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 (indeed in the physics literature the distinction is often elided).
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Representations on a complex finite-dimensional vector space
Let us first discuss representations acting on finite-dimensional complex vector spaces. A representation of a Lie group G on a finite-dimensional complex vector space V is a smooth group homomorphism <math>\Psi</math>: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 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 <math>\Psi</math> is smooth, in the definition above, means that <math>\Psi</math> is a smooth map from the smooth manifold G to the smooth manifold Aut(V).
If a basis for the complex vector space V is chosen, the representation can be expressed as a homomorphism into GL(n,C). This is known as a matrix representation.
Representations on a finite-dimensional vector space over an arbitrary field
A representation of a Lie group G on a vector space V (over a field K) is a smooth (i.e. respecting the differential structure) group homomorphism G→Aut(V) from G to the automorphism group of V. If a basis for the vector space V is chosen, the representation can be expressed as a homomorphism into 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.
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.
If the homomorphism is in fact an monomorphism, the representation is said to be faithful.
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.
Representations on Hilbert spaces
A representation of a Lie group G on a complex Hilbert space V is a group homomorphism <math>\Psi</math>: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 x V → V given by (g,v) → <math>\Psi</math>(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 L^2(R). We define the representation <math>\Psi</math>:R → B(L^2(R)) by (<math>\Psi</math>(r)f)(x) → f(r^{-1} x).
See also Wigner's classification for representations of the Poincaré group.
Classification
If G is a semisimple group, its finite-dimensional representations can be decomposed as direct sums of irreducible representations. The irreducibles are indexed by highest weight; the allowable (dominant) highest weights satisfy a suitable positivity condition. In particular, there exists a set of fundamental weights, indexed by the vertices of the Dynkin diagram of G, such that dominant weights are simply non-negative integer linear combinations of the fundamental weights.
If G is a commutative compact Lie group, then its irreducible representations are simply the continuous characters of G: see Pontryagin duality for this case.
A quotient representation is a quotient module of the group ring.
Formulaic examples
Let <math>\mathbb{F}_q</math> be a finite field of order q and characteristic p. Let <math>G</math> be a finite group of Lie type, that is, <math>G</math> is the <math>\mathbb{F}_q</math>-rational points of a connected reductive group <math>\mathbb{G}</math> defined over <math>\mathbb{F}_q</math>. For example, if n is a positive integer <math>GL_n(\mathbb{F}_q)</math> and <math>SL_n(\mathbb{F}_q)</math> are finite groups of Lie type. Let <math>J = \begin{pmatrix}0 & I_n \\ -I_n & 0\end{pmatrix}</math>, where <math>I_n\,\!</math> is the <math>\,\!n \times n</math> identity matrix. Let
Then <math>Sp_2(\mathbb{F}_q)</math> is a symplectic group of rank n and is a finite group of Lie type. For <math>G = GL_n(\mathbb{F}_q)</math> or <math>SL_n(\mathbb{F}_q)</math> (and some other examples), the standard Borel subgroup <math>B\,\!</math> of <math>G\,\!</math> is the subgroup of <math>G\,\!</math> consisting of the upper triangular elements in <math>G\,\!</math>. A standard parabolic subgroup of <math>G\,\!</math> is a subgroup of <math>G\,\!</math> which contains the standard Borel subgroup <math>B\,\!</math>. If <math>P\,\!</math> is a standard parabolic subgroup of <math>GL_n(\mathbb{F}_q)</math>, then there exists a partition <math>(n_1,\ldots,n_r)\,\!</math> of <math>n\,\!</math> (a set of positive integers <math>n_j\,\!</math> such that <math>n_1 + \ldots + n_r = n\,\!</math>) such that <math>P = P_{(n_1,\ldots,n_r)} = M \times N</math>, where <math>M \simeq GL_{n_1}(\mathbb{F}_q) \times \ldots \times GL_{n_r}(\mathbb{F}_q)</math> has the form
and
where <math>*\,\!</math> denotes arbitrary entries in <math>\mathbb{F}_q</math>.
References
Knapp, A. W. (2002). Lie groups beyond an introduction (2nd ed.) Boston:Birkhäuser.
Rossmann, W. (2002). Lie groups : an introduction through linear groups. Oxford:Oxford Univ. Press.