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Real Forms of Complex Semi-Simple Lie Algebras

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Real Forms of Complex Semi-Simple Lie Algebras Real forms of complex semi-simple Lie algebras Gurkeerat Chhina We give a survey of real forms on complex semi-simple Lie algebras. This is a big piece of the puzzle in understanding real forms on complex semi-simple Lie groups. The material is from Onishchik and Vinberg Chapter 5 x1. 1 Real forms and real structures ∼ Recall from Max's talk that a real form h ⊂ g is subalgebra such that h ⊗R C = g. In other words, and R-basis of h is a C basis for g. The key example is gln(R) ⊂ gln(C) (this is the split real form). There is a 1-to-1 correspondence between real forms of a complex Lie algebra g and anti-linear involutive automorphisms σ. Given h ⊂ g, we produce the automorphism σ given by complex conjugation with respect to h. Given an anti-linear involutive automorphism σ we produce h = gσ = fx 2 g : σ(x) = xg. For simplicity, we call such an automorphism a real structure on g. −1 Proposition 1.1. Let σ; σ0 be real structures and φ 2 Aut(g). Then φσφ−1 is a real structure, and gφσφ = 0 φ(gσ). Furthermore, gσ =∼ gσ if and only if σ0 = φσφ−1 for some φ 2 Aut(g) This shows us that we need to classify involutions up to conjugacy in Aut(g). We can make similar definitions on the group side. Let H be a Lie subgroup of G considered as a real Lie group. H is a real form if the tangent algebra h ⊂ g is a real form, and G = HG0. A real structure is a real differentiable, involutive homomorphism S such that dS is a real structure on the tangent algebra g. Alternatively, it is an involutive, anti-holomorphic automorphism. If G is connected, then GS is a real form of G with tangent algebra gdS. The same holds for algebraic groups if we assume irreducibility. Our example ∗ n in this case is the torus (C ) , with the real structure (z1; : : : ; zn) 7! (z1;:::; zn), which determines the real ∗ n −1 −1 form (R ) . The other example we have is (z1; : : : ; zn) 7! (z1 ;:::; zn ) which determines the real form f(z1; : : : ; zn): jzij = 1g. As an example of why the theory of real forms of complex semi-simple Lie groups is more complicated than ∼ that of the Lie algebras, we consider the following example. It is a fact that π1(SL2(R)) = Z. Thus SL2 admits a simply connected cover, which we shall call G, but G cannot be embedded as a real form of any complex Lie group, G0. In fact, G does not admit a real algebraic group structure, nor is the identity com- ponent of an irreducible real algebraic group. Now back to the Lie algebras. Let g be a complex Lie algebra, and let gR be the same algebra considered over R. In gR, multiplication by i is a linear transformation I satisfying I2 = −Id (1) I[X; Y ] = [X; IY ] (2) This leads us to define a complex structure on a real Lie algebra g~ to be a linear transformation satisfying (1) and (2). This lets us turn g~ into a complex Lie algebra via (a + bi)X = aX + bIX. Of course, the linear transformation −I is also a complex structure, which allows us to construct another complex Lie algebra denoted g¯. An isomorphism g =∼ g¯ is a anti-linear automorphism of g. Thus if g has a real form, then g =∼ g¯. 1 Proposition 1.2. Any complex semi-simple Lie algebra admits a real form. To construct it, consider the canconical set of generators Xi;Yi;Hi for g. Consider the real subalgebra h generated by these. This is a real form with corresponding real structure τ, which sends each of Xi;Yi;Hi to itself. Proposition 1.3. If g is a complex semi-simple Lie algebra, then gR(C) =∼ g ⊕ g. If h is another complex semi-simple Lie algebra such that hR = gR, then h = g. First, by Prop 1.2, we know that g =∼ g¯. Then one considers the Lie algebra g ⊕ g¯, and shows that σ(x; y) = (y; x) is a real form, and that (x; x) 7! x is an isomorphism (g ⊕ g¯)σ =∼ gR. Theorem 1.4. A non-commutative real Lie algebra is simple if and only if it is isomorphic to a real form of a complex simple Lie algebra or the realification gR of a complex simple Lie algebra. Thus to classify real simple Lie algebras, one classifies complex simple Lie algebras (which is done via Dynkin diagrams), and then