From this question, we have some non-matrix Lie groups, but all of them are real Lie groups, is there a non-matrix complex Lie group? Or a complex Lie group which is not a linear algebraic group?
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How about $\mathbb{C}^{n}$ with addition? – Seewoo Lee Apr 26 '18 at 06:09
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@See-Woo Lee $\mathbb{C}$ is naturally isomorphism with the 2×2 upper triangular unipotent group I+$\mathbb{C}$e$_{12}$, do the direct sum by n copies, we get $\mathbb{C}$$^n$ as a matrix Lie group. – Strongart Apr 27 '18 at 13:59
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Take an elliptic curve over the complex numbers. It is a complex Lie group, however it can not be a matrix Lie group because any matrix Lie group is affine and elliptic curves are projective.
Jesko Hüttenhain
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Very nice example! Can we say that all holomorphic maps from a connected compact complex manifold to $\mathbb{C}$ are constant? – orangeskid Sep 19 '21 at 05:09
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Dear @orangeskid, I think the best idea would be to ask that as a separate question. Cheers! – Jesko Hüttenhain Sep 19 '21 at 17:59