Complex Lie groups

Question

A text states every complex connected Lie group must be abelian. Now surely this must be an error and has to be wrong because the unitary group is certainly (has to be complex in general) complex except for the real subgroups such as real orthogonal group and it is NOT abelian. For an easy example $SU_2$ for which the generators are the 2 dim. Pauli spin matrices in which the usual representation has $\sigma_2=\sigma_y$ say as imaginary and then taking $\exp(\sqrt{-1}t_i\sigma_i)$ for arbitrary real scalars $t_i$,implicit repeated sum over $i$ $1$ to $3$ and all possible infinite products of such is certainly a complex Lie group and not abelian. Eg obviously the subgroups $\exp(\sqrt{-1}\sigma_{1\text{ or }3})$ has imaginary generators $\sqrt{-1}\sigma_{1\text{ or }3}$ in a familiar basis. So would we not all agree the statement is false?

Answer 1

It is not true that every complex connected Lie group must be Abelian; for instance, $SL(2,\Bbb C)$ is not Abelian. However, it is true that every complex, connected and compact Lie group must be Abelian (this is a consequence of the maximum modulus principle). And, yes, $SU(2)$ is not abelian. There is no contradiction here, since $SU(2)$ is a real Lie group but not a complex one (notice that its dimension as a real differentiable manifold is $3$; if it had the structure a complex differentiable manifold, its dimension as a real differentiable manifold should be even).