# Complexification (Lie group)

__: Universal construction of a complex Lie group from a real Lie group__

**Short description**Group theory → Lie groupsLie groups |
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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 unique 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.

## Universal complexification

### Definition

If *G* is a Lie group, a **universal complexification** is given by a complex Lie group *G*_{C} and a continuous homomorphism *φ*: *G* → *G*_{C} 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*: *G*_{C} → *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).

### Existence

If *G* is connected with Lie algebra 𝖌, then its universal covering group **G** is simply connected. Let **G**_{C} be the simply connected complex Lie group with Lie algebra 𝖌_{C} = 𝖌 ⊗ **C**, let Φ: **G** → **G**_{C} be the natural homomorphism (the unique morphism such that Φ_{*}: 𝖌 ↪ 𝖌 ⊗ **C** is the canonical inclusion) and suppose *π*: **G** → *G* is the universal covering map, so that ker *π* is the fundamental group of *G*. We have the inclusion Φ(ker *π*) ⊂ Z(**G**_{C}), which follows from the fact that the kernel of the adjoint representation of **G**_{C} equals its centre, combined with the equality

- [math]\displaystyle{ (C_{\Phi(k)})_*\circ \Phi_* = \Phi_* \circ (C_k)_* = \Phi_* }[/math]

which holds for any *k* ∈ ker *π*. Denoting by Φ(ker *π*)^{*} the smallest closed normal Lie subgroup of **G**_{C} that contains Φ(ker *π*), we must now also have the inclusion Φ(ker *π*)^{*} ⊂ Z(**G**_{C}). We define the universal complexification of *G* as

- [math]\displaystyle{ G_{\mathbf C}=\frac{\mathbf G_{\mathbf C}}{\Phi(\ker \pi)^*}. }[/math]

In particular, if *G* is simply connected, its universal complexification is just **G**_{C}.^{[1]}

The map *φ*: *G* → *G*_{C} is obtained by passing to the quotient. Since *π* is a surjective submersion, smoothness of the map *π*_{C} ∘ Φ implies smoothness of *φ*.

For non-connected Lie groups *G* with identity component *G*^{o} and component group Γ = *G* / *G*^{o}, the extension

- [math]\displaystyle{ \{1\} \rightarrow G^o \rightarrow G \rightarrow \Gamma \rightarrow \{1\} }[/math]

induces an extension

- [math]\displaystyle{ \{1\} \rightarrow (G^o)_{\mathbf{C}} \rightarrow G_{\mathbf{C}} \rightarrow \Gamma \rightarrow \{1\} }[/math]

and the complex Lie group *G*_{C} is a complexification of *G*.^{[2]}

#### Proof of the universal property

The map *φ*: *G* → *G*_{C} indeed possesses the universal property which appears in the above definition of complexification. The proof of this statement naturally follows from considering the following instructive diagram.

Here, [math]\displaystyle{ f\colon G\rightarrow H }[/math] is an arbitrary smooth homomorphism of Lie groups with a complex Lie group as the codomain.

For simplicity, we assume [math]\displaystyle{ G }[/math] is connected. To establish the existence of [math]\displaystyle{ F }[/math], we first naturally extend the morphism of Lie algebras [math]\displaystyle{ f_*\colon \mathfrak g\rightarrow \mathfrak h }[/math] to the unique morphism [math]\displaystyle{ \overline f_*\colon \mathfrak g_{\mathbf C}\rightarrow \mathfrak h }[/math] of complex Lie algebras. Since [math]\displaystyle{ \mathbf G_{\mathbf C} }[/math] is simply connected, Lie's second fundamental theorem now provides us with a unique complex analytic morphism [math]\displaystyle{ \overline F\colon \mathbf G_{\mathbf C}\rightarrow H }[/math] between complex Lie groups, such that [math]\displaystyle{ (\overline F)_*=\overline f_* }[/math]. We define [math]\displaystyle{ F\colon G_{\mathbf C}\rightarrow H }[/math] as the map induced by [math]\displaystyle{ \overline F }[/math], that is: [math]\displaystyle{ F(g\,\Phi(\ker \pi)^*)=\overline F(g) }[/math] for any [math]\displaystyle{ g\in\mathbf G_{\mathbf C} }[/math]. To show well-definedness of this map (i.e. [math]\displaystyle{ \Phi(\ker\pi)^*\subset \ker \overline F }[/math]), consider the derivative of the map [math]\displaystyle{ \overline F\circ \Phi }[/math]. For any [math]\displaystyle{ v\in T_e \mathbf G\cong \mathfrak g }[/math], we have

