# complexification

Let $G$ be a real Lie group. Then the complexification ${G}_{\u2102}$ of $G$ is the unique complex Lie group equipped with a map $\phi :G\to {G}_{\u2102}$ such that any map $G\to H$ where $H$ is a complex Lie group, extends to a holomorphic map ${G}_{\u2102}\to H$. If $\U0001d524$ and ${\U0001d524}_{\u2102}$ are the respective Lie algebras, ${\U0001d524}_{\u2102}\cong \U0001d524{\otimes}_{\mathbb{R}}\u2102$.

For simply connected groups, the construction is obvious: we simply take the simply connected complex group with Lie algebra ${\U0001d524}_{\u2102}$, and $\phi $ to be the map induced by the inclusion $\U0001d524\to {\U0001d524}_{\u2102}$.

If $\gamma \in G$ is central, then its image is in central in ${G}_{\u2102}$ since
$g\mapsto \gamma g{\gamma}^{-1}$ is a map extending $\phi $, and thus must be the
identity^{} by uniqueness half of the universal property^{}. Thus, if
$\mathrm{\Gamma}\subset G$ is a discrete central subgroup, then we get a map
$G/\mathrm{\Gamma}\to {G}_{\u2102}/\phi (\mathrm{\Gamma})$, which gives a complexification for
$G/\mathrm{\Gamma}$. Since every Lie group is of this form, this shows existence.

Some easy examples: the complexification both of ${\mathrm{SL}}_{n}\mathbb{R}$ and $\mathrm{SU}(n)$ is ${\mathrm{SL}}_{n}\u2102$. The complexification of $\mathbb{R}$ is $\u2102$ and of ${S}^{1}$ is ${\u2102}^{*}$.

The map $\phi :G\to {G}_{\u2102}$ is not always injective^{}. For example, if $G$ is
the universal cover of ${\mathrm{SL}}_{n}\mathbb{R}$ (which has fundamental group^{} $\mathbb{Z}$), then
${G}_{\u2102}\cong {\mathrm{SL}}_{n}\u2102$, and $\phi $ factors through the covering $G\to {\mathrm{SL}}_{n}\mathbb{R}$.

Title | complexification |
---|---|

Canonical name | Complexification |

Date of creation | 2013-03-22 13:53:55 |

Last modified on | 2013-03-22 13:53:55 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 7 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 22E15 |