Complexification (Lie group)
|Group theory → Lie groups|
In mathematics, the complexification or universal complexification of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that every continuous homomorphism of the original group into another complex Lie group extends compatibly to a complex analytic homomorphism between the complex Lie groups. The complexification, which always exists, is unique up to isomorphism. Its Lie algebra is a quotient of the complexification of the Lie algebra of the original group. They are isomorphic if the original group has a quotient by a discrete normal subgroup which is linear.
For compact Lie groups, the complexification, sometimes called the Chevalley complexification after Claude Chevalley, can be defined as the group of complex characters of the Hopf algebra of representative functions, i.e. the matrix coefficients of finite-dimensional representations of the group. In any finite-dimensional faithful unitary representation of the compact group it can be realized concretely as a closed subgroup of the complex general linear group. It consists of operators with polar decomposition g = u • exp iX, where u is a unitary operator in the compact group and X is a skew-adjoint operator in its Lie algebra. In this case the complexification is a complex algebraic group and its Lie algebra is the complexification of the Lie algebra of the compact Lie group.
- 1 Universal complexification
- 2 Chevalley complexification
- 3 Decompositions in the Chevalley complexification
- 4 Complex structures on homogeneous spaces
- 5 Noncompact real forms
- 6 See also
- 7 Notes
- 8 References
If G is a Lie group, a universal complexification is given by a complex Lie group GC and a continuous homomorphism φ: G → GC with the universal property that, if f: G → H is an arbitrary continuous homomorphism into a complex Lie group H, then there is a unique complex analytic homomorphism F: GC → H such that f = F ∘ φ.
Universal complexifications always exist and are unique up to a unique complex analytic isomorphism (preserving inclusion of the original group).
If G is connected with Lie algebra 𝖌, then its universal covering group G is simply connected. Let GC be the simply connected complex Lie group with Lie algebra 𝖌 ⊗ C. Let Φ: G → GC be the natural homomorphism and π:G → G the natural covering map. Then given a homomorphism f: G → H, there is a unique complex analytic homomorphism E: GC → H such that f ∘ π = E ∘ Φ. Let K be the intersection of the kernels of the homomorphisms E as f varies over all possibilities. Then K is a closed normal complex Lie subgroup of GC and the quotient group is a universal complexification. In particular if G is simply connected, its universal complexification is just GC.
For non-connected Lie groups G with identity component Go and component group Γ = G / Go, the extension
induces an extension
and the complex Lie group GC is a complexification of G.
The universal property implies that the universal complexification is unique up to complex analytic isomorphism.
If the original group is linear, so too is the universal complexification and the homomorphism between the two is an inclusion. Onishchik & Vinberg (1994) give an example of a connected real Lie group for which the homomorphism is not injective even at the Lie algebra level: they take the product of T by the universal covering group of SL(2,R) and quotient out by the discrete cyclic subgroup generated by an irrational rotation in the first factor and a generator of the center in the second.
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
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 GC 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 GC.
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 AN = EndG 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 GC of G consists of all operators g in GL(V) such that g⊗N commutes with AN 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 AN is generated by unitaries, an invertible operator g lies in GC if the unitary operator u and positive operator p in its polar decomposition g = u ⋅ p both lie in GC. 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 AN 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 GC and thus every finite-dimensional representation of G extends uniquely to GC. The extension is compatible with the polar decomposition. Finally the polar decomposition implies that G is a maximal compact subgroup of GC, 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
The decomposition derived from the polar decomposition
where 𝖌 is the Lie algebra of G, is called the Cartan decomposition of GC. 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.
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, ..., en an element g of GL(V) can be factorized in the form
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 ( eA eB ) 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
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 𝖙C ⊕ 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 𝖙C ⊕ 𝖓+ shows that there is an orthonormal basis e1, ..., en of V consisting of eigenvectors of 𝖙C with 𝖓+ acting as upper triangular matrices with zeros on the diagonal.
If N± and TC are the complex Lie groups corresponding to 𝖓+ and 𝖙C, then the Gauss decomposition states that the subset
is a direct product and consists of the elements in GC for which the principal minors are non-vanishing. It is open and dense. Moreover, if T denotes the maximal torus in U(V),
These results are an immediate consequence of the corresponding results for GL(V).
decomposing GC 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 GC.
The Bruhat decomposition is easy to prove for SL(n,C). Let B be the Borel subgroup of upper triangular matrices and TC the subgroup of diagonal matrices. So N(TC) / TC = Sn. 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(TC), 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 TC, 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 Nσ = N+ ∩ σ N− σ−1. In fact, if Mσ = N+ ∩ σ N+ σ−1, this follows from the identity
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 Nσ(k) and Mσ(k) to be the intersections with N+(k), it follows by decreasing induction on k that N+(k) = Nσ(k) ⋅ Mσ(k). Indeed, Nσ(k)N+(k + 1) and Mσ(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
Then Sp(n,C) is the fixed point subgroup of the involution θ(g) = A (gt)−1 A−1 of SL(2n,C). It leaves the subgroups N±, TC 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, TC 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 (gt)−1 B−1 in SL(n,C) where B = J.
gives a decomposition for GC 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.
