# Tag Info

• 7,168

### G abelian Lie group, $(\pi, V)$ a finite-dim unitary rep. $\exists$ mutually orthog. 1-dim invariant linear subspaces s.t $V= \bigoplus_i V_i$

Let $\{\pi(g):g\in G\}\subset GL(V)$. These are unitary operatots that also commute ($\pi(g_1)\pi(g_2)=\pi(g_1g_2)=\pi(g_2g_1)=\pi(g_2)\pi(g_1))$. Moreover, as they are unitary, they are also ...
• 4,386
Accepted

### details in the proof: G a commutative Lie group and $(\pi, V)$ a finite-dim unitary representation of G. Then $\pi$ is irreducible iff dim $V = 1$

A representation assumes that the map $\pi:G\times V\to V$ is continuous. So the composition $V\to \{g\}\times V\subseteq G\times V \xrightarrow{\pi} V$ is continuous. The topologies taken are the ...
• 2,674
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### Why is $V \longrightarrow V: v \mapsto \pi(g)(v)$ an endomorphism of $G$-modules?

Lets write down explicitly: $$f:V\to V$$ $$f(v)=\pi(g)(v)$$ Or simply $f=\pi(g)$. It is linear because $\pi(g)$ is. It is a bijection because $\pi(g^{-1})$ is inverse of $\pi(g)$. So the only question ...
• 39.9k

### How do projective representations map elements? Are they mulivalued?

I have rewritten my answer. Let $V$ be a vector space on a field $k$. Let us recall that if $V$ is a vector space, $\mathbb{P}(V)$ is the set of equivalence classes of vectors up to the multiplication ...
• 2,232

• 23.4k
1 vote
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### dual representation map as a linear map

Every linear map $f:V\to W$ induces a linear map $W^*\to V^*$. This linear map is denoted by any of the following symbols: $f^*,f^t,f’$, and is called the dual map or adjoint map or transpose map or ...
• 51.1k
1 vote
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### How can I show this property about lie algebras and lie subgroups?

Continuing off of your approach, it suffices to show that $\overline{\mathfrak h}$ is commutative. Consider any $X,Y \in \overline{\mathfrak h}$. $e^{sX}$ and $e^{tY}$ commute for all $s,t \in \Bbb R$ ...
• 222k
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### Extensions of simple Lie groups with a topological splitting

Yes, indeed, if $H$ is abelian then the Lie group central extensions of $G$ by $H$ are classified by the 2ned continuous cohomology $H^2_c(G; H)$. A good survey can be found in Stasheff, James D., ...
• 94k
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### Is there a way to convert an element of $\mathfrak{so}(3)/\mathfrak{so}(2)$ into a "geometric" tangent vector on $S^2$?

Consider $\Bbb S^n\subseteq \Bbb R^{n+1}$, write $\Bbb R^{n+1} = \Bbb R^n \times \Bbb R$, and fix the north pole $p = (0,1)$. Note that $T_p(\Bbb S^n) = \Bbb R^n\times \{0\}$. Then ${\rm SO}(n+1)$ ...
• 76.3k

### What are the invariant polynomials of representations of the Lorentz group $SO^+(3,1)$ and $SL(2,\mathbb{C})$?

The case $SL(2)$ is the well known problem of the classical invariant theory. You may find lists of invariants for example here
• 7,899
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• 2,487
1 vote
Accepted

1 vote

### Weyl Group of Parabolic subgroups

In this case the Weyl group is trivial. For $SL(n,\Bbb R)$, if we set $A_o$ equal to the usual diagonal split component of $B$ then the Weyl group of $A_o$ is $S_n$ acting by permuting the diagonal ...
• 11
Accepted

### Uniform open subgroups of $p$-adic Lie group

No: there is no guarantee that the subgroup generated by $H_1$ and $H_2$ is still torsion-free. For a simple example, let $\Gamma=\mathbb{Z}_p\times\mathbb{Z}/(p)$, let $H_1$ be the closed subgroup ...
• 326k
$p$ is a fiber bundle without assuming the $G$ action on $X$ has closed orbits. In fact we can write down a local trivialization of $p$ in terms of a local trivialization of $G \to G/H$. Consider a ...