Appendix Complexifications of Real Lie Algebras and the Tensor Product Decomposition of sl(2,C)R Representations The goal of this appendix is to prove Proposition 5.8 about the tensor product decomposition of two sl(2, C)R representations. The proof is long but will intro- duce some useful notions, like the direct sum and complexification of a Lie al- gebra. We will use these notions to show that the representations of sl(2, C)R are in 1–1 correspondence with certain representations of the complex Lie alge- bra sl(2, C) ⊕ sl(2, C), and the fact that this complex Lie algebra is a direct sum will imply certain properties about its representations, which in turn will allow us to prove Proposition 5.8. A.1 Direct Sums and Complexifications of Lie Algebras In this text we have dealt only with real Lie algebras, as that is the case of greatest interest for physicists. From a more mathematical point of view, however, it actually simplifies matters to focus on the complex case, and we will need that approach to prove Proposition 5.8. With that in mind, we make the following definition (in total analogy to the real case): A complex Lie algebra is a complex vector space g equipped with a complex-linear Lie bracket [·,·] : g × g → g which satisfies the usual axioms of antisymmetry [X, Y ]=−[Y,X]∀X, Y ∈ g, and the Jacobi identity [X, Y ],Z + [Y,Z],X + [Z,X],Y = 0 ∀X, Y, Z ∈ g. Examples of complex Lie algebras are Mn(C) = gl(n, C), the set of all complex n × n matrices, and sl(n, C), the set of all complex, traceless n × n matrices. In both cases the bracket is just given by the commutator of matrices. For our application we will be interested in turning real Lie algebras into complex Lie algebras. We already know how to complexify vector spaces, so to turn a real Lie algebra into a complex one we just have to extend the Lie bracket to the complexified vector space. This is done in the obvious way: given a real Lie algebra g with bracket N. Jeevanjee, An Introduction to Tensors and Group Theory for Physicists, 227 DOI 10.1007/978-0-8176-4715-5, © Springer Science+Business Media, LLC 2011 228 Complexifications of Real Lie Algebras [·,·], we define its complexification to be the complexified vector space gC = C ⊗ g with Lie bracket [·,·]C defined by [1 ⊗ X1 + i ⊗ X2, 1 ⊗ Y1 + i ⊗ Y2]C ≡ 1 ⊗[X1,Y1]−1 ⊗[X2,Y2]+i ⊗[X1,Y2]+i ⊗[X2,Y1] where Xi,Yi ∈ g,i= 1, 2. If we abbreviate i ⊗ X as iX and 1 ⊗ X as X, this tidies up and becomes [X1 + iX2,Y1 + iY2]C ≡[X1,Y1]−[X2,Y2]+i [X1,Y2]+[X2,Y1] . This formula defines [·,·]C in terms of [·,·], and is also exactly what you would get by naively using complex-linearity to expand the left hand side. What does this process yield in familiar cases? For su(2) we define a map φ : su(2)C → sl(2, C) i 0 (A.1) 1 ⊗ X + i ⊗ X → X + X ,X,X ∈ su(2). 1 2 1 0 i 2 1 2 You will show below that this is a Lie algebra isomorphism, and hence su(2) complexifies to become sl(2, C). You will also use similar maps to show that u(n)C gl(n, R)C gl(n, C). If we complexify the real Lie algebra sl(2, C)R,we also get something nice. The complexified Lie algebra (sl(2, C)R)C has complex basis 1 ˜ Mi ≡ (1 ⊗ Si − i ⊗ Ki), i = 1, 2, 3 2 (A.2) 1 ˜ Ni ≡ (1 ⊗ Si + i ⊗ Ki), i = 1, 2, 3 2 which we can again abbreviate1 as 1 ˜ Mi ≡ (Si − iKi), i = 1, 2, 3 2 1 ˜ Ni ≡ (Si + iKi), i = 1, 2, 3. 2 These expressions are very similar to ones found in our discussion of sl(2, C)R rep- resentations, and using the bracket on (sl(2, C)R)C one can verify that the analogs of (5.83) hold, i.e. that [Mi, Nj ]C = 0 3 [Mi, Mj ]C = ij k Mk k=1 (A.3) 3 [Ni, Nj ]C = ij k Nk. k=1 1 ˜ ˜ Careful here! When we write iKi ,thisisnot to be interpreted as i times the matrix Ki ,asthis ˜ would make the Ni identically zero (check!); it is merely shorthand for i ⊗ Ki . A.2 Representations of Complexified Lie Algebras 229 Notice that both Span{Mi} and Span{Ni} (over the complex numbers) are Lie sub- algebras of (sl(2, C)R)C and that both are isomorphic to sl(2, C), since Span{Mi} Span{Ni} su(2)C sl(2, C). Furthermore, the bracket between an element of Span{Mi} and an element of Span{Ni} is 0, by (A.3). Also, as a (complex) vector