Representation of a Lie group

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

Let us first discuss representations acting on finite-dimensional vector spaces over a field K, where K is usually taken to be the field of complex numbers, or occasionally the field of real numbers. A representation of a Lie group G on a finite-dimensional vector space V over K is a smooth group homomorphism Ψ:G→Aut(V) from G to the automorphism group of V. For n-dimensional V, the automorphism group of V is identified with a subset of the complex square matrices of order n. The automorphism group of V is given the structure of a smooth manifold using this identification. The condition that Ψ is smooth, in the definition above, means that Ψ is a smooth map from the smooth manifold G to the smooth manifold Aut(V). If the homomorphism is in fact 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 GL(n,K). 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 Ψ:G→Aut(V), we say that a subspace W of V is invariant if $\psi (g)w\in W$ for all $g\in G$ and $w\in W$ . 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^[1] and semisimple^[2] 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.)

On the Lie algebra level, there is a corresponding linear mapping from the Lie algebra of G to End(V) preserving the Lie bracket [ , ]. See representation of Lie algebras for the Lie algebra theory.

A unitary representation is defined in the same way, except that G maps to unitary matrices; the Lie algebra will then map to skew-hermitian matrices. If G is a compact Lie group, every finite-dimensional representation is equivalent to a unitary one.^[3]

An example: The rotation group SO(3)

In quantum mechanics, the time-independent Schrödinger equation equation, ${\hat {H}}\psi =E\psi$ plays an important role. In the three-dimensional case, if ${\hat {H}}$ has rotational symmetry, then the space of solutions to ${\hat {H}}\psi =E\psi$ 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 ${\hat {H}}\psi =E\psi$ , 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.^[4] For each non-negative integer $k$ , the irreducible representation of dimension $2k+1$ can be realized as the space $V_{k}$ of homogeneous harmonic polynomials on $\mathbb {R} ^{3}$ of degree $k$ .^[5] Here, SO(3) acts on $V_{k}$ in the usual way that rotations act on functions on $\mathbb {R} ^{3}$ :

(\psi (R)f)(x)=f(R^{-1}x)\quad R\in \mathrm {SO} (3)

.

The restriction to the unit sphere $S^{2}$ of the elements of $V_{k}$ are the spherical harmonics of degree $k$ .

If, say, $k=1$ , then all polynomials that are homogeneous of degree one are harmonic, and we obtain a three-dimensional space $V_{1}$ spanned by the linear polynomials $x$ , $y$ , and $z$ . If $k=2$ , the space $V_{2}$ is spanned by the polynomials $xy$ , $xz$ , $yz$ , $x^{2}-y^{2}$ , and $x^{2}-z^{2}$ .

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).^[6]

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

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^[7] says that we do, in fact, get a one-to-one correspondence between the group and Lie algebra representations.

Matrix Lie group representations from Lie algebra representations, projective representations

Let $G$ be a matrix Lie group with Lie algebra ${\mathfrak {g}}$ , and assume that a representation $\pi$ of ${\mathfrak {g}}$ is at hand. The Lie correspondence may be employed for obtaining group representations of the connected component of the $G$ . 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 $G$ .

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 Lie correspondence

The Lie correspondence for linear groups and Lie algebras is stated for reference. If $G$ denotes a linear Lie group (i.e. a group of matrices)^{[nb 1]} and ${\mathfrak {g}}$ a linear Lie algebra (again a set of matrices),^{[nb 2]} let $\Gamma ({\mathfrak {g}})$ denote the group generated by $\exp({\mathfrak {g}}),$ the image of the Lie algebra under the exponential mapping (which is the matrix exponential in this case),^{[nb 3]} and let $Lie(G)$ denote the Lie algebra of $G$ (interpreted as the set of matrices $X$ such that $e tX \in G$ for all $t\in \mathbb {R}$ ). The Lie correspondence reads in modern language, here specialized to linear Lie groups, as follows:

There is a one-to-one correspondence between connected linear Lie groups and linear Lie algebras given by

G\leftrightarrow {\mathfrak {g}}

with

{\mathfrak {g}}={\text{Lie}}(G)

or, equivalently

G=\Gamma ({\mathfrak {g}}),

expressed as

\Gamma ({\text{Lie}}(G))=G,

respectively

{\text{Lie}}(\Gamma ({\mathfrak {g}}))={\mathfrak {g}}.

