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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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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27 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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37 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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1answer
98 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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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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65 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 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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77 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 ...
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28 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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49 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
32 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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40 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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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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29 views

Length of fundamental weights equal to length of the corresponding simple roots?

Let $R$ be a root system with simple roots $\Delta$. For all $\alpha\in\Delta$ let $\omega_\alpha$ be the fundamental weight associated to $\alpha$. Is then the length of $\alpha$ equal to the length ...
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38 views

positive roots remain positive

Let $W$ be the Weyl group of $SL_{n+1}$ and $w \in W$. Let $R^+$ denote the set of positive roots with respect to the Borel subgroup of upper triangular matrices. Define $R^+(w)=\{\alpha \in R^+: ...
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2answers
24 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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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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37 views

Reflection in the highest root

Is there some canonical (reduced) expression (in terms of simple reflections) for the reflection associated to the highest (short) root? Is this the same as the longest element of the Weyl group? ...
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62 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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92 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, ...
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39 views

Relation between simple roots and fundamental weights.

Let $\alpha_1, \ldots, \alpha_n$ be simple roots of a semisimple complex Lie algebra. Let $\omega_1, \ldots, \omega_n$ be the fundamental weights. We have $$ \alpha_i = \sum_{s} k_s \omega_s, $$ for ...
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24 views

how to compute the norm of a root from the Cartan matrix?

As far as I understand, the Cartan matrix is associated with a unique semi simple algebra. How can we compute the norm of a root $\alpha$ from it since its components are invariant under rescaling? ...
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Length function

Let $W$ be a Coxeter group with simple system $S$, positive system $P$ and root system $R$. Then $S\subset P\subset R$. Let $\lambda:R\rightarrow\{0,1\}$ be the characteristiv function of $P$, in ...
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Root Systems and Dynkin diagrams.

On page 142, the textbook An Introduction to Lie Groups and Lie Algebras (by Kirillov) determines the fundamental group of the root system $A_2$. Basically, the author says we have two simple roots ...
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26 views

Standard set of Generators

A standard set of generators for a semisimple Lie algebra $ L $ is defined as: {${x_\alpha}, {y_\alpha}, {h_\alpha} $} Where: $ x_\alpha \in L_\alpha, $ $ y_\alpha \in L_{-\alpha}, $ $ ...
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27 views

Weyl Chambers of $ B_2$

How many Weyl Chambers/bases does $ B_2$ have? I thought it was 8, but if instead of for bases using obtuse root pairs you use orthogonal pairs, you get 8 different chambers intersect partially ...
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29 views

Codimension of $\ker $ $\alpha $

Can someone explain why the codimension of $\ker $ $\alpha $ is $1$ in $ H $, with complement $ Fh_\alpha $? Is this because $ h_\alpha $ when $ \alpha $ is simple is part of the dual basis to ...
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Dual spaces: Roots and Cartan subalgebra

Can someone show that the roots and the Cartan subalgebra are dual vector spaces? I don't see how simple roots acting on non-corresponding indices of a Cartan basis produce 0 and a simple root ...
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44 views

Do involutions suffice to find reflected vectors in a reflection group representation?

Consider a reflection group $W$ acting by isometries on a Euclidean space $V$. I want to understand the union of $(-1)$-eigenspaces for this action, the set $$\{v \in V : \exists w \in W\ (w\cdot v = ...
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29 views

two Roots questions

Just two questions on roots... 1) Can the length of roots only be defined relatively? And does length only come about because of the dot product and cartan integers? 2) This might be a weird ...
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2answers
76 views

Closure of a Fundamental Weyl Chamber

Can someone explain what a "closure" of a Fundamental Weyl Chamber means? I assume it is related to an algebraic closure, but I don't see how. In addition, how does the Weyl group act on it and why ...
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89 views

killing form and the dot product

When going from talking about roots as functionals to talking about roots as vectors in a Euclidian space (root system), does the killing form become the dot product? Are the killing form and dot ...
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39 views

Killing form and Roots

I know that the roots of a Lie Algebra are functionals such that if $\alpha$ is a root and $h \in \mathfrak h$ is an element of the Cartan subalgebra, then $\alpha(h)$ is an eigenvalue. I'm looking ...
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Angles between adjacent roots in a reduced root system.

Let $R$ be a reduced root system. ($R$ is a finite set spanning $V$, $\alpha \in R \rightarrow -k\alpha \in R$ iff $k=1$, $s_{\alpha}(R)=R$, $s_{\alpha}(\beta)-\beta=k\alpha$ whit $k$ integer). ...
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What are the root systems for the n-dimensional torus?

My question may seem silly at first, but currently I am not able to work out the question of finding all roots for the n-dimensional torus. At first, it seemed obvious to me that there are no roots at ...
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310 views

Proving that there exists a saturated set with given highest weight

This is an question about an exercise in Humphreys book on Lie algebras. First of all a bunch of definitions and notation, see §13 in Humphreys for details. Let $\Phi$ be a root system, $\Delta$ a ...
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46 views

coxeter graph and root system

I want to show that a coxeter graph $\Gamma$ is connected if and only if its root system $\Phi$ is irreducible. So let $\Delta$ be a simple system of $\Phi$, and $\Delta$ is also our simple system. ...
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1answer
89 views

Dynkin Diagram $SU(n)$

The goal is to give the Dynkin diagram of $SU(n)$. One can show that the complexification of the Lie algebra $\mathfrak{g}$ of $G$ is given by $\mathfrak{G}_{\mathbb{C}}=\mathfrak{sl}(n,\mathbb{C})$ ...
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109 views

angles between simple roots are obtuse, problem with proof

Let $\Phi$ be a root system in the following sense: (1) $\Phi \subset \mathbb{R}^n$ consists of a finite number of nonzero vectors, (2) for each $\alpha \in \Phi$, $\Phi \cap \mathbb{R} \alpha = ...
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1answer
78 views

Angle between roots in a root system

Let $\Phi$ be a root system in the following sense: (1) $\Phi \subset \mathbb{R}^n$ consists of a finite number of nonzero vectors, (2) for each $\alpha \in \Phi$, $\Phi \cap \mathbb{R} \alpha = ...
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1answer
79 views

vectors with the same mutual angles

Let $S = \{v_1,\ldots,v_n\} \subset \mathbb{R}^n$ and let $T = \{w_1,\ldots,w_n\} \subset \mathbb{R}^n$ be such that the angle between $v_i$ and $v_j$ is equal to the angles between $w_i$ and $w_j$. ...
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262 views

Is there a definition of a dual Lie algebra?

Let $L$ be a Lie algebra. For vector spaces, modules, Banach spaces, etc. we have the notion of a dual. Question: Is it possible to define naturally a Lie algebra $L^*$ that is in some sense dual to ...
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Conjugacy of simple system in a root system

I'll set up the problem, then ask the question. Let $V$ be a finite dimension vector space over $\mathbb{R}$ and let $\Phi$ be a root system in $V$, i.e. (1) $\Phi \cap \mathbb{R} \alpha = ...
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How does a root datum determine a root system?

A root datum is given by: A subset $R$ of a free abelian group $M$ A subset $C$ of the dual free abelian group Hom$(M,\mathbf{Z})$ A bijection between $R$ and $C$ subject to conditions. A root ...
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A construction of $\mathfrak{e}_8$ in Fulton and Harris

In section $22.4$ of "Representation Theory: A First Course" by Fulton and Harris, the exceptional Lie algebra $\mathfrak{e}_8$ is constructed using a method of Freudenthal. For background, I will ...
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127 views

Inverse of Cartan matrix

The Cartan matrix of the root system $A_n$ looks like, denote it by $A'_n$ $$A'_n= \begin{bmatrix} 2 & -1 & 0 & 0&\ldots & 0 \\[0.3em] -1 & 2 & -1 ...
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Roots and Weights

I use a Mathematica package to compute roots and weights (and other things) but the package gives me only the expression of the roots in $\omega$-basis (basis of fundamental weights) and in the ...
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31 views

Kac-Moody root datum introductory text?

I have been given a project to describe the construction of the Lie algebra associated to a Kac-Moody root datum $D=(I,A,\Lambda, (c_i)_{i\in I}, (h_i)_{i\in I})$. I only know basic definitions: that ...
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How to prove that $B^\vee$ is a base for coroots?

Let $\Phi$ be a root system in a real inner product space $E$. Define $\alpha^\vee = \frac{2\alpha}{(\alpha, \alpha)}$. Then the set $\Phi^\vee = \{\alpha^\vee: \alpha \in \Phi \}$ is also a root ...
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31 views

Any explicit pictures of root datum

Consider the root datum $(X^*, \Delta,X_*, \Delta^{\vee})$ of a reductive algebraic group, where $X^*$ is the lattice of characters of a maximal torus, $X_*$ the dual lattice (given by the 1-parameter ...
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the real roots of a connected tame quiver

Consider a tame quiver $Q$ whose underlying undirected graph is connected. So that undirected graph is one of the extended, simply laced Dynkin graphs; it's either $\tilde A_n$ for some $n\ge 1$ or ...