non-isomorphic real forms of them. We give one example of a genuinely non-trivial real form, before discussing an important real form in detail. Example 1.5. Consider the quadratic form 2 2 2 2 x1 + ··· + x` − x`+1 − · · · − x`+k (3) Let O`;k be the group of pseudo-orthogonal matrices (they preserve the quadratic form (3)), and let SO`;k E` 0 be the subgroup of unimodular matrices. Let I`;k = be the matrix of of the form (3), and 0 −Ek E` 0 2 ¯ let L`;k = so that L`;k = I`;k. The transformation S(A) = I`;kAI`;k is a real structure on 0 iEk −1 O`+k(C) and SO`+k(C). The corresponding real forms are L`;kO`;kL`;k and L`;kSO`;kL`;k. The real form X iY on the tangent algebra is L so L−1 which consists of matrices of the form for X; Y; Z real `;k `;k `;k −iY T Z matrices, XT = −X, ZT = −Z. 2 The compact form Onishchik and Vinberg define a Lie algebra over R to be compact if there is a positive definite invariant scalar product. The more useful definitions are given as exercises. Proposition 2.1. The killing form on a compact Lie algebra is negative-semi definite. A real Lie algebra is semi-simple and compact if and only if the killing form is negative-definite. Proposition 2.2. The tangent algebra of a compact Lie group is compact. Conversely, given a compact Lie algebra g, there is a connected compact Lie group with tangent algebra g. If g is semi-simple, we can take G = Intg. Definition 2.3. Given a complex Lie algebra g with real structure σ, The Hermitian form on g is hσ(x; y) = −κ(x; σ(y)) Proposition 2.4. Now let G be a connected complex semi-simple Lie group with tangent algebra g. Let S be a real structure on G and σ = dS the tangent real structure. Then the following are equivalent: 1. GS is compact. 2. gσ is compact. 3. hσ is positive definite. 2 Thus given G, if we can find such an s and show that hσ is positive definite, then we conclude that G has a compact real form. This turns out to be the case, using the construction of a Cartan involution (it turns out that any two Cartan involutions on a real semi-simple Lie algebra are equivalent). To start, we use the real form τ constructed in proposition 1.2. This is an anti-linear automorphism that sends each of Xi;Yi;Hi to itself. However, as we saw, this gives us the split form, and not the compact form like we want. Thus we have to modify it. Consider another choice of generators for g: {−Yi; −Xi; −Hig. There is a unique isomorphism of g that sends one set of generators to the other. Call this isomorphism µ. 2 We observe that µ = id. We have µ(Xi) = −Yi, µ(Yi) = −Xi, and µ(Hi) = −Hi. The real structure we want is σ = τµ = µτ, and it is known as the Cartan involution. It turns out that σ arises as dS for some S, and that hσ is positive definite. Hence we have the following: Theorem 2.5. Any connected complex semi-simple Lie group G has a compact real form H. The corre- sponding tangent algebra h is a compact real form of g. Example 2.6. The compact real form of sln is sun, and the corresponding lie group we produce is SUn - these are real lie algebras/groups. 3 Involutive automorphisms Recall that we wish to classify real forms of complex Lie algebras, and to do so, we need to classify the anti-linear involutive automorphisms up to conjugacy in Aut(g). For this reason we make the following definition. Definition 3.1. Let σ; τ be two real structures. The real forms gσ and gτ are said to be compatible if any of the following are satisfied: 1. στ = τσ 2. τ(gσ) = gσ 3. σ(gτ ) = gτ 4. θ = στ is an involutive automorphism. Let g be a semi-simple Lie algebra, with compact form u, produced by Section 2. Let τ be the corresponding real structure, and let σ be any other real structure. Consider θ = στ and the Hermetian form hτ . 2 Proposition 3.2. hτ (θx; y) = hτ (x; θy). Consequently, p = θ is a positive definite self-adjoint (with respect to hτ ) operator. Now we can define logs and exponentials of operators, so we can also define pt = exp(t log(p)) for any t 2 R. Because the image of exp will be a positive definite operator, we have that pt defines a one parameter family inside Aut(g). Proposition 3.3. σptσ = τptτ = p−t Proposition 3.4. Taking φ = p1=4, we have σ(φτφ−1) = (φτφ−1)σ.
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