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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If $\lambda \prec \eta$ are positive roots, then there exists another root $\zeta$ such that $\lambda \prec \zeta \prec \eta$.

Let $\Phi$ be an irreducible root system with a base $\Delta$ and $\lambda, \eta$ be positive roots such that $\lambda \prec \eta$ and $ht(\eta)-ht(\lambda)\geq 2$. Question: Does there exist another ...
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1answer
20 views

Constructing the root diagram for $B_2$

I'm trying to self-teach some Lie theory, and in particular I'm trying to construct the root diagram for $B_2$. I've found 8 roots, labelled $\pi_1,\pi_2,-\pi_1,-\pi_2,\pi_1+\pi_2,-(\pi_1+\pi_2),\pi_1+...
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0answers
22 views

$SO(p,q)$ Fundamental Weights?

The weights in the $D^{n-1}$ and $D^{n}$ spinor representations of $SO(2n)$ are of the form $$\frac{1}{2}(\pm e_1 \pm e_2 \pm ... \pm e_{n-1} \pm e_n)$$ such that the products of all the $\pm 1$'s are ...
2
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1answer
33 views

Lie Algebras: Root algorithm for the positive roots of $G_2$

I am trying to compute the positive roots of $G_2$ starting from its simple roots. Our instructor has given us an algorithm to do that which goes as follows: First of all, if $\Sigma=\{\alpha_1,\dots,\...
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49 views

Meanings of root system: Show definitions/intuitions and $Λ_{root} ⊂ Λ_{char} ⊂ Λ_{weight}?$

Let G be a compact Lie group. Let $g_{\mathbb{C}}$ be the associated complex Lie algebra. There is a root system associated to $g_{\mathbb{C}}$. Denote this by a 4-tuple $$R = \{Λ_{root}, Λ_{weight}, ...
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1answer
96 views

There are no semisimple Lie algebras of dimension $4$, $5$, or $7$

I came across the claim here which states that there are no complex semisimple Lie algebras of dimension $4$, $5$, or $7$. As the problem suggests, we can take a Cartan subalgebra $H$ and root system $...
2
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1answer
43 views

Isomorphism between root systems: why not assume isometry?

In the definition of an isomorphism of root systems, Humphreys emphasizes that it is not assumed that the map is an isometry, it should just preserve the Cartan integers. I don't understand why we ...
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0answers
11 views

Are the following real roots in the root system $\widetilde{E}_6$?

The Dynkin diagram of type $\widetilde{E}_6$ root system is \begin{align} & \ \circ \\ & \ | \\ & \ \circ \\ & \ | \\ \circ - \circ - & \circ - \circ - \circ \end{align} Let $\...
2
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1answer
27 views

Classification of non-simple Lie algebras

Over the past couple of months I had the chance to study the classification of compact simple Lie algebras. During this time I've always been wondering if these results can be extended to more general ...
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26 views

Extending isomorphism theorem about simple Lie algebras to semisimple case

My question is from Humphreys Introduction to Lie Algebras and Representation Theory. The theorem (in Sec. 14.2) states that if $L, L'$ are simple Lie algebras with maximal toral subalgebras $H, H'$ ...
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1answer
47 views

If the Weyl group $\mathcal W$ is a normal subgroup of $\mathrm{Aut}(\Phi)$, how can the Weyl group of $A_2$ be dihedral of order $6$?

