Complexification (Lie group)

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

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

Existence

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

$(C_{\mathrm{\Phi }(k)}{)}_{*}\circ {\mathrm{\Phi }}_{*}={\mathrm{\Phi }}_{*}\circ (C_k{)}_{*}={\mathrm{\Phi }}_{*}$

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

$G_{𝐂}=\frac{{𝐆}_{𝐂}}{\mathrm{\Phi }(\ker \pi {)}^{*}}.$

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

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

Construction of the complexification map

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

$\{1\}\to G^o\to G\to \mathrm{\Gamma }\to \{1\}$

induces an extension

$\{1\}\to (G^o{)}_{𝐂}\to G_{𝐂}\to \mathrm{\Gamma }\to \{1\}$

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

Proof of the universal property

The map Template: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.

Here, $f:G\to H$ is an arbitrary smooth homomorphism of Lie groups with a complex Lie group as the codomain.

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

$(\bar{F}{)}_{*}{\mathrm{\Phi }}_{*}v=(\bar{F}{)}_{*}(v\otimes 1)=f_{*}{\pi }_{*}v$,

which (by simple connectedness of $𝐆$) implies $\bar{F}\circ \mathrm{\Phi }=f\circ \pi$. This equality finally implies $\mathrm{\Phi }(\ker \pi )\subset \ker \bar{F}$, and since $\ker \bar{F}$ is a closed normal Lie subgroup of ${𝐆}_{𝐂}$, we also have $\mathrm{\Phi }(\ker \pi {)}^{*}\subset \ker \bar{F}$. Since ${\pi }_{ℂ}$ is a complex analytic surjective submersion, the map $F$ is complex analytic since $\bar{F}$ is. The desired equality $F\circ \phi =f$ is imminent. Template:Hidden end

Template:Hidden begin To show uniqueness of $F$, suppose that $F_1,F_2$ are two maps with $F_1\circ \phi =F_2\circ \phi =f$. Composing with $\pi$ from the right and differentiating, we get $(F_1{)}_{*}({\pi }_{𝐂}{)}_{*}{\mathrm{\Phi }}_{*}=(F_2{)}_{*}({\pi }_{𝐂}{)}_{*}{\mathrm{\Phi }}_{*}$, and since ${\mathrm{\Phi }}_{*}$ is the inclusion $𝔤\hookrightarrow {𝔤}_{𝐂}$, we get $(F_1{)}_{*}({\pi }_{𝐂}{)}_{*}=(F_2{)}_{*}({\pi }_{𝐂}{)}_{*}$. But ${\pi }_{𝐂}$ is a submersion, so $(F_1{)}_{*}=(F_2{)}_{*}$, thus connectedness of $G$ implies $F_1=F_2$. Template: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. Template: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 Template:Math by the universal covering group of Template: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

$\mathrm{S}\mathrm{U}(2{)}_{𝐂}\cong \mathrm{S}\mathrm{L}(2,𝐂)$.

This follows from the isomorphism of Lie algebras

$𝔰𝔲(2{)}_{𝐂}\cong 𝔰𝔩(2,𝐂)$,

together with the fact that $\mathrm{S}\mathrm{U}(2)$ is simply connected.

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

$\mathrm{S}\mathrm{L}(2,𝐂{)}_{𝐂}\cong \mathrm{S}\mathrm{L}(2,𝐂)\times \mathrm{S}\mathrm{L}(2,𝐂)$.

This follows from the isomorphism of Lie algebras

$𝔰𝔩(2,𝐂{)}_{𝐂}\cong 𝔰𝔩(2,𝐂)\oplus 𝔰𝔩(2,𝐂)$,

together with the fact that $\mathrm{S}\mathrm{L}(2,𝐂)$ is simply connected.

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

$\mathrm{S}\mathrm{O}(3{)}_{𝐂}\cong \frac{\mathrm{S}\mathrm{L}(2,𝐂)}{{𝐙}_2}\cong {\mathrm{S}\mathrm{O}}^{+}(1,3)$,

where ${\mathrm{S}\mathrm{O}}^{+}(1,3)$ denotes the proper orthochronous Lorentz group. This follows from the fact that $\mathrm{S}\mathrm{U}(2)$ is the universal (double) cover of $\mathrm{S}\mathrm{O}(3)$, hence:

$𝔰𝔬(3{)}_{𝐂}\cong 𝔰𝔲(2{)}_{𝐂}\cong 𝔰𝔩(2,𝐂)$.

