Английская Википедия:Complexification (Lie group)

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Шаблон:Short description Шаблон: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 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 Шаблон:Math, where Шаблон:Math is a unitary operator in the compact group and Шаблон:Math 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 Шаблон:Math is a Lie group, a universal complexification is given by a complex Lie group Шаблон:Math and a continuous homomorphism Шаблон:Math with the universal property that, if Шаблон:Math is an arbitrary continuous homomorphism into a complex Lie group Шаблон:Math, then there is a unique complex analytic homomorphism Шаблон:Math such that Шаблон:Math.

Universal complexifications always exist and are unique up to a unique complex analytic isomorphism (preserving inclusion of the original group).

Existence

If Шаблон:Math is connected with Lie algebra Шаблон:Math, then its universal covering group Шаблон:Math is simply connected. Let Шаблон:Math be the simply connected complex Lie group with Lie algebra Шаблон:Math, let Шаблон:Math be the natural homomorphism (the unique morphism such that Шаблон:Math is the canonical inclusion) and suppose Шаблон:Math is the universal covering map, so that Шаблон:Math is the fundamental group of Шаблон:Math. We have the inclusion Шаблон:Math, which follows from the fact that the kernel of the adjoint representation of Шаблон:Math equals its centre, combined with the equality

<math>(C_{\Phi(k)})_*\circ \Phi_* = \Phi_* \circ (C_k)_* = \Phi_*</math>

which holds for any Шаблон:Math. Denoting by Шаблон:Math the smallest closed normal Lie subgroup of Шаблон:Math that contains Шаблон:Math, we must now also have the inclusion Шаблон:Math. We define the universal complexification of Шаблон:Math as

<math>G_{\mathbf C}=\frac{\mathbf G_{\mathbf C}}{\Phi(\ker \pi)^*}.</math>

In particular, if Шаблон:Math is simply connected, its universal complexification is just Шаблон:Math.

The map Шаблон:Math is obtained by passing to the quotient. Since Шаблон:Math is a surjective submersion, smoothness of the map Шаблон:Math implies smoothness of Шаблон:Math.

Construction of the complexification map

Construction of the complexification map

For non-connected Lie groups Шаблон:Math with identity component Шаблон:Math and component group Шаблон:Math, the extension

<math> \{1\} \rightarrow G^o \rightarrow G \rightarrow \Gamma \rightarrow \{1\} </math>

induces an extension

<math>\{1\} \rightarrow (G^o)_{\mathbf{C}} \rightarrow G_{\mathbf{C}} \rightarrow \Gamma \rightarrow \{1\} </math>

and the complex Lie group Шаблон:Math is a complexification of Шаблон:Math.

Proof of the universal property

The map Шаблон:Math 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.

Universal property of complexification

Here, <math>f\colon G\rightarrow H</math> is an arbitrary smooth homomorphism of Lie groups with a complex Lie group as the codomain.

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

<math>(\overline F)_*\Phi_*v=(\overline F)_*(v\otimes 1)=f_*\pi_*v</math>,

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

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

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. Шаблон:Harvtxt 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 Шаблон:Math by the universal covering group of Шаблон:Math 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>\mathrm{SU}(2)_{\mathbf C}\cong \mathrm{SL}(2,\mathbf C)</math>.

This follows from the isomorphism of Lie algebras

<math>\mathfrak{su}(2)_{\mathbf C}\cong \mathfrak{sl}(2,\mathbf C)</math>,

together with the fact that <math>\mathrm{SU}(2)</math> is simply connected.

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

<math>\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>\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>\mathrm{SL}(2,\mathbf C)</math> is simply connected.

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

<math>\mathrm{SO}(3)_{\mathbf C}\cong \frac{\mathrm{SL}(2,\mathbf C)}{\mathbf Z_2}\cong \mathrm{SO}^+(1,3)</math>,

where <math>\mathrm{SO}^+(1,3)</math> denotes the proper orthochronous Lorentz group. This follows from the fact that <math>\mathrm{SU}(2)</math> is the universal (double) cover of <math>\mathrm{SO}(3)</math>, hence:

<math>\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>\mathrm{SL}(2,\mathbf C)</math> is the universal (double) cover of <math>\mathrm{SO}^+(1,3)</math>.

