(adsbygoogle = window.adsbygoogle || []).push({}); [SOLVED] Exercises on Lie-algebras

I'm doing some homework on Lie-algebra's, and there are several things I am missing. Let me list them below in one thread.

- The statement is "If
Lis a classical linear Lie-algebra, thenLis semisimple." I have reduced this to proving Rad L [itex]\subseteq[/itex] Z(L) -- then I have that Rad L = Z(L), so L is reductive and I can use an earlier result to conclude that Rad L = 0 so L is semisimple. But I don't know how to show this. Since what I need to prove does not hold in general, I should plug in some (common) property of classical linear Lie-algebras, but I can't really think of anything useful here... any ideas?

- Finally, I am required to prove that
If any Lie-algebra L satisfies Rad L = Z(L), then all finite-dimensional representations of L in which Z(L) is represented by semisimple endomorphisms are completely reducible(Humphreys' book on Introduction to Lie-algebras and Representation theory, exercise 6.5d). So I took such a representation and found that I can simultaneously diagonalize all elements in the center, but I don't really see how this will give me any information on the reducibility of the entire representation.

I know that I haven't really posted a lot of work, but I did try to reduce the problem to the points I'm really stuck at. I obviously don't expect a fully worked answer, but I'd really appreciate some hints or general steps to solve these problems. Thanks you very much!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Exercises on Lie-algebras

**Physics Forums | Science Articles, Homework Help, Discussion**