We also use the fact that $\mathrm{S}\mathrm{L}(2,𝐂)$ is the universal (double) cover of ${\mathrm{S}\mathrm{O}}^{+}(1,3)$.

• The complexification of the proper orthochronous Lorentz group is

${\mathrm{S}\mathrm{O}}^{+}(1,3{)}_{𝐂}\cong \frac{\mathrm{S}\mathrm{L}(2,𝐂)\times \mathrm{S}\mathrm{L}(2,𝐂)}{{𝐙}_2}$.

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

$\mathrm{S}\mathrm{O}(4{)}_{𝐂}\cong \frac{\mathrm{S}\mathrm{L}(2,𝐂)\times \mathrm{S}\mathrm{L}(2,𝐂)}{{𝐙}_2}$.

This follows from the fact that $\mathrm{S}\mathrm{U}(2)\times \mathrm{S}\mathrm{U}(2)$ is the universal (double) cover of $\mathrm{S}\mathrm{O}(4)$, hence $𝔰𝔬(4)\cong 𝔰𝔲(2)\oplus 𝔰𝔲(2)$ and so $𝔰𝔬(4{)}_{𝐂}\cong 𝔰𝔩(2,𝐂)\oplus 𝔰𝔩(2,𝐂)$.

The last two examples show that Lie groups with isomorphic complexifications may not be isomorphic. Furthermore, the complexifications of Lie groups $\mathrm{S}\mathrm{U}(2)$ and $\mathrm{S}\mathrm{L}(2,𝐂)$ show that complexification is not an idempotent operation, i.e. $(G_{𝐂}{)}_{𝐂}\cong ̸G_{𝐂}$ (this is also shown by complexifications of $\mathrm{S}\mathrm{O}(3)$ and ${\mathrm{S}\mathrm{O}}^{+}(1,3)$).

Chevalley complexification

Hopf algebra of matrix coefficients

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

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

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

Invariant theory

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

Decompositions in the Chevalley complexification

Cartan decomposition

The decomposition derived from the polar decomposition

$G_{𝐂}=G\cdot P=G\cdot \exp i𝔤,$

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

$g=XDY$

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

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

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

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

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

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

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

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

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

$N_{-}{T}_{𝐂}{N}_{+}$

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

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

These results are an immediate consequence of the corresponding results for Template:Math.

Bruhat decomposition

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

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

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

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

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

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

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

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

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

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

Iwasawa decomposition

The Iwasawa decomposition

$G_{𝐂}=G\cdot A\cdot N$

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

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

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

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

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

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

$A=\exp i𝔱=𝐀\cap G_{𝐂},N=\exp {𝔫}_{+}=𝐍\cap G_{𝐂}.$

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

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

Complex structures on homogeneous spaces

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

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

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

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

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

$G/H=G_{𝐂}/P.$

In particular

$G/T=G_{𝐂}/B.$

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

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

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

where Template:Math is the stabilizer of Template:Mvar in the Weyl group Template:Math. It is generated by the reflections corresponding to the simple roots orthogonal to Template: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 $𝔱$ onto T is a lattice Λ with a Z-basis indexed by simple roots, which σ permutes. Splitting up according to orbits, T can be written as a product of terms T on which σ acts trivially or terms T² where σ interchanges the factors. The fixed point subgroup just corresponds to taking the diagonal subgroups in the second case, so is connected.

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

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

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

Conjugations on the complexification

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

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

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

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

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

for X, Y in $𝔤$.

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

$(X,Y)=-B(X,c(Y)),$

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

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

It satisfies

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

Cartan decomposition in a real form

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

$G_{𝐂}=G\cdot \exp i𝔤=G\cdot P=P\cdot G.$

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

$𝔤=𝔨\oplus 𝔭$

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

$𝔤=𝔨\oplus 𝔭$

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

$G_0=K\cdot \exp i𝔭=K\cdot P_0=P_0\cdot K$

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

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

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

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

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

See also

• Real form (Lie theory)

References

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