Semisimple Lie subalgebra closed under JC decomposion

Asked Jun 04 '17 at 10:21

Active Jun 05 '17 at 18:04

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I have a question about the tagged argument in Prop. 8.5.1: Let $g$ be a semisimple Lie subalgebra of $gl(V )$.

For any $x \in g$ we have J-C decomposion $ x= x_s +x_n$ but that's not clear to me why $ x_s ,x_n \in n_{gl(V)}(g) $ where $ n_{gl(V)}(g) = \{y \in gl(V) | [y,x] \in g $ for every $ x \in g \} $.

By the way: it would be enough to prove $ad(x_s)(g) ⊆ g, ad(x_n)(g) ⊆ g$, but I don't now how...

edited Jun 05 '17 at 18:04

asked Jun 04 '17 at 10:21

user267839

7,293

Unfortunately, I have no answer, but what book is that? – I am not Paul Erdos Jun 04 '17 at 12:58
1

Hi. That's from an online script. Here is the link to the whole pdf document: relaunch.hcm.uni-bonn.de/fileadmin/perrin/chap8.pdf – user267839 Jun 05 '17 at 16:38
Interesting. Thanks – I am not Paul Erdos Jun 05 '17 at 20:22
If you still stuck on that question you can take a look at the book Introduction to Lie algebras and Representation Theory" of Humphreys (page 29) he exaplains this there. – I am not Paul Erdos Jun 12 '17 at 10:43

Semisimple Lie subalgebra closed under JC decomposion

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