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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$\mathfrak{sl}_2$ has the root lattice of type $A_1$.

Let $L$ be a Lie algebra over $\mathbb{Z}$ constructed from a root lattice $R$. It is well-known that if $R=A_1$, then $L \cong \mathfrak{sl}_2$ and this is widely used example in many books on Lie ...
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59 views

Prove that the sum of all simple roots is a root

Let $\Delta$ be an indecomposable root system in a real inner product space $E$, and suppose that $\Phi$ is a simple system of roots in $\Delta$, with respect to an ordering of $E$. If $\Phi = ...
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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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25 views

Definition of coroots

I'm having a bit of trouble with the definition of coroots. From textbooks, we know that given a root system $\Phi$ with $\alpha \in \Phi$, there exists a coroot system $\Phi^{\vee}$ with ...
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9 views

Lenght of the affine transformation $s_{\varphi_n, \, 1} \cdot \dots \cdot s_{\varphi_1, \, 1}$

Let $\{\varphi_1, \, \dots , \, \varphi_n\}$ be a subset of positive roots of a root system $\varPhi$ and consider the affine reflections $\{s_{\varphi_n, \, 1}, \, \dots , \, s_{\varphi_1, \, 1} \}$ ...
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34 views

Root systems and the possible angles between roots.

On page 4 of these notes by John Dusel (http://math.ucr.edu/~jmd/Root_Systems.pdf) it reminds us that if we have a symmetric positive bilinear form (pageg 2) we can define an angle between vectors ...
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64 views

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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1answer
16 views

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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1answer
34 views

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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30 views

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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1answer
89 views

$\alpha$ 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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1answer
74 views

Action of $\text{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)$ on $X(T)$ permutes root system

I got these questions while reading Chapter 3 of "Representations of Finite Groups of Lie Type" by Digne-Michel. Let $T$ be a torus defined over $\mathbb{F}_q$. Then ...
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54 views

Root system of an abelian lie subalgebra.

Let $L$ be a lie algebra and $H$ an abelian subalgebra of $L$ such that each element of $h \in H$ is diagonalizable under the adjoint representation. So there exists a basis of common eigenvectors for ...
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1answer
36 views

Two questions on roots of finite, simple, complex lie algebra

Why are there at most two root lengths for a finite, simple, complex lie algebra? I know it is from the constraint that the $2(\alpha,\beta)/(\alpha,\alpha)$ is integer, but what is the argument? ...
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1answer
18 views

If $w'(\beta)<0$ and $\ell(w)+\ell(w')=\ell(ww')$, then $ww'(\beta)<0$?

There's a small step in a computation with root systems that eludes me. Suppose $w,w'$ are elements of the Weyl group (which is a Coxeter group) such that $\ell(w)+\ell(w')=\ell(ww')$. Suppose you ...
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47 views

Sum of traces over Weyl group

I'm interested in computing sums like $\sum_{\sigma \in W} tr(\sigma ^3)$ , where $W$ is the Weyl group of $SO(2n+1)$, i.e. $W = (\mathbb{Z}/2\mathbb{Z})^n \rtimes S_n$. I tried to figure out what an ...
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28 views

A chain between two positive roots having same length

Let $\Phi$ be an irreducible root system and $\Phi^+$ a system of positive roots. Denote by $\Delta=\{\beta_1, \beta_2,\ldots, \beta_n\}$ the corresponding base. My question concerns Jyrki's answer ...
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1answer
20 views

Roots of a simple Lie algebra as elements of a Cartan subalgebra

Consider a Lie algebra $L$ over $\mathbb{C}$ and consider its Cartan decomposition: $$L = H \oplus L_1 \oplus ... \oplus L_k.$$ For each $1 \leqslant i \leqslant k$, take a non-zero element $e_i$ in a ...
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38 views

Definitions of length function on a Weyl group

Let $\Phi$ be an irreducible root system and $W$ the Weyl group of $\Phi$. Denote by $\Delta=\{\alpha_1, \alpha_2,\ldots,\alpha_l\}$ the corresponding base. Can anyone give me the standard definition ...
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1answer
31 views

Roots of height 1 are necessarily simple.

Suppose $\Phi$ is a root system, $\Pi \subset \Phi$ is a fundamental system (let $\Pi = \{r_1,...,r_l\}$). Now any root $r \in \Phi$ is a linear combination of the elements of $\Pi$ with all the ...
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16 views

Understanding Weights and Roots

I'm refering to this book called Semi-Simple lie algebras in Particle Phsics by Cahn for understanding weights and roots as given by our instructor. It has a definition of weights on Pg.33 which is ...
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1answer
26 views

Reflection transforms positive root into a positive root.