In fact let e1, ..., en be an orthonormal basis of V and let g be an element in GL(V). Applying the Gram–Schmidt process to ge1, ..., gen, there is a unique orthonormal basis f1, ..., fn and positive constants ai such that
If k is the unitary taking (ei) to (fi), it follows that g−1k lies in the subgroup AN, where A is the subgroup of positive diagonal matrices with respect to (ei) and N is the subgroup of upper unitriangular matrices.
Using the notation for the Gauss decomposition, the subgroups in the Iwasawa decomposition for GC are defined by 
Since the decomposition is direct for GL(V), it is enough to check that GC = GAN. From the properties of the Iwasawa decomposition for GL(V), the map G × A × N is a diffeomorphism onto its image in GC, which is closed. On the other hand, the dimension of the image is the same as the dimension of GC, so it is also open. So GC = GAN because GC is connected.
Zhelobenko (1973) gives a method for explicitly computing the elements in the decomposition. For g in GC 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 TC 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 GC / 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 CG(X) of elements X in 𝖙. By a result of Hopf CG(x) is always connected: indeed any element y is along with S contained in some maximal torus, necessarily contained in CG(x).
Given an irreducible finite-dimensional representation Vλ 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 GC also acts on V and the stabilizer is a closed complex subgroup P containing TC. 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 𝖙C and root space vectors annihilating v, so that
The Lie algebra of H = P ∩ G is given by p ∩ 𝖌. By the Iwasawa decomposition GC = GAN. Since AN fixes C v, the G-orbit of v in the complex projective space of Vλ coincides with the GC orbit and
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 Po is the identity component of P, GC / P has GC / Po as a covering space, so that P = Po. The homogeneous space GC / 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
where Wλ 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 the complexified Lie algebra. These are the other real forms of GC.
Involutions of simply connected compact Lie groups
If G is a simply connected compact Lie group and σ is an automorphism of period 2, then the fixed point subgroup K = Gσ 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 Gσ. 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 Gσ 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 Tσ is connected. In fact the kernel of the exponential map from 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 CG(x)σ and let T be the identity component of CG(x, S). Then T is a maximal torus in G containing x and S. It is invariant under σ and the identity component of Tσ 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 Tσ contains S, lies in CG(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 , so it and therefore also are Abelian.
The proof is completed by showing that σ preserves a Weyl chamber associated with T. For then Tσ is connected so must equal S. Hence x lies in S. Since x was arbitrary, Gσ must therefore be connected.
To produce a Weyl chamber invariant under σ, note that there is no root space on which both x and S acted trivially, for this would contradict the fact that CG(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 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 GC. The map c(g) = (g*)−1 defines an automorphism of GC as a real Lie group with G as fixed point subgroup. It is conjugate-linear on and satisfies c2 = id. Such automorphisms of either GC or are called conjugations. Since GC is also simply connected any conjugation c1 on corresponds to a unique automorphism c1 of GC.
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 GC such that
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
for X, Y in .
To prove the existence of φ let ψ = c1c an automorphism of the complex group GC. On the Lie algebra level it defines a self-adjoint operator for the complex inner product
where B is the Killing form on . Thus ψ2 is a positive operator and an automorphism along with all its real powers. In particular take
Cartan decomposition in a real form
On GC 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 GC and therefore a Lie group. The involution σ acts on both G and G0. For the Lie algebra of G there is a decomposition
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 . The Lie algebra of G0 is given by
and the fixed point subgroup of σ is again K, so that G ∩ G0 = K. In G0, there is a Cartan decomposition
which is again a diffeomorphism onto the direct and corresponds to the polar decomposition of matrices. It is the restriction of the decomposition on GC. 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 in , A = exp is a toral subgroup such that σ(a) = a−1 on A; and any two such '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 and hence equals by maximality. A can be generated topologically by a single element exp X, so is the centralizer of X in . In the K-orbit of any element of there is an element Y such that (X,Ad k Y) is minimized at k = 1. Setting k = exp tT with T in , it follows that (X,[T,Y]) = 0 and hence [X,Y] = 0, so that Y must lie in . Thus is the union of the conjugates of . In particular some conjugate of X lies in any other choice of , which centralizes that conjugate; so by maximality the only possibilities are conjugates of .
A similar statements hold for the action of K on in . Morevoer, from the Cartan decomposition for G0, if A0 = exp , then
Iwasawa decomposition in a real form
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- Hochschild 1966
- Zhelobenko 1973, p. 28
- Bump 2001, pp. 202–203
- Zhelobenko 1973
- Gelfand & Naimark 1950, section 18, for SL(n,C)
- Bruhat 1956, p. 187 for SO(n,C) and Sp(n,C)
- Chevalley 1955 for complexifications of simple compact Lie groups
- Helgason 1978, pp. 403–406 for Harish-Chandra's method
- Humphreys 1981 for a treatment using algebraic groups
- Carter 1972, Chapter 8
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