= Complex Lie group
= Complex analytic Lie group
{synonym}
A complex <Lie group> is a <complex manifold> whose multiplication and inversion are <holomorphic maps between complex manifolds>. Its <tangent space> at the identity is a complex <Lie algebra>; the <Adjoint representation of a Lie group> and the <Lie exponential map> are holomorphic. Compatibility with a <complex manifold> structure is essential: an arbitrary even-dimensional real <Lie group> need not have holomorphic multiplication.
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