Complexification (Lie group)

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 , where is a unitary operator in the compact group and 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 is a Lie group, a universal complexification is given by a complex Lie group and a continuous homomorphism with the universal property that, if is an arbitrary continuous homomorphism into a complex Lie group , then there is a unique complex analytic homomorphism such that .
Universal complexifications always exist and are unique up to a unique complex analytic isomorphism (preserving inclusion of the original group).
Existence.
If is connected with Lie algebra , then its universal covering group is simply connected. Let be the simply connected complex Lie group with Lie algebra . Let be the natural homomorphism and the natural covering map. Then given a homomorphism , there is a unique complex analytic homomorphism such that . Let be the intersection of the kernels of the homomorphisms as varies over all possibilities. Then is a closed normal complex Lie subgroup of and the quotient group is a universal complexification. In particular if is simply connected, its universal complexification is just .
For non-connected Lie groups with identity component and component group , the extension
induces an extension
and the complex Lie group is a complexification of .
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. 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 by the universal covering group of 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.
Chevalley complexification.
Hopf algebra of matrix coefficients.
If is a compact Lie group, the *-algebra of matrix coefficients of finite-dimensional unitary representations is a uniformly dense *-subalgebra of , the *-algebra of complex-valued continuous functions on . It is naturally a Hopf algebra with comultiplication given by
The characters of are the *-homomorphisms of into . They can be identified with the point evaluations for in and the comultiplication allows the group structure on to be recovered. The homomorphisms of into also form a group. It is a complex Lie group and can be identified with the complexification of . The *-algebra is generated by the matrix coefficients of any faithful representation of . It follows that defines a faithful complex analytic representation of .
Invariant theory.
The original approach of to the complexification of a compact Lie group can be concisely stated within the language of classical invariant theory, described in . Let be a closed subgroup of the unitary group where is a finite-dimensional complex inner product space. Its Lie algebra consists of all skew-adjoint operators such that lies in for all real . Set with the trivial action of on the second summand. The group acts on , with an element acting as . The commutant (or centralizer algebra) is denoted by . It is generated as a *-algebra by its unitary operators and its commutant is the *-algebra spanned by the operators . The complexification of consists of all operators in such that commutes with and acts trivially on the second summand in . By definition it is a closed subgroup of . The defining relations (as a commutant) show that is an algebraic subgroup. Its intersection with coincides with , since it is "a priori" a larger compact group for which the irreducible representations stay irreducible and inequivalent when restricted to . Since is generated by unitaries, an invertible operator lies in if the unitary operator and positive operator in its polar decomposition both lie in . Thus lies in and the operator can be written uniquely as with a self-adjoint operator. By the functional calculus for polynomial functions it follows that lies in the commutant of if with in . In particular taking purely imaginary, must have the form with in the Lie algebra of . Since every finite-dimensional representation of occurs as a direct summand of , it is left invariant by and thus every finite-dimensional representation of extends uniquely to . The extension is compatible with the polar decomposition. Finally the polar decomposition implies that is a maximal compact subgroup of , since a strictly larger compact subgroup would contain all integer powers of a positive operator , a closed infinite discrete subgroup.
Decompositions in the Chevalley complexification.
Cartan decomposition.
The decomposition derived from the polar decomposition
where is the Lie algebra of , is called the Cartan decomposition of . The exponential factor is invariant under conjugation by but is not a subgroup. The complexification is invariant under taking adjoints, since consists of unitary operators and 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 it states that with respect to a given orthonormal basis an element of can be factorized in the form
with lower unitriangular, upper unitriangular and diagonal if and only if all the principal minors of are non-vanishing. In this case and are uniquely determined.
In fact Gaussian elimination shows there is a unique such that is upper triangular.
The upper and lower unitriangular matrices, and , 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 and subalgebras of their Lie algebras. The exponential map is onto in each case, since the polynomial function lies in a given Lie subalgebra if and do and are sufficiently small.
The Gauss decomposition can be extended to complexifications of other closed connected subgroups of by using the root decomposition to write the complexified Lie algebra as
where is the Lie algebra of a maximal torus of and are the direct sum of the corresponding positive and negative root spaces. In the weight space decomposition of as eigenspaces of acts as diagonally, acts as lowering operators and as raising operators. are nilpotent Lie algebras acting as nilpotent operators; they are each other's adjoints on . In particular acts by conjugation of , so that is a semidirect product of a nilpotent Lie algebra by an abelian Lie algebra.
By Engel's theorem, if is a semidirect product, with abelian and nilpotent, acting on a finite-dimensional vector space with operators in diagonalizable and operators in nilpotent, there is a vector that is an eigenvector for and is annihilated by . In fact it is enough to show there is a vector annihilated by , which follows by induction on , since the derived algebra annihilates a non-zero subspace of vectors on which and act with the same hypotheses.
Applying this argument repeatedly to shows that there is an orthonormal basis of consisting of eigenvectors of with acting as upper triangular matrices with zeros on the diagonal.
If and are the complex Lie groups corresponding to and , then the Gauss decomposition states that the subset
is a direct product and consists of the elements in for which the principal minors are non-vanishing. It is open and dense. Moreover, if denotes the maximal torus in ,
These results are an immediate consequence of the corresponding results for .
Bruhat decomposition.
If denotes the Weyl group of and denotes the Borel subgroup , the Gauss decomposition is also a consequence of the more precise Bruhat decomposition
decomposing into a disjoint union of double cosets of . The complex dimension of a double coset is determined by the length of as an element of . The dimension is maximized at the Coxeter element and gives the unique open dense double coset. Its inverse conjugates into the Borel subgroup of lower triangular matrices in .
