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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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Identification of cartan subalgebra and its dual - root space decomposition

For a semisimple lie algebra $\mathfrak{g}$ we have a root space decomposition with Cartan Subalgebra $\mathfrak{t}$. We often consider the dual $\mathfrak{t}^*$ and want to identify it with $\...
Rick's user avatar
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About elements in affine extended Weyl group

I'm not sure how to prove a statement about extended Weyl groups. Let $V$ be a finite vector space over $\mathbb{R}$, with a positive definite symmetric bilinear form (·,·), R ⊂ V be a reduced ...
linofiore's user avatar
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If $\alpha$ and $c \alpha$ are roots of a semisimple Lie algebra, then $c \in \{-1, 1\}$ [duplicate]

Let $L$ be a semisimple Lie algebra over a field $F$ (which is algebraically closed with characteristic 0) and $\alpha$ and $c \alpha$ for $c \in F$ are roots of $L$. I want to prove that $c \in \{-1, ...
Flynn Fehre's user avatar
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Can Borel subgroups be partitioned into sets of roots?

I'm doing a project in algebraic geometry where Borel subgroups play a very important role, but my supervisor made a comment that confused me. Let $G$ be a semisimple algebraic group and let $T\subset ...
nspace's user avatar
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Connection between maximal tori and roots

I've been studying algebraic groups and there is a confusion that I have been unable to resolve. Let $G$ be a semisimple algebraic group and $T\subset G$ a maximal torus. If we let $T$ act on ${\rm ...
nspace's user avatar
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Is the Lie-subalgebra generated by the root-spaces of a semisimple Lie-algebra semisimple?

Let $\mathfrak{g}$ be a finite-dimensional real semisimple Lie algebra and $\Sigma$ its root system (not necessarily reduced). Let $\alpha \in \Sigma$. Then $$ \mathfrak{l} := ( \mathfrak{g}_\alpha \...
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Relationships between the representations of $G$ and the representations of $G^{\text{der}}$

I'm struggling with something in roots datum theory and representations. This can be summarized in the following two questions (take $G$ a real reductive groupe and $T$ a maximal torus) 1- What are ...
Marsault Chabat's user avatar
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4 answers
206 views

How do people compute the root systems of the classical Lie algebras?

I am curious about computing the root system (as linear functionals) using the minimum prerequisite knowledge and straightforward computation. For example, for Type A the eigenvalues of $ad_{H}$ can ...
Dinoman's user avatar
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Weyl group acting on integral lattice

Suppose we have a connected compact Lie group $G$ with root system $\Sigma$ with respect to a maximal torus $T$. As usual, we identify $\mathfrak t=Lie(T)$ with its dual $\mathfrak t^*$ using the ...
GhostAmarth's user avatar
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Elementary consequences of the root system axioms

On Root_system: Elementary_consequences_of_the_root_system_axioms (wikipedia), from the relation $\langle \alpha, \beta \rangle = (2\cos(\theta))^2 \in \mathbb Z$, the value $\cos(\theta)$ can only be ...
Iris's user avatar
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Two alternative descriptions of $\mathrm{so}_{2n}(\mathbb{C})$

It seems that there are two ways to define the Lie algebra $\mathrm{so}_{2n}(\mathbb{C})$. The first one is $\mathrm{so}_{2n}(\mathbb{C})_{(1)}:=\{M \in \mathrm{gl}_{2n}(\mathbb{C}) \ | \ M + M^t = 0\}...
Grabovsky's user avatar
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Number of type $A_{n-1}$ sub-systems of root system $A_n$

If I am not mistaken, it should be true that a type $A_n$ root system contains $n+1$ different sub-systems of type $A_{n-1}$. In my geometric application these do appear quite naturally, but is there ...
Bipolar Minds's user avatar
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Explicit expression for simple roots of the root systems $A_2$, $B_2$ and $G_2$ in 2D

I often find explicit expression for rank-2 root systems as $A_2$, $B_2$ and $G_2$ in a 3D Euclidean space. Does anybody have an explicit expression for the simple roots in terms of $e_1,e_2$ in the ...
Dac0's user avatar
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3 votes
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119 views

Extending torus of split reductive group

Let $G$ be a split reductive group and $T$ its maximal torus. Is it always possible to embed $G$ into some split reductive group $G'$ such that $G,G'$ have the same unipotent $U$, same Weyl group $W$, ...
idocomb's user avatar
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Are the weights of an irreducible representation of a complex semisimple algebra induced by the weights of a representation of $SL(N)$ for some $N$?

