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Sep
30
revised Are there simple examples of Banach spaces with no non-trivial Clifford Isometries?
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Sep
30
revised Are there simple examples of Banach spaces with no non-trivial Clifford Isometries?
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Sep
30
awarded  Cleanup
Sep
30
revised Are there simple examples of Banach spaces with no non-trivial Clifford Isometries?
rolled back to a previous revision
Sep
30
revised Are there simple examples of Banach spaces with no non-trivial Clifford Isometries?
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Sep
25
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.
Sep
25
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.
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awarded  Promoter
Sep
20
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?
Sep
19
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 $GL_n\mathbb{C}$.
Sep
19
comment Complex algebraic group is reductive $\iff$ it is the complexification of a compact Lie group?
@studiosus I am most concerned with $\implies$.
Sep
19
revised Complex algebraic group is reductive $\iff$ it is the complexification of a compact Lie group?
deleted 10 characters in body
Sep
19
asked Complex algebraic group is reductive $\iff$ it is the complexification of a compact Lie group?
Jun
17
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?
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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 $s_i=\frac{z_j}{z_i}s_j$ ?