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 representationsEdit

RepresentationsEdit

Let us first discuss representations of groups acting on a finite-dimensional vector space over the field  . (Occasionally representations regarding spaces over the field of real numbers are also considered.) 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   matrices, generally called   Smoothness of the map   can be regarded as a technicality, in that any continuous homomorphism will automatically be smooth.[1]

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 definitionsEdit

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[2] and semisimple[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.[4]

Lie algebra representationsEdit

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)Edit

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 representationsEdit

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.[5] For each non-negative integer  , the irreducible representation of dimension   can be realized as the space   of homogeneous harmonic polynomials on   of degree  .[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 representationsEdit

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).[7] 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 representationsEdit

In this section, we describe three basic operations on representations.[8] See also the corresponding constructions for representations of a Lie algebra.

Direct sumsEdit

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.[9] In such cases, the classification of representations reduces to the classification of irreducible representations. See Weyl's theorem on complete reducibility.

Tensor products of representationsEdit

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:[10]

 

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 representationsEdit

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,[11] 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  .[12]

Lie group versus Lie algebra representationsEdit

OverviewEdit

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[13] 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  , and assume that a representation   of   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 mappingEdit

 
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.

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 representationsEdit

It is always possible to pass from a representation of a Lie group G to a representation of its Lie algebra   If Π : G → GL(V) is a group representation for some vector space V, then its pushforward (differential) at the identity, or Lie map,   is a Lie algebra representation. It is explicitly computed using[14]

 

 

 

 

 

(G6)

A basic property relating   and   involves the exponential map:[15]

 

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 representationsEdit

The main result of this section is the following:[16]

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  .

A projective representation is one in which each   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   is a vector in the quantum Hilbert space, then   represents the same physical state for any constant  .) Every finite-dimensional projective representation of a connected Lie group   comes from an ordinary representation of the universal cover   of  .[17] Conversely, as we will discuss below, every irreducible ordinary representation of   descends to a projective representation of  . In the physics literature, projective representations are often described as multi-valued representations (i.e., each   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.

 
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.

We now outline the proof of the main results above. Suppose   is a representation of   on a vector space V. If there is going to be an associated Lie group representation  , 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   from a neighborhood   of the identity in   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,   would be a globally defined map, but it is not obvious why   would be a homomorphism.) The answer to this question is yes:   is a local homomorphism, and this can be established using the Baker–Campbell–Hausdorff formula.[18]

If   is connected, then every element of   is at least a product of exponentials of elements of  . Thus, we can tentatively define   globally as follows.

 

 

 

 

 

(G2)

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   is actually well defined.

To address the question of whether   is well defined, we connect each group element   to the identity using a continuous path. It is then possible to define   along the path, and to show that the value of   is unchanged under continuous deformation of the path with endpoints fixed. If   is simply connected, any path starting at the identity and ending at   can be continuously deformed into any other such path, showing that   is fully independent of the choice of path. Given that the initial definition of   near the identity was a local homomorphism, it is not difficult to show that the globally defined map is also a homomorphism satisfying (G2).[19]

If   is not simply connected, we may apply the above procedure to the universal cover   of  . Let   be the covering map. If it should happen that the kernel of   contains the kernel of  , then   descends to a representation of the original group  . Even if this is not the case, note that the kernel of   is a discrete normal subgroup of  , which is therefore in the center of  . Thus, if   is irreducible, Schur's lemma implies that the kernel of   will act by scalar multiples of the identity. Thus,   descends to a projective representation of  , 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 SO(3, 1)+, the universal covering group is  , and whether its corresponding representation is faithful decides whether Π is projective.

Classification in the compact caseEdit

If G is a connected compact Lie group, its finite-dimensional representations can be decomposed as direct sums of irreducible representations.[20] 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   is an irreducible representation of G, the restriction of   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  , 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   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 spacesEdit

Let V be a complex Hilbert space, which may be infinite dimensional, and let   denote the group of unitary operators on V. A unitary representation of a Lie group G on V is a group homomorphism   with the property that for each fixed  , the map

 

is a continuous map of G into V.

Finite-dimensional unitary representationsEdit

If the Hilbert space V is finite-dimensional, there is an associated representation   of the Lie algebra   of  . If   is connected, then the representation   of   is unitary if and only if   is skew-self-adjoint for each  .[21]

If   is compact, then every representation   of   on a finite-dimensional vector space V is "unitarizable," meaning that it is possible to choose an inner product on V so that each   is unitary.[22]

Infinite-dimensional unitary representationsEdit

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   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  , and let the complex Hilbert space V be  . We define the representation   by

 

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

Projective representationsEdit

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  —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   into the unitary group  —but rather in projective unitary representations—that is, homomorphisms of   into the projective unitary group

 

To put it differently, for a projective representation, we construct a family of unitary operators  , where it is understood that changing   by a constant of absolute value 1 is counted as "the same" operator. The operators   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.

Bargmann's theorem states that for certain types of Lie groups  , irreducible projective unitary representations of   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.

One example where Bargmann's theorem does not apply is the group  . The set of translations in position and momentum on   form a projective unitary representation of   but they do not come from an ordinary representation of the universal cover of  —which is just   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  . (See the discussion here.)

The commutative caseEdit

If   is a commutative Lie group, then every irreducible unitary representation of   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   are simply continuous homomorphisms of   into the unit circle group, U(1). For example, if  , the irreducible unitary representations have the form

 ,

for some real number  .

See also Pontryagin duality for this case.

See alsoEdit

RemarksEdit

NotesEdit

  1. ^ Hall 2015 Corollary 3.51
  2. ^ Hall 2015 Theorem 4.28
  3. ^ Hall 2015 Section 10.3
  4. ^ Hall 2015 Theorem 4.28
  5. ^ Hall 2015 Section 4.7
  6. ^ Hall 2013 Section 17.6
  7. ^ Hall 2015 Proposition 4.35
  8. ^ Hall 2015, Section 4.3
  9. ^ Hall 2015 Theorem 4.28
  10. ^ Hall 2015, Proposition 4.18
  11. ^ Hall 2015 Proposition 4.22
  12. ^ Hall 2015 Chapter 6, Exercise 3. See also Chapter 10, Exercise 10
  13. ^ Hall 2015 Theorem 5.6
  14. ^ Hall 2015, Theorem 3.28
  15. ^ Hall 2015, Theorem 3.28
  16. ^ Hall 2015, Theorem 5.6
  17. ^ Hall 2013, Section 16.7.3
  18. ^ Hall 2015, Proposition 5.9
  19. ^ Hall 2015, Theorem 5.10
  20. ^ Hall 2015 Theorems 4.28
  21. ^ Hall 2015 Proposition 4.8
  22. ^ Hall 2015 proof of Proposition 4.28

ReferencesEdit

  • Fulton, W.; Harris, J. (1991). Representation theory. A first course. Graduate Texts in Mathematics. 129. New York: Springer-Verlag. ISBN 978-0-387-97495-8. MR 1153249.CS1 maint: ref=harv (link)
  • Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, 267, Springer, ISBN 978-1461471158.
  • Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN 978-3319134666.
  • Knapp, Anthony W. (2002), Lie Groups Beyond an Introduction, Progress in Mathematics, 140 (2nd ed.), Boston: Birkhäuser.
  • Rossmann, Wulf (2001), Lie Groups: An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, Oxford University Press, ISBN 978-0-19-859683-7. The 2003 reprint corrects several typographical mistakes.
  • Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, 1, Cambridge: Cambridge University Press, ISBN 0-521-55001-7