• The complexification of the proper orthochronous Lorentz group is

<math>\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>\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>\mathrm{SU}(2)\times\mathrm{SU}(2)</math> is the universal (double) cover of <math>\mathrm{SO}(4)</math>, hence <math>\mathfrak{so}(4)\cong \mathfrak{su}(2)\oplus\mathfrak{su}(2)</math> and so <math>\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>\mathrm{SU}(2)</math> and <math>\mathrm{SL}(2,\mathbf C)</math> show that complexification is not an idempotent operation, i.e. <math>(G_{\mathbf C})_{\mathbf C}\not\cong G_{\mathbf C}</math> (this is also shown by complexifications of <math>\mathrm{SO}(3)</math> and <math>\mathrm{SO}^+(1,3)</math>).

Chevalley complexification

Hopf algebra of matrix coefficients

If Шаблон:Mathis a compact Lie group, the *-algebra Шаблон:Math of matrix coefficients of finite-dimensional unitary representations is a uniformly dense *-subalgebra of Шаблон:Math, the *-algebra of complex-valued continuous functions on Шаблон:Math. It is naturally a Hopf algebra with comultiplication given by

<math>\displaystyle{\Delta f(g,h)= f(gh).}</math>

The characters of Шаблон:Math are the *-homomorphisms of Шаблон:Math into Шаблон:Math. They can be identified with the point evaluations Шаблон:Math for Шаблон:Math in Шаблон:Math and the comultiplication allows the group structure on Шаблон:Math to be recovered. The homomorphisms of Шаблон:Math into Шаблон:Math also form a group. It is a complex Lie group and can be identified with the complexification Шаблон:Math of Шаблон:Math. The *-algebra Шаблон:Math is generated by the matrix coefficients of any faithful representation Шаблон:Mvar of Шаблон:Math. It follows that Шаблон:Mvar defines a faithful complex analytic representation of Шаблон:Math.

Invariant theory

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

Decompositions in the Chevalley complexification

Cartan decomposition

The decomposition derived from the polar decomposition

<math>\displaystyle{G_{\mathbf{C}} = G\cdot P =G \cdot \exp i\mathfrak{g},}</math>

where Шаблон:Math is the Lie algebra of Шаблон:Math, is called the Cartan decomposition of Шаблон:Math. The exponential factor Шаблон:Math is invariant under conjugation by Шаблон:Math but is not a subgroup. The complexification is invariant under taking adjoints, since Шаблон:Math consists of unitary operators and Шаблон:Math 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 Шаблон:Math it states that with respect to a given orthonormal basis Шаблон:Math an element Шаблон:Math of Шаблон:Math can be factorized in the form

<math>\displaystyle{g=XDY}</math>

with Шаблон:Math lower unitriangular, Шаблон:Math upper unitriangular and Шаблон:Math diagonal if and only if all the principal minors of Шаблон:Math are non-vanishing. In this case Шаблон:Math and Шаблон:Math are uniquely determined.

In fact Gaussian elimination shows there is a unique Шаблон:Math such that Шаблон:Math is upper triangular.

The upper and lower unitriangular matrices, Шаблон:Math and Шаблон:Math, 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 Шаблон:Math and subalgebras of their Lie algebras. The exponential map is onto in each case, since the polynomial function Шаблон:Math lies in a given Lie subalgebra if Шаблон:Math and Шаблон:Math do and are sufficiently small.

The Gauss decomposition can be extended to complexifications of other closed connected subgroups Шаблон:Math of Шаблон:Math by using the root decomposition to write the complexified Lie algebra as

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

where Шаблон:Math is the Lie algebra of a maximal torus Шаблон:Math of Шаблон:Math and Шаблон:Math are the direct sum of the corresponding positive and negative root spaces. In the weight space decomposition of Шаблон:Math as eigenspaces of Шаблон:Math acts as diagonally, Шаблон:Math acts as lowering operators and Шаблон:Math as raising operators. Шаблон:Math are nilpotent Lie algebras acting as nilpotent operators; they are each other's adjoints on Шаблон:Math. In particular Шаблон:Math acts by conjugation of Шаблон:Math, so that Шаблон:Math is a semidirect product of a nilpotent Lie algebra by an abelian Lie algebra.

