PlanetMath: weight (Lie algebras)

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weight (Lie algebras)

(Definition)

Let $\mathfrak{h}$ be an abelian Lie algebra, a vector space and $\rho\colon\mathfrak{h}\to{\mathrm{End}}V$ a representation. Then the representation is said to be diagonalisable, if can be written as a direct sum

$\displaystyle V=\bigoplus\limits _{\lambda\in\mathfrak{h}^*}V_\lambda$

where $\mathfrak{h}^*$ is the dual space of $\mathfrak{h}$ and

$\displaystyle V_\lambda=\{v\in V\mid\rho(h)v=\lambda(h)v$ for all $\displaystyle h\in\mathfrak{h}\}.$

Now let $\mathfrak{g}$ be a semi-simple Lie algebra. Fix a Cartan subalgebra $\mathfrak{h}$ , then $\mathfrak{h}$ is abelian. Let $\rho\colon\mathfrak{g}\to{\mathrm{End}} V$ be a representation whose restriction to $\mathfrak{h}$ is diagonalisable. Then for any $\lambda\in\mathfrak{h}^*$ , the space $V_\lambda$ is the weight space of $\lambda$ with respect to $\rho$ . The multiplicity of $\lambda$ with respect to $\rho$ is the dimension of $V_\lambda$ :