If we consider the weights of a complex irreducible representation $V$ of a complex semisimple Lie algebra $\mathfrak{g}$, with respect to a chosen Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$, ...
Malkoun's user avatar
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4 votes
1 answer
180 views

Root space decomposition of Lie subgroups

Let $\varphi \colon G \to \tilde{G}$ be a group homomorphism between two real semisimple Lie groups. For example, $\varphi$ could be an inclusion of a subgroup $G \subseteq \tilde{G}$. Let $\mathfrak{...
Strichcoder's user avatar
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The Weyl group of type $D_2$, as a subset of $S_4$

My question is related to this post. I am trying to compare the Weyl groups associated to root systems of type $D_2$ and $A_3$ respectively. I know that the simple roots of $D_2$ are $e_1-e_2$ and $...
EJB's user avatar
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1 answer
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Dynkin diagram of a semisimple Lie algebra

Let $J=\begin{pmatrix}0&0&0&0&0&1 \\ 0&0&0&0&1&0 \\ 0&0&0&1&0&0 \\ 0&0&1&0&0&0 \\ 0&1&0&0&0&0 \\ 1&...
Mario's user avatar
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The sum of two roots when their pairing is zero

Let $(\Delta,(-,-))$ be a root system and $\alpha,\beta \in \Delta$ be s.t. $(\alpha,\beta) = 0$. Is it always true that $$ \alpha + \beta, ~ \alpha - \beta \notin \Delta? $$ This works for the $A$-...
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A question related to maximally non-compact Cartan subalgebra.

I was looking into the classification of real simple lie algebras from Araki (1962). In page $4$, proposition $1.1$, a criteria is given as to when $\mathfrak{h}^-$ is maximal abelian in $\mathfrak p$ ...
Soumyadip Sarkar's user avatar
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How to calculate the trace of $ad(H)$ as a linear operator acting on $\mathfrak{g}_{n\alpha}$?

The convention is set as follows: we have a semi-simple complex Lie algebra $\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_\alpha\mathfrak{g}_\alpha$ as its root space decomposition. If $\alpha\notin\Delta$...
Rescy_'s user avatar
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2 votes
1 answer
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Length function on Weyl group $S_n$ gives part of the Pascal's triangle

$$ \begin{matrix} \ell&\mathbf 0&\mathbf 1&\mathbf 2&\mathbf 3&\mathbf 4&\mathbf 5&\mathbf 6&\mathbf 7&\mathbf 8&\cdots\\ \hline \color{gray}{A_0|}&\color{...
Drinzjeng Triang's user avatar
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41 views

Irreducible factors of universal enveloping algebra

Let $\mathfrak{g}$ be a simple complex Lie algebra and $U$ be its universal enveloping algebra. We have an action $\mathfrak{g} \to \mathfrak{gl}(U)$ by extending the adjoint action of $\mathfrak{g}$ ...
Henrique Augusto Souza's user avatar
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83 views

The closure of the Tits' cone of a Kac-Moody Lie algebra

The question is based on Proposition $5.8$ in 'Infinite dimensional Lie algebras' by Victor G. Kac. The Proposition describes the closure of the tits cone in the $\textit{metric topology}$ of $\...
Irfan's user avatar
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2 votes
0 answers
46 views

When is a sum of two real roots of a symmetrizable Kac-Moody algebra is again a real root?

