Complexification (Lie group)

Chevalley complexification

Hopf algebra of matrix coefficients

If G is a compact Lie group, the *-algebra A of matrix coefficients of finite-dimensional unitary representations is a uniformly dense *-subalgebra of C(G), the *-algebra of complex-valued continuous functions on G. It is naturally a Hopf algebra with comultiplication given by

${\displaystyle {\displaystyle \mathrm{\Delta }f(g,h)=f(gh).}}$

The characters of A are the *-homomorphisms of A into C. They can be identified with the point evaluations f ↦ f(g) for g in G and the comultiplication allows the group structure on G to be recovered. The homomorphisms of A into C also form a group. It is a complex Lie group and can be identified with the complexification G_C of G. The *-algebra A is generated by the matrix coefficients of any faithful representation σ of G. It follows that σ defines a faithful complex analytic representation of G_C.

Invariant theory

The original approach of Chevalley (1946) to the complexification of a compact Lie group can be concisely stated within the language of classical invariant theory, described in Weyl (1946). Let G be a closed subgroup of the unitary group U(V) where V is a finite-dimensional complex inner product space. Its Lie algebra consists of all skew-adjoint operators X such that exp tX lies in G for all real t. Set W = V ⊕ C with the trivial action of G on the second summand. The group G acts on W^⊗N , with an element u acting as u^⊗N. The commutant (or centralizer algebra) is denoted by A_N = End_G W^⊗N. It is generated as a *-algebra by its unitary operators and its commutant is the *-algebra spanned by the operators u^⊗N. The complexification G_C of G consists of all operators g in GL(V) such that g^⊗N commutes with A_N and g acts trivially on the second summand in C. By definition it is a closed subgroup of GL(V). The defining relations (as a commutant) show that G is an algebraic subgroup. Its intersection with U(V) coincides with G, since it is a priori a larger compact group for which the irreducible representations stay irreducible and inequivalent when restricted to G. Since A_N is generated by unitaries, an invertible operator g lies in G_C if the unitary operator u and positive operator p in its polar decomposition g = u ⋅ p both lie in G_C. Thus u lies in G and the operator p can be written uniquely as p = exp T with T a self-adjoint operator. By the functional calculus for polynomial functions it follows that h^⊗N lies in the commutant of A_N if h = exp z T with z in C. In particular taking z purely imaginary, T must have the form iX with X in the Lie algebra of G. Since every finite-dimensional representation of G occurs as a direct summand of W^⊗N, it is left invariant by G_C and thus every finite-dimensional representation of G extends uniquely to G_C. The extension is compatible with the polar decomposition. Finally the polar decomposition implies that G is a maximal compact subgroup of G_C, since a strictly larger compact subgroup would contain all integer powers of a positive operator p, a closed infinite discrete subgroup.

Decompositions in the Chevalley complexification

Cartan decomposition

The decomposition derived from the polar decomposition

${\displaystyle {\displaystyle G_{\mathbf{C}}=G\cdot P=G\cdot \exp i\mathfrak{g},}}$

where 𝖌 is the Lie algebra of G, is called the Cartan decomposition of G_C. The exponential factor P is invariant under conjugation by G but is not a subgroup. The complexification is invariant under taking adjoints, since G consists of unitary operators and P of positive operators.

Gauss decomposition

The Gauss decomposition is a generalization of the LU decomposition for the general linear group and a specialization of the Bruhat decomposition. For GL(V) it states that with respect to a given orthonormal basis e₁, ..., e_n an element g of GL(V) can be factorized in the form

${\displaystyle {\displaystyle g=XDY}}$

with X lower unitriangular, Y upper unitriangular and D diagonal if and only if all the principal minors of g are non-vanishing. In this case X, Y and D are uniquely determined.

In fact Gaussian elimination shows there is a unique X such that X⁻¹ g is upper triangular.

The upper and lower unitriangular matrices, N₊ and N₋, are closed unipotent subgroups of GL(V). Their Lie algebras consist of upper and lower strictly triangular matrices. The exponential mapping is a polynomial mapping from the Lie algebra to the corresponding subgroup by nilpotence. The inverse is given by the logarithm mapping which by unipotence is also a polynomial mapping. In particular there is a correspondence between closed connected subgroups of N_± and subalgebras of their Lie algebras. The exponential map is onto in each case, since the polynomial function log ( e^A e^B ) lies in a given Lie subalgebra if A and B do and are sufficiently small.