- [math]\displaystyle{ (\overline F)_*\Phi_*v=(\overline F)_*(v\otimes 1)=f_*\pi_*v }[/math],

which (by simple connectedness of [math]\displaystyle{ \mathbf G }[/math]) implies [math]\displaystyle{ \overline F\circ\Phi=f\circ\pi }[/math]. This equality finally implies [math]\displaystyle{ \Phi(\ker\pi)\subset \ker \overline F }[/math], and since [math]\displaystyle{ \ker \overline F }[/math] is a closed normal Lie subgroup of [math]\displaystyle{ \mathbf G_{\mathbf C} }[/math], we also have [math]\displaystyle{ \Phi(\ker \pi)^*\subset \ker \overline F }[/math]. Since [math]\displaystyle{ \pi_{\mathbb C} }[/math] is a complex analytic surjective submersion, the map [math]\displaystyle{ F }[/math] is complex analytic since [math]\displaystyle{ \overline F }[/math] is. The desired equality [math]\displaystyle{ F\circ\varphi=f }[/math] is imminent.

To show uniqueness of [math]\displaystyle{ F }[/math], suppose that [math]\displaystyle{ F_1,F_2 }[/math] are two maps with [math]\displaystyle{ F_1\circ\varphi=F_2\circ\varphi=f }[/math]. Composing with [math]\displaystyle{ \pi }[/math] from the right and differentiating, we get [math]\displaystyle{ (F_1)_*(\pi_{\mathbf C})_*\Phi_*=(F_2)_*(\pi_{\mathbf C})_*\Phi_* }[/math], and since [math]\displaystyle{ \Phi_* }[/math] is the inclusion [math]\displaystyle{ \mathfrak g\hookrightarrow \mathfrak g_{\mathbf C} }[/math], we get [math]\displaystyle{ (F_1)_*(\pi_{\mathbf C})_*=(F_2)_*(\pi_{\mathbf C})_* }[/math]. But [math]\displaystyle{ \pi_{\mathbf C} }[/math] is a submersion, so [math]\displaystyle{ (F_1)_*=(F_2)_* }[/math], thus connectedness of [math]\displaystyle{ G }[/math] implies [math]\displaystyle{ F_1=F_2 }[/math].

### Uniqueness

The universal property implies that the universal complexification is unique up to complex analytic isomorphism.

### Injectivity

If the original group is linear, so too is the universal complexification and the homomorphism between the two is an inclusion.^{[3]} (Onishchik Vinberg) 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.

### Basic examples

The following isomorphisms of complexifications of Lie groups with known Lie groups can be constructed directly from the general construction of the complexification.

- The complexification of the special unitary group of 2x2 matrices is

- [math]\displaystyle{ \mathrm{SU}(2)_{\mathbf C}\cong \mathrm{SL}(2,\mathbf C) }[/math].

- This follows from the isomorphism of Lie algebras
- [math]\displaystyle{ \mathfrak{su}(2)_{\mathbf C}\cong \mathfrak{sl}(2,\mathbf C) }[/math],

- together with the fact that [math]\displaystyle{ \mathrm{SU}(2) }[/math] is simply connected.

- The complexification of the special linear group of 2x2 matrices is

- [math]\displaystyle{ \mathrm{SL}(2,\mathbf C)_{\mathbf C}\cong \mathrm{SL}(2,\mathbf C)\times \mathrm{SL}(2,\mathbf C) }[/math].

- This follows from the isomorphism of Lie algebras
- [math]\displaystyle{ \mathfrak{sl}(2,\mathbf C)_{\mathbf C}\cong \mathfrak{sl}(2,\mathbf C) \oplus \mathfrak{sl}(2,\mathbf C) }[/math],

- together with the fact that [math]\displaystyle{ \mathrm{SL}(2,\mathbf C) }[/math] is simply connected.