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

Applying this argument repeatedly to Шаблон:Math shows that there is an orthonormal basis Шаблон:Math of Шаблон:Math consisting of eigenvectors of Шаблон:Math with Шаблон:Math acting as upper triangular matrices with zeros on the diagonal.

If Шаблон:Math and Шаблон:Math are the complex Lie groups corresponding to Шаблон:Math and Шаблон:Math, then the Gauss decomposition states that the subset

<math>\displaystyle{N_- T_{\mathbf{C}} N_+}</math>

is a direct product and consists of the elements in Шаблон:Math for which the principal minors are non-vanishing. It is open and dense. Moreover, if Шаблон:Math denotes the maximal torus in Шаблон:Math,

<math>\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 Шаблон:Math.

Bruhat decomposition

If Шаблон:Math denotes the Weyl group of Шаблон:Math and Шаблон:Math denotes the Borel subgroup Шаблон:Math, the Gauss decomposition is also a consequence of the more precise Bruhat decomposition

<math>\displaystyle{G_{\mathbf{C}} =\bigcup_{\sigma\in W} B\sigma B,}</math>

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

The Bruhat decomposition is easy to prove for Шаблон:Math. Let Шаблон:Math be the Borel subgroup of upper triangular matrices and Шаблон:Math the subgroup of diagonal matrices. So Шаблон:Math. For Шаблон:Math in Шаблон:Math, take Шаблон:Math in Шаблон:Math so that Шаблон:Math 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 Шаблон:Math in Шаблон:Math, it follows that Шаблон:Math lies in Шаблон:Math. For uniqueness, if Шаблон:Math, then the entries of Шаблон:Math vanish below the diagonal. So the product lies in Шаблон:Math, proving uniqueness.

Шаблон:Harvtxt showed that the expression of an element Шаблон:Math as Шаблон:Math becomes unique if Шаблон:Math is restricted to lie in the upper unitriangular subgroup Шаблон:Math. In fact, if Шаблон:Math, this follows from the identity

<math>\displaystyle{N_+=N_\sigma\cdot M_\sigma.}</math>

The group Шаблон:Math has a natural filtration by normal subgroups Шаблон:Math with zeros in the first Шаблон:Math superdiagonals and the successive quotients are Abelian. Defining Шаблон:Math and Шаблон:Math to be the intersections with Шаблон:Math, it follows by decreasing induction on Шаблон:Math that Шаблон:Math. Indeed, Шаблон:Math and Шаблон:Math are specified in Шаблон:Math by the vanishing of complementary entries Шаблон:Math on the Шаблон:Mathth superdiagonal according to whether Шаблон:Mvar preserves the order Шаблон:Math 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 Шаблон:Math. For Шаблон:Math, let Шаблон:Math be the Шаблон:Math matrix with Шаблон:Math's on the antidiagonal and Шаблон:Math's elsewhere and set

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

Then Шаблон:Math is the fixed point subgroup of the involution Шаблон:Math. It leaves the subgroups Шаблон:Math and Шаблон:Math invariant. If the basis elements are indexed by Шаблон:Math, then the Weyl group of Шаблон:Math consists of Шаблон:Mvar satisfying Шаблон:Math, i.e. commuting with Шаблон:Mvar. Analogues of Шаблон:Math and Шаблон:Math are defined by intersection with Шаблон:Math, i.e. as fixed points of Шаблон:Mvar. The uniqueness of the decomposition Шаблон:Math implies the Bruhat decomposition for Шаблон:Math.

The same argument works for Шаблон:Math. It can be realised as the fixed points of Шаблон:Math in Шаблон:Math where Шаблон:Math.

