# Tag Info

Accepted

### Endomorphisms of a Lie algebra representation

Things are not quite as easy. If you have non-zero eigenvectors for different weights, then the sum of these spaces is not an eigenspace. Moreover, any homomorphism of $\mathfrak g$-modules preserves ...
• 17.7k
Accepted

### Why should I expect the generators of Lie Groups to be closed under the commutator?

So the proper formal answer should be that the vector fields on a Lie group have a natural Lie bracket (as do all vector fields on a general manifold). Then the Lie algebra can be identified with the ...
• 2,220

• 368k
Accepted

### Image of the adjoint representation of a Lie algebra

No, this is not true. It would say that a discrete subgroup $K\subset G$ of a connected Lie group would automatically lie in the kernel of $Ad_G$ and hence in the center of $G$. But this is not true ...
• 17.7k
1 vote

If someone is interested here an answer based on Qiaochu Yuan‘s suggestions and definitions: „Exercise 1a“: $\forall g \in G \ \ \forall w \in W \ \ \forall X \in \mathfrak{g}$ $\rho(g)w \in W \... • 660 1 vote Accepted ### Prove that this action admits three orbits You have already identified the 3 orbits of$T$on$C$by the way you have written$C$. Namely, the two points of$N(T)/T$are both$T$-fixed, and so constitute 2 orbits. For the 3rd, note that$\$ \...
• 2,295

Only top scored, non community-wiki answers of a minimum length are eligible