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.
404
questions
1
vote
0
answers
28
views
On axiomatic definition of affine root systems
In Macdonald's book Affine Hecke Algebras and Orthogonal Polynomials, chapter 1 introduces affine root systems. I will recall the definition here: Let $E$ be a non-zero real Euclidean space (finite ...
- 197
0
votes
1
answer
56
views
Set of roots such that sum of them with given positive root is not a root
Let $\Phi$ be a indecomposable root system of rank greater than 1 and fix a positive root $\alpha$. How can one describe the set $ B = \{ \beta \in \Phi \;| \; \alpha + \beta \notin \Phi \}$?
It's ...
1
vote
1
answer
112
views
The effect of a Cayley transform on a Cartan subalgebra
I'm trying to understand equation 6.65b from Knapp's 'Lie groups,' 2ed.
Setup: Let $\mathfrak{g}_0$ be a real semisimple Lie algebra with an involution $\theta$. Let $B$ be a bilinear, symmetric, non-...
0
votes
1
answer
39
views
Root system for Lie algebras: If $\alpha, \beta, \alpha - \beta \in \Phi$ are all roots, is $(\alpha, \beta) > 0$?
Exercise 8.11 of Humphreys' Introduction to Lie Algebras and Representation Theory asks to prove that if $\alpha, \beta \in \Phi$ and $(\alpha,\beta) > 0$, then $\alpha - \beta \in \Phi$ (...
- 300
1
vote
0
answers
16
views
Angles of the Fundamental Alcove (Chamber?)
I am trying to calculate the angles of the fundamental alcove (chamber?) for the root systems of type $B_2$, $C_2$, and $G_2$; the fundamental alcove (chamber?) forms a triangle in these cases, so I ...
- 7,082
1
vote
1
answer
41
views
Reflection Group of Type $D_n$
Here is the description of the reflection group of type $D_n$ in Humphreys' book Reflection Groups and Coxeter Groups:
($D_n$, $n \ge 4)$ Let $V = \Bbb{R}^n$, and define $\Phi$ to be the set of ...
- 7,082
1
vote
1
answer
31
views
Reflection Group of Type $C_n$
In Humphreys' book Reflection Groups and Coexeter groups, he defines a group of type $B_n$ (for $n \ge 2$) in the following way:
Let $V = \Bbb{R}^n$, and define $\Phi$ to be the set of all vectors of ...
- 7,082
6
votes
2
answers
218
views
Explicit matrices $h_\alpha$ that correspond to the long roots $\alpha$ in a classical compact simple Lie algebra over the reals
While it is easy to find many sources that give expressions for the (co)roots of an abstract root system, it is less easy to find a reference that gives explicit matrices that are the "coroots&...
- 2,204
1
vote
1
answer
121
views
How are weights and roots connected? (In the context of the semisimple Lie algebras)
I am studying representations of complex semisimple Lie algebras, their root system, weight spaces etc.
I am a very beginner so this question is about relating key definitions to one another.
Question ...
- 1,962
0
votes
0
answers
29
views
Reduced adapted expression for the longest element given an orientation for a Dynkin graph
In Kirillov's book "Quiver representation and quiver varieties" in page 45 there's a Theorem 3.33 that he say is due to Lusztig that says:
Given an orientation $\Omega$ of a Dynkin graph $Q$,...
- 175
1
vote
1
answer
32
views
About imaginary root of a symmetrizable Kac Moody algebra
The question is from Infinite dimensional Lie algebras by Victor G Kac (ex 5.17, Page 74)
Let $\mathfrak{g}=\mathfrak{g}(A)$ be a symmetrizable Kac Moody algebra and let $\Delta$ be its set of roots. ...
- 172
1
vote
0
answers
36
views
If $\Delta$ is decomposable, why does $\mathfrak{g}$ have a corresponding decomposition into direct sum of ideals.
Assume that $\mathfrak{g}$ is a finite-dimensional semisimple Lie algebra over an algebraically closed field $F$ of characteristic $0, \mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$ and $\...
