# 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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### Sum of cubes of positive roots, $\alpha(H)^3$, of arbitrary Cartan element

Let $\mathfrak{h}$ the Cartan subalgebra of a simply-laced (ADE type) Lie algebra of rank $r$, $\mathfrak{g}$, and $\Phi$ the associated root system. Furthermore, let $\Phi^+$, $\Pi$, denote the ...
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### Exponentiating Lie algebra element does not result in algebraic group element

This question is about a particular case of the special orthogonal groups considered in section 23.4 of Borel's Linear Algebraic Groups. Let $k$ be a field of characteristic $\neq 2$. Let $F$ be a ...
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### Decomposition of finite-dimensional representation of semisimple Lie-Algebra into irreducible subrepresentations.

This is my first post on MathSE, since I could not find a helpful answer to my question on here yet. Let $(L,[\cdot,\cdot])$ be a finite-dimensional semisimple complex Lie-Algebra, $H\subseteq L$ a ...
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### Question regarding root systems of a complex Lie algebra

Let $\mathfrak g$ be a semi-simple Lie algebra with Cartan subalgebra $\mathfrak h$ and root system $\Phi$. I have seen it stated that for any $\alpha \in \Phi$, $-\alpha \in \Phi$, and that this is a ...
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### Root system of group of symplectic similitudes $GSp_{2n}$

Let $G = \operatorname{GSp}_{2n}$ be the group of symplectic similitudes. I am trying to work out certain cocharacters and have a question on the root system of this group: A maximal torus $T$ in $G$ ...
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### Centre of the algebra $\mathbb{Z} [\hat{W}]$ of the affine Weyl group

For $W$ the Weyl group of some root datum $(P, R, P^\vee, R^\vee)$ (here $P$ is the weight lattice, $R$ the root lattice, and I will write the group structure additively) associated to an algebraic ...
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### Levi subgroups and subsystems of root systems

Let $G$ be a connected reductive algebraic group over a local field $F$ with fixed maximal torus $T$, and denote by $R = R(G,T)$ the set of roots of $T$ in $G$, namely, the set of all nontrivial ...
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### Integrality property of the root system of a semisimple Lie algebra

For $\mathfrak{g}$ a semisimple Lie algebra, $\mathfrak{h}$ a choice of Cartan subalgebra, $\Phi$ the set of roots, there is the integrality property that if $\alpha$ is a root then $c \alpha \in \Phi$...
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### Maximally compact Cartan subalgebra of $SO(p,q)$ and its adjoint representation

This is a somewhat technical question so please bear with me. However, apparently my understanding leads to a contradiction, so I must be missing something basic, and I would be grateful to gain ...
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### How to prove that the root systems of Coxeter groups are discrete?

First I want to know the definition of "discrete" here. I guess it can be inferred from the bilinear form on the Coxeter datum. But it is still hard to organize the prove by finding the ...
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Let $e_i$ be the standard unit basis vector in $\mathbb{R}^{l+1}$ I'm trying to show that the $R = \{ \pm(e_i-e_j) : 1 \leq i < j \leq l+1\}$ is a root system for $E = spanR = \{ \Sigma \alpha_i ... 1answer 90 views ### Cartan Matrix from commutation relations Let a set of elements,$T^i_j$, with$i,j=1,\cdots,n$satisfying the$\mathcal{su}(n)$algebra $$[T^i_j, T^k_l] = \delta^k_j T^i_l - \delta^i_l T^k_j\,,\qquad (T^i_j)^\dagger = T^j_i.$$ There are$n^...
Let $\Phi$ be an irreducible root system of $\mathrm{rank}(\Phi) = n$. I am allowing the case where $\Phi$ is not reduced. Say that a subset $\Psi \subset \Phi$ is a root subsystem if it is a root ...