# Tagged Questions

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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### Is the assignment of a root system to a semisimple Lie algebra functorial?

As described here, we have a category of root systems, where a morphism from a root system $\Phi$ in a Euclidean space $E$ to a root system $\Phi'$ in $E'$ is given by a linear map $f: E \to E'$ such ...
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### How does the Weyl group act on the root system of type $B_n$?

Suppose $\{e_1,\dots,e_n\}$ are the standard unit vectors in $\mathbb{R}^n$. Then the root system of type $B_n$ consists of $\pm e_i$, and $\pm(e_i\pm e_j)$ for $i\neq j$. I know the Weyl group $W$ ...
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### Equivalent definitions of a root system.

For studying root systems many authors start from a vector space $V$ over $\mathbb{R}$ with a positive definite scalar product $(\cdot,\cdot)$, in which a reflection $\sigma_\alpha$ is a linear ...
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### Springer's proof existence of system of positive roots w.r.t. a Borel subgroup

Let $G$ be an algebraic group, $T$ a maximal torus and $B$ a Borel subgroup containing $T$. For every root $\alpha$ define $$G_\alpha:=C_G(\ker\alpha^0)$$ and $H_\alpha:=G_\alpha/R_u(G_\alpha)$. The ...
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### Selfnormalizing sub-algebra and direct sum decomposition

I got the following setting: Consider the decomposition $L=H+\sum_{\alpha\in \phi}L_{\alpha}$, where the sum is a direct sum, the $L_{\alpha}$ are the root spaces and $H$ is nilpotent (because its ...
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### Connection between quivers and representations of Lie algebras

Can anyone recommend a reference to study the connection between quiver theory and representation theory of Lie algebras? Supposedly those two things have something to do with each other, with the ...
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### How to prove that a certain type of isogeny of a reductive group is a Frobenius for some $\mathbb{F}_q$-structure

It is well known that a connected reductive linear algebraic group $G$ over $\mathbb{F} = \overline{\mathbb{F}_p}$ can be classified via its root datum $\Psi(G,T) = (X(T),\Phi, Y(T), \Phi^\vee)$. ...
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### $\alpha$ is a root $\implies -\alpha$ is a root

Let $\mathfrak{h}$ be a Cartan subalgebra of Lie algebra $\mathfrak{g}$. I want to prove: $\alpha\in \mathfrak{h}^*$ be a root of $\frak g$, $\implies$ so is $-\alpha$. Let $\mathfrak{h}$ ...
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### Find all possibles values. Looks Hard. Is it hard?

I need some help, (or advise) how I can solve this problem and in which category I need to put it. The problem state: Let m be a solution of the equation $y^{2015}-15 y+ 14=0$. Find all possible ...
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### How to calculate total number of roots in an A-type root system?

In particular, I am interested in $A_4$ root system. Considering simpler cases of $A_2$ and $A_3$ my guess would be $(n+1)^2-(n+1)$ (where n is rank of the root system), but I'm not certain if it's ...
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### Lie algebra-like structure corresponding to noncrystallographic root systems

In the classification of Coxeter groups, or equivalently root systems: $$A_n, B_n/C_n, D_n, E_6, E_7, E_8, F_4, G_2, H_2, H_3, H_4, I_2(p)$$ with $p \geq 7$, the last four fail to generate any ...
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### Affine Weyl group as Coxeter group

How do you write the affine Weyl group corresponding to type $A_n$ as a Coxeter group ? The generators are $s_0,s_1,\dots,s_n$ where $s_0$ corresponds to the highest root. What are all the relations ? ...
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### Does there exist $\gamma \in \Phi^+$ such that $(\gamma,\alpha)$ and $(\gamma,\beta)$ are non-zero?

Let's denote by $\Phi^+$ a positive root system of an irreducible root system $\Phi$. I have a quick question that: for any distinct positive roots $\alpha, \beta$, does there exist $\gamma \in \Phi^+$...
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### For a root system, why does $\beta\in\Delta_+\setminus\{\alpha_i\}$ imply $(\beta+\mathbb{Z}\alpha_i)\cap\Delta\subset\Delta_+$?

Let $\mathfrak{g}(A)$ be a Kac-Moody algebra for a matrix $A$, with root basis $\{\alpha_1,\dots,\alpha_n\}$. There is a remark on the bottom of page 6 of Kac's Infinite Dimensional Lie Algebras ...
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### How can one assume that an isomorphism of root spaces $\Phi\to\Phi'$ comes from an isometry?

By definition, if $\Phi$ and $\Phi'$ are root systems of the Euclidean spaces $E$ and $E'$, respectively, then an isomorphism $\Phi\to\Phi'$ is one that is induced by an isomorphism $E\to E'$ which ...
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### Coefficients of positive roots in term of simple roots

Let $\Phi$ be an irreducible root system and $\Phi^+$ be positive root system and $\Delta$ be base. For every positive root $\beta=\sum_{\alpha \in \Delta}m_\alpha\alpha$, the numbers $m_\alpha$ are ...
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### Adding tori to semi-simple groups

Let $G$ be a complex, connected, semi-simple Lie group (throw in simply connected if you like) with Lie algebra $\mathfrak g$. Let $T \subseteq B$ be a maximal torus and choice of Borel, respectively. ...
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### Write every element of a nilpotent Lie subgroup as product of exponentials of simple generators

I have a question about Lie groups. Let $G$ be a simply connected semi-simple complex Lie group and $\mathfrak{g}$ its Lie algebra. We consider a Cartan-Weyl basis of $\mathfrak{g}$, giving the usual ...
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### Equivalent definitions of positive root system

I begin with a definition of positive root systems of a root system over Euclidean space. A subset $\Delta$ of root system $\Phi$ is called a simple root system (or base) in $\Phi$ if (1) $\Delta$ ...
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### How to understand Weyl chambers? [duplicate]

Recall the definition of the Weyl Chambers: A Weyl Chamber is a region of $V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}$, where $V$ is underlying Euclidean space, and $H_\alpha$ the hyperplane ...
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### Fundamental group of a Root System and determinant of the Cartan matrix

This is the first time I am posting, so I hope I didn´t get the formatting wrong. I am currently reading J. E. Humphreys "Introduction to Lie Algebras and Representation Theory" and got stuck at ...
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187 views

### How to understand Weyl chambers?

Recall the definition of the Weyl Chambers: A Weyl Chamber is a region of $V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}$, where $V$ is underlying Euclidean space, and $H_\alpha$ the hyperplane ...