Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'm sorry to bother but i'm having some problems in proving that, given a simple Lie Algebra L of finite dimension $n$ (equipped with the Killing form) and its enveloping universal algebra U(L), then the element (Casimir): c = $\sum x_iy_i$ where $(x_i)_i$ is a basis and $(y_i)_i$ is its dual basis (with respect to the Killing form) doesn't depend on the choice of a particular basis.

The argument should be just related to linear algebra i guess. I tried to take another basis and the change-of-coordinates-matrix but i can't get to the solution.

Can someone help me? Am I missing something? Thank you

share|cite|improve this question

The idea is that, given a finite dimensional vector space, $V$ we have a natural isomorphism $V\otimes V^*\cong \hom(V,V)$ given by $v\otimes \phi \mapsto \phi(\cdot)v$ (and extending linearly). Thus, we have a unique element corresponding to the identity map, which we can give explicitly as follows. Let $v_1, \ldots, v_n$ be a basis of $V$, and let $\phi_1, \ldots, \phi_n$ be the corresponding dual basis. The identity corresponds to $\sum v_i \otimes \phi_i$. This is the casimir element, after we use the Killing form to establish an isomorphism $V\cong V^*$.

share|cite|improve this answer
Thank you very much. I'm sorry to ask a (probably stupid) question, but formally (once we've identified $V$ and $V^*$ the element $\sum v_i \times \phi_i$ lies in the tensor algebra of $L$, correct? And it's a homogeneus tensor of order 2.. that's why it's ok to say $\sum v_i \phi_i \in U(L)$? – claudia Jun 8 '11 at 18:42
@claudia It does lie in the tensor algebra. While it is homogenous of order $2$, we do not need this: there is a surjective map of rings $T(L)\to \mathcal{U}(L)$, and we simply identify the element with its image. – Aaron Jun 8 '11 at 18:50
The isomorphism map should be: $v\otimes \phi \mapsto \phi(\cdot)v$. – AlbertH Jun 8 '11 at 19:04
@AlbertH Fixed. Thank you. – Aaron Jun 8 '11 at 19:11
You're perfectly right. I'm not very familiar with these things. Thank you very much for you help. – claudia Jun 8 '11 at 19:29

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.