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By Kantor W.M.

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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 Examples Let G be a Lie group. A representation of G on a ﬁnite-dimensional vector space V smoothly assigns to each g ∈ G an invertible linear transformation of V satisfying π(g)π(g ) = π(gg ) for all g, g ∈ G.

In other words, once a basis is chosen and the G action is realized by matrix multiplication, the action of G on V is obtained from the action of G on V simply by taking the conjugate of the matrix. It should also be noted that few of these constructions are independent of each other. For instance, the action in number (7) on V ∗ is just the special case of the action in number (3) on Hom(V, W ) in which W = C is the trivial representation. Also the actions in (4), (5), and (6) really only make repeated use of number (2).

However, W1 ∩ (W2 ⊕ · · · ⊕ Wn ) is a G-invariant submodule of W1 , so the initial hypothesis and irreducibility ﬁnish the argument. If V, W are representations of a Lie group G and V ∼ = W ⊕ W , note this decomposition is not canonical. For example, if c ∈ C\{0}, then W = {(w, cw) | w ∈ W } and W = {(w, −cw) | w ∈ W } are two other submodules both equivalent to W and satisfying V ∼ = W ⊕ W . 18. 24 (Canonical Decomposition). Let V be a ﬁnite-dimensional representation of a compact Lie group G.

### 4-Homogeneous groups by Kantor W.M.

by Charles

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