INTRODUCTION TO THE STRUCTURE OF SEMISIMPLE LIE ALGEBRAS AND THEIR REPRESENTATION THEORY GEORGE H. SEELINGER 1. Introduction The representation theory of semisimple Lie algebras is an incredibly beautiful model of a representation theory, since the simple representations of semisimple Lie algebras are completely classified. Furthermore, such a tool is very powerful, since it can be used to • completely classify all simple Lie algebras • completely understand the representation theory of simply connected Lie groups • serve as a guide for understanding the reprsentation theory of other algebraic objects In this monograph, we will be more concerned with the former application and then touch on the general representation theory of semisimple Lie alge- bras. Note that, in this monograph, a linear Lie algebra is a Lie algebra that is viewed a subalgebra of gl(V ), although Ado's theorem shows that any Lie algebra can be realized in such a way. Furthermore, a field F will always be of characteristic 0 and algebraically closed unless otherwise stated. 2. Weyl's Theorem In this section, we seek to prove Weyl's theorem, which provides us with a characterization of the structure of representations of semisimple Lie alge- bras. This section is mostly a rewriting and expanding of [Hum72] chapter 6. 2.1. Theorem (Weyl's Theorem). Let g be a semisimple Lie algebra. If φ: g ! gl(V ) is a representation of g, then φ is completely reducible. To prove this theorem, we will need to show that, for any g-module V and any submodule W ≤ V , we get V = W ⊕ W 0, where W 0 is a complement of W . The easiest way to do this will be to find some module homomorphism Date: June 2018. 1 f : V ! W with ker f \ W = f0g and W + ker f = V . To construct such a homomorphism, we will need the following. 2.2. Definition. Let φ: g ! gl(V ) be a faithful representation of a Lie algebra g and let β(x; y) := tr(φ(x)φ(y)) be a bilinear form on g. Then, for basis fx1; : : : ; xng of g and fy1; : : : ; yng of g such that β(xi; yj) = δij, we define the Casimir element of φ to be X cφ(β) = φ(xi)φ(yi) 2 End(V ) i 2.3. Example. Let g = sl2 with standard basis fe; h; fg. Then, as shown 1 1 1 in [See18], we have dual basis f 4 f; 8 h; 4 eg. Now, if φ is the adjoint repre- sentation, we get 1 1 1 c (fe; f; hg) = φ(e)φ( f) + φ(h)φ( h) + φ(f)φ( e) φ 4 8 4 1 1 1 = ad ad + ad2 + ad f ad 4 e f 8 h 4 e 1 1 1 = (2e + 2e ) + (4e + 4e ) + (2e + 2e ) 4 1;1 2;2 8 1;1 3;3 4 2;2 3;3 = e1;1 + e2;2 + e3;3 = Id3 2.4. Lemma. Take a basis fx1; : : : ; xng of g and a basis fy1; : : : ; yng of g such that, for nondegenerate symmetric associative bilinear form β on g, β(xi; yj) = δij. For x 2 g, if X X [x; xi] = aijxj; [x; yi] = bijyj j j then aik = −bki Proof. We compute X aik = aijδjk j X = aijβ(xj; yk) by bilinearity j = β([x; xi]; yk) = β(−[xi; x]; yk) = β(xi; −[x; yk]) by associativity X = − bkjβ(xi; yj) j = −bki 2 2.5. Example. Continuing the example above, consider basis fe; h; fg of 1 1 1 sl2 and dual basis with respect to the Killing form, f 4 f; 8 h; 4 eg. Then, for x = ae + bh + cf, we check that [x; e] = 2be − ch =) a1;1 = 2b; a1;2 = −c; a1;3 = 0 [x; h] = −2ae + 2cf =) a2;1 = −2a; a2;2 = 0; a2;3 = 2c [x; f] = ah − 2bf =) a3;1 = 0; a3;2 = a; a3;3 = −2b 1 1 1 [x; f] = − bf + ah =) b = −2b; b = 2a; b = 0 4 2 4 1;1 1;2 1;3 1 1 1 [x; h] = cf − ae =) b = c; b = 0; b = −a 8 4 4 2;1 2;2 2;3 1 1 1 [x; e] = − ch + be =) b = 0; b = −2c; b = 2b 4 4 2 3;1 3;2 3;3 2.6. Lemma. For x; y; z 2 End(V ), [x; yz] = [x; y]z + y[x; z] Proof. [x; y]z + y[x; z] = (xy − yx)z + y(xz − zx) = xyz − yxz + yxz − yzx = xyz − yzx = [x; yz] 2.7. Proposition. cφ(β) commutes with φ(g) in End(V ). Proof. For x 2 g, consider [φ(x); cφ(β)]. Using the lemma immediately above, we get X X [φ(x); cφ(β)] = [φ(x); φ(xi)φ(yi)] = [φ(x); φ(xi)]φ(yi)+φ(xi)[φ(x); φ(yi)] i i P Moreover, we know that [φ(x); φ(xi)] = j aijφ(xj) and [φ(x); φ(yi)] = P j bijφ(xj), with aik = −bki for any k using the same decomposition as in the first of the two lemmas. Thus, X X [φ(x); cφ(β)] = aijφ(xj)φ(yi) + bi;jφ(xi)φ(yj) = 0 i;j i;j This proposition is vitally important for proving Weyl's theorem. With this element, we now have an object that must \behave like a scalar" with respect to φ(g) in End(V ). Indeed, in the sl2 example, our Casimir element is exactly the identity, although this need not always be true: 2.8. Proposition. If φ is faithful and irreducible, then cφ(β) is a scalar in End(V ). 3 Proof. Since φ is irreducible, we can invoke Schur's Lemma to get that cφ(β) must be a scalar multiple of the identity map. 2.9. Remark. Note that cφ(β) is independent of choice of basis when φ is faithful, so we will often denote cφ := cφ(β). For more details, see [Hum72, p 27]. 2.10. Proposition. Given Casimir element cφ of faithful representation φ: g ! gl(V ), we get that tr(cφ) = dim g Proof. This follows from simply applying the trace operator. X X tr(cφ) = tr(φ(xi)φ(yi)) = β(xi; yi) = dim g i i 2.11. Example. The Casimir element of sl2 under the adjoint representation was precisely Id3 and tr(Id3) = 3 = dim sl2. 2.12. Lemma. Let φ: g ! gl(V ) be a representation of a semisimple Lie algebra g. Then, φ(g) ⊆ sl(V ). Proof. We know that, for a semisimple Lie algebra, [g; g] = g by the fact that g is a direct sum of simple Lie algebras. Furthermore, [gl(V ); gl(V )] = sl(V ) by simple computation. Thus, φ(g) = [φ(g); φ(g)] ⊆ [gl(V ); gl(V )] = sl(V ). 2.13. Lemma. Let φ: g ! gl(V ) be a representation of a semisimple Lie algebra and let W ≤ V be irreducible with codimension 1. Then, φ is com- pletely reducible. Proof. Without loss of generality, we may assume φ is faithful . Now, con- why? sider cφ commutes with φ(g), so cφ(g:v) = (cφ ◦φ(g))(v) = (φ(g)◦cφ)(v) tells us that cφ is actually a g-module homormophism on V . Thus, cφ(W ) ⊆ W and ker cφ is a submodule of V . Now, by the lemma above, g must send all of V into W and thus act trivially on the quotient module V=W . There- fore, cφ must also act trivially since it is simply a linear combination of actions in g. So, it must be that the trace of cφ on V=W is 0. However, we know trV (c) = dim g 6= 0. Now, since cφ acts as a (must be non-zero) scalar on W , it must be that ker cφ \ W = f0g and ker cφ + W = V . Thus, since ker cφ is a submodule of V , it must be that ker cφ is 1-dimensional and V = W ⊕ ker cφ. 2.14. Lemma. Let φ: g ! gl(V ) be a representation of a semisimple Lie algebra and let W ≤ V with codimension 1. Then, φ is completely reducible. Proof. Note that this is the same as the lemma above, but now W need not be irreducible. We will show that we can reduce this lemma to the case of 4 the previous lemma 2.13. By 2.12, we know g must act trivially on V=W . Thus, V=W =∼ F and we have short exact sequence 0 ! W ! V ! F ! 0 Now, if dim W = 1, W is irreducible and we can appeal to the previous lemma. So, now we proceed by induction and assume the result it true for dim W < n. Let dim W = n and W have proper submodule W 0. Then, we have short exact sequence 0 ! W=W 0 ! V=W 0 ! F ! 0 Now, by induction, V=W 0 = W=W 0 ⊕ W 00=W 0. However, this lifts to a short exact sequence 0 ! W 0 ! W 00 ! F ! 0 However, dim W 0 < dim W =) dim W 00 = dim W 0+1 < dim W +1 = dim V , so we can apply induction to get a one dimensional submodule complemen- tary to W 0 in W 00, say X, so that W 00 = W 0 ⊕ X. Thus, we get V=W 0 = W=W 0 ⊕ W 00=W 0 = W=W 0 ⊕ (W 0 ⊕ X)=W 0 =) V = W ⊕ X since W \ X = 0 and dim W + dim X = dim V . Thus, we can repeat this process until we get an irreducible W and appeal to the lemma above (2.13) to get the result.