The Gauss decomposition can be extended to complexifications of other closed connected subgroups G of U(V) by using the root decomposition to write the complexified Lie algebra as

${\displaystyle {\displaystyle {\mathfrak{g}}_{\mathbf{C}}={\mathfrak{n}}_{-}\oplus {\mathfrak{t}}_{\mathbf{C}}\oplus {\mathfrak{n}}_{+},}}$

where 𝖙 is the Lie algebra of a maximal torus T of G and 𝖓_± are the direct sum of the corresponding positive and negative root spaces. In the weight space decomposition of V as eigenspaces of T, 𝖙 acts as diagonally, 𝖓₊ acts as lowering operators and 𝖓₋ as raising operators. 𝖓_± are nilpotent Lie algebras acting as nilpotent operators; they are each other's adjoints on V. In particular T acts by conjugation of 𝖓₊, so that 𝖙_C ⊕ 𝖓₊ is a semidirect product of a nilpotent Lie algebra by an abelian Lie algebra.

By Engel's theorem, if 𝖆 ⊕ 𝖓 is a semidirect product, with 𝖆 abelian and 𝖓 nilpotent, acting on a finite-dimensional vector space W with operators in 𝖆 diagonalizable and operators in 𝖓 nilpotent, there is a vector w that is an eigenvector for 𝖆 and is annihilated by 𝖓. In fact it is enough to show there is a vector annihilated by 𝖓, which follows by induction on dim 𝖓, since the derived algebra 𝖓' annihilates a non-zero subspace of vectors on which 𝖓 / 𝖓' and 𝖆 act with the same hypotheses.

Applying this argument repeatedly to 𝖙_C ⊕ 𝖓₊ shows that there is an orthonormal basis e₁, ..., e_n of V consisting of eigenvectors of 𝖙_C with 𝖓₊ acting as upper triangular matrices with zeros on the diagonal.

If N_± and T_C are the complex Lie groups corresponding to 𝖓₊ and 𝖙_C, then the Gauss decomposition states that the subset

${\displaystyle {\displaystyle N_{-}{T}_{\mathbf{C}}{N}_{+}}}$

is a direct product and consists of the elements in G_C for which the principal minors are non-vanishing. It is open and dense. Moreover, if T denotes the maximal torus in U(V),

${\displaystyle {\displaystyle N_{\pm }={\mathbf{N}}_{\pm }\cap G_{\mathbf{C}},T_{\mathbf{C}}={\mathbf{T}}_{\mathbf{C}}\cap G_{\mathbf{C}}.}}$

These results are an immediate consequence of the corresponding results for GL(V).

Bruhat decomposition

If W = N_G(T) / T denotes the Weyl group of T and B denotes the Borel subgroup T_C N₊, the Gauss decomposition is also a consequence of the more precise Bruhat decomposition

${\displaystyle {\displaystyle G_{\mathbf{C}}=\bigcup _{\sigma \in W}B\sigma B,}}$

decomposing G_C into a disjoint union of double cosets of B. The complex dimension of a double coset BσB is determined by the length of σ as an element of W. The dimension is maximized at the Coxeter element and gives the unique open dense double coset. Its inverse conjugates B into the Borel subgroup of lower triangular matrices in G_C.

The Bruhat decomposition is easy to prove for SL(n,C). Let B be the Borel subgroup of upper triangular matrices and T_C the subgroup of diagonal matrices. So N(T_C) / T_C = S_n. For g in SL(n,C), take b in B so that bg maximizes the number of zeros appearing at the beginning of its rows. Because a multiple of one row can be added to another, each row has a different number of zeros in it. Multiplying by a matrix w in N(T_C), it follows that wbg lies in B. For uniqueness, if w₁b w₂ = b₀, then the entries of w₁w₂ vanish below the diagonal. So the product lies in T_C, proving uniqueness.