^[8] Lie

Lie algebra representations from group representations

Using the above theorem it is always possible to pass from a representation of a Lie group $G$ to a representation of its Lie algebra ${\mathfrak {g}}.$ If $Π : G \to GL(V)$ is a group representation for some vector space $V$ , then its pushforward (differential) at the identity, or Lie map, $\pi :{\mathfrak {g}}\to {\text{End}}V$ is a Lie algebra representation. It is explicitly computed using^[9]

\pi (X)=\left.{\frac {d}{dt}}\Pi (e^{tX})\right|_{t=0},\quad X\in {\mathfrak {g}}.

(G6)

Not all Lie algebra representations arise this way because their corresponding group representations may not exist as proper representations, i.e. they are projective, see below.

Group representations from Lie algebra representations

If $\pi :{\mathfrak {g}}\to {\mathfrak {gl}}(V)$ for some vector space $V$ is a representation, a representation $Π$ of the connected component of $G$ is tentatively defined by setting

{\begin{aligned}\Pi (g=e^{X})&\equiv e^{\pi (X)},&&X\in {\mathfrak {g}},\quad g=e^{X}\in \mathrm {im} (\exp ),\\\Pi (g=g_{1}g_{2}\cdots g_{n})&\equiv \Pi (g_{1})\Pi (g_{2})\cdots \Pi (g_{n}),&&g\notin \mathrm {im} (\exp ),\quad g_{1},g_{2},\ldots ,g_{n}\in \mathrm {im} (\exp ).\end{aligned}}

(G2)

It can be shown that simple connectedness of $G$ is a sufficient condition for (G2) to yield a representation, but it is not a necessary condition.

The simply connected case is the statement of the following theorem: If $\pi :{\mathfrak {g}}\to {\mathfrak {h}}$ is a Lie algebra homomorphism and $G$ is simply connected, then there is a unique Lie group homomorphism $Π : G \to H$ satisfying the first line in (G2).^[10]
If G is not simply connected, then there is a unique representation Π_c of the universal covering group G_c of G satisfying the same equation as the first equation in (G2). It is a consequence of the above theorem.
- If the kernel of the covering map is included in the kernel of $Π c$ , then the representation of $Π c$ descends to a unique representation of $G$ . This is essentially a consequence of a variant of the first isomorphism theorem.
- If the kernel of the covering map is not included in the kernel of $Π c$ , then a (non-unique) projective representation of $G$ results.

All representations have the following properties:

Near the identity i.e. for $X$ in a small enough open neighborhood, (G2) yields, by the Baker–Campbell–Hausdorff formula and that $exp$ is one-to-one on that neighborhood, a unique local homomorphism.^[11]
Representatives of elements g far off the identity are defined by selecting a path from the identity to g, partitioning it finely enough so that the above property can be used. The result using (G2) can then only depend on the homotopy class (in the standard representation of G) of the path used in the (attempted) definition of Π.^[12]^[13] In turn, this depends only on which X in the Lie algebra is used to represent an element g in the standard representation (and is used in (G2).
- In the simply connected case, there is only one homotopy class, and (G2) is unambiguous even far off the identity.
- In the non-simply connected case, there are $card π 1$ homotopy classes, and (G2) is to this extent ambiguous far off 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 ${\text{SL}}(2,\mathbb {C} )$ , and whether its corresponding representation $Π c$ is faithful decides whether $Π$ is projective.

Unitary representations on Hilbert spaces

Let V be a complex Hilbert space, which may be infinite dimensional, and let $U(V)$ denote the group of unitary operators on V. A unitary representation of a Lie group G on V is a group homomorphism $\psi :G\rightarrow U(V)$ with the property that for each fixed $v\in V$ , the map

g\mapsto \psi (g)v

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 $G=\mathbb {R}$ , and let the complex Hilbert space V be $L^{2}(\mathbb {R} )$ . We define the representation $\psi :\mathbb {R} \rightarrow U(L^{2}(\mathbb {R} ))$ by

[\psi (a)(f)](x)=f(x-a)

.

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

The Stone–von Neumann theorem can be understood as giving a classification of the irreducible unitary representations of the Heisenberg group.
Wigner's classification for representations of the Poincaré group plays a major conceptual role in quantum field theory by showing how the mass and spin of particles can be understood in group-theoretic terms.
The representation theory of SL(2,R) was worked out by V. Bargmann and serves as the prototype for the study of unitary representations of noncompact semisimple Lie groups.