- 95
1
vote
1
answer
43
views
Simple Reflections on Simple Roots
I have two related questions concerning simple reflections and simple roots. Let $\Phi$ be a root system for a reflection group $W$, let $\Pi \subset \Phi$ be a positive system, and let $\Delta$ be a ...
- 7,082
1
vote
2
answers
120
views
Property of order 2 automorphism of simple Lie algebra
If $\sigma$ is an automorphism of simple Lie algebra, of order 2, such that we fixed : $\sigma (x_\beta)=-y_\beta$ for each simple root of a given base $\Delta$, is it also necessarily true that $\...
- 186
1
vote
0
answers
66
views
Difference of two simple roots is zero (?)
I am studying simple Lie algebras and, on page 129 of the book "Group Theory A physicist's Survey", by Pierre Ramond, it is claimed at the bottom of the page that << Hence the ...
- 31
1
vote
1
answer
36
views
Determinie the multiplicity of a root in $\mathfrak{g}(A)$ where $2\times 2$-matrix $A$ given (or arbitrary)
I'm trying to solve the exercises on Infinite-dimensional Lie algebras by Victor G.Kac.
Exercise 1.6.:
Let $A = \begin{pmatrix}2&-3\\-3&2\end{pmatrix}$.
Show that $\operatorname{mult} (2\...
- 107
1
vote
0
answers
44
views
Counting and finding root subsystems
Let $\Phi$ be an irreducible root system. A root subsystem of $\Phi$ is a subset $\Psi \subseteq \Phi$ which is a root system. One can find the possible types of root subsystems of $\Phi$ by deleting ...
- 11
1
vote
1
answer
108
views
Type of a root (sub)-system
Let $E:=\{x\in \mathbb{R}^{l+1}:x_1+x_2+\cdots + x_{l+1}=0\}$ and let $\Phi\subseteq E$ denote its root system of type $A_l$ given the basis $\Delta=\{e_i-e_{i+1}, 1\leq i \leq l\}$ and with $\{e_i\}$ ...
- 237
1
vote
1
answer
89
views
When is the Lie bracket of two root spaces nonzero?
Let $L$ be a semisimple Lie algebra with root space decomposition $L = L_0 \oplus \bigoplus_{\alpha \in \Phi} L_\alpha$. For roots $\alpha, \beta \in \Phi$, we always have $[L_\alpha, L_\beta] \...
- 642
3
votes
0
answers
71
views
Determining the Weyl group from a given root system
1. Definitions
For $V$ a vector space over $\mathbb C$ we call a subset $R \subset V$ an abstract root system if:
(1) The set $R$ is finite, spans $V$ and $0 \notin R$.
(2) For every $\alpha$ in $R$ ...
- 277
0
votes
0
answers
39
views
Intuition behind calculating weight systems using SU(2) strings of roots?
I am in a weird sitution in my Groups course where I can 'blindly follow the rules' in, for instance, finding a weight system from its highest weight by representing the weights in their Dynkin basis, ...
- 523
0
votes
0
answers
44
views
How to derive the mentioned relations (in part a-b-c of example 9) from the Weyl denominator formula?
I want to understand Example 9 (a-b-c) of section 1.5 of the book "Symmetric functions and Hall polynomials (https://math.berkeley.edu/~corteel/MATH249/macdonald.pdf)" (pages 78-79):
In ...
0
votes
1
answer
56
views
A simple question on root systems
I read the Wikipedia page about root systems. It is very easy to verify that the 2 following images are showing some root systems.
https://en.wikipedia.org/wiki/Root_system
These 2 pictures are from ...
2
votes
0
answers
19
views
System of representatives in reflection groups and subgroups
I'm working on a paper from Steinberg (1974), "On a theorem of Pittie". The paper is mainly about roots and reflection groups. I'm having trouble understanding the proof of lemma 2.5(a) ...
- 21
0
votes
0
answers
24
views
Change of basis in root systems
It is for sure s really easy question, but I need to ask.
Let $\Phi \subset \mathbb R^n$ be an abstract root system. We denote by $\{e_1,\ldots,e_n\}$ the canonical basis of $\mathbb R^n$ and by $\{\...