Chevalley (1955) showed that the expression of an element g as g = b₁σb₂ becomes unique if b₁ is restricted to lie in the upper unitriangular subgroup N_σ = N₊ ∩ σ N₋ σ⁻¹. In fact, if M_σ = N₊ ∩ σ N₊ σ⁻¹, this follows from the identity

${\displaystyle {\displaystyle N_{+}=N_{\sigma }\cdot M_{\sigma }.}}$

The group N₊ has a natural filtration by normal subgroups N₊(k) with zeros in the first k − 1 superdiagonals and the successive quotients are Abelian. Defining N_σ(k) and M_σ(k) to be the intersections with N₊(k), it follows by decreasing induction on k that N₊(k) = N_σ(k) ⋅ M_σ(k). Indeed, N_σ(k)N₊(k + 1) and M_σ(k)N₊(k + 1) are specified in N₊(k) by the vanishing of complementary entries (i, j) on the kth superdiagonal according to whether σ preserves the order i < j or not.

The Bruhat decomposition for the other classical simple groups can be deduced from the above decomposition using the fact that they are fixed point subgroups of folding automorphisms of SL(n,C). For Sp(n,C), let J be the n × n matrix with 1's on the antidiagonal and 0's elsewhere and set

${\displaystyle {\displaystyle A=\left(\begin{array}{cc}0& J\\ -J& 0\end{array}\right).}}$

Then Sp(n,C) is the fixed point subgroup of the involution θ(g) = A (g^t)⁻¹ A⁻¹ of SL(2n,C). It leaves the subgroups N_±, T_C and B invariant. If the basis elements are indexed by n, n−1, ..., 1, −1, ..., −n, then the Weyl group of Sp(n,C) consists of σ satisfying σ(j) = −j, i.e. commuting with θ. Analogues of B, T_C and N_± are defined by intersection with Sp(n,C), i.e. as fixed points of θ. The uniqueness of the decomposition g = nσb = θ(n) θ(σ) θ(b) implies the Bruhat decomposition for Sp(n,C).

The same argument works for SO(n,C). It can be realised as the fixed points of ψ(g) = B (g^t)⁻¹ B⁻¹ in SL(n,C) where B = J.

Iwasawa decomposition

The Iwasawa decomposition

${\displaystyle {\displaystyle G_{\mathbf{C}}=G\cdot A\cdot N}}$

gives a decomposition for G_C for which, unlike the Cartan decomposition, the direct factor A ⋅ N is a closed subgroup, but it is no longer invariant under conjugation by G. It is the semidirect product of the nilpotent subgroup N by the Abelian subgroup A.

For U(V) and its complexification GL(V), this decomposition can be derived as a restatement of the Gram–Schmidt orthonormalization process.

In fact let e₁, ..., e_n be an orthonormal basis of V and let g be an element in GL(V). Applying the Gram–Schmidt process to ge₁, ..., ge_n, there is a unique orthonormal basis f₁, ..., f_n and positive constants a_i such that

${\displaystyle {\displaystyle f_i=a_{i}g{e}_i+\sum _{j<i}{n}_{ji}g{e}_j.}}$

If k is the unitary taking (e_i) to (f_i), it follows that g⁻¹k lies in the subgroup AN, where A is the subgroup of positive diagonal matrices with respect to (e_i) and N is the subgroup of upper unitriangular matrices.

Using the notation for the Gauss decomposition, the subgroups in the Iwasawa decomposition for G_C are defined by

${\displaystyle {\displaystyle A=\exp i\mathfrak{t}=\mathbf{A}\cap G_{\mathbf{C}},N=\exp {\mathfrak{n}}_{+}=\mathbf{N}\cap G_{\mathbf{C}}.}}$

Since the decomposition is direct for GL(V), it is enough to check that G_C = GAN. From the properties of the Iwasawa decomposition for GL(V), the map G × A × N is a diffeomorphism onto its image in G_C, which is closed. On the other hand, the dimension of the image is the same as the dimension of G_C, so it is also open. So G_C = GAN because G_C is connected.

Zhelobenko (1973) gives a method for explicitly computing the elements in the decomposition. For g in G_C set h = g*g. This is a positive self-adjoint operator so its principal minors do not vanish. By the Gauss decomposition, it can therefore be written uniquely in the form h = XDY with X in N₋, D in T_C and Y in N₊. Since h is self-adjoint, uniqueness forces Y = X*. Since it is also positive D must lie in A and have the form D = exp iT for some unique T in 𝖙. Let a = exp iT/2 be its unique square root in A. Set n = Y and k = g n⁻¹ a⁻¹. Then k is unitary, so is in G, and g = kan.

