Representation of a Lie group

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 
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 generally Smoothness of the map formula_7 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_8 as a "linear action" of formula_8 on a vector space formula_5. Notationally, we would then write formula_11 in place of formula_12 for the way a group element formula_13 acts on the vector formula_14.
A typical example in which representations arise in physics would be the study of a linear partial differential equation having symmetry group formula_8. Although the individual solutions of the equation may not be invariant under the action of formula_8, the "space" formula_5 of all solutions is invariant under the action of formula_8. Thus, formula_5 constitutes a representation of formula_8. See the example of SO(3), discussed below.
Basic definitions.
If the homomorphism formula_7 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_22. 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_24 for all formula_13 and formula_26. 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_7 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_28 plays an important role. In the three-dimensional case, if formula_29 has rotational symmetry, then the space formula_30 of solutions to formula_28 will be invariant under the action of SO(3). Thus, formula_30 will—for each fixed value of formula_33—constitute a representation of SO(3), which is typically finite dimensional. In trying to solve formula_28, 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_35 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_36, the irreducible representation of dimension formula_37 can be realized as the space formula_38 of homogeneous harmonic polynomials on formula_39 of degree formula_36. Here, SO(3) acts on formula_38 in the usual way that rotations act on functions on formula_39:
The restriction to the unit sphere formula_44 of the elements of formula_38 are the spherical harmonics of degree formula_36.
If, say, formula_47, then all polynomials that are homogeneous of degree one are harmonic, and we obtain a three-dimensional space formula_48 spanned by the linear polynomials formula_49, formula_50, and formula_51. If formula_52, the space formula_53 is spanned by the polynomials formula_54, formula_55, formula_56, formula_57, and formula_58.
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_35 of SO(3), this Lie algebra is isomorphic to the Lie algebra formula_60 of SU(2). By the representation theory of formula_60, there is then one irreducible representation of formula_35 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_8, formula_64 and formula_65, then the direct sum would have formula_66 as the underlying vector space, with the action of the group given by
for all formula_68 formula_69, and formula_13.
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_8, formula_64 and formula_65, then the tensor product of the representations would have the tensor product vector space formula_74 as the underlying vector space, with the action of formula_8 uniquely determined by the assumption that
for all formula_77 and formula_69. That is to say, formula_79.
The Lie algebra representation formula_80 associated to the tensor product representation formula_7 is given by the formula:
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_8 be a Lie group and formula_84 be a representation of G. Let formula_85 be the dual space, that is, the space of linear functionals on formula_5. Then we can define a representation formula_87 by the formula
where for any operator formula_89, the transpose operator formula_90 is defined as the "composition with formula_91" operator:
(If we work in a basis, then formula_93 is just the usual matrix transpose of formula_91.) The inverse in the definition of formula_95 is needed to ensure that formula_95 is actually a representation of formula_8, in light of the identity formula_98.
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_99 of non-negative integers. The dual of the representation associated to formula_99 is the representation associated to formula_101.
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 be a Lie group with Lie algebra formula_102, and assume that a representation formula_80 of formula_102 is at hand. The Lie correspondence may be employed for obtaining group representations of the connected component of the . 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 formula_8 is a Lie group with Lie algebra formula_102, then we have the exponential map from formula_102 to formula_8, written as
If formula_8 is a matrix Lie group, the expression formula_111 can be computed by the usual power series for the exponential. In any Lie group, there exist neighborhoods formula_112 of the identity in formula_8 and formula_5 of the origin in formula_102 with the property that every formula_116 in formula_112 can be written uniquely as formula_118 with formula_119. 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 formula_120 If is a group representation for some vector space , then its pushforward (differential) at the identity, or Lie map, formula_121 is a Lie algebra representation. It is explicitly computed using
A basic property relating formula_7 and formula_80 involves the exponential map:
The question we wish to investigate is whether every representation of formula_102 arises in this way from representations of the group formula_8. As we shall see, this is the case when formula_8 is simply connected.
Group representations from Lie algebra representations.
The main result of this section is the following:
From this we easily deduce the following:
A projective representation is one in which each formula_143 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_144 is a vector in the quantum Hilbert space, then formula_145 represents the same physical state for any constant formula_146.) Every "finite-dimensional" projective representation of a connected Lie group formula_8 comes from an ordinary representation of the universal cover formula_138 of formula_8. Conversely, as we will discuss below, every irreducible ordinary representation of formula_138 descends to a projective representation of formula_8. In the physics literature, projective representations are often described as multi-valued representations (i.e., each formula_152 does not have a single value but a whole family of values). This phenomenon is important to the study of fractional spin in quantum mechanics.
We now outline the proof of the main results above. Suppose formula_153 is a representation of formula_102 on a vector space . If there is going to be an associated Lie group representation formula_7, it must satisfy the exponential relation of the previous subsection. Now, in light of the local invertibility of the exponential, we can "define" a map formula_7 from a neighborhood formula_112 of the identity in formula_8 by this relation:
A key question is then this: Is this locally defined map a "local homomorphism"? (This question would apply even in the special case where the exponential mapping is globally one-to-one and onto; in that case, formula_7 would be a globally defined map, but it is not obvious why formula_7 would be a homomorphism.) The answer to this question is yes: formula_7 is a local homomorphism, and this can be established using the Baker–Campbell–Hausdorff formula.
If formula_8 is connected, then every element of formula_8 is at least a "product" of exponentials of elements of formula_102. Thus, we can tentatively define formula_7 globally as follows.
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 formula_7 is actually well defined.
