Representation of a Lie group

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In mathematics and theoretical physics, the idea of a representation of a Lie group plays 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. The physics literature sometimes passes over the distinction between Lie groups and Lie algebras.

Finite-dimensional representations[edit]

Representations[edit]

Let us first discuss representations acting on finite-dimensional vector spaces over a field . (Occasionally representations over the field of real numbers are also considered.) A representation of a Lie group G on a finite-dimensional vector space V over is a smooth group homomorphism

,

where is the group of all invertible linear transformations of . For n-dimensional V, the group is identified with the , the group of invertible matrices. Smoothness of the map should be regarded as a technicality, in that any homomorphism that is continuous will automatically be smooth.[1]

Basic definitions[edit]

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 invariant 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 representations[edit]

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 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) and will, therefore 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. 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 so(3) 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.

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 .

If we look at the Lie algebra so(3) of SO(3), this Lie algebra is isomorphic to the Lie algebra su(2) of SU(2). By the representation theory of su(2), there is then one irreducible representation of so(3) in every dimension. The even-dimensional representations, however, do not correspond to representations of the group SO(3).[7]

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.)

Lie group versus Lie algebra representations[edit]

Overview[edit]

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[8] 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, multivalued-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[edit]

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 representations[edit]

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[9]

 

 

 

 

(G6)

A basic property relating and involves the exponential map:[10]

.

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[edit]

The main result of this section is the following:[11]

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 .[12] 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).

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.[13]

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).[14]

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.

Unitary representations on Hilbert spaces[edit]

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.

Since 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. 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.

Classification in the compact case[edit]

If G is a connected compact Lie group, its finite-dimensional representations can be decomposed as direct sums of irreducible representations.[15] 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.

The commutative case[edit]

If G is a commutative Lie group, then its irreducible representations are simply the continuous characters of G: see Pontryagin duality for this case.

See also[edit]

Remarks[edit]

Notes[edit]

  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 Theorem 5.6
  9. ^ Hall 2015, Theorem 3.28
  10. ^ Hall 2015, Theorem 3.28
  11. ^ Hall 2015, Theorem 5.6
  12. ^ Hall 2013, Section 16.7.3
  13. ^ Hall 2015, Proposition 5.9
  14. ^ Hall 2015, Theorem 5.10
  15. ^ Hall 2015 Theorems 4.28

References[edit]