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 Jul2 awarded Curious Sep30 awarded Benefactor Sep30 revised Are there simple examples of Banach spaces with no non-trivial Clifford Isometries? rolled back to a previous revision Sep30 revised Are there simple examples of Banach spaces with no non-trivial Clifford Isometries? edited body Sep30 awarded Cleanup Sep30 revised Are there simple examples of Banach spaces with no non-trivial Clifford Isometries? rolled back to a previous revision Sep30 revised Are there simple examples of Banach spaces with no non-trivial Clifford Isometries? added 14 characters in body Sep25 comment Complex algebraic group is reductive $\iff$ it is the complexification of a compact Lie group? (part 2) I would be happy with this, $except$ they seem to define complex reductive groups in terms of their Lie algebra. They define a complex Lie algebra $\mathfrak{g}$ to be reductive if it splits as $\mathfrak{g}=\mathfrak{z}(\mathfrak{g})\oplus[\mathfrak{g},\mathfrak{g}]$ and define a complex algebraic group $G$ to be reductive if its tangent algebra is reductive. But in this case, isn't the additive group $\mathbb{C}$ a counterexample? I feel like I'm missing a basic point here. Sep25 comment Complex algebraic group is reductive $\iff$ it is the complexification of a compact Lie group? (Part 1) Thank you for your answer. One of the lines in the UofT notes is a really good summary of the type of statement found throughout the literature on this question: "I think this is an equivalence of categories". After a lot more searching, I think I may have found what I want in the book "Lie Groups and Algebraic Groups" by Onishchik & Vinberg but there's a problem. In Chapter 5, Section 2, they establish a 1-1 correspondence between compact Lie groups (up to differentiable isomorphism) and reductive complex algebraic groups (up to polynomial isomorphism) given by complexification. Sep23 awarded Promoter Sep20 comment Complex algebraic group is reductive $\iff$ it is the complexification of a compact Lie group? I'm not quite sure what you mean. Could you please expand your answer? Sep19 comment Complex algebraic group is reductive $\iff$ it is the complexification of a compact Lie group? @TobiasKildetoft I believe you can realize a compact Lie group as a real algebraic group in $GL_n\mathbb{R}$ and then complexification can be interpreted as taking the complex zeroes of its defining polynomials in $GL_n\mathbb{C}$. Sep19 comment Complex algebraic group is reductive $\iff$ it is the complexification of a compact Lie group? @studiosus I am most concerned with $\implies$. Sep19 revised Complex algebraic group is reductive $\iff$ it is the complexification of a compact Lie group? deleted 10 characters in body Sep19 asked Complex algebraic group is reductive $\iff$ it is the complexification of a compact Lie group? Jun17 comment What is reductive group intuitively? @Brad I can't seem to find a source for the statement that every complex reductive algebraic group is the complexification of a compact Lie group. Do you happen to know one? May12 awarded Self-Learner Mar18 awarded Yearling Feb15 awarded Critic Jan29 comment Global sections of $\mathcal{O}(-1)$ and $\mathcal{O}(1)$, understanding structure sheaves and twisting. There seems to be a typo on the second line of the important example $(1)$: Should $s_i=\frac{z_j}{z_j}s_i$ read $s_i=\frac{z_j}{z_i}s_j$ ?