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

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.

Representations on Hilbert spaces

A representation of a Lie group G on a complex Hilbert space V is a group homomorphism Ψ:G → B(V) from G to B(V), the group of bounded linear operators of V which have a bounded inverse, such that the map G×V → V given by (g,v) → Ψ(g)v is continuous.

This definition can handle representations on infinite-dimensional Hilbert spaces. Such representations can be found in e.g. quantum mechanics, but also in Fourier analysis as shown in the following example.

Let G=R, and let the complex Hilbert space V be L²(R). We define the representation Ψ:R → B(L²(R)) by Ψ(r){f(x)} → f(r⁻¹x).

See also Wigner's classification for representations of the Poincaré group.

Classification in the compact case

If G is a connected compact Lie group compact Lie group, its finite-dimensional representations can be decomposed as direct sums of irreducible representations.^[8] 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.

References

↑ 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
↑ Hall 2015 Theorems 4.28

Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. ISBN 978-0-387-97495-8. MR 1153249.
Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN 0-387-40122-9 .
Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, 267, Springer, ISBN 978-1461471158 .
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.

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