If the Weyl group $\mathcal W$ is a normal subgroup of $\mathrm{Aut}(\Phi)$, how can the Weyl group of $A_2$ be dihedral of order $6$? The roots of $A_2$ are $\{\pm \alpha, \pm \beta, \pm(\alpha+\...
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1answer
54 views

Construction of the $E_8$ root system

The following construction of $E_8$ root system is introduced in J.Humphreys' Introduction to Lie Algebras and Representation Theory(3rd printing, p.65). Some notations are slightly changed from the ...
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1answer
48 views

Brent's Method convergence criteria

I am using Brent's method to solve the BEM equations for a wind turbine model. I have run into a scenario where Brent's method has converged i.e., abs(m) is below set tolerance of 1e-8 but the value ...
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0answers
28 views

A remark about $\rho$ and $G$ being simply connected

Let $G$ be a reductive complex algebraic group, $H \subset G$ a Cartan subgroup and $R^+$ a set of positive roots, and $X_+(H)$ the set of dominant weights. Let us also assume that $G$ is simply ...
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1answer
102 views

Quotient manifold of a finite group action

Let R be a (crystallographic) root system on an Euclidian space $(E,⟨−,−⟩)$ and $$W:=gen\{σ_r∣r∈R\}=gen\{σ_r∣r∈R^+\}$$ its associated reflection group. Taking $$M=E-\cup_{r\in R^+} H_r,$$ where $H_r$ ...
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2answers
45 views

Differing definitions of Cartan subalgebras

I'm quite new to Lie algebras in general and I have recently come across two differing definitions of a Cartan subalgebra. The first is from J-P. Serre's book 'Complex Semisimple Lie Algebras' wherein ...
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1answer
60 views

Weyl group of type $A_n$

Let $E$ be the subspace of $\mathbb{R}^{n+1}$ for which the coordinates sum to $0$ and let $\Phi$ be the set of vectors in $E$ of length $\sqrt{2}$ and which are integers vectors. It is known that $\...
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1answer
60 views

Finding Lie algebra from relation of roots in root system

Let $\Phi$ be the corresponding root system of a finite dimensional semisimple Lie algebra. Let $\alpha_i,\alpha_j\in \Phi$ be simple roots. I want to find all Lie algebras which satisfy $$\langle \...
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1answer
62 views

Orientation in reflection groups

Let R be a (crystallographic) root system on an Euclidian space $(E,\langle-,-\rangle)$ and $$W:=gen\{\sigma_r \mid r\in R\}=gen\{\sigma_r \mid r\in R^+\}$$ its associated reflection group. If $r_1$, $...
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0answers
24 views

Values of $b(h_a)-a(h_b)$ with $a,b$ being roots in root system of a Lie algebra

Let $a,b$ be roots in a root system of a finite dimensional complex semisimple Lie algebra. I want to determine the possible values of $b(h_a)-a(h_b)$. The difference equals zero when $a=b$ (clearly) ...
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0answers
32 views

Weyl group of a compact Lie group vsWeyl group of a root system

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$ and $T$ be a maximal torus with Lie algebra $\mathfrak{t}$. I read that the Weyl group $W$ of $G$ is "the group of automorphisms of $T$ ...
2
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1answer
28 views

Determine Cartan matrix from conditions

Let $\{a,b,c\}$ be a set of simple roots of the Lie algebra $B_3$ and suppose $|a|=|c|$ and $\langle b,c\rangle=0$. I want to find the corresponding Cartan matrix. I know that it's a $3\times3$ ...
2
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1answer
29 views

The relative weights form a basis of $\mathfrak a_P^{G \ast}$

Let $G$ be a connected, reductive group over $\mathbb Q$ with minimal parabolic $P_0 = M_0 N_0$. For $P = MN$ a standard parabolic subgroup of $G$, let $A_P$ be the split component of $M$ and let $\...
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0answers
20 views

Why is a specific subalgebra of semi-simple Lie algebra a $\mathfrak{sl}(2,\mathbb{F})$ submodule?

I'm currently struggeling to understand a certain step in a proof. Let $L$ be a semi-simple Lie algebra and let $H$ be a maximal toral subalgebra. Define $$L_\alpha = \{x\in L\mid [xh]=\alpha(h)x\...
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1answer
28 views

Generalised eigenvectors of a finite dimensional vector space

I'm currently reading Knapp's 'Lie Groups, Beyond an Introduction', and I can't seem to understand this point. Let $V$, be a finite dimensional $\mathbb{C}$-vector space, $\pi: \mathfrak{h} \...
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1answer
24 views

Does one get any set of positive roots by “cutting” with a hyperplane?