- The complexification of the special orthogonal group of 3x3 matrices is

- [math]\displaystyle{ \mathrm{SO}(3)_{\mathbf C}\cong \frac{\mathrm{SL}(2,\mathbf C)}{\mathbf Z_2}\cong \mathrm{SO}^+(1,3) }[/math],

- where [math]\displaystyle{ \mathrm{SO}^+(1,3) }[/math] denotes the proper orthochronous Lorentz group. This follows from the fact that [math]\displaystyle{ \mathrm{SU}(2) }[/math] is the universal (double) cover of [math]\displaystyle{ \mathrm{SO}(3) }[/math], hence:
- [math]\displaystyle{ \mathfrak{so}(3)_{\mathbf C}\cong \mathfrak{su}(2)_{\mathbf C} \cong\mathfrak{sl}(2,\mathbf C) }[/math].

- We also use the fact that [math]\displaystyle{ \mathrm{SL}(2,\mathbf C) }[/math] is the universal (double) cover of [math]\displaystyle{ \mathrm{SO}^+(1,3) }[/math].

- The complexification of the proper orthochronous Lorentz group is

- [math]\displaystyle{ \mathrm{SO}^+(1,3)_{\mathbf C}\cong \frac{\mathrm{SL}(2,\mathbf C)\times \mathrm{SL}(2,\mathbf C)}{\mathbf Z_2} }[/math].

- This follows from the same isomorphism of Lie algebras as in the second example, again using the universal (double) cover of the proper orthochronous Lorentz group.

- The complexification of the special orthogonal group of 4x4 matrices is

- [math]\displaystyle{ \mathrm{SO}(4)_{\mathbf C}\cong \frac{\mathrm{SL}(2,\mathbf C)\times \mathrm{SL}(2,\mathbf C)}{\mathbf Z_2} }[/math].

- This follows from the fact that [math]\displaystyle{ \mathrm{SU}(2)\times\mathrm{SU}(2) }[/math] is the universal (double) cover of [math]\displaystyle{ \mathrm{SO}(4) }[/math], hence [math]\displaystyle{ \mathfrak{so}(4)\cong \mathfrak{su}(2)\oplus\mathfrak{su}(2) }[/math] and so [math]\displaystyle{ \mathfrak{so}(4)_{\mathbf C}\cong \mathfrak{sl}(2,\mathbf C)\oplus\mathfrak{sl}(2,\mathbf C) }[/math].

The last two examples show that Lie groups with isomorphic complexifications may not be isomorphic. Furthermore, the complexifications of Lie groups [math]\displaystyle{ \mathrm{SU}(2) }[/math] and [math]\displaystyle{ \mathrm{SL}(2,\mathbf C) }[/math] show that complexification is not an idempotent operation, i.e. [math]\displaystyle{ (G_{\mathbf C})_{\mathbf C}\not\cong G_{\mathbf C} }[/math] (this is also shown by complexifications of [math]\displaystyle{ \mathrm{SO}(3) }[/math] and [math]\displaystyle{ \mathrm{SO}^+(1,3) }[/math]).

## 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

- [math]\displaystyle{ \displaystyle{\Delta f(g,h)= f(gh).} }[/math]

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}.^{[4]}

### 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.^{[5]}

## Decompositions in the Chevalley complexification

### Cartan decomposition

The decomposition derived from the polar decomposition

- [math]\displaystyle{ \displaystyle{G_{\mathbf{C}} = G\cdot P =G \cdot \exp i\mathfrak{g},} }[/math]

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 *e*_{1}, ..., *e*_{n} an element *g* of GL(*V*) can be factorized in the form

- [math]\displaystyle{ \displaystyle{g=XDY} }[/math]

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.^{[6]}

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.^{[7]}

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^{[8]}

- [math]\displaystyle{ \displaystyle{\mathfrak{g}_{\mathbf{C}} = \mathfrak{n}_- \oplus \mathfrak{t}_{\mathbf{C}} \oplus \mathfrak{n}_+,} }[/math]

