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. π ( X ) = d d t Π ( e t X ) | t = 0 , X ∈ g . {displaystyle pi (X)=left.{frac {d}{dt}}Pi (e^{tX}) ight|_{t=0},quad Xin {mathfrak {g}}.}     (G6) Π ( g = e X ) ≡ e π ( X ) , X ∈ g , g = e X ∈ im ⁡ ( exp ) , Π ( g = g 1 g 2 ⋯ g n ) ≡ Π ( g 1 ) Π ( g 2 ) ⋯ Π ( g n ) , g ∉ im ⁡ ( exp ) , g 1 , g 2 , … , g n ∈ im ⁡ ( exp ) . {displaystyle {egin{aligned}Pi (g=e^{X})&equiv e^{pi (X)},&&Xin {mathfrak {g}},quad g=e^{X}in operatorname {im} (exp ),\Pi (g=g_{1}g_{2}cdots g_{n})&equiv Pi (g_{1})Pi (g_{2})cdots Pi (g_{n}),&&g otin operatorname {im} (exp ),quad g_{1},g_{2},ldots ,g_{n}in operatorname {im} (exp ).end{aligned}}}     (G2) 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. Let us first discuss representations of groups acting on a finite-dimensional vector space over the field C {displaystyle mathbb {C} } . (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 C {displaystyle mathbb {C} } is then a smooth group homomorphism where GL ⁡ ( V ) {displaystyle operatorname {GL} (V)} is the general linear group of all invertible linear transformations of V {displaystyle V} under their composition. Since all n-dimensional spaces are isomorphic, the group GL ⁡ ( V ) {displaystyle operatorname {GL} (V)} can be identified with the group of the invertible, complex n × n {displaystyle n imes n} matrices, generally called GL ⁡ ( n ; C ) . {displaystyle operatorname {GL} (n;mathbb {C} ).} Smoothness of the map Π {displaystyle Pi } can be regarded as a technicality, in that any continuous homomorphism will automatically be smooth. We can alternatively describe a representation of a Lie group G {displaystyle G} as a linear action of G {displaystyle G} on a vector space V {displaystyle V} . Notationally, we would then write g ⋅ v {displaystyle gcdot v} in place of Π ( g ) v {displaystyle Pi (g)v} for the way a group element g ∈ G {displaystyle gin G} acts on the vector v ∈ V {displaystyle vin V} . A typical example in which representations arise in physics would be the study of a linear partial differential equation having symmetry group G {displaystyle G} . Although the individual solutions of the equation may not be invariant under the action of G {displaystyle G} , the space V {displaystyle V} of all solutions is invariant under the action of G {displaystyle G} . Thus, V {displaystyle V} constitutes a representation of G {displaystyle G} . See the example of SO(3), discussed below. If the homomorphism Π {displaystyle Pi } 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 GL ⁡ ( n ; C ) {displaystyle operatorname {GL} (n;mathbb {C} )} . 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 Π : G → GL ⁡ ( V ) {displaystyle Pi :G ightarrow operatorname {GL} (V)} , we say that a subspace W of V is an invariant subspace if Π ( g ) w ∈ W {displaystyle Pi (g)win W} for all g ∈ G {displaystyle gin G} and w ∈ W {displaystyle win 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 and semisimple 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 Π {displaystyle Pi } 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.