Symmetry And Group

Linear ODEs and group theory from Riemann to Poincare by J.J.Gray

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By J.J.Gray

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11), although the changes will not affect the finite-dimensional case. Two representations will be called equivalent if they are the same up to, basically, a change of basis. Recall that Hom(V, V ) is the set of all linear maps from V to V . 2. Let (π, V ) and (π , V ) be finite-dimensional representations of a Lie group G. (1) T ∈ Hom(V, V ) is called an intertwining operator or G-map if T ◦ π = π ◦ T . (2) The set of all G-maps is denoted by HomG (V, V ). (3) The representations V and V are equivalent, V ∼ = V , if there exists a bijective G-map from V to V .

2, this defines a representation. As fine and natural as this representation is, it actually contains a smaller, even nicer, representation. Write = ∂x21 + · · · + ∂x2n for the Laplacian on Rn . 5). 5. , Hm (Rn ) = {P ∈ Vm (Rn ) | P = 0}. If P ∈ Hm (Rn ) and g ∈ O(n), then (g · P) = g · ( P) = 0 so that g · P ∈ Hm (Rn ). In particular, the action of O(n) on Vm (Rn ) descends to a representation of O(n) (or S O(n), of course) on Hm (Rn ). It will turn out that these representations do not break into any smaller pieces.

44 to show that the invariant integral on S O(3) is given by f (g) dg = S O(3) 1 8π 2 π 2π 0 0 2π f (α(θ )β(φ)α(ψ)) sin φ dθdφdψ 0 for integrable f on S O(3). 48 Let θ α(θ) = ei 2 0 θ 0 e−i 2 and β (θ) = cos θ2 − sin θ2 sin θ2 cos θ2 . 47, show that the invariant integral on SU (2) is given by SU (2) f (g) dg = 1 8π 2 for integrable f on SU (2). π 2π 0 0 2π 0 f (α(θ )β(φ)α(ψ)) sin φ dθdφdψ 2 Representations Lie groups are often the abstract embodiment of symmetry. However, most frequently they manifest themselves through an action on a vector space which will be called a representation.

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