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 *e*_{1}, ..., *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

- [math]\displaystyle{ \displaystyle{N_- T_{\mathbf{C}} N_+} }[/math]

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*),

- [math]\displaystyle{ \displaystyle{N_\pm=\mathbf{N}_\pm\cap G_{\mathbf{C}},\,\,\, T_{\mathbf{C}} = \mathbf{T}_{\mathbf{C}}\cap G_{\mathbf{C}}.} }[/math]

These results are an immediate consequence of the corresponding results for GL(*V*).^{[9]}

### 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**

- [math]\displaystyle{ \displaystyle{G_{\mathbf{C}} =\bigcup_{\sigma\in W} B\sigma B,} }[/math]

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}.^{[10]}

The Bruhat decomposition is easy to prove for SL(*n*,**C**).^{[11]} 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 *w*_{1}*b* *w*_{2} = *b*_{0}, then the entries of *w*_{1}*w*_{2} 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* = *b*_{1}*σb*_{2} becomes unique if *b*_{1} 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

- [math]\displaystyle{ \displaystyle{N_+=N_\sigma\cdot M_\sigma.} }[/math]

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 *k*th superdiagonal according to whether σ preserves the order *i* < *j* or not.^{[12]}

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**).^{[13]} For Sp(*n*,**C**), let *J* be the *n* × *n* matrix with 1's on the antidiagonal and 0's elsewhere and set

- [math]\displaystyle{ \displaystyle{A=\begin{pmatrix} 0 & J\\ -J & 0\end{pmatrix}.} }[/math]

Then Sp(*n*,**C**) is the fixed point subgroup of the involution *θ*(*g*) = *A* (*g*^{t})^{−1} *A*^{−1} of SL(2*n*,**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

- [math]\displaystyle{ \displaystyle{G_{\mathbf{C}} = G\cdot A \cdot N} }[/math]

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.^{[14]}

In fact let *e*_{1}, ..., *e*_{n} be an orthonormal basis of *V* and let *g* be an element in GL(*V*). Applying the Gram–Schmidt process to *ge*_{1}, ..., *ge*_{n}, there is a unique orthonormal basis *f*_{1}, ..., *f*_{n} and positive constants *a*_{i} such that

- [math]\displaystyle{ \displaystyle{f_i= a_i ge_i + \sum_{j\lt i} n_{ji} ge_j.} }[/math]

If *k* is the unitary taking (*e*_{i}) to (*f*_{i}), it follows that *g*^{−1}*k* 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.^{[15]}

Using the notation for the Gauss decomposition, the subgroups in the Iwasawa decomposition for *G*_{C} are defined by
^{[16]}

- [math]\displaystyle{ \displaystyle{A=\exp i\mathfrak{t} = \mathbf{A} \cap G_{\mathbf{C}}, \,\,\, N=\exp \mathfrak{n}_+=\mathbf{N} \cap G_{\mathbf{C}}.} }[/math]

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.^{[17]}

(Zhelobenko 1973) gives a method for explicitly computing the elements in the decomposition.^{[18]} 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 **sl**_{2} 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

- [math]\displaystyle{ \displaystyle{\mathfrak{p}=\mathfrak{b}\oplus \bigoplus_{(\alpha,\lambda)=0} \mathfrak{g}_{-\alpha}.} }[/math]

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

- [math]\displaystyle{ \displaystyle{G/H=G_{\mathbf{C}}/P.} }[/math]

In particular

- [math]\displaystyle{ \displaystyle{G/T=G_{\mathbf{C}}/B.} }[/math]

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*.^{[19]} 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) and (Sepanski 2007).

