Lie algebra homomorphism

In this section we introduce Lie algebras, Lie algebra homomorphisms and Lie algebra Cartesian sums. In the examples we introduce all the Lie algebras we will need in our study of the hydrogen atom. 

The notions of Lie algebra homomorphism and Lie algebra isomorphism will be important to us. 

To define a Lie algebra homomorphism, it suffices to define it on basis elements of 01 and check that the commutation relations are satisfied. Because the homomorphism is linear, it is defined uifiquely by its value at basis elements. Because the bracket is linear, if the brackets of basis elements satisfy the equality in Definition 8.7, then any linear combination of basis elements will satisfy equality in Definition 8.7. 

The reader should check that Definition 8.7 is satisfied and notice that those factors of 1 /2 are necessary. Thus Ti is a Lie algebra homomorphism. To see that it is an isomorphism, note that the matrices on the three right-hand sides of the defining equations for Ti form a basis of su(2). Similarly, defining 72 0Q 5o(3) by... 

Physicists call L the total angular momentum. To check that it is a Lie algebra homomorphism, we must check that the Lie brackets behave properly. They... 

To confirm that this is an isomorphism of Lie algebras, note first that 5 is a well-defined linear transformation (by Proposition 2,3). Then check that it is a homomorphism of Lie algebras by checking all bracket relations between the matrices above. We leave this verification mostly to the reader, giving just one example ... 

Then we say that T is a homomorphism of (Lie algebra) representations. If in addition T is injective and surjective then we say that T is an isomorphism of (Lie algebra) representations and that p is isomorphic fo p. 

All of the results of Section 6.1 apply, mutatis mutandis, to irreducible Lie algebra representations. For example, if T is a homomorphism of Lie algebra representations, then the kernel of T and the image of T are both invariant subspaces. This leads to Schur s Lemma for Lie algebra representations. 

Proposition 8.4 (Schur s Lemma) Suppose (fl, Vi, pf) and (fl, Pf) are irreducible representations of the Lie algebra fl. Suppose that T Vi V2 is a homomorphism of representations. Then there are only two possible cases ... 

Corollary. A homomorphism G- H induces a Lie algebra map, injective if G-> H is a closed embedding. 

Lemma (2. 3.1) The above correspondence 0 I— u establishes an isomo rphism between Horn (M, Lie fUl D I U) and the group of bi-algebra homomorphisms, u, such that u(r+(M))cl-U and such that u is compatible with divided powers. 

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