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What is the difference between algebraic Lie groups and complex Lie groups? quelle est la différence entre les groupes de Lie algébriques et les groupes Lie complexes?

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  • $\begingroup$ Some complex Lie groups are not linear algebraic, for example complex tori, as they admit no holomorphic faithful representation. Others, for example many solvables, admit such, but sit inside the matrices as a complex analytic subgroup, cut out by complex analytic functions which are not algebraic. I am not sure what kind of answer you could hope for. $\endgroup$ – Ben McKay Dec 26 '19 at 20:41
  • $\begingroup$ Does there exist a notion of algebraic Lie groups at the first place? When the definitions are ready, then one can tell the difference. $\endgroup$ – Yuhang Chen Dec 26 '19 at 20:48
  • $\begingroup$ Perhaps I should say that every algebraic group has a maximal abelian variety subgroup, which is normal, with linear algebraic quotient. So you might want to picture how abelian varieties differ from generic complex tori, and how linear algebraic groups differ from generic linear analytic groups. $\endgroup$ – Ben McKay Dec 26 '19 at 20:49
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    $\begingroup$ @YuhangChen: I think the question is about algebraic groups defined over the complex numbers, in the sense of algebraic geometry. $\endgroup$ – Ben McKay Dec 26 '19 at 20:50
  • $\begingroup$ therefore the Lie group which has subgroups of dimension 1 (copy of \ C) it is an algebraic Lie group the case of a complex Lie group is not always true? $\endgroup$ – Samir Dec 26 '19 at 20:54
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The question is discussed in Chapter III of these notes https://www.jmilne.org/math/CourseNotes/LAG.pdf

As the comments indicate, the answer is complicated.

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