In general I would like to know when the sum $\alpha+\beta$ of two real roots $\alpha$ and $\beta$ of a symmetrizable Kac-Moody algebra $\mathfrak{g}(A)$ is again a real root (e.g. if $A$ is of finite ...
Irfan's user avatar
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Understanding Dynkin Labels

I'm currently studying a Lie algebra and I'm struggling to understand the concept of Dynkin labels. I know they're used to label certain irreps in the algebra, but I'm not entirely sure what they ...
iron's user avatar
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One question about reductive group

I'm reading Rapoport's paper "On the classification and specialization of Fisocrystals with additional structure", and I'm confused for one detail. Here G is a reductive group over $F$, $F$ ...
user572258's user avatar
1 vote
1 answer
122 views

Dimension of flag variety and sum of coefficients of anticanonical divisor

Added: I have created some code that verifies the inequality is true for Dynkin diagrams of type $A_n,B_n,C_n,D_n$ for $n\leq 10$, and it has verified the special type $E_6,E_7,E_8,F_4,G_2$. I'll post ...
Dave's user avatar
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2 votes
1 answer
123 views

Roots of direct sum of Lie Algebras

Assume this is all over $\mathbb{C}$. Given (semisimple, finite-dimensional) Lie algebras $\mathfrak{g}_1$ and $\mathfrak{g}_2$ with Cartan subalgebras $\mathfrak{h}_1$ and $\mathfrak{h}_2$, why is ...
SomeCallMeTim's user avatar
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0 answers
115 views

Simple roots of E6 root system

The simple roots of root system E6 are six 8-dimensional vetors. As I understand, simple roots should form a basis for the underlying vector space(in this case it is $\mathbb{R}^8$). For example, in ...
user777's user avatar
  • 75
2 votes
0 answers
65 views

Problem in understanding a fact about Lie algebra.

Let $L$ be a simple complex Lie algebra. Let $\Phi$ be the root set of $L$ and $\Gamma$ be the set of all simple roots of $L.$ We know that for every root $\alpha \in \Phi$ there is a copy $S_{\alpha}$...
Anacardium's user avatar
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1 vote
0 answers
100 views

Sublattice of the weight lattice

Recently, I am studying Lie algebra and have some question for the root system. Let $\mathfrak g$ be a simple Lie algebra and $\mathfrak h$ a cartan subalgebra and $\Delta$ the root system of $\...
KS M's user avatar
  • 333
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0 answers
132 views

If $\alpha, \beta, \alpha + \beta \in \Phi$ then $[L_{\alpha} , L_{\beta}] = L_{\alpha + \beta}.$

Let $L$ be a semisimple Lie algebra and $\Phi$ be the root set of $L.$ Let $L_{\alpha}$ be the root subspaces of $L$ corresponding to the root $\alpha \in \Phi.$ Show that if $\alpha, \beta, \alpha + \...
Anil Bagchi.'s user avatar
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0 votes
1 answer
50 views

Show that roots are preserved by $\tau.$

Let $L$ be a finite dimensional semisimple Lie algebra with Cartan subalgebra $H.$ Let $\Gamma$ be the set of all fundamental roots or simple roots of $L.$ Let $\Gamma_1, \Gamma_2 \subset \Gamma$ and $...
Anil Bagchi.'s user avatar
  • 2,912
1 vote
2 answers
88 views

Verifying that the angle between $\alpha$ and $\beta$ is $\frac {2 \pi} {3}.$

Consider the Lie algebra $sl_{3} (\mathbb C)$ of dimension $8.$ Let $$\begin{align*} H : & = \text {span} \left \{H_1 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 ...
Anil Bagchi.'s user avatar
  • 2,912
1 vote
0 answers
221 views

Action of the absolute Galois group on the based root datum

Let $G$ be a connected reductive group over a field $k$ with separable closure $k^s$ and $\Gamma=\text{Gal}(k^s/k)$. Pick $T\subset B$ maximal split torus and Borel subgroup for $G_{k^s}$. I am aware ...
Tengu's user avatar
  • 4,092
2 votes
1 answer
187 views

Root space decomposition of a semisimple Lie algebra.

Let $L$ be a semisimple Lie algebra. I am trying to understand root space decomposition of $L$ on my own. Since $L$ is semisimple, $L$ possesses an abelian maximal toral subalgebra i.e. an abelian ...
Anil Bagchi.'s user avatar
  • 2,912
0 votes
1 answer
47 views

What is the Lie subalgebra generated by $\mathfrak g_{\pm \alpha}\ $?