Iwasawa decomposition

The Iwasawa decomposition

<math>\displaystyle{G_{\mathbf{C}} = G\cdot A \cdot N}</math>

gives a decomposition for Шаблон:Math for which, unlike the Cartan decomposition, the direct factor Шаблон:Math is a closed subgroup, but it is no longer invariant under conjugation by Шаблон:Math. It is the semidirect product of the nilpotent subgroup Шаблон:Math by the Abelian subgroup Шаблон:Math.

For Шаблон:Math and its complexification Шаблон:Math, this decomposition can be derived as a restatement of the Gram–Schmidt orthonormalization process.

In fact let Шаблон:Math be an orthonormal basis of Шаблон:Math and let Шаблон:Math be an element in Шаблон:Math. Applying the Gram–Schmidt process to Шаблон:Math, there is a unique orthonormal basis Шаблон:Math and positive constants Шаблон:Math such that

<math>\displaystyle{f_i= a_i ge_i + \sum_{j<i} n_{ji} ge_j.}</math>

If Шаблон:Math is the unitary taking Шаблон:Math to Шаблон:Math, it follows that Шаблон:Math lies in the subgroup Шаблон:Math, where Шаблон:Math is the subgroup of positive diagonal matrices with respect to Шаблон:Math and Шаблон:Math is the subgroup of upper unitriangular matrices.

Using the notation for the Gauss decomposition, the subgroups in the Iwasawa decomposition for Шаблон:Math are defined by

<math>\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 Шаблон:Math, it is enough to check that Шаблон:Math. From the properties of the Iwasawa decomposition for Шаблон:Math, the map Шаблон:Math is a diffeomorphism onto its image in Шаблон:Math, which is closed. On the other hand, the dimension of the image is the same as the dimension of Шаблон:Math, so it is also open. So Шаблон:Math because Шаблон:Math is connected.

Шаблон:Harvtxt gives a method for explicitly computing the elements in the decomposition. For Шаблон:Math in Шаблон:Math set Шаблон:Math. 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 Шаблон:Math with Шаблон:Math in Шаблон:Math, Шаблон:Math in Шаблон:Math and Шаблон:Math in Шаблон:Math. Since Шаблон:Math is self-adjoint, uniqueness forces Шаблон:Math. Since it is also positive Шаблон:Math must lie in Шаблон:Math and have the form Шаблон:Math for some unique Шаблон:Math in Шаблон:Math. Let Шаблон:Math be its unique square root in Шаблон:Math. Set Шаблон:Math and Шаблон:Math. Then Шаблон:Math is unitary, so is in Шаблон:Math, and Шаблон:Math.

Complex structures on homogeneous spaces

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

The closed connected subgroups of Шаблон:Math containing Шаблон:Math are described by Borel–de Siebenthal theory. They are exactly the centralizers of tori Шаблон:Math. Since every torus is generated topologically by a single element Шаблон:Math, these are the same as centralizers Шаблон:Math of elements Шаблон:Math in Шаблон:Math. By a result of Hopf Шаблон:Math is always connected: indeed any element Шаблон:Math is along with Шаблон:Math contained in some maximal torus, necessarily contained in Шаблон:Math.

Given an irreducible finite-dimensional representation Шаблон:Math with highest weight vector Шаблон:Math of weight Шаблон:Math, the stabilizer of Шаблон:Math in Шаблон:Math is a closed subgroup Шаблон:Math. Since Шаблон:Math is an eigenvector of Шаблон:Math, Шаблон:Math contains Шаблон:Math. The complexification Шаблон:Math also acts on Шаблон:Math and the stabilizer is a closed complex subgroup Шаблон:Math containing Шаблон:Math. Since Шаблон:Math is annihilated by every raising operator corresponding to a positive root Шаблон:Math, Шаблон:Math contains the Borel subgroup Шаблон:Math. The vector Шаблон:Math is also a highest weight vector for the copy of Шаблон:Math corresponding to Шаблон:Math, so it is annihilated by the lowering operator generating Шаблон:Math if Шаблон:Math. The Lie algebra Шаблон:Math of Шаблон:Math is the direct sum of Шаблон:Math and root space vectors annihilating Шаблон:Math, so that

