1
vote
1answer
25 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}, $ $ ...
0
votes
0answers
25 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 ...
0
votes
0answers
37 views

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 ...
1
vote
0answers
29 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 = ...
0
votes
0answers
24 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 ...
2
votes
2answers
43 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 ...
1
vote
1answer
36 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 ...
0
votes
1answer
25 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 ...
2
votes
0answers
30 views

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). ...
1
vote
0answers
28 views

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 ...
0
votes
0answers
54 views

Which linear combinations of simple roots are roots?

An answer to the following question should be well known to any specialist on Lie theory. Since I don't have time to go through textbooks, I post it here. Let $\Delta$ be a root system, $\Delta^+$ ...
8
votes
1answer
244 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 ยง13 in Humphreys for details. Let $\Phi$ be a root system, $\Delta$ a ...
1
vote
1answer
40 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 $\mathfrak{G}_{\mathbb{C}}=\mathfrak{sl}(n,\mathbb{C})$ ...
4
votes
2answers
74 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 ...
3
votes
2answers
44 views

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 = ...
0
votes
0answers
15 views

Dual basis to $e_{i+1}-e_{i} \in \ker ((1,1,…1)^\vee\in(\Bbb E^{n+1})^\vee)$

Studying the root system $A_n$ given by the simple roots $v_i:=e_{i+1}-e_i \in \Bbb E^{n+1}/\Bbb R(1,1,...,1)$ for $i = 1,...,n$, I came across the following dual basis: $v_i^\vee:= ...
2
votes
0answers
41 views

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 ...
0
votes
0answers
29 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 ...
0
votes
1answer
24 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 ...
4
votes
3answers
142 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 ...
2
votes
1answer
53 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 ...
1
vote
1answer
37 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 $W_1 \times W_2$? ...
1
vote
0answers
58 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 $U(n)\simeq SU(n)\times U(1)$?
1
vote
1answer
34 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 ...
1
vote
1answer
52 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 ...
0
votes
1answer
59 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 ...
3
votes
1answer
55 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
votes
1answer
62 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 ...
5
votes
0answers
92 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 ...
1
vote
1answer
94 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 ...
2
votes
0answers
22 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 ...
2
votes
0answers
53 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
votes
0answers
72 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 ...
2
votes
0answers
73 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 ...
1
vote
0answers
24 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 ...
0
votes
0answers
28 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
votes
1answer
64 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 ...
2
votes
0answers
43 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, ...
0
votes
1answer
122 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 ...
1
vote
0answers
88 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 ...
3
votes
1answer
94 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
votes
2answers
268 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 ...
4
votes
1answer
97 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 ...
0
votes
1answer
127 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 ...
1
vote
1answer
73 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 ...
1
vote
1answer
75 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 ...
1
vote
0answers
94 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 ...
0
votes
1answer
150 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
votes
2answers
221 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 ...
9
votes
0answers
112 views

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

(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 ...