View source for Complexification of a Lie group

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The ''complexification of a Lie group $G$ over $\R$'' is
a complex Lie group $G_\C$ containing $G$ as a real Lie subgroup such that the Lie algebra $\def\fg{ {\mathfrak g}}\fg$ of $G$ is a real form of the Lie algebra $\fg_\C$ of $G_\C$ (see
[[Complexification of a Lie algebra|Complexification of a Lie algebra]]). One then says that the group $G$ is a real form of the Lie group $G_\C$. For example, the group $\def\U{ {\rm U}}\U(n)$ of all unitary matrices of order $n$ is a real form of the group $\def\GL{ {\rm GL}}\GL(n,\C)$ of all non-singular matrices of order $n$ with complex entries.

There is a one-to-one correspondence between the complex-analytic linear representations of a connected simply-connected complex Lie group $G_\C$ and the real-analytic representations of its connected real form $G$, under which irreducible representations correspond to each other. This correspondence is set up in the following way: If $\rho$ is an (irreducible) finite-dimensional complex-analytic representation of $G_\C$, then the restriction of $\rho$ to $G$ is an (irreducible) real-analytic representation of $G$.

Not every real Lie group has a complexification. In particular, a connected semi-simple Lie group has a complexification if and only if $G$ is linear, that is, is isomorphic to a subgroup of some group $\GL(n,\C)$. For example, the universal covering of the group of real second-order matrices with determinant 1 does not have a complexification. On the other hand, every compact Lie group has a complexification.

The non-existence of complexifications for certain real Lie groups inspired the introduction of the more general notion of a universal complexification $(\tilde G,\tau)$ of a real Lie group $G$. Here $\tilde G$ is a complex Lie group and $\tau : G\to \tilde G$ is a real-analytic homomorphism such that for every complex Lie group $H$ and every real-analytic homomorphism $\alpha : G\to H$ there exists a unique complex-analytic homomorphism $\beta : \tilde G\to H$ such that $\alpha=\beta\circ \tau$. The universal complexification of a Lie group always exists and is uniquely defined {{Cite|Bo}}. Uniqueness means that if $(\tilde G',\tau')$ is another universal complexification of $G$, then there is a natural isomorphism  $\lambda : \tilde G\to \tilde G'$, such that $\lambda\circ\tau = \tau'$. In general, $\dim_\C\;\tilde G \le \dim_\R G$, but if $G$ is simply connected, then $\dim_\C \tilde G = \dim_\R G$ and the kernel of $\tau$ is discrete.

See also
[[Form of an algebraic group|Form of an algebraic group]].

====References====
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|valign="top"|{{Ref|Bo}}||valign="top"|  N. Bourbaki,   "Elements of mathematics. Lie groups and Lie algebras", Addison-Wesley  (1975)  (Translated from French)  {{MR|0682756}} {{ZBL|0319.17002}}
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|valign="top"|{{Ref|Na}}||valign="top"|  M.A. Naimark,   "Theory of group representations", Springer  (1982)  (Translated from Russian)  {{MR|0793377}} {{ZBL|0484.22018}}
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|valign="top"|{{Ref|Zh}}||valign="top"|  D.P. Zhelobenko,   "Compact Lie groups and their representations", Amer. Math. Soc.  (1973)  (Translated from Russian)  {{MR|0473097}} {{MR|0473098}} {{ZBL|0228.22013}}
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