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

- [math]\displaystyle{ \displaystyle{P=\bigcup_{\sigma\in W_\lambda} B\sigma B,} }[/math]

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.^{[20]}

## 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}.^{[21]}

### 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.) ^{[22]}

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*.^{[23]}

For a general involution σ, the connectedness of *G*^{σ} can be seen as follows.^{[24]}

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 [math]\displaystyle{ \mathfrak{t} }[/math] 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 **T**^{2} 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 [math]\displaystyle{ \mathfrak{t}\ominus \mathfrak{s} }[/math], so it and therefore also [math]\displaystyle{ \mathfrak{t} }[/math] 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 [math]\displaystyle{ \mathfrak{g}_\alpha }[/math] 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 [math]\displaystyle{ \mathfrak{t} }[/math] 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 [math]\displaystyle{ \mathfrak{g}_{\mathbf{C}} }[/math] and satisfies *c*^{2} = id. Such automorphisms of either *G*_{C} or [math]\displaystyle{ \mathfrak{g}_{\mathbf{C}} }[/math] are called **conjugations**.
Since *G*_{C} is also simply connected any conjugation *c*_{1} on [math]\displaystyle{ \mathfrak{g}_{\mathbf{C}} }[/math] corresponds to a unique automorphism *c*_{1} of *G*_{C}.

The classification of conjugations *c*_{0} reduces to that of involutions σ of *G* because
given a *c*_{1} there is an automorphism φ of the complex group *G*_{C} such that

- [math]\displaystyle{ \displaystyle{c_0=\varphi\circ c_1\circ \varphi^{-1}} }[/math]

commutes with *c*. The conjugation *c*_{0} 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 *c*_{0} can be recovered from σ by the formula

- [math]\displaystyle{ \displaystyle{c_0(X+iY)=\sigma(X)- i\sigma(Y)} }[/math]

for *X*, *Y* in [math]\displaystyle{ \mathfrak{g} }[/math].

To prove the existence of φ let ψ = *c*_{1}*c* an automorphism of the complex group *G*_{C}. On the Lie algebra level it defines a self-adjoint operator for the complex inner product

- [math]\displaystyle{ \displaystyle{(X,Y)=-B(X,c(Y)),} }[/math]

where *B* is the Killing form on [math]\displaystyle{ \mathfrak{g}_{\mathbf{C}} }[/math]. Thus ψ^{2} is a positive operator and an automorphism along with all its real powers. In particular take

- [math]\displaystyle{ \displaystyle{\varphi=(\psi^2)^{1/4}} }[/math]

It satisfies

- [math]\displaystyle{ \displaystyle{c_0c=\varphi c_1 \varphi^{-1} c=\varphi cc_1 \varphi=(\psi^2)^{1/2} \psi^{-1} =\varphi^{-1} cc_1 \varphi^{-1}=c \varphi c_1\varphi^{-1}=cc_0.} }[/math]

### 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

- [math]\displaystyle{ \displaystyle{G_{\mathbf{C}} = G\cdot \exp i\mathfrak{g} = G\cdot P = P\cdot G.} }[/math]

On *G*_{C} there is a conjugation operator *c* corresponding to *G* as well as an involution σ commuting with *c*. Let *c*_{0} = *c* σ and let *G*_{0} 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 *G*_{0}. For the Lie algebra of *G* there is a decomposition

- [math]\displaystyle{ \displaystyle{\mathfrak{g}=\mathfrak{k} \oplus \mathfrak{p}} }[/math]

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 [math]\displaystyle{ \mathfrak{k} }[/math]. The Lie algebra of *G*_{0} is given by

- [math]\displaystyle{ \displaystyle{\mathfrak{g}=\mathfrak{k} \oplus \mathfrak{p}} }[/math]

and the fixed point subgroup of σ is again *K*, so that *G* ∩ *G*_{0} = *K*. In *G*_{0}, there is a Cartan decomposition

- [math]\displaystyle{ \displaystyle{G_0=K\cdot \exp i\mathfrak{p} =K\cdot P_0 = P_0\cdot K} }[/math]

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 *G*_{0}. To check that it is surjective, for *g* in *G*_{0} write *g* = *u* ⋅ *p* with *u* in *G* and *p* in *P*. Since *c*_{0} *g* = *g*, uniqueness implies that σ*u* = *u* and σ*p* = *p*^{−1}. Hence *u* lies in *K* and *p* in *P*_{0}.