To address the question of whether formula_7 is well defined, we connect each group element formula_13 to the identity using a continuous path. It is then possible to define formula_7 along the path, and to show that the value of formula_152 is unchanged under continuous deformation of the path with endpoints fixed. If formula_8 is simply connected, any path starting at the identity and ending at formula_116 can be continuously deformed into any other such path, showing that formula_152 is fully independent of the choice of path. Given that the initial definition of formula_7 near the identity was a local homomorphism, it is not difficult to show that the globally defined map is also a homomorphism satisfying .
If formula_8 is not simply connected, we may apply the above procedure to the universal cover formula_138 of formula_8. Let formula_179 be the covering map. If it should happen that the kernel of formula_180 contains the kernel of formula_181, then formula_7 descends to a representation of the original group formula_8. Even if this is not the case, note that the kernel of formula_181 is a discrete normal subgroup of formula_138, which is therefore in the center of formula_138. Thus, if formula_80 is irreducible, Schur's lemma implies that the kernel of formula_181 will act by scalar multiples of the identity. Thus, formula_7 descends to a "projective" representation of formula_8, that is, one that is defined only modulo scalar multiples of the identity.
A pictorial view of how the universal covering group contains "all" such homotopy classes, and a technical definition of it (as a set and as a group) is given in geometric view.
For example, when this is specialized to the doubly connected , the universal covering group is formula_191, and whether its corresponding representation is faithful decides whether is projective.
Classification in the compact case.
If "G" is a connected compact Lie group, its finite-dimensional representations can be decomposed as direct sums of irreducible representations. The irreducibles are classified by a "theorem of the highest weight." We give a brief description of this theory here; for more details, see the articles on representation theory of a connected compact Lie group and the parallel theory classifying representations of semisimple Lie algebras.
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." If formula_192 is an irreducible representation of "G", the restriction of formula_192 to "T" will usually not be irreducible, but it will decompose as a direct sum of irreducible representations of "T", labeled by the associated weights. (The same weight can occur more than once.) For a fixed formula_192, one can identify one of the weights as "highest" and the representations are then classified by this highest weight.
An important aspect of the representation theory is the associated theory of characters. Here, for a representation formula_192 of "G", the character is the function
given by
Two representations with the same character turn out to be isomorphic. Furthermore, the Weyl character formula gives a remarkable formula for the character of a representation in terms of its highest weight. 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.
Unitary representations on Hilbert spaces.
Let "V" be a complex Hilbert space, which may be infinite dimensional, and let formula_198 denote the group of unitary operators on "V". A unitary representation of a Lie group "G" on "V" is a group homomorphism formula_199 with the property that for each fixed formula_14, the map
is a continuous map of "G" into "V".
Finite-dimensional unitary representations.
If the Hilbert space "V" is finite-dimensional, there is an associated representation formula_80 of the Lie algebra formula_102 of formula_8. If formula_8 is connected, then the representation formula_7 of formula_8 is unitary if and only if formula_208 is skew-self-adjoint for each formula_209.
If formula_8 is compact, then every representation formula_7 of formula_8 on a finite-dimensional vector space "V" is "unitarizable," meaning that it is possible to choose an inner product on "V" so that each formula_213 is unitary.
Infinite-dimensional unitary representations.
If the Hilbert space "V" is allowed to be infinite dimensional, the study of unitary representations involves a number of interesting features that are not present in the finite dimensional case. For example, the construction of an appropriate representation of the Lie algebra formula_102 becomes technically challenging. One setting in which the Lie algebra representation is well understood is that of semisimple (or reductive) Lie groups, where the associated Lie algebra representation forms a (g,K)-module.
Examples of unitary representations arise in quantum mechanics and quantum field theory, but also in Fourier analysis as shown in the following example. Let formula_215, and let the complex Hilbert space "V" be formula_216. We define the representation formula_217 by
Here are some important examples in which unitary representations of a Lie group have been analyzed. 
Projective representations.
In quantum physics, one is often interested in "projective" unitary representations of a Lie group formula_8. The reason for this interest is that states of a quantum system are represented by vectors in a Hilbert space formula_220—but with the understanding that two states differing by a constant are actually the same physical state. The symmetries of the Hilbert space are then described by unitary operators, but a unitary operator that is a multiple of the identity does not change the physical state of the system. Thus, we are interested not in ordinary unitary representations—that is, homomorphisms of formula_8 into the unitary group formula_222—but rather in projective unitary representations—that is, homomorphisms of formula_8 into the projective unitary group 
To put it differently, for a projective representation, we construct a family of unitary operators formula_225, where it is understood that changing formula_226 by a constant of absolute value 1 is counted as "the same" operator. The operators formula_226 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.
 states that for certain types of Lie groups formula_8, irreducible projective unitary representations of formula_8 are in one-to-one correspondence with ordinary unitary representations of the universal cover of formula_8. 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.
One example where Bargmann's theorem does "not" apply is the group formula_232. The set of translations in position and momentum on formula_233 form a projective unitary representation of formula_232 but they do not come from an ordinary representation of the universal cover of formula_232—which is just formula_232 itself. In this case, to get an ordinary representation, one has to pass to the Heisenberg group, which is a one-dimensional central extension of formula_232. (See the discussion .)
The commutative case.
If formula_8 is a commutative Lie group, then every irreducible unitary representation of formula_8 on complex vector spaces is one dimensional. (This claim follows from Schur's lemma and holds even if the representations are not assumed ahead of time to be finite dimensional.) Thus, the irreducible unitary representations of formula_8 are simply continuous homomorphisms of formula_8 into the unit circle group, U(1). For example, if formula_215, the irreducible unitary representations have the form 
for some real number formula_244.
See also Pontryagin duality for this case.