Real Forms of Complex Semi-Simple Lie Algebras

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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 §1.

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 = {x ∈ g : σ(x) = x}. For simplicity, we call such an automorphism a real structure on g.

−1 Proposition 1.1. Let σ, σ0 be real structures and φ ∈ 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 ∈ 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 {(z1, . . . , zn): |zi| = 1}.

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, −Hi}. 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 corresponding 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 ∈ 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. Thus φ(u) is a real form compatible with gσ, and if ψ commutes with σ and τ, it commutes with φ as well. These propositions together give us the following theorem. Theorem 3.5. Let g be a complex semi-simple Lie algebra. Any real form of g is compatible with a compact real form. Any two compact real forms are conjugate. If a real form h is compatible with two compact real forms u1 and u2, then there is an automorphism φ ∈ Int(g) such that φ(u1) = u2 and φ(h) = h. Theorem 3.6. There is a bijection between the set of isomorphism classes of real forms of g and conjugacy classes of involutive automorphisms of g. Let σ be a real structure of g. Then by the previous theorem, there exists a compact real structure τ commuting with σ such that στ is an involutive automorphism. If τ 0 is another such compact real structure, then στ and στ 0 are conjugate in Aut(g).

3 To prove this theorem we need to show that the aforementioned map is surjective and injective. For sur- jectivity, let θ be an involution, and let τ be any compact real structure, and consider q = (θτ)2. Let 1/4 −1/4 τ1 = q τq , so that τ1 commutes with θ. τ1 is determined up to conjugacy by some φ commuting with θ. Then we can ﬁnd some σ such that θ = στ1 = τ1σ.

For injectivity, suppose θ1 = σ1τ1 and θ2 = σ2τ2. Because τ1 is conjugate to τ2, we can assume τ1 = τ2. −1 −1 −1 −1 −1 Suppose θ2 = ψθ1ψ . Then ψ τψ = φτφ , φθ1 = θ1φ. Then ψφσ1φ ψ = σ2.