Complex structures on homogeneous spaces

The Iwasawa decomposition can be used to describe complex structures on the G-orbits in complex projective space of highest weight vectors of finite-dimensional irreducible representations of G. In particular the identification between G / T and G_C / B can be used to formulate the Borel–Weil theorem. It states that each irreducible representation of G can be obtained by holomorphic induction from a character of T, or equivalently that it is realized in the space of sections of a holomorphic line bundle on G / T.

The closed connected subgroups of G containing T are described by Borel–de Siebenthal theory. They are exactly the centralizers of tori S ⊆ T. Since every torus is generated topologically by a single element x, these are the same as centralizers C_G(X) of elements X in 𝖙. By a result of Hopf C_G(x) is always connected: indeed any element y is along with S contained in some maximal torus, necessarily contained in C_G(x).

Given an irreducible finite-dimensional representation V_λ with highest weight vector v of weight λ, the stabilizer of C v in G is a closed subgroup H. Since v is an eigenvector of T, H contains T. The complexification G_C also acts on V and the stabilizer is a closed complex subgroup P containing T_C. Since v is annihilated by every raising operator corresponding to a positive root α, P contains the Borel subgroup B. The vector v is also a highest weight vector for the copy of sl₂ corresponding to α, so it is annihilated by the lowering operator generating 𝖌_−α if (λ, α) = 0. The Lie algebra p of P is the direct sum of 𝖙_C and root space vectors annihilating v, so that

${\displaystyle {\displaystyle \mathfrak{p}=\mathfrak{b}\oplus \underset{(\alpha ,\lambda )=0}{⨁}{\mathfrak{g}}_{-\alpha }.}}$

The Lie algebra of H = P ∩ G is given by p ∩ 𝖌. By the Iwasawa decomposition G_C = GAN. Since AN fixes C v, the G-orbit of v in the complex projective space of V_λ coincides with the G_C orbit and

${\displaystyle {\displaystyle G/H=G_{\mathbf{C}}/P.}}$

In particular

${\displaystyle {\displaystyle G/T=G_{\mathbf{C}}/B.}}$

Using the identification of the Lie algebra of T with its dual, H equals the centralizer of λ in G, and hence is connected. The group P is also connected. In fact the space G / H is simply connected, since it can be written as the quotient of the (compact) universal covering group of the compact semisimple group G / Z by a connected subgroup, where Z is the center of G. If P^o is the identity component of P, G_C / P has G_C / P^o as a covering space, so that P = P^o. The homogeneous space G_C / P has a complex structure, because P is a complex subgroup. The orbit in complex projective space is closed in the Zariski topology by Chow's theorem, so is a smooth projective variety. The Borel–Weil theorem and its generalizations are discussed in this context in Serre (1954), Helgason (1994), Duistermaat & Kolk (2000) and Sepanski (2007).

The parabolic subgroup P can also be written as a union of double cosets of B

${\displaystyle {\displaystyle P=\bigcup _{\sigma \in W_{\lambda }}B\sigma B,}}$

where W_λ is the stabilizer of λ in the Weyl group W. It is generated by the reflections corresponding to the simple roots orthogonal to λ.

Noncompact real forms

There are other closed subgroups of the complexification of a compact connected Lie group G which have the same complexified Lie algebra. These are the other real forms of G_C.

Involutions of simply connected compact Lie groups

If G is a simply connected compact Lie group and σ is an automorphism of order 2, then the fixed point subgroup K = G^σ is automatically connected. (In fact this is true for any automorphism of G, as shown for inner automorphisms by Steinberg and in general by Borel.)

This can be seen most directly when the involution σ corresponds to a Hermitian symmetric space. In that case σ is inner and implemented by an element in a one-parameter subgroup exp tT contained in the center of G^σ. The innerness of σ implies that K contains a maximal torus of G, so has maximal rank. On the other hand, the centralizer of the subgroup generated by the torus S of elements exp tT is connected, since if x is any element in K there is a maximal torus containing x and S, which lies in the centralizer. On the other hand, it contains K since S is central in K and is contained in K since z lies in S. So K is the centralizer of S and hence connected. In particular K contains the center of G.

