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Questions tagged [dynkin-diagrams]

A Dynkin diagram, named after the russian mathematician Eugen B. Dynkin, is a member of a small family of directed graph originally used as a shorthand to classify and describe the structure of semi-simple Lie algebras. They are increasingly used and generalized for other mathematical objects having similar combinatorial properties. A related and earlier concept is the Coxeter diagram used to classify reflextion groups.

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For which graphs does this “+1 game”, the Sponsor Game, terminate?

Consider this game on simple graphs described by Allen Knutson: Begin by assigning a $1$ to a single node and a $0$ to each other node in the graph. Then, while such a node exists, choose a node with ...
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Subgraphs of Dynkin Diagrams

Am I right in thinking that if we have two semisimple Lie Algebras $\mathfrak{g} $ and $\mathfrak{h}$ with respective Dynkin Diagrams $A$ and $B$, we may find an injective homomorphism of Lie Algebras ...
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Subgroups of $E_8$ by using extended Dynkin diagrams

I need to show that the following are subgroups of $E_8$ using extended Dynkin diagrams. $$SU\left(5\right)\times SU\left(5\right)$$ $$SU\left(3\right)\times E_6$$ $$SU\left(4\right)\times SO\left(...
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Structure constants from Dynkin diagram [closed]

Is there an easy way to obtain the structure constants of a Lie Algebra starting from its Dynkin diagram?
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Why do $SU(2)$ and $SL(2,\mathