# Complexification of a Lie group

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The complex Lie group containing as a real Lie subgroup such that the Lie algebra of is a real form of the Lie algebra of (see Complexification of a Lie algebra). One then says that the group is a real form of the Lie group . For example, the group of all unitary matrices of order is a real form of the group of all non-singular matrices of order with complex entries.

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

Not every real Lie group has a complexification. In particular, a connected semi-simple Lie group has a complexification if and only if is linear, that is, is isomorphic to a subgroup of some group . 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 of a real Lie group . Here is a complex Lie group and is a real-analytic homomorphism such that for every complex Lie group and every real-analytic homomorphism there exists a unique complex-analytic homomorphism such that . The universal complexification of a Lie group always exists and is uniquely defined [3]. Uniqueness means that if is another universal complexification of , then there is a natural isomorphism such that . In general, , but if is simply connected, then and the kernel of is discrete.