If $\Phi$ is the root system of a complex semisimple Lie algebra $\mathfrak{g}$, then one way to choose a set $\Phi^+$ of positive roots is by choosing a hyperplane which does not contain any root, ...
2
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1answer
98 views

Basic question regarding roots from a Lie algebra in $\mathbb{R}^2$

I'm trying to reconstruct a root diagram of a Lie algebra akin to the attached image. I've constructed all the root vectors but I'm struggling to see how one practically can view the roots as being in ...
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0answers
29 views

Finding the simple roots of the classical Lie algebra $G_2$ using some kind of diagram

Given the Cartan Matrix $$\begin{pmatrix}2&-1 \\ -3& 2\end{pmatrix}$$ one can find all the positive roots via the following algorithm. Let $\beta$ be the long root and $\alpha$ the short root....
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38 views

Exercise on the highest root

Problem : Let $\phi$ be an irreducible root system and $\phi^{+}$ a choice of positive roots. Prove that if $(\alpha, \beta) \geq 0$, $\forall \beta \in \phi^{+}$, then $\alpha$ is the highest root ...
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1answer
30 views

There is an element of every possible length in $[W_{\theta} \backslash W]$

Let $(W,S)$ be the Weyl group of a root system with base $\Delta$, and let $\theta \subset \Delta$. Let $W_{\theta}$ be the group generated by $\theta$. It is a general result that in every right ...
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0answers
32 views

Reference for application of the theory of weights and weight-spaces to $\mathfrak{su}(2)$ and $\mathfrak{su}(3)$

In the last couple of weeks I've been reading J.E. Humphreys' "Introdcution to Lie Algebras and Representation Theory" and after finishing Chapter 3 I had a chat with my TA about the topics*. He told ...
2
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1answer
115 views

Extending base vector field via tensorproduct from $\mathbb{Q}$ to $\mathbb{R}$

Let $L$ be a finite dimensional vector space over some field $\mathbb{F}$ ($\mathrm{char}(\mathbb{F})=0$), $H\subset L$ a subspace of $L$ with $\mathrm{dim}_{\mathbb{F}}(H)=h$ and let $H^*$ be its ...
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1answer
108 views

Exercise on root system of type $A_n$

Problem Let $n$ be a positive integer and let $\phi$ be a root system of type $A_n$. Let $\Delta = \{ \alpha_1, .. , \alpha_n \}$ be a base, such that the Dynkin diagram is a string enumerated from $...
2
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0answers
99 views

How can we explain this symmetry in counting the number of positive roots?

I thought I could figure this out just be reading up on root systems and weights, but I couldn't find an easy way to explain it. I'll ask a specific version of the question, but this observation might ...
2
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1answer
26 views

What is the basis of the root space $\mathfrak g_\theta$, where $\theta = \alpha_1+\cdots+\alpha_n$?

Let $\mathfrak g = \mathfrak{sl}_{n+1}$ with its canonical basis $\{x^\pm_i, h_i\: i=1,\cdots, n\},$ where $x_i^+ = e_{i,i+1}, x_i^- = e_{i+1,i}, h_i = e_{i,i}-e_{i+1,i+1}$ and $e_{ij}$ denotes the ...
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1answer
37 views

Show that every dominant fundamental weight is decomposed as a positive rational sum of roots.

I'm studying Humphreys's Introduction to Lie Algebras and Representation Theory, and I'm working on the following problem from the book (Chapter 13, problem 8): Let $\Phi$ be irreducible. Prove ...
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0answers
30 views

Unique maximal short root

Let $\Phi$ be irreducible. Prove that $\Phi^\vee$ is also irreducible. If $\Phi$ has all roots of equal length, so does $\Phi^\vee$ (and then $\Phi^\vee$ is isomorphic to $\Phi$). On the other hand, ...
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0answers
24 views

When is the group action transitive?

