Representation Theory

Let G be a matrix Lie group. Then, a finite-dimensional complex representation of G is a Lie group homomorphism: Π : G --> GL(V), where V is a finite-dimensional complex vector space (with dim(V) ≥ 1). We may think of a representation as a linear action of a group on a vector space, i.e. to every g ∈ G, there is an operator Π(g) acts on the vector space. 

If g is a real or complex Lie algebra, then a finite-dimensional complex representation of g is a Lie algebra homomorphism π of g into gl(V). It is a unique representation π of g acting on the same space V such that Π(e^X) = e^π(X) for all X ∈ g. The representation π can be computed as π(X) = d/dt  Π(e^tX) |_t=0 and satisfies π(AXA^-1) =  Π(A) π(X) Π(A)^-1 for all X ∈ g  and all A ∈ G.  

If Π or π is a one-to-one homomorphism, then the representation is called faithful.

Let Π be a finite-dimensional real or complex representation of a matrix Lie group G, acting on a space V. A subspace W of V is called invariant if  Π(A)w ∈ W for all w ∈ W and all A ∈ G. An invariant subspace W is called nontrivial if W ≠ {0} and W ≠ V. A representation with no nontrivial invariant subspaces is called irreducible.

Let G be a connected matrix Lie group with Lie algebra g. Let Π be a representation of G and π the associated representation of g. Then, Π is irreducible if and only if π is irreducible.

Let G be a matrix Lie group, let H be a Hilbert space, and let U(H) denote the group of unitary operators on H. Then, a homomorphism Π : G --> U(H) is called a unitary representation of G if Π satisfies the following continuity condition (strong) condition: If A_n, A ∈ G and A --> A_n, then Π(A_n)v --> Π(A)v for all v ∈ H. A unitary representation with no nontrivial closed invariant subspace is called irreducible.  

The terms invariant, nontrivial, and irreducible are defined analogously for representations of Lie algebra.

Equivalence

Let G be a matrix Lie group, let  Π be a representation of G acting on the space V, and let ∑ be a representation of G acting on the space W. A linear map ϕ : V --> W is called an intertwining map of representation if ϕ(Π(A)v) = ∑(A)ϕ(v) for all A ∈ G and v ∈ V. The analogous property defines intertwining maps of representations of a Lie algebra. 

If ϕ is an intertwining map of representations and, in addition, ϕ is invertible, then ϕ is said to be an equivalence of representations. If there exists an isomorphism between V and W, then the representations are said to be equivalent. 

Let G be a connected matrix Lie group, let Π₁ and Π₂ be represenation of G, and let π₁and π₂ be the associated Lie algebra representations. Then, π₁and π₂ is equivalent if and only if Π₁ and Π₂ are equivalent.

Schur's Lemma

1. Let V and W be irreducible real or complex representations of a group or Lie algebra and let  ϕ : V --> W be an intertwining map. Then either ϕ = 0 or ϕ is an isomorphism.

2. Let V be an irreducible complex representation of a group or Lie algebra and let ϕ : V --> V be an intertwining map of V with itself. Then ϕ = λI, for some λ ∈ C.

3. Let V and W be irreducible complex representations of a group or Lie algebra and let ϕ₁ , ϕ₂  : V --> W , be nonzero intertwining maps. Then, ϕ₁ = λϕ₂, for some λ ∈ C. 

Applications of Representation Theory

Studying the representations of a group G (or of a Lie algebra) can give information about the group (or Lie algebra) itself. For example, if G is a finite group, then associated to G is something called the group algebra. The structure of this group algebra can be described very nicely in terms of the irreducible representations of G. 

One of the chief applications of representation theory is to exploit symmetry. If a system has symmetry, then the set of symmetries will form a group, and understanding the representations of the symmetry group allows one to use symmetry to simplify the problem. For example, if the equation has rotational symmetry, then the space of solutions will be invariant under rotations. Thus, the space of solutions will constitute a representation of the rotation group SO(3). If one knows what all of the representation of SO(3) are, this can help in narrowing down what the space of solutions can be.