Complexification of a Lie group

2020 Mathematics Subject Classification: Primary: 22E [MSN][ZBL]

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). 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 [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.

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