Let $\Gamma$ be the set of all simple roots of a simple Lie algebra $\mathfrak g.$ Let $\Gamma_1 \subset \Gamma$ be an arbitrary subset. Then what is Lie subalgebra generated by $\mathfrak g_{\pm \...
Anil Bagchi.'s user avatar
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0 votes
0 answers
37 views

Classifying simple roots of $sl_n (\mathbb C).$ [duplicate]

Is there any way to classify all the simple roots of $sl_n (\mathbb C)\ $? I know that there $\frac {n^2 - n} {2}$ many positive roots and hence the same number of negative roots. But what is the ...
Anil Bagchi.'s user avatar
  • 2,912
3 votes
1 answer
67 views

Not all nonnegative integer combinations of simple roots are dominant weights

Let $\frak{g}$ be a complex semi-simple Lie algebra with a choice of Cartan, and hence with an associated root system $\Delta$. As is well known, not all nonnegative integer combinations of simple ...
Jake Wetlock's user avatar
2 votes
0 answers
82 views

A question on root system projected down orthogonally onto a subspace

A reduced root system $\Delta$ on a finite dimensional Euclidean space $V$ is called a normal $\sigma$-system of roots if $\exists$ a linear involutive isometry $\sigma:V\to V$ such that $\sigma\...
Soumyadip Sarkar's user avatar
1 vote
1 answer
89 views

A doubt from Araki's 1962 paper on classification of irreducible symmetric spaces

I am looking at Soho Araki's 1962 paper for the classification of real semisimple lie algebras.Here's the link to the paper: Araki's paper.In page $9$, proposition $2.2$,there is a criteria for $\psi\...
Soumyadip Sarkar's user avatar
2 votes
0 answers
21 views

Sum of weights of a representation of $U(N)$

Let $R$ be a finite-dimensional irreducible representation of $U(N)$, with the set of weights $W_R$. Each element of $W_R$ is a vector of length $N$ with integer entries. Firstly, I would like to know ...
Blind Miner's user avatar
0 votes
1 answer
66 views

How do roots of a root system correspond to symmetric points of the associated Weyl group

So I've read that the Weyl group of the $A_3$ root system is the full symmetry group of a tetrahedron $T_h$, $C_3$ is $O_h$, and $H_3$ is $I_h$. I've read some complicated proofs of this but I don't ...
Mathematical Lie's user avatar
2 votes
0 answers
106 views

Can this linear algebra / geometry lemma in Humphreys be proven by induction?

Many questions have been asked on this site about the proof of Lemma 9.1 on page 42 of Humphreys's Lie algebra book. The full proof is posted here. I understand the proof, but I don't like it and ...
WillG's user avatar
  • 6,611
5 votes
1 answer
99 views

Connection between exponents of a root system and solutions to linear systems over finite fields

Let $h_1, \ldots, h_r$ be linear forms in variables $x_1, \ldots, x_n$ with integer coefficients. Let $\mathbb F_q$ denote the finite field with $q = p^e$ elements. I am asked to prove that except in ...
Haydn Gwyn's user avatar
  • 1,362
2 votes
0 answers
95 views

Computation of roots and Borel subgroups

I have a question about the computation of positive roots/coroots and Borel subgroups of the indefinite group $G = \operatorname{GSO}(n, 2)$ for some $n \in \mathbb{Z}^{+}$, where the form defining ...
user1128433's user avatar
3 votes
0 answers
39 views

$w(\Delta) = \Delta \implies w = \mathrm{id}$ in Weyl group

Consider $\Phi$ a root system (definition of Erdmann) with basis $\Delta \subseteq \Phi$ (subset which is a basis of $E$ and each element of $\Phi$ is a linear combination with only non negative ...
raisinsec's user avatar
  • 449
3 votes
1 answer
177 views

Why isn't the Weyl group of a root system defined as the isometry group of that system?

I know that the Weyl group of a root system is a subgroup of its isometry group, but (as in the case of $A_2$) it isn't always the whole isometry group. Why isn't the Weyl group defined as the ...
mathlander's user avatar
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0 votes
1 answer
54 views

Standard reference for a fundamental theorem on classification of root lattices

In Schuett-Shioda's Mordell-Weil Lattices, the authors refer to a fundamental theorem on root lattices: Theorem 2.25 Any positive-definite even integral root lattice is isometric to an orthogonal sum ...
rmdmc89's user avatar
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