The Cartan decomposition in *G*_{0} shows that *G*_{0} is connected, simply connected and noncompact, because of the direct factor *P*_{0}. Thus *G*_{0} is a noncompact real semisimple Lie group.^{[25]}

Moreover, given a maximal Abelian subalgebra [math]\displaystyle{ \mathfrak{a} }[/math] in [math]\displaystyle{ \mathfrak{p} }[/math], *A* = exp [math]\displaystyle{ \mathfrak{a} }[/math] is a toral subgroup such that σ(*a*) = *a*^{−1} on *A*; and any two such [math]\displaystyle{ \mathfrak{a} }[/math]'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 [math]\displaystyle{ \mathfrak{m} }[/math] and hence equals [math]\displaystyle{ \mathfrak{a} }[/math] by maximality. *A* can be generated topologically by a single element exp *X*, so [math]\displaystyle{ \mathfrak{a} }[/math] is the centralizer of *X* in [math]\displaystyle{ \mathfrak{m} }[/math]. In the *K*-orbit of any element of [math]\displaystyle{ \mathfrak{m} }[/math] there is an element *Y* such that (X,Ad *k* Y) is minimized at *k* = 1. Setting *k* = exp *tT* with *T* in [math]\displaystyle{ \mathfrak{k} }[/math], it follows that (*X*,[*T*,*Y*]) = 0 and hence [*X*,*Y*] = 0, so that *Y* must lie in [math]\displaystyle{ \mathfrak{a} }[/math]. Thus [math]\displaystyle{ \mathfrak{m} }[/math] is the union of the conjugates of [math]\displaystyle{ \mathfrak{a} }[/math]. In particular some conjugate of *X* lies in any other choice of [math]\displaystyle{ \mathfrak{a} }[/math], which centralizes that conjugate; so by maximality the only possibilities are conjugates of [math]\displaystyle{ \mathfrak{a} }[/math].^{[26]}

A similar statements hold for the action of *K* on [math]\displaystyle{ \mathfrak{a}_0=i\mathfrak{a} }[/math] in [math]\displaystyle{ \mathfrak{p}_0 }[/math]. Morevoer, from the Cartan decomposition for *G*_{0}, if *A*_{0} = exp [math]\displaystyle{ \mathfrak{a}_0 }[/math], then

- [math]\displaystyle{ \displaystyle{G_0=KA_0K.} }[/math]

### Iwasawa decomposition in a real form

## See also

## Notes

- ↑ See:
- Hochschild 1965
- Bourbaki 1981, pp. 212–214

- ↑ Bourbaki 1981, pp. 210–214
- ↑ Hochschild 1966
- ↑ See:
- ↑ See:
- ↑ Zhelobenko 1973, p. 28
- ↑ Bump 2004, pp. 202–203
- ↑ See:
- ↑ Zhelobenko 1973
- ↑ See:
- 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
- Dieudonné 1977, pp. 216–217
- Bump 2004, pp. 205–211

- Gelfand & Naimark 1950, section 18, for SL(
- ↑ Steinberg 1974, p. 73
- ↑ Chevalley 1955, p. 41
- ↑ See:
- Steinberg 1974, pp. 73–74
- Bourbaki 1981a, pp. 53–54

- ↑ Sepanski 2007, p. 8
- ↑ Knapp 2001, p. 117
- ↑ See:
- Zhelobenko 1973, pp. 288–290
- Dieudonné 1977, pp. 197–207
- Helgason 1978, pp. 257–262
- Bump 2004, pp. 197–204

- ↑ Bump 2004, pp. 203–204
- ↑ Zhelobenko 1973, p. 289
- ↑ Helgason 1978
- ↑ See:
- ↑ Dieudonné 1977, pp. 164–173
- ↑ See:
- Helgason 1978, pp. 320–321
- Bourbaki 1982, pp. 46–48
- Duistermaat & Kolk 2000, pp. 194–195
- Dieudonné 1977, p. 151, Exercise 11

- ↑ Wolf 2010
- ↑ See: Bourbaki 1982, pp. 46–48
- ↑ Dieudonné 1977, pp. 166–168
- ↑ Helgason 1978, p. 248

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Original source: https://en.wikipedia.org/wiki/Complexification (Lie group).
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