The Bruhat decomposition is easy to prove for . Let be the Borel subgroup of upper triangular matrices and the subgroup of diagonal matrices. So . For in , take in so that 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 in , it follows that lies in . For uniqueness, if , then the entries of vanish below the diagonal. So the product lies in , proving uniqueness.
 showed that the expression of an element as becomes unique if is restricted to lie in the upper unitriangular subgroup . In fact, if , this follows from the identity
The group has a natural filtration by normal subgroups with zeros in the first superdiagonals and the successive quotients are Abelian. Defining and to be the intersections with , it follows by decreasing induction on that . Indeed, and are specified in by the vanishing of complementary entries on the th superdiagonal according to whether preserves the order 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 . For , let be the matrix with 's on the antidiagonal and 's elsewhere and set
Then is the fixed point subgroup of the involution . It leaves the subgroups and invariant. If the basis elements are indexed by , then the Weyl group of consists of satisfying
, i.e. commuting with . Analogues of and are defined by intersection with , i.e. as fixed points of . The uniqueness of the decomposition implies the Bruhat decomposition for .
The same argument works for . It can be realised as the fixed points of in where .
Iwasawa decomposition.
The Iwasawa decomposition
gives a decomposition for for which, unlike the Cartan decomposition, the direct factor is a closed subgroup, but it is no longer invariant under conjugation by . It is the semidirect product of the nilpotent subgroup by the Abelian subgroup .
For and its complexification , this decomposition can be derived as a restatement of the Gram–Schmidt orthonormalization process.
In fact let be an orthonormal basis of and let be an element in . Applying the Gram–Schmidt process to , there is a unique orthonormal basis and positive constants such that
Since the decomposition is direct for , it is enough to check that . From the properties of the Iwasawa decomposition for , the map is a diffeomorphism onto its image in , which is closed. On the other hand, the dimension of the image is the same as the dimension of , so it is also open. So because is connected.
 gives a method for explicitly computing the elements in the decomposition. For in set . 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
 with in , in and in . Since is self-adjoint, uniqueness forces . Since it is also positive must lie in and have the form for some unique in . Let be its unique square root in . Set and . Then is unitary, so is in , and .
Complex structures on homogeneous spaces.
The Iwasawa decomposition can be used to describe complex structures on the s in complex projective space of highest weight vectors of finite-dimensional irreducible representations of . In particular the identification between and can be used to formulate the Borel–Weil theorem. It states that each irreducible representation
of can be obtained by holomorphic induction from a character of , or equivalently that it is realized in the space of sections of a holomorphic line bundle on .
The closed connected subgroups of containing are described by Borel–de Siebenthal theory. They are exactly the centralizers of tori . Since every torus is generated topologically by a single element , these are the same as centralizers of elements in . By a result of Hopf is always connected: indeed any element is along with contained in some maximal torus, necessarily contained in .
Given an irreducible finite-dimensional representation with highest weight vector of weight , the stabilizer of in is a closed subgroup . Since is an eigenvector of , contains . The complexification also acts on and the stabilizer is a closed complex subgroup containing . Since is annihilated by every raising operator corresponding to a positive root , contains the Borel subgroup . The vector is also a highest weight vector for the copy of corresponding to , so it is annihilated by the lowering operator generating if . The Lie algebra of is the direct sum of and root space vectors annihilating , so that
The Lie algebra of is given by . By the Iwasawa decomposition . Since fixes , the -orbit of in the complex projective space of coincides with the orbit and
In particular
Using the identification of the Lie algebra of with its dual, equals the centralizer of in , and hence is connected. The group is also connected. In fact the space is simply connected,
since it can be written as the quotient of the (compact) universal covering group of the compact semisimple group by a connected subgroup, where is the center of . If is the identity component of , has as a covering space, so that . The homogeneous space has a complex structure, because 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 , , and .
The parabolic subgroup can also be written as a union of double cosets of 
where is the stabilizer of in the Weyl group . It is generated by the reflections corresponding to the simple roots orthogonal to .
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 period 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 formula_18 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 formula_19, so it and therefore also formula_18 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 formula_21 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 formula_18 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 formula_23 and satisfies "c"2 = id. Such automorphisms of either "G"C or formula_23 are called conjugations.
Since "G"C is also simply connected any conjugation "c"1 on formula_23 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
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
for "X", "Y" in formula_28.
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
where "B" is the Killing form on formula_23. Thus ψ2 is a positive operator and an automorphism along with all its real powers. In particular take
It satisfies
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
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
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 formula_35. The Lie algebra of "G"0 is given by
and the fixed point subgroup of σ is again "K", so that "G" ∩ "G"0 = "K". In "G"0, there is a Cartan decomposition
which is again a diffeomorphism onto the direct and corresponds to the polar decomposition of matrices.
It is the restriction of the decomposition on "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.
Moreover, given a maximal Abelian subalgebra formula_38 in formula_39, "A" = exp formula_38 is a toral subgroup such that σ("a") = "a"−1 on "A"; and any two such formula_38'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 formula_42 and hence equals formula_38 by maximality. "A" can be generated topologically by a single element exp "X", so formula_38 is the centralizer of "X" in formula_42. In the "K"-orbit of any element of formula_42 there is an element "Y" such that (X,Ad "k" Y) is minimized at "k" = 1. Setting "k" = exp "tT" with "T" in formula_35, it follows that ("X",["T","Y"]) = 0 and hence ["X","Y"] = 0, so that "Y" must lie in formula_38. Thus formula_42 is the union of the conjugates of formula_38. In particular some conjugate of "X" lies in any other choice of formula_38, which centralizes that conjugate; so by maximality the only possibilities are conjugates of formula_38.
A similar statements hold for the action of "K" on formula_53 in formula_54. Morevoer, from the Cartan decomposition for "G"0, if "A"0 = exp formula_55, then