12a - Lecture 12 Theorem 1 Let X Y be a nonzero...

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Lecture 12 Theorem 1. Let φ : X Y be a nonzero homomorphism between two g -modules. 1. If X is simple then φ is injective. 2. If Y is simple then φ is surjective. 3. If both are simple then φ is an isomorphism. Proof. Ker φ is a submodule of X . Im φ is a submodule of Y . Remark: In the last case, φ is unique up to a scalar multiple (given ﬁnite dimension, algebraically closed ﬁeld). If ψ : X Y is another, then ψ - 1 φ : X X is a linear map such that ψ - 1 φ ( z ) = z · ψ - 1 φ ( x ) for all z g , x X , so ψ - 1 φ is a scalar by Schur’s Lemma. Theorem 2. L ( V M ,V N ) V N + M V N + M - 2 ... V | N - M | Remark: This is the ’Clebsch-Gordan’ theorem (usually stated for V M V N ). L(V M ,V N ) V * M V N , V * M V M Proof to follow. 1

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Example 1. L ( sl 2 , sl 2 ) ( sl 2 module via ad) sl 2 V 2 then from Clebsch-Gordan L ( sl 2 , sl 2 ) L ( V 2 ,V 2 ) V 4 V 2 V 0 where Dim ( V 4 ) = 5 ,Dim ( V 2 ) = 3
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This note was uploaded on 07/08/2011 for the course MAT 6932 taught by Professor Staff during the Summer '10 term at University of Florida.

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12a - Lecture 12 Theorem 1 Let X Y be a nonzero...

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