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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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 ...
2
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1answer
20 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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0answers
21 views

Root spaces of Lie Algebras — semisimple vs. general

(I am mainly following the notation of Roger Carter's Lie Algebras of Finite and Affine Type). Letting $L$ denote a (finite-dimensional) Lie algebra with roots $\Phi$ and Cartan subalgebra $H$, we ...
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0answers
16 views

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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1answer
18 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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2answers
114 views

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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1answer
26 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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1answer
14 views

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 ...
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1answer
27 views

Auto-Langlands dual gruops.

Consider a semisimple Lie group $G$. We define the Langlands dual $\hat{G}$ of $G$ as the group which has as a root system, the root system generated by the coroots of $G$. Recall that given a root ...
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1answer
26 views

Weyl group and weight lattice chambers.

Consider two simple Lie groups $G_1$ and $G_2$. Let $G_1$ have $W_1$ as a Weyl group and $G_2$ have $W_2$ as a Weyl group. Is it true that the Weyl group of $G_1 \times G_2$ is $W_1 \times W_2$? ...
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0answers
36 views

Which is the Weyl group of $U(n)$

Consider the unitary group $U(n)$. How does one compute its Weyl group? Is it the same as the Weyl group of $SU(n)$ since $U(n)\simeq SU(n)\times U(1)$?
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1answer
30 views

Why is that an automorphism that preserves $B$ and $H$ an automorphism of $\Phi$ that leaves $\Delta$ invariant?

Let $L$ be a semisimple finite dimensional Lie algebra, $H$ its CSA and $\Phi$ its root system with base $\Delta$ and $B = B(\Delta) = H\bigoplus_{\alpha \succ 0}L_\alpha$. If we have an automorphism ...
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1answer
47 views

How does a semisimple Lie algebra determine its root space?

I understand that given a root system $\Phi$, by Serre's theorem there exists a Lie algebra $L$ with root system $\Phi$. Also isomorphism theorem implies that any two such $L$ are isomorphic. That ...
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1answer
44 views

Simple Lie algebras have irreducible root systems?

I was unable to see why $(\alpha+\beta,\alpha) \ne 0$ and $(\alpha+\beta,\beta)\ne0$ implies $\alpha+\beta \not\in\Phi$. Everything else is fine. $\quad$*Proposition.* Let $L$ be a simple Lie ...
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1answer
50 views

— Cartan matrix for a semisimple Lie algebra with an extension

The question is a modified one inspired by this post: What is the Cartan matrix for this Lie algebra below? (for this semisimple Lie algebra $g(X) \oplus h(Y)$,) $$ [X_i, X_j] = f_{ij}{}^k X_k ...
2
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1answer
51 views

Regarding root space decomposition

In Humphreys, given a finite dimensional semisimple Lie algebra $L$ and a maximal toral subalgebra $H$, $$L_\alpha := \{x\in L|[hx] = \alpha(h)x\;\forall h\in H\}$$ Then since $ad_L\;H$ is a commuting ...
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84 views

— Cartan matrix for an exotic type of Lie algebra --

(1) Is there a notion of Cartan matrix for non-semisimple Lie algebra? For example, consider this Lie algebra: $$ [X_i, X_j] = f_{ij}{}^k X_k \qquad\qquad [X_i,Y^j] = - f_{ik}{}^j Y^k \qquad\qquad ...
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26 views

Question about closed subsets of a root system of a vector space?

Let $S$ be a root system of an euclidean space $V$. A subset $T \subseteq S$ is called closed if the following holds: $a,b \in T$ and $a+b \in S$ implies $a+b \in T$. Now let $T$ be a closed subset ...
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1answer
82 views

Problem 9.7 - Lie Algebras - Humphreys

Let $\alpha,\beta\in\Phi$ span a subspace $E'$ of $E$. Prove that $E'\cap\Phi$ is a root system in $E'$. Prove similarly that $\Phi\cap(\mathbb{Z}\alpha+\mathbb{Z}\beta)$ is a root system in E' (must ...
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0answers
19 views

Maximal noncompact forms in classical Lie algebra?