Let $W$ be a finite reflection group with irreducible root system $\Phi$. Let $\Delta$ be the set consisting of reflections $s_{\alpha}$ corresponding to pairs $\{\alpha, -\alpha \}$ of roots in $\...
5
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1answer
40 views

Is the sum of coroots the coroot of the sum?

While studying the book Introduction to Lie Algebras and Representation Theory, by Humphreys, I've came across a problem that seems simple, but I just cannot figure out: If $\alpha, \beta \in \Phi$,...
2
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1answer
115 views

Given Cartan matrix, find root and Dynkin diagram

Given: Cartan matrix of $\tilde{\mathfrak{so}}=B_2=\begin{pmatrix}2 & -2 \\ -1 & 2 \end{pmatrix}$ The formula for the components of a Cartan matrix $\textbf{A}$ is $A_{ij}=2\frac{\alpha_i\...
2
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1answer
122 views

Highest Roots of Classical Lie Algebra

I want to compute the length of the highest root of the classical lie algebra. The classical lie algebra are $\mathfrak{sl}(r+1)$, $\mathfrak{so}(2r+1)$, $\mathfrak{sp}(2r)$, and $\mathfrak{so}(2r)$. ...
2
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1answer
112 views

Where does non-reduced root system come up?

A reduced root system $R$ (over $\mathbb{R}$) is one which satisfies the condition that $\mathbb{R}\alpha \cap R $ consists of only $\alpha$ and $-\alpha$ for every root $\alpha$ (following Bourbaki's ...
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2answers
68 views

What does Bourbaki mean by displacement in Lie Groups and Lie Algebras chapter 4-6

In Bourbaki Lie Groups and Lie Algebras chapter 4-6 the term displacement is used a lot. For example groups generated by displacements. But I can not find a definition of the term displacement given ...
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1answer
60 views

Why the commutator of two elements is in the Cartan subalgebra?

I was reading Fulton and Harris' book when I found the following theorem: In the proof of (ii) they say that $[X, Y] \in \mathfrak{h}$. Why is it true? (It is an obvious corollary of this statement:...
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1answer
38 views

About expressing $w\rho-\rho$ as a sum of roots

Let $\Phi$ be a root system, $\Phi^+$ be the positive system, $\rho$ be the half sum of positive roots, and $W$ be the Weyl group of $\Phi$. I remember that there is a way to express $w\rho-\rho$ ...
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1answer
75 views

Length of a root string - proof clarification

I am trying to understand the proof In Fulton's and Harris's Representation Theory book where they show that the length of the root is at most 4: Theorem If $\alpha,\beta$ are roots with $\beta \neq \...
2
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2answers
125 views

Reflection reverses a root string

I am trying to understand the part of the proof In Fulton's and Harris's Represantation Theory book where he shows that the length of the root is at most 4: Theorem If $\alpha,\beta$ are roots with $\...
3
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0answers
48 views

Intersection of Levi subgroups via intersection of their Weyl groups

Let $G$ be a connected reductive group over $\mathbb{C}$. We fix a maximal torus $T\subset G$. Let $M,L \subset G$ be its Levi subgroups containing $T$ (note that we do note assume that $M,L$ are ...
3
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1answer
156 views

Picture of Root System of $\mathfrak{sl}_{3}(\mathbb{C})$

Let $\mathfrak{h} \subseteq \mathfrak{sl}_{3}(\mathbb{C})$ be the CSA consisting of the diagonal matrices and R the corresponding roots. Then R is a root system in $\mathfrak{h}^{\ast}$. I always see ...
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1answer
44 views

$x^4-6x^3+(13-m^2)x^2 - 12x+4$ [closed]

How to use Vieta's formula? Or is there Simpler way to do? What i know is $x_1 + x_2 = \frac{-b}{a}$, $x_1 . x_2 = \frac{c}{a}$

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