For a general involution σ, the connectedness of G^σ can be seen as follows.

The starting point is the Abelian version of the result: if T is a maximal torus of a simply connected group G and σ is an involution leaving invariant T and a choice of positive roots (or equivalently a Weyl chamber), then the fixed point subgroup T^σ is connected. In fact the kernel of the exponential map from ${\displaystyle \mathfrak{t}}$ onto T is a lattice Λ with a Z-basis indexed by simple roots, which σ permutes. Splitting up according to orbits, T can be written as a product of terms T on which σ acts trivially or terms T² where σ interchanges the factors. The fixed point subgroup just corresponds to taking the diagonal subgroups in the second case, so is connected.

Now let x be any element fixed by σ, let S be a maximal torus in C_G(x)^σ and let T be the identity component of C_G(x, S). Then T is a maximal torus in G containing x and S. It is invariant under σ and the identity component of T^σ is S. In fact since x and S commute, they are contained in a maximal torus which, because it is connected, must lie in T. By construction T is invariant under σ. The identity component of T^σ contains S, lies in C_G(x)^σ and centralizes S, so it equals S. But S is central in T, to T must be Abelian and hence a maximal torus. For σ acts as multiplication by −1 on the Lie algebra ${\displaystyle \mathfrak{t}\ominus \mathfrak{s}}$, so it and therefore also ${\displaystyle \mathfrak{t}}$ are Abelian.

The proof is completed by showing that σ preserves a Weyl chamber associated with T. For then T^σ is connected so must equal S. Hence x lies in S. Since x was arbitrary, G^σ must therefore be connected.

To produce a Weyl chamber invariant under σ, note that there is no root space ${\displaystyle {\mathfrak{g}}_{\alpha }}$ on which both x and S acted trivially, for this would contradict the fact that C_G(x, S) has the same Lie algebra as T. Hence there must be an element s in S such that t = xs acts non-trivially on each root space. In this case t is a regular element of T—the identity component of its centralizer in G equals T. There is a unique Weyl alcove A in ${\displaystyle \mathfrak{t}}$ such that t lies in exp A and 0 lies in the closure of A. Since t is fixed by σ, the alcove is left invariant by σ and hence so also is the Weyl chamber C containing it.

Conjugations on the complexification

Let G be a simply connected compact Lie group with complexification G_C. The map c(g) = (g*)⁻¹ defines an automorphism of G_C as a real Lie group with G as fixed point subgroup. It is conjugate-linear on ${\displaystyle {\mathfrak{g}}_{\mathbf{C}}}$ and satisfies c² = id. Such automorphisms of either G_C or ${\displaystyle {\mathfrak{g}}_{\mathbf{C}}}$ are called conjugations. Since G_C is also simply connected any conjugation c₁ on ${\displaystyle {\mathfrak{g}}_{\mathbf{C}}}$ corresponds to a unique automorphism c₁ of G_C.

The classification of conjugations c₀ reduces to that of involutions σ of G because given a c₁ there is an automorphism φ of the complex group G_C such that

${\displaystyle {\displaystyle c_0=\phi \circ c_1\circ {\phi }^{-1}}}$

commutes with c. The conjugation c₀ then leaves G invariant and restricts to an involutive automorphism σ. By simple connectivity the same is true at the level of Lie algebras. At the Lie algebra level c₀ can be recovered from σ by the formula

${\displaystyle {\displaystyle c_0(X+iY)=\sigma (X)-i\sigma (Y)}}$

for X, Y in ${\displaystyle \mathfrak{g}}$.