In this short note on Lie algebra, discussing about classical Lie algebra A,B,C,D class, in page 4 after Eq.(7), on the part of B,D class of O(2n,F) and O(2n+1,F) group (or algebra?), there is a ...
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0answers
41 views

Quaternion, Dihedral groups and A-D-E classification

$\bullet$ What is the role of Quaternion group $H$ and dihedral groups $D_n$ in A-D-E classification? $\bullet$ Is Quaternion group $H$ in $A$ (special linear Lie algebra of traceless operators) or ...
4
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0answers
62 views

Weyl orbits of integral dominant weights and convex polytopes

Let $\xi$ be an integral dominant weight of a root system $\Delta$, and let $\mathcal{O}_{\xi}$ be its orbit under the action of the Weyl group. The elements of the orbit are the vertices of an ...
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0answers
47 views

Coxeter numbers for semisimple and reductive algebraic groups

I'd like to know how to define the coxeter number for semisimple and reductive algebraic groups. I know that for a simple algebraic group $G$, we can fix a maximal torus $T\subset G$, which acts on ...
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0answers
20 views

Base of a root system

Let $R \subset V$ be a reduced root system, and $R' \subset R$. Assume that: (i) $\alpha \in R' \ \to \ - \alpha \notin R'$, (ii) $ \alpha, \beta \in R'$ and $\alpha + \beta \in R$ implies $\alpha ...
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3answers
57 views

Examples of root, parabolic, and borel subgroups corresponding to roots

I'm interested in seeing a few examples of root, parabolic, and Borel subgroups given a specific reductive group $G$. Here is what I know. Let $G$ be a reductive algebraic group over an ...
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0answers
25 views

parabolic subalgebra

Let $G$ be a semisimple lie group, a parabolic subgroup of $P$ is a connected subgroup that contains a conjugate of $B$, (which $B$ is Borel subgroup of $G$) then I can not see why lie algebra of $P$ ...
2
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1answer
46 views

Equation On Root Systems (Humphreys Exercise 9.10)

I am stuck in the following problem from Humphreys. Let $\alpha, \beta$ be roots in a root system $\Phi$. Let the $\alpha$-string through $\beta$ be $\beta - r\alpha, \ldots, \beta + q\alpha$ and let ...
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0answers
31 views

Bounding the inner product in root systems.

Let $R$ be a root system (irreducible if that makes this easier) in the real vectorspace $E$. Let $\lambda$ and $\mu$ in $E$ with $w_0(\lambda)\leq \mu \leq \lambda$ where $w_0$ is the longest ...
2
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0answers
58 views

Weyl group of orthogonal group

My question is why a particular element of the Weyl group of $O(8)$ seems to contradict a theorem about root systems. But to tell you the particular element I have to tell you specifically how I'm ...
2
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0answers
39 views

Relationship between invariants of a simple algebraic group

Let $G$ be a simple algebraic group over an algebraically closed field $k$. I believe all of the following invariants are well-defined. Besides the coxeter number, I haven't read about the others, ...
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1answer
100 views

Why is a root system irreducible iff its Dynkin diagram is connected?

I think this shouldn't be a difficult proof, but I have problems in proving the implication $\Phi$ irreducible root system $\Longrightarrow$ Dynkin diagram associated $D_\Phi$ connected. I tried ...
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0answers
60 views

Dimension of the root spaces of a semisimple complex Lie algebra

I have problems in understanding the proof that the root spaces of a semisimple Lie algebra are all 1-dimensional and that the only multiples of a root $\alpha \in \Phi$ which occur in $\Phi$ are $\pm ...
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0answers
45 views

What is the structure of the Coxeter groups of type $\text{D}_n$

I am curious on the structure of the Coxeter group $G$ of type $\text{D}_n$. Here I let $\{e_1,\cdots,e_n\}$ be the standard basis of the vector space $\mathbb{R}^n$. Then I choose ...
2
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0answers
53 views

Weyl group of a restricted root system

What is the order of the Weyl group of the restricted root system of the real Lie algebra $\mathfrak g= \mathfrak{so}(p,q)$? More precisely, $\mathfrak g= \mathfrak k \oplus \mathfrak p$ and let ...
4
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2answers
196 views

Relation between root systems and representations of complex semisimple Lie algebras

I'm trying to understand the machinery of root systems for the purpose of classifying complex semisimple Lie algebras. During this process i lost the overview, espacially when it came to highest ...
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0answers
62 views

the orbit of a root under operations of irreducible crystallographic group?

