All Questions
4 questions
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Dual of short exact sequence of Lie algebras exact?
Let $\varphi:\mathfrak{g}\to\mathfrak{q}$ be a morphism of finite-dimensional Lie algebras over a field $K$. Define $\varphi^\vee:\mathfrak{q}^\vee\to\mathfrak{g}^\vee$, $l\mapsto l\circ\varphi$, ...
2
votes
1
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184
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Complete reducibility of a Lie algebra $\mathfrak{g}$ and splitting of short exact sequence of $\mathfrak{g}$-modules.
Suppose that $\mathfrak{g}$ is a semisimple Lie algebra. By Weyl's theorem on complete reducibility, $\mathfrak{g}$ is completely reducible. Now my book says that the following is equivalent:
For ...
- 1,237
2
votes
1
answer
279
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Splitting of Lie algebra extensions: why does this linear map exist?
Let $\mathfrak{b}$ and $\mathfrak{g}$ be finite dimensional Lie algebras and let $(\tilde{\mathfrak{g}},j,\phi)$ be a Lie algebra extension of $\mathfrak{g}$ by $\mathfrak{b}$, so we have a short ...
- 1,237
0
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227
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Why do linear splitting maps of Lie Algebra central extensions induce cocycles?
If we consider a central extension $\mathfrak h$ of a Lie algebra $\mathfrak{g}$ by the abelian $\mathfrak a$:
$$0 \longrightarrow \mathfrak a \longrightarrow \mathfrak h \stackrel{\pi}\...
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