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

The Lie algebra of Шаблон:Math is given by Шаблон:Math. By the Iwasawa decomposition Шаблон:Math. Since Шаблон:Math fixes Шаблон:Math, the Шаблон:Math-orbit of Шаблон:Math in the complex projective space of Шаблон:Math coincides with the Шаблон:Math orbit and

<math>\displaystyle{G/H=G_{\mathbf{C}}/P.}</math>

In particular

<math>\displaystyle{G/T=G_{\mathbf{C}}/B.}</math>

Using the identification of the Lie algebra of Шаблон:Math with its dual, Шаблон:Math equals the centralizer of Шаблон:Mvar in Шаблон:Math, and hence is connected. The group Шаблон:Math is also connected. In fact the space Шаблон:Math is simply connected, since it can be written as the quotient of the (compact) universal covering group of the compact semisimple group Шаблон:Math by a connected subgroup, where Шаблон:Math is the center of Шаблон:Math. If Шаблон:Math is the identity component of Шаблон:Math, Шаблон:Math has Шаблон:Math as a covering space, so that Шаблон:Math. The homogeneous space Шаблон:Math has a complex structure, because Шаблон:Math 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 Шаблон:Harvtxt, Шаблон:Harvtxt, Шаблон:Harvtxt and Шаблон:Harvtxt.

The parabolic subgroup Шаблон:Math can also be written as a union of double cosets of Шаблон:Math

<math>\displaystyle{P=\bigcup_{\sigma\in W_\lambda} B\sigma B,}</math>

where Шаблон:Math is the stabilizer of Шаблон:Mvar in the Weyl group Шаблон:Math. It is generated by the reflections corresponding to the simple roots orthogonal to Шаблон:Mvar.

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 <math>\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 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 <math>\mathfrak{t}\ominus \mathfrak{s}</math>, so it and therefore also <math>\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>\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>\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>\mathfrak{g}_{\mathbf{C}}</math> and satisfies c² = id. Such automorphisms of either G_C or <math>\mathfrak{g}_{\mathbf{C}}</math> are called conjugations. Since G_C is also simply connected any conjugation c₁ on <math>\mathfrak{g}_{\mathbf{C}}</math> 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

<math>\displaystyle{c_0=\varphi\circ c_1\circ \varphi^{-1}}</math>

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

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

for X, Y in <math>\mathfrak{g}</math>.

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

<math>\displaystyle{(X,Y)=-B(X,c(Y)),}</math>

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

<math>\displaystyle{\varphi=(\psi^2)^{1/4}}</math>

It satisfies

<math>\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{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 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

<math>\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>\mathfrak{k}</math>. The Lie algebra of G₀ is given by

<math>\displaystyle{\mathfrak{g}=\mathfrak{k} \oplus \mathfrak{p}}</math>

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

<math>\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 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 <math>\mathfrak{a}</math> in <math>\mathfrak{p}</math>, A = exp <math>\mathfrak{a}</math> is a toral subgroup such that σ(a) = a⁻¹ on A; and any two such <math>\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>\mathfrak{m}</math> and hence equals <math>\mathfrak{a}</math> by maximality. A can be generated topologically by a single element exp X, so <math>\mathfrak{a}</math> is the centralizer of X in <math>\mathfrak{m}</math>. In the K-orbit of any element of <math>\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>\mathfrak{k}</math>, it follows that (X,[T,Y]) = 0 and hence [X,Y] = 0, so that Y must lie in <math>\mathfrak{a}</math>. Thus <math>\mathfrak{m}</math> is the union of the conjugates of <math>\mathfrak{a}</math>. In particular some conjugate of X lies in any other choice of <math>\mathfrak{a}</math>, which centralizes that conjugate; so by maximality the only possibilities are conjugates of <math>\mathfrak{a}</math>.

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

<math>\displaystyle{G_0=KA_0K.}</math>

See also

• Real form (Lie theory)

References

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