weight (Lie algebras)
WeightLieAlgebras
2002-12-04 16:31:00
2005-06-22 11:05:21
Definition
diagonalisable
diagonalizable
multiplicity
weight space
representation
Cartan
Lie
abelian
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Let $\mf{h}$ be an abelian Lie algebra, $V$ a vector space and
$\rho\colon\mf{h}\to\End V$ a representation. Then the representation
is said to be \emph{diagonalisable}, if $V$ can be written as a direct
sum
\begin{equation*}
V=\DirectSum_{\lambda\in\mf{h}^*}V_\lambda
\end{equation*}
where $\mf{h}^*$ is the dual space of $\mf{h}$ and
\begin{equation*}
V_\lambda=\{v\in V\mid\rho(h)v=\lambda(h)v\text{ for all
}h\in\mf{h}\}.
\end{equation*}
Now let $\mf{g}$ be a semi-simple Lie algebra. Fix a Cartan subalgebra
$\mf{h}$, then $\mf{h}$ is abelian. Let $\rho\colon\mf{g}\to\End
V$ be a representation whose restriction to $\mf{h}$ is
diagonalisable. Then for any $\lambda\in\mf{h}^*$, the space
$V_\lambda$ is the \emph{weight space} of $\lambda$ with respect to
$\rho$. The \emph{multiplicity} of
$\lambda$ with respect to $\rho$ is the dimension of $V_\lambda$:
\begin{equation*}
\mult_\rho(\lambda):=\dim V_\lambda.
\end{equation*}
If the multiplicity of $\lambda$ is greater than zero, then $\lambda$
is called a \emph{weight} of the representation $\rho$.
A representation of a semi-simple Lie algebra is determined by the
multiplicities of its weights.