To prove the existence of φ let ψ = c₁c an automorphism of the complex group G_C. On the Lie algebra level it defines a self-adjoint operator for the complex inner product

${\displaystyle {\displaystyle (X,Y)=-B(X,c(Y)),}}$

where B is the Killing form on ${\displaystyle {\mathfrak{g}}_{\mathbf{C}}}$. Thus ψ² is a positive operator and an automorphism along with all its real powers. In particular take

${\displaystyle {\displaystyle \phi =({\psi }^2{)}^{1/4}}}$

It satisfies

${\displaystyle {\displaystyle c_{0}c=\phi c_1{\phi }^{-1}c=\phi c{c}_1\phi =({\psi }^2{)}^{1/2}{\psi }^{-1}={\phi }^{-1}c{c}_1{\phi }^{-1}=c\phi c_1{\phi }^{-1}=c{c}_0.}}$

Cartan decomposition in a real form

For the complexification G_C, the Cartan decomposition is described above. Derived from the polar decomposition in the complex general linear group, it gives a diffeomorphism

${\displaystyle {\displaystyle G_{\mathbf{C}}=G\cdot \exp i\mathfrak{g}=G\cdot P=P\cdot G.}}$

On G_C there is a conjugation operator c corresponding to G as well as an involution σ commuting with c. Let c₀ = c σ and let G₀ be the fixed point subgroup of c. It is closed in the matrix group G_C and therefore a Lie group. The involution σ acts on both G and G₀. For the Lie algebra of G there is a decomposition

${\displaystyle {\displaystyle \mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}}}$

into the +1 and −1 eigenspaces of σ. The fixed point subgroup K of σ in G is connected since G is simply connected. Its Lie algebra is the +1 eigenspace ${\displaystyle \mathfrak{k}}$. The Lie algebra of G₀ is given by

${\displaystyle {\displaystyle \mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}}}$

and the fixed point subgroup of σ is again K, so that G ∩ G₀ = K. In G₀, there is a Cartan decomposition

${\displaystyle {\displaystyle G_0=K\cdot \exp i\mathfrak{p}=K\cdot P_0=P_0\cdot K}}$

which is again a diffeomorphism onto the direct and corresponds to the polar decomposition of matrices. It is the restriction of the decomposition on G_C. The product gives a diffeomorphism onto a closed subset of G₀. To check that it is surjective, for g in G₀ write g = u ⋅ p with u in G and p in P. Since c₀ g = g, uniqueness implies that σu = u and σp = p⁻¹. Hence u lies in K and p in P₀.

The Cartan decomposition in G₀ shows that G₀ is connected, simply connected and noncompact, because of the direct factor P₀. Thus G₀ is a noncompact real semisimple Lie group.

Moreover, given a maximal Abelian subalgebra ${\displaystyle \mathfrak{a}}$ in ${\displaystyle \mathfrak{p}}$, A = exp ${\displaystyle \mathfrak{a}}$ is a toral subgroup such that σ(a) = a⁻¹ on A; and any two such ${\displaystyle \mathfrak{a}}$'s are conjugate by an element of K. The properties of A can be shown directly. A is closed because the closure of A is a toral subgroup satisfying σ(a) = a⁻¹, so its Lie algebra lies in ${\displaystyle \mathfrak{m}}$ and hence equals ${\displaystyle \mathfrak{a}}$ by maximality. A can be generated topologically by a single element exp X, so ${\displaystyle \mathfrak{a}}$ is the centralizer of X in ${\displaystyle \mathfrak{m}}$. In the K-orbit of any element of ${\displaystyle \mathfrak{m}}$ there is an element Y such that (X,Ad k Y) is minimized at k = 1. Setting k = exp tT with T in ${\displaystyle \mathfrak{k}}$, it follows that (X,[T,Y]) = 0 and hence [X,Y] = 0, so that Y must lie in ${\displaystyle \mathfrak{a}}$. Thus ${\displaystyle \mathfrak{m}}$ is the union of the conjugates of ${\displaystyle \mathfrak{a}}$. In particular some conjugate of X lies in any other choice of ${\displaystyle \mathfrak{a}}$, which centralizes that conjugate; so by maximality the only possibilities are conjugates of ${\displaystyle \mathfrak{a}}$.

A similar statements hold for the action of K on ${\displaystyle {\mathfrak{a}}_0=i\mathfrak{a}}$ in ${\displaystyle {\mathfrak{p}}_0}$. Morevoer, from the Cartan decomposition for G₀, if A₀ = exp ${\displaystyle {\mathfrak{a}}_0}$, then

${\displaystyle {\displaystyle G_0=K{A}_{0}K.}}$

Iwasawa decomposition in a real form