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3 votes
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Irreducible complex representations of some abelian Lie groups

These groups are all products of the groups $\mathbb{R}, S^1, C_2$. Using the fact that if the group of characters of $A$ is $\widehat{A}$ and the group of characters of $B$ is $\widehat{B}$ then the ...
Qiaochu Yuan's user avatar
2 votes
Accepted

What is the completion with respect to a topology

Define a function $$ |\ |_\infty:\mathbb{C}[t,t^{-1}]\longrightarrow\mathbb{R} $$ by $$ |P(z)|_\infty=2^N\qquad\text{if $P(z)=az^{-N}+\sum_{k>-N}a_kz^k$ and $a\neq0$} $$ and $|0|_\infty=0$. Then $|\...
Andrea Mori's user avatar
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1 vote
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example of non-compactly supported vector field.

For a compactly supported example, as suggested in the comments just take any vector field on a compact manifold. E.g. consider the vector field $X_{(x,y)} = x\partial_y - y\partial_x$ on the unit ...
Alekos Robotis's user avatar
1 vote
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The Lie algebra of an algebraic group

Let $X = \text{Spec } R$ be an affine scheme over a field $K$ and let $x \in X$ be a $K$-point, represented by an evaluation homomorphism $e_x : R \to K$, whose kernel is a maximal ideal $m_x$. The ...
Qiaochu Yuan's user avatar
1 vote
Accepted

What is the semidirect product we use in Levi Decomposition

$Rad(\mathfrak g)$ is an ideal; the only derivation in sight, and the only way that $\mathfrak{s}$ can act on this, is the adjoint action. Formally, take $$\pi: \mathfrak s \rightarrow Der(Rad(\...
Torsten Schoeneberg's user avatar
1 vote

What is the semidirect product we use in Levi Decomposition

$\operatorname{Rad}(\mathfrak{g})$ is an ideal of $\mathfrak{g}$ and $\mathfrak{s}\cong \mathfrak{g}/\operatorname{Rad}(\mathfrak{g})$ is semisimple with the multiplication $$ [X+\operatorname{Rad}(\...
Marius S.L.'s user avatar
  • 2,303

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