- 402
0
votes
0
answers
33
views
Simple Lie algebras have irreducible root systems
Lemma 2.1. Let $g$ be a simple Lie algebra with Cartan subalgebra $h$. Then
the root system $\Delta$ corresponding to $h$ is irreducible.
Proof. Suppose $\Delta$ decomposes as $\Delta_1\cup\Delta_2$ ...
- 1
0
votes
1
answer
56
views
Isomorphisms of irreducible root systems
Let $E,E'$ be two euclidean vector spaces and $\Phi,\Phi'$ two root systems of $E$ and $E'$, respectively. Let $\varphi:E\to E'$ be an isomorphism of root systems. Applying the definition of root ...
- 509
0
votes
0
answers
31
views
Uniqueness of roots system of a semisimple Lie algebra
Take $L$ a $\mathbb{C}$-Lie algebra with finite dimension and semisimple. Now take $H \subset L$ a maximal toral subalgebra we can define the roots system associated to $(L,H)$ as the set of maps $H\...
- 107
5
votes
1
answer
74
views
An Exercise on Abstract Weight Theory (Humphrey Exercise 13.8)
The exercise goes like this:
Let $\Phi$ be an irreducible root system. Prove that each $\lambda_i$ is of the form $\sum_j q_{ij}\alpha_j$, where all $q_{ij}$ are positive rational numbers.
Here, as ...
- 1,399
3
votes
0
answers
53
views
What does it mean for a root to be an element of the Lie algebra?
My understanding of roots in Lie algebra theory is as follows: given a Cartan subalgebra $\mathfrak{h}\subset\mathfrak{g}$ with basis $\{H_i\}_{i=1}^{r}$, and root vectors $\{X_\alpha\}_\alpha$, the ...
- 987
0
votes
0
answers
24
views
Condition for a sum of images of fundamental dominant weights to lie on a wall
Let $\Delta$ be a base of a root system with Weyl group $W$. For $\alpha\in\Delta$, let $\varpi_\alpha$ be the corresponding fundamental dominant weight. Let $w\neq r$ be elements of $W$. I would like ...
- 2,048
1
vote
1
answer
72
views
Cartan integers are preserved by isomorphism
I have this proposition and the demonstration, but there is some parts I don't understand, $E$ is a euclidean space and $\Phi$ is a root system with base $\Delta$.
Proposition: Let $\Phi' \subset E'$ ...
- 175
1
vote
2
answers
233
views
I don´t understand root systems
I don´t understand root systems. The Wikipedia (and my university lectures) say it is some configuration of vectors with certan properties. The root vectors should span the whole space, which I ...
- 1,962
1
vote
1
answer
54
views
A question about roots and root spaces of semisimple Lie algebras
In the book "Lie Groups: Beyond and Introduction" by Anthony Knapp, the author describes roots of a semisimple (A Lie algebra without any non-zero solvable ideal) Lie algebra as elements of ...
- 3,635
2
votes
1
answer
104
views
A description of the $A_n$-root system in terms of $\mathbb{R}^{n}$
The root system $(V,\Delta)$ of type $A$ is usually defined as a subset of $\mathbb{R}^{n+1}$:
$$
\{ e_i - e_j ;\, 1 \leq i,j \leq n+1,\, i \neq j\}.
$$
However these vectors span an $n$-dimensional ...
1
vote
0
answers
55
views
Dominance of $w\mu$ for dominant cocharacter $\mu$
NOTE: The question has now been posted on
MathOverflow: Dominance of $w\mu$ for dominant cocharacter
Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$ ...
- 268
0
votes
0
answers
46
views
Dominance order on cocharacter group $X_*(T)$
Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$ of rank $n$ and a Borel $B \supset T$ defining a set of simple roots $\Delta$. By $X_*(T)$ we denote ...
- 268
2
votes
1
answer
99
views
Understanding how to find roots of Lie algebra $\mathfrak{su}(N)$
I am having difficulties understanding why the roots of $\mathfrak{su}(N)$ algebra are $\alpha_{ij}=e_i-e_j$. This is (roughly) what my professor said:
We define the ladder operators in $\mathfrak{su}...