Suppose we have an irreducible crystallographic coxeter group G acting in a vector space V, how can we show that the orbit of an ...
4
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1answer
87 views

Same Dynkin diagrams $\Longrightarrow$ Isomorphic root systems.

I am studying the book Introduction to Lie algebras. In page 122 there is something I don't understand and I am looking for some help. In the beginning of that page the authors give the definition of ...
3
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1answer
132 views

How to find root subgroups

$\newcommand{\GL}{\text{GL}}\newcommand{\diag}{\text{diag}}$For $G = \GL_n(k)$ let $B$ be the upper triangular matrices and $T$ be the diagonal matrices in $G$. In this case I understand that the ...
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1answer
103 views

Classifying all rank 2 and 3 root systems

I am working with the representation theory of complex simple Lie algebras, and have a question: It is intuitively clear that the root systems $A_1\times A_1$, $A_2$, $B_2$, and $G_2$ comprise all ...
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0answers
50 views

Why is the dual space of Cartan subalgebra an irreducible representation of Weyl group

it is proposition 14.31 in Fulton-Harris book. The proof goes like this. Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$, and assume $\mathfrak{z}\subseteq\mathfrak{h}^*$ were preserved ...
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1answer
67 views

The Weyl group preserves inner product

Show that the Weyl group $W$ preserves the inner product: $(w(\lambda)\,,\, w(\mu)) = (\lambda, \mu)$ for all $w\in W$ and $\lambda, \mu\in E$. I know it suffices to check this on reflections ...
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0answers
81 views

Root system of a Lie Algebra

Could anybody help me to solve this problem with roots system? Be $\Phi$ an irreducible root system. $\Phi^{+}$ a choice of positives roots in $\Phi$. Prove that if $(\alpha,\beta)\ge0$ $\forall ...
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1answer
119 views

Dimension of Lie algebra according to root system

I was wondering how is it possible to find the dimension of a semi-simple lie algebra $L$ if its corresponding root system is (lets make it simple) of type $B_2$. We can find the number of roots and ...
2
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2answers
178 views

Length of root strings is at most 4

Let $\Phi$ be a root system. In his Introduction to Lie algebras and Representation Theory, J. Humphreys proves that if $$\beta-p\alpha,\dots,\beta,\dots,\beta+q\alpha$$ is the $\alpha$-root string ...
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0answers
85 views

$\Delta \subset \Phi$ is a base in a root system imples $\Delta^\vee \subset \Phi^\vee$ is a base in a root system

(the notation here is compatible with J.E. Humphrey's "Introduction to Lie Algebras and Representation Theory") Let $\Phi \subset E$ be a root system. Let $\Delta \subset \Phi$ be a base. I already ...
2
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2answers
123 views

Greatest elements in crystallographic root systems

I have a question regarding a remark in the book "Reflection Groups And Coxeter Groups" by James E. Humphreys (unfortunately the book is not to be found as a whole on google books or such). In ...
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1answer
89 views

Weyl group of this root system is $S_n$?

Let $E=\{(x_1,\dots,x_n)\in \mathbb{R}^n:\sum x_i=0\}$ is of dimension $n-1$ take root system $R=\{e_1-e_j:1\ge i,j\le n\}$, I know that it is of rank $n-1$ root system, but why its weyl group is ...
0
votes
1answer
54 views

weyl group act by conjugation?

let $W$ be a weyl group and $\alpha\in R$ we have $s_{w(\alpha)}=ws_{\alpha}w^{-1}$, to prove this the author says $ws_{\alpha}w^{-1}$ acts as identity on $wL_{\alpha}=L_{w(\alpha)}$ , and ...
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1answer
105 views

not all automorphisms of a root system are elements of weyl group

could any one tell me why not all automorphisms of a root system are elements of weyl group? For example in $A_n, n>2$ the automorphism $\alpha\mapsto -\alpha$ is not in the weyl group. I do not ...