# Questions tagged [root-systems]

For questions involving abstract root systems, their associated Weyl groups and Dynkin diagrams, as well as their applications to Lie theory, graph theory, or other related fields.

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### Irreducible factors of universal enveloping algebra

Let $\mathfrak{g}$ be a simple complex Lie algebra and $U$ be its universal enveloping algebra. We have an action $\mathfrak{g} \to \mathfrak{gl}(U)$ by extending the adjoint action of $\mathfrak{g}$ ...
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### Verifying that the angle between $\alpha$ and $\beta$ is $\frac {2 \pi} {3}.$

Consider the Lie algebra $sl_{3} (\mathbb C)$ of dimension $8.$ Let \begin{align*} H : & = \text {span} \left \{H_1 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 ...
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### Action of the absolute Galois group on the based root datum

Let $G$ be a connected reductive group over a field $k$ with separable closure $k^s$ and $\Gamma=\text{Gal}(k^s/k)$. Pick $T\subset B$ maximal split torus and Borel subgroup for $G_{k^s}$. I am aware ...
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### Root space decomposition of a semisimple Lie algebra.

Let $L$ be a semisimple Lie algebra. I am trying to understand root space decomposition of $L$ on my own. Since $L$ is semisimple, $L$ possesses an abelian maximal toral subalgebra i.e. an abelian ...
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