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.^[14] 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. The representations can be classified easily and are labeled by certain "analytically integral elements" or "weights." If $\Sigma$ is an irreducible representation of G, the restriction of $\Sigma$ 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 $\Sigma$ , 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 $\Sigma$ of G, the character is the function

\chi _{G}:G\rightarrow \mathbb {C}

given by

\chi _{G}(g)=\mathrm {trace} (\chi (g))

.

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

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

Remarks

^ Not all groups have faithful matrix representations. For example, the universal covering group of the linear Lie group SL(2, R) has none. See Hall (2015, Proposition 5.16.) A quotient of a matrix Lie group need not be linear. This is e.g. the case for the quotient of the Heisenberg group by a discrete subgroup of its center. See Hall (2015, Section 4.8.)

However, if $G$ is a compact Lie group, it is representable as a matrix Lie group. This is a consequence of the Peter–Weyl theorem. See Rossmann (2002, Section 6.2.)
^ It's a rather deep fact that all finite-dimensional Lie algebras have faithful matrix representations. This is the content of Ado's theorem. See Hall (2015, Section 5.10.)
^ The exponential mapping need not be onto and the image is in those cases not a group. Therefore one takes all finite products of elements in the image in order to obtain a group, which necessarily must be closed under multiplication.

Notes

^ Hall 2015 Theorem 4.28
^ Hall 2015 Section 10.3
^ Hall 2015 Theorem 4.28
^ Hall 2015 Section 4.7
^ Hall 2013 Section 17.6
^ Hall 2015 Proposition 4.35
^ Hall 2015 Theorem 5.6
^ Rossmann 2002 Theorem 1, Paragraph 2.5.
^ Hall 2003, Equation 2.16.
^ Hall 2015, Theorem 5.6.
^ Hall 2015, Proposition 5.9.
^ Weinberg 2002, Appendix B.
^ Hall 2015, Step 3 in proof of theorem 5.10.
^ Hall 2015 Theorems 4.28

References

Fulton, W.; Harris, J. (1991). Representation theory. A first course. Graduate Texts in Mathematics. Vol. 129. New York: Springer-Verlag. ISBN 978-0-387-97495-8. MR 1153249. {{cite book}}: Invalid |ref=harv (help)
Hall, Brian C. (2003). Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics. Vol. 222 (1st ed.). Springer. ISBN 0-387-40122-9. {{cite book}}: Invalid |ref=harv (help)
Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 0-387-40122-9.
Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer, ISBN 978-1461471158.
Knapp, Anthony W. (2002), Lie Groups Beyond an Introduction, Progress in Mathematics, vol. 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, vol. 1, Cambridge: Cambridge University Press, ISBN 0-521-55001-7

[8] Not all groups have faithful matrix representations. For example, the universal covering group of the linear Lie group SL(2, R) has none. See Hall (2015, Proposition 5.16.) A quotient of a matrix Lie group need not be linear. This is e.g. the case for the quotient of the Heisenberg group by a discrete subgroup of its center. See Hall (2015, Section 4.8.)

However, if $G$ is a compact Lie group, it is representable as a matrix Lie group. This is a consequence of the Peter–Weyl theorem. See Rossmann (2002, Section 6.2.)

[9] It's a rather deep fact that all finite-dimensional Lie algebras have faithful matrix representations. This is the content of Ado's theorem. See Hall (2015, Section 5.10.)

[10] The exponential mapping need not be onto and the image is in those cases not a group. Therefore one takes all finite products of elements in the image in order to obtain a group, which necessarily must be closed under multiplication.

[1] Hall 2015 Theorem 4.28

[2] Hall 2015 Section 10.3

[3] Hall 2015 Theorem 4.28

[4] Hall 2015 Section 4.7

[5] Hall 2013 Section 17.6

[6] Hall 2015 Proposition 4.35

[7] Hall 2015 Theorem 5.6

[11] Rossmann 2002 Theorem 1, Paragraph 2.5.

[12] Hall 2003, Equation 2.16.

[13] Hall 2015, Theorem 5.6.

[14] Hall 2015, Proposition 5.9.

[15] Weinberg 2002, Appendix B.

[16] Hall 2015, Step 3 in proof of theorem 5.10.

[17] Hall 2015 Theorems 4.28

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