- 2,232
1
vote
1
answer
87
views
Help with a proof of a lemma from Humphreys about reflections [closed]
I want to understand this proof, I don't get why he says that the minimal polinomial of $\tau$ divides $(T-1)^l$. And at the end how can I explain why gcd$(T^k-1,(T-1)^l)=T-1$?
- 175
3
votes
0
answers
42
views
Sum of tensor products of positive roots of $\mathfrak{su}(n)$
Let $\Delta^+$ denote a choice of positive roots for the Lie algebra $\mathfrak{su}(n)$.
The Weyl vector, $\rho$, is half the sum of the positive roots
\begin{equation}
2\rho = \sum_{\beta \in \...
- 131
2
votes
0
answers
43
views
Sum of cubes of positive roots, $\alpha(H)^3$, of arbitrary Cartan element
Let $\mathfrak{h}$ the Cartan subalgebra of a simply-laced (ADE type) Lie algebra of rank $r$, $\mathfrak{g}$, and $\Phi$ the associated root system. Furthermore, let $\Phi^+$, $\Pi$, denote the ...
- 257
1
vote
1
answer
57
views
How do I prove a set is a root system?
I will start by setting up the problem then I will ask my question regarding it.
Problem:
Let $E=\{x=\sum_{i=1}^{n+1} x_{i}\epsilon_{i} \in \mathbb{R}^{n+1}|\sum_{i=1}^{n+1}x_{i}=0 \}$, where $\{\...
- 13
1
vote
0
answers
26
views
Constructing a surjective intertwining map onto Verma module
I want to prove the following result:
Theorem: Let $\mathfrak{g}$ be a semi-simple Lie algebra, $\mathfrak{h}$ be a Cartan subalgebra, $\mu \in \mathfrak{h}$, and $W_{\mu} = \mathfrak{U}_{\mathfrak{g}...
- 3,635
2
votes
1
answer
49
views
Exponentiating Lie algebra element does not result in algebraic group element
This question is about a particular case of the special orthogonal groups considered in section 23.4 of Borel's Linear Algebraic Groups. Let $k$ be a field of characteristic $\neq 2$. Let $F$ be a ...
- 2,812
1
vote
1
answer
123
views
Decomposition of finite-dimensional representation of semisimple Lie-Algebra into irreducible subrepresentations.
This is my first post on MathSE, since I could not find a helpful answer to my question on here yet.
Let $(L,[\cdot,\cdot])$ be a finite-dimensional semisimple complex Lie-Algebra, $H\subseteq L$
a ...
- 33
1
vote
0
answers
30
views
Weyl groups sends Weyl Chamber onto another:
I came across two statements while studying Weyl group of root Systems.
First one: The Weyl group say $W$ sends one Weyl Chamber onto another. If $\gamma$ is regular in a Euclidean Space $E$, we have $...
- 203
1
vote
1
answer
36
views
Relation between product of reflections and angle
I am currently reading Chapter 3 Root Systems of John Humphreys book on Lie Algebras.
It's known that Weyl Group $W$ is generated by set of reflections.
If I consider an arbitrary element of Weyl ...
- 203
7
votes
0
answers
128
views
Automorphism groups of which lattices act irreducibly on the ambient Euclidean space
Let $V$ be a finite-dimensional real inner product space and let $L \subset V$ be a lattice of full rank. Consider the finite group
$$
\mathrm{Aut}(L) = \{ f \in \mathrm{O}(V) \mid f(L) = L \},
$$
...
- 215
0
votes
0
answers
47
views
Recovering roots from Cartan integers
I am currently reading the chapter 3- Root Systems of John Humphreys book on Lie Algebra.
I am trying to recover my set of roots from knowledge of Cartan integers.
(I am considering the base $\...
- 203
0
votes
0
answers
42
views
Question regarding root systems of a complex Lie algebra
Let $\mathfrak g$ be a semi-simple Lie algebra with Cartan subalgebra $\mathfrak h$ and root system $\Phi$. I have seen it stated that for any $\alpha \in \Phi$, $-\alpha \in \Phi$, and that this is a ...
- 420