composition. Since all "n"-dimensional spaces are isomorphic, the group formula_4 can be identified with the group of the invertible, complex formula_7 matrices, generally called formula_8 Smoothness of the map formula_9 can be regarded as a technicality, in that any continuous homomorphism will automatically be smooth.
We can alternatively describe a representation of a Lie group formula_10 as a "linear action" of formula_10 on a vector space formula_5. Notationally, we would then write formula_13 in place of formula_14 for the way a group element formula_15 acts on the vector formula_16.
A typical example in which representations arise in physics would be the study of a linear partial differential equation having symmetry group formula_10. Although the individual solutions of the equation may not be invariant under the action of formula_10, the "space" formula_5 of all solutions is invariant under the action of formula_10. Thus, formula_5 constitutes a representation of formula_10. See the example of SO(3), discussed below.
Basic definitions.
If the homomorphism formula_9 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 formula_24. 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 formula_3, we say that a subspace "W" of "V" is an invariant subspace if formula_26 for all formula_15 and formula_28. 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 and semisimple 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 formula_9 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, formula_30 plays an important role. In the three-dimensional case, if formula_31 has rotational symmetry, then the space formula_32 of solutions to formula_30 will be invariant under the action of SO(3). Thus, formula_32 will—for each fixed value of formula_35—constitute a representation of SO(3), which is typically finite dimensional. In trying to solve formula_30, 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 formula_37 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. For each non-negative integer formula_38, the irreducible representation of dimension formula_39 can be realized as the space formula_40 of homogeneous harmonic polynomials on formula_41 of degree formula_38. Here, SO(3) acts on formula_40 in the usual way that rotations act on functions on formula_41:
formula_45
The restriction to the unit sphere formula_46 of the elements of formula_40 are the spherical harmonics of degree formula_38.
If, say, formula_49, then all polynomials that are homogeneous of degree one are harmonic, and we obtain a three-dimensional space formula_50 spanned by the linear polynomials formula_51, formula_52, and formula_53. If formula_54, the space formula_55 is spanned by the polynomials formula_56, formula_57, formula_58, formula_59, and formula_60.
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 formula_37 of SO(3), this Lie algebra is isomorphic to the Lie algebra formula_62 of SU(2). By the representation theory of formula_62, there is then one irreducible representation of formula_37 in "every" dimension. The even-dimensional representations, however, do not correspond to representations of the "group" SO(3). 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. See also the corresponding constructions for representations of a Lie algebra.
Direct sums.
If we have two representations of a group formula_10, formula_66 and formula_67, then the direct sum would have formula_68 as the underlying vector space, with the action of the group given by
formula_69
for all formula_70 formula_71, and formula_15.
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 formula_10, formula_66 and formula_67, then the tensor product of the representations would have the tensor product vector space formula_76 as the underlying vector space, with the action of formula_10 uniquely determined by the assumption that
formula_78
for all formula_79 and formula_71. That is to say, formula_81.
The Lie algebra representation formula_82 associated to the tensor product representation formula_9 is given by the formula:
formula_84
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 formula_10 be a Lie group and formula_86 be a representation of G. Let formula_87 be the dual space, that is, the space of linear functionals on formula_5. Then we can define a representation formula_89 by the formula
formula_90
where for any operator formula_91, the transpose operator formula_92 is defined as the "composition with formula_93" operator:
formula_94
(If we work in a basis, then formula_95 is just the usual matrix transpose of formula_93.) The inverse in the definition of formula_97 is needed to ensure that formula_97 is actually a representation of formula_10, in light of the identity formula_100.
The dual of an irreducible representation is always irreducible, 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 formula_101 of non-negative integers. The dual of the representation associated to formula_101 is the representation associated to formula_103.
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. This fact is, for example, lying behind 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 says that we do, in fact, get a one-to-one correspondence between the group and Lie algebra representations.
Let G be a Lie group with Lie algebra formula_104, and assume that a representation formula_82 of formula_104 is at hand. 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. A subtlety arises if G is not simply connected. This may result in projective representations or, in physics parlance, multi-valued representations of G. These are actually representations of the universal covering group of G.
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 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. Further illustrations will be drawn from the representation theory of the Lorentz group below.
The exponential mapping.
If formula_10 is a Lie group with Lie algebra formula_104, then we have the exponential map from formula_104 to formula_10, written as
formula_111
If formula_10 is a matrix Lie group, the expression formula_113 can be computed by the usual power series for the exponential. In any Lie group, there exist neighborhoods formula_114 of the identity in formula_10 and formula_5 of the origin in formula_104 with the property that every formula_118 in formula_114 can be written uniquely as formula_120 with formula_121. 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 G to a representation of its Lie algebra formula_122 If Π : "G" → GL("V") is a group representation for some vector space V, then its pushforward (differential) at the identity, or Lie map, formula_123 is a Lie algebra representation. It is explicitly computed using
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A basic property relating formula_9 and formula_82 involves the exponential map:
formula_126
The question we wish to investigate is whether every representation of formula_104 arises in this way from representations of the group formula_10. As we shall see, this is the case when formula_10 is simply connected.
Group representations from Lie algebra representations.
The main result of this section is the following:
Theorem: If formula_10 is simply connected, then every representation formula_82 of the Lie algebra formula_104 of formula_10 comes from a representation formula_9 of formula_10 itself.
From this we easily deduce the following:
Corollary: If formula_10 is connected but not simply connected, every representation formula_82 of formula_104 comes from a representation formula_9 of formula_140, the universal cover of formula_10. If formula_82 is irreducible, then formula_9 descends to a "projective representation" of formula_10. 
A projective representation is one in which each formula_145 is defined only up to multiplication by a constant. 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. (That is to say, if formula_146 is a vector in the quantum Hilbert space, then formula_147 represents the same physical state for any constant formula_148.) Every "finite-dimensional" projective representation of a connected Lie group formula_10 comes from an ordinary representation of the universal cover formula_140 of formula_10. Conversely, as we will discuss below, every irreducible ordinary representation of formula_140 descends to a projective representation of formula_10. In the physics literature, projective representations are often described as multi-valued representations (i.e., each formula_154