Questions tagged [dynkin-diagrams]

Question 1

I am wondering about how to write out an explicit form of a Lie Algebra automorphism from an automorphism of the associated Dynkin Diagram of the root system. Within this question are some other ...

Question 2

Let $V$ be a real vector space and let $\Phi \subset V$ be an irreducible root system of rank $r \geq 2$. Let $\Phi_1, \Phi_2 \subset \Phi$ be irreducible root subsystems (that is, root systems in ...

Question 3

It is just a few steps of algebra to show that the projections of vertices of an $E8$ polytope (Gosset polytope $4_{21}$ to the Coxeter plane can be brought in congruence with the projections of the ...

Question 4

Consider the $\mathfrak{so}_{2n}$ Lie algebra over $\mathbb{C}$, $n\geq 2$. Consider $U=\mathbb{R}^n$, and a basis of $U$ given by the roots \begin{equation} e_1-e_2,e_2-e_3,\ldots,e_{n-2}-e_{n-1},e_{...

Question 5

Given the Dynkin diagram B4 and a Basis $\{ \alpha_1, \alpha_2, \alpha_3, \alpha_4 \}$ of the root system $\phi$, show that the element $\beta= \alpha_1+\alpha_2+\alpha_3+2\alpha_4$ is a root. I tried ...

Question 6

For some time now, I have been looking for references that provide concrete models of the Lie algebras associated with the exceptional algebras $E_6, E_7, E_8, F_4$ e $G_2$. However, I have not been ...

Question 7

When reading Fulton-Harris's book on representation theory, in section 21.2 they classify all the connected Dynkin Diagrams leading to the classification of the complex simple Lie algebras. I am ...

Question 8

I am a school teacher of Mathematics who self study advanced topics as an interest. I am trying to calculate the Cartan matrix of $F_4 $ assuming it's dynkin diagram is given to me and I have made the ...

Question 9

In the book of Introduction to Lie Algebras by Karin Erdmann and Mark J. Wildon, in Chapter 13 The Classification of Root Systems, there is this Lemma 13.10, and later proposition 13.11, that ...

Question 10

I wish to consider the automorphism of a Lie algebra induced by considering an automorphism on its Dynkin diagram, in particular the order 3 automorphism on $D_4$. I thought to extend linearly on the ...

Question 11

Consider $\Delta$ a root system. Let $\alpha, \beta \in \Delta$. Denote by $s_\alpha$ and $s_\beta$ the reflections in the Weyl group associated with $\alpha$ and $\beta$, respectively. Show that $(s_\...

Question 12

In his talk "Split Octonions and the Rolling Ball," John Baez says that a Dynkin diagram describes a type of geometry, with vertices representing types of objects, and edges representing ...

Question 13

I am searching in the literature (so far unsuccessfully) the correct notion of "abstract Dynkin diagram" as the one that characterises intrinsically graphs that arise as the Dynkin diagram ...

Question 14

EDIT: So, my question I believe can be broken down to: for a Lie Algebra $\mathfrak{g}$, and an outer automorphism on $\mathfrak{g}$ named $\hat{O}$: if I have a representation $\rho$, the question is ...

Question 15

This is sort of a follow up question to this question on Satake-Tits diagrams. In the question, user Callum comments that for, $\mathfrak{so}(n, \mathbb{C})$, the real forms (potentially with double ...

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Questions tagged [dynkin-diagrams]

A Principled(?) Way to Determine a Lie Algebra Automorphism from a Dynkin Diagram Automorphism (and invariant subalgebra)

Rank of root subsystem

Align two 600 cellls with the E8 (Gosset $4_{21}$) polytope

Dynkin diagram of $\mathfrak{so}_{2n}$

Question on the dynkin diagram B4 and to show that an element is still a root.

A reference for the realization of the exceptional Lie algebras

Connection Between Triangle Groups and Exceptional Lie Algebras $E_n$

Assume that the Dynkin diagram of $F_4$ is given, calculate the Cartan Matrix of $F_4$ using this information.

Why do I need an $i$ in Lemma 13.10 of Book Introduction to Lie Algebras for the vector inner product [closed]

Inducing a Lie Algebra Automorphism From a Dynkin Diagram Automorphism

Root system and reflections

In what way exactly does a Dynkin diagram encode a type of geometrical incidence structure?

What graphs arise as the Dynkin diagrams of semisimple Lie algebras?

When does a Lie algebra's outer automorphism group 'inherit' a representation?

How to be certain one has found all possible real forms of a semi-simple Lie algebra?

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