Lie groups and Lie algebras
Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)
Simple Lie groups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8
Other Lie groups Circle Lorentz Poincaré Conformal group Diffeomorphism Loop Euclidean
Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebra Loop algebra Affine Lie algebra
Semisimple Lie algebra Dynkin diagrams Cartan subalgebra Root system Weyl group Real form Complexification Split Lie algebra Compact Lie algebra
Representation theory Lie group representation Lie algebra representation Representation theory of semisimple Lie algebras Representations of classical Lie groups Theorem of the highest weight Borel–Weil–Bott theorem
Lie groups in physics Particle physics and representation theory Lorentz group representations Poincaré group representations Galilean group representations
Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel
Glossary Table of Lie groups
v t e

In mathematics, the complexification or universal complexification of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that every continuous homomorphism of the original group into another complex Lie group extends compatibly to a complex analytic homomorphism between the complex Lie groups. The complexification, which always exists, is unique up to unique isomorphism. Its Lie algebra is a quotient of the complexification of the Lie algebra of the original group. They are isomorphic if the original group has a quotient by a discrete normal subgroup which is linear.

For compact Lie groups, the complexification, sometimes called the Chevalley complexification after Claude Chevalley, can be defined as the group of complex characters of the Hopf algebra of representative functions, i.e. the matrix coefficients of finite-dimensional representations of the group. In any finite-dimensional faithful unitary representation of the compact group it can be realized concretely as a closed subgroup of the complex general linear group. It consists of operators with polar decomposition g = u • exp iX, where u is a unitary operator in the compact group and X is a skew-adjoint operator in its Lie algebra. In this case the complexification is a complex algebraic group and its Lie algebra is the complexification of the Lie algebra of the compact Lie group.

Universal complexification

Definition

If G is a Lie group, a universal complexification is given by a complex Lie group G_C and a continuous homomorphism φ: G → G_C with the universal property that, if f: G → H is an arbitrary continuous homomorphism into a complex Lie group H, then there is a unique complex analytic homomorphism F: G_C → H such that f = F ∘ φ.

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

Existence

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

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

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

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

In particular, if G is simply connected, its universal complexification is just G_C.^[1]

The map φ: G → G_C is obtained by passing to the quotient. Since π is a surjective submersion, smoothness of the map π_C ∘ Φ implies smoothness of φ.

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

${\displaystyle \{1\}\rightarrow G^{o}\rightarrow G\rightarrow \Gamma \rightarrow \{1\}}$

induces an extension

${\displaystyle \{1\}\rightarrow (G^{o})_{\mathbf {C} }\rightarrow G_{\mathbf {C} }\rightarrow \Gamma \rightarrow \{1\}}$

and the complex Lie group G_C is a complexification of G.^[2]

Proof of the universal property

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

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