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.

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 suppose is the universal covering map, so that is the fundamental group of. We have the inclusion, which follows from the fact that the kernel of the adjoint representation of equals its centre, combined with the equality
which holds for any. Denoting by the smallest closed normal Lie subgroup of that contains, we must now also have the inclusion. We define the universal complexification of as
In particular, if is simply connected, its universal complexification is just.
The map is obtained by passing to the quotient. Since is a surjective submersion, smoothness of the map implies smoothness of.
Image:Complexification-quotient-map.svg|frameless|none|alt=Construction of the complexification map|Construction of the complexification map|350px
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.

Proof of the universal property

The map 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, is an arbitrary smooth homomorphism of Lie groups with a complex Lie group as the codomain.
For simplicity, we assume is connected. To establish the existence of, we first naturally extend the morphism of Lie algebras to the unique morphism of complex Lie algebras. Since is simply connected, Lie's second fundamental theorem now provides us with a unique complex analytic morphism between complex Lie groups, such that. We define as the map induced by, that is: for any. To show well-definedness of this map, consider the derivative of the map. For any, we have
which implies. This equality finally implies, and since is a closed normal Lie subgroup of, we also have. Since is a complex analytic surjective submersion, the map is complex analytic since is. The desired equality is imminent.
To show uniqueness of, suppose that are two maps with. Composing with from the right and differentiating, we get, and since is the inclusion, we get. But is a submersion, so, thus connectedness of implies.

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.

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
  • The complexification of the special linear group of 2x2 matrices is
  • The complexification of the special orthogonal group of 3x3 matrices is
  • The complexification of the proper orthochronous Lorentz group is
  • The complexification of the special orthogonal group of 4x4 matrices is
The last two examples show that Lie groups with isomorphic complexifications may not be isomorphic. Furthermore, the complexifications of Lie groups and show that complexification is not an idempotent operation, i.e. .

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 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 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. 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.