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I am learning about weights of subalgefbras of $\mathfrak{gl}(V)$ where $V$ is a vector space over some field $\mathbb{F}$.

The definition that I have read states

Let $M$ be a subalgebra of $\mathfrak{gl}(V)$. A weight of $M$ is a linear map $\lambda : M\rightarrow\mathbb{F}$ such that $V_\lambda := \left\{ v\in V\mid A(v)=\lambda(A)\cdot v \quad \forall A\in M \right\}$ is a non-zero subspace of $V$. Then $V_\lambda$ is called a weight subspace associated with the weight $\lambda$.

So I was thinking about this definition a bit, and it reminded me of eigenvectors and eigenvalues in linear algebra. I then thought a bit more and came up with that, in a way, the weight subspace is a set of "global eigenvectors" for $M$. By which I mean all vectors in the weight subspace get mapped to scalar multiples of themselves by all elements of $M$. Then, the weight $\lambda:M\rightarrow \mathbb{F}$ gives the "associated eigenvalue" corresponding to $A\in M$.

I was wondering whether this view is correct (or indeed helpful) in understanding this particular notion, or have I got the wrong end of the stick?

Thanks,

Andy.

Kenny Wong
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andrew1601
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1 Answers1

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Yes, that is exactly correct. Each $V_\lambda$ is a simultaneous eigenspace for every linear map in $M$. $\lambda$ is the function that assigns to every linear map in $M$ the eigenvalue associated to its action on vectors in $V_\lambda$.

This example may be helpful: Take $V = \mathbb C^3$ and take $$M = \left\{ \left( \begin{array}{ccc} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array} \right)\in \mathfrak{gl}(V) \mid a+ b+ c = 0 \right\} .$$ (This is actually a Cartan subalgebra of $\mathfrak{sl}_3 \mathbb C$, by the way.)

The three simultaneous eigenspaces are $$ V_{\lambda_1} = \mathbb C \left( \begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right) , \ \ \ V_{\lambda_2} = \mathbb C \left( \begin{array}{c} 0 \\ 1 \\ 0 \end{array} \right), V_{\lambda_3} = \mathbb C \left( \begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right), $$ and the weights, which are functions assigning each matrix in $M$ to its eigenvalue when acting on the respective space, are $$ \lambda_1 : \left( \begin{array}{ccc} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array} \right) \mapsto a,$$ $$ \lambda_2 : \left( \begin{array}{ccc} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array} \right) \mapsto b,$$ $$ \lambda_3 : \left( \begin{array}{ccc} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array} \right) \mapsto c,$$

Note that in this example, $a + b+ c = 0$ for all matrices in $M$. Therefore, $$ \lambda_1 + \lambda_2 + \lambda_3 = 0.$$ so only two of the three weights are linearly independent. This "dependency between weights" is quite a common feature of weights of Lie algebras.

Kenny Wong
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