I am reading the book "Simple groups of Lie type" by R.Carter, and stuck with the following lemma: Let $r \in \Pi$. Then $w_r$ transforms $r$ into $-r$ but every other positive root into a positive ...
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24 views

What is a weight vectors weight really?

What does a weight vector with weight $(\lambda_1,\cdots,\lambda_n)$ actually mean? Let $V$ be a $gl(n)$-module and I have that $v\in V$ is a weight vector if it is an eigenvector for all elements of ...
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1answer
80 views

A subset of roots whose mutual angles agree with those of a simple system

I would appreciate help/hints solving the following exercise from Humphreys book "Reflection Groups and Coxeter Groups", page 11, exercise 1. Let $\Phi$ be a root system of rank $n$ of unit ...
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27 views

Root system for Lie algebras, Does $t_{\alpha+\beta}=t_\alpha+t_\beta$?

For a root system, where $\alpha,\beta \in \Delta$ and $\alpha+\beta\in \Delta$ Does $t_{\alpha+\beta}=t_\alpha+t_\beta$? Where $t_\alpha$ I know is denoted in different ways depending on author, ...
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2answers
88 views

Every positive system of roots contains a unique simple system.

The following question is in effort to understand a proof to a theorem appearing in "Reflection Groups and Coxeter Groups" by Humphreys on page 8. Let $\Phi$ be a root system in the euclidean space ...
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1answer
87 views

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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38 views

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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1answer
36 views

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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36 views

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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95 views

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 ...
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1answer
32 views

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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1answer
116 views

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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1answer
48 views

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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45 views

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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137 views

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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34 views

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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1answer
129 views

Fundamental group of a Root System and determinant of the Cartan matrix

This is the first time I am posting, so I hope I didnĀ“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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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 ...
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48 views

Possible angles between roots in a root system

Given a Root System $\Phi$ let $\alpha,\beta \in \Phi$ with $\alpha \neq \pm \beta$ and $||\beta||\geq ||\alpha||$. Let $\theta$ be the angle between $\alpha$ and $\beta$. Since $<\alpha,\beta> ...
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95 views

Killing forms and Hermitian inner products

Let $K$ be a compact, connected, simply connected Lie group with Lie algebra $\mathfrak k$ and Killing from $B_{\mathfrak k}$. It is well known that $B_{\mathfrak k}$ is a negative definite symmetric ...
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1answer
77 views

How to show that set of diagonal matrices is the maximal toral subalgebra of sl(n)

sl(n) is the set of nxn matrices with trace=0. i know that sl(n) is a finite dimensional simple lie algebra and the maximal toral subalgebra of a finite dimensional semi simple lie algebra is abelian. ...
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77 views

Dominant Weight

I am reading a paper which begin by a reminder about root system associated to a simple lie algebra $\mathfrak g$. let $\mathfrak h\subset \mathfrak g$ a cartan subalgebra. Question 1: It says that ...
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32 views

Questions about the indivisible imaginary root in affine root system.

I am reading the paper. On page 5, $\delta$ is defined as the indivisible imaginary root in $\widehat{\Delta_+}$. $\Lambda_0 \in \widehat{\mathfrak{h}^*}$ is the unique element satisfying $\langle K, ...
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2answers
53 views

Positivity of a particular vector

How to prove that, for any $w$ $\in$ $W$ (Weyl group), $ \delta - w \delta $ is in positive part (non negative part) of the root lattice $\mathbb{Z}[\Delta]$ ? where $\Delta$ is a simple system in ...
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42 views

Root system of a simple lie algebra is irreducible

The proposition is from Humphreys. I don't understand how to prove the highlighted statements. How can I express a general element of K? I tried using Cartan decomposition of L but it doesn't work. ...
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1answer
77 views

Representing Petersen graph in root system $E_6$

It is well-known that Petersen graph is an strongly regular graph with parameters (10,3,0,1) and can be considered as complement graph of $L(K_5)$ and its spectrum is $\{3,1^5,(-2)^4\}$. Also, It is ...
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211 views

Why are root spaces of root decomposition of semisimple Lie algebra 1 dimensional?

I'm trying to understand root system of semisimple Lie algebra but having trouble following one of the step in the note which explain why each root spaces are 1-dimensional. According to the note, ...