space (sl(2, C)R)C is the direct sum Span{Mi}⊕Span{Ni}. When a Lie algebra g can be written as a direct sum of subspaces W1 and W2, where the Wi are each subalgebras and [w1,w2]=0for all w1 ∈ W1, w2 ∈ W2, we say that the original Lie algebra g is a Lie algebra direct 2 sum of W1 and W2, and we write g = W1 ⊕ W2. Thus, we have the Lie algebra direct sum decomposition C C ⊕ C sl(2, )R C sl(2, ) sl(2, ). (A.4) This decomposition will be crucial in our proof of Proposition 5.8. Exercise A.1 Prove that (A.1) is a Lie algebra isomorphism. Remember, this consists of showing that φ is a vector space isomorphism, and then showing that φ preserves brackets. Then find similar Lie algebra isomorphisms to prove that u(n)C gl(n, C) gl(n, R)C gl(n, C). A.2 Representations of Complexified Lie Algebras and the Tensor Product Decomposition of sl(2,C)R Representations In order for (A.4) to be of any use, we must know how the representations of a real Lie algebra relate to the representations of its complexification. First off, we should clarify that when we speak of a representation of a complex Lie algebra g we are ignoring the complex vector space structure of g; in particular, the Lie algebra homomorphism π : g → gl(V ) is only required to be real-linear,inthe sense that π(cX) = cπ(X) for all real numbers c.Ifg is a complex Lie algebra, the representation space V is complex and π(cX) = cπ(X) for all complex numbers c, then we say that π is complex-linear. Not all complex representations of complex Lie algebras are complex-linear. For instance, all of the sl(2, C)R representations described in Sect. 5.10 can be thought of as representations of the complex Lie algebra sl(2, C), but only those of the form (j, 0) are complex-linear, as you will show below. 2 Notice that this notations is ambiguous, since it could mean either that g is the direct sum of W1 and W2 as vector spaces (which would then tell you nothing about how the direct sum decomposi- tion interacts with the Lie bracket), or it could mean Lie algebra direct sum. We will be explicit if there is any possibility of confusion. 230 Complexifications of Real Lie Algebras Exercise A.2 Consider all the representations of sl(2, C)R as representations of sl(2, C) as 1 C well. Show directly that the fundamental representation ( 2 , 0) of sl(2, ) is complex-linear, and use this to prove that all sl(2, C) representations of the form (j, 0) are complex-linear. Furthermore, by considering the operators Ni defined in (5.82), show that these are the only complex-linear representations of sl(2, C). Now let g be a real Lie algebra and let (π, V ) be a complex representation of g. Then we can extend (π, V ) to a complex-linear representation of the complexifica- tion gC in the obvious way, by setting π(X1 + iX2) ≡ π(X1) + iπ(X2), X1,X2 ∈ g. (A.5) (Notice that this representation is complex-linear by definition, and that the operator iπ(X2) is only well-defined because V is a complex vector space.) Furthermore, this extension operation is reversible: that is, given a complex-linear representation (π, V ) of gC, we can get a representation of g by simply restricting π to the subspace {1 ⊗ X + i ⊗ 0 | X ∈ g} g, and this restriction reverses the extension just defined. Furthermore, you will show below that (π, V ) is an irrep of gC if and only if it corresponds to an irrep of g. We thus have Proposition A.1 The irreducible complex representations of a real Lie algebra g are in one-to-one correspondence with the irreducible complex-linear representa- tions of its complexification gC. This means that we can identify the irreducible complex representations of g with the irreducible complex-linear representations of gC, and we will freely make this identification from now on. Note the contrast between what we are doing here and what we did in Sect. 5.9; there, we complexified real representations to get complex representations which we could then classify; here, the representation space is fixed (and is always complex!) and we are complexifying the Lie algebra itself, to get a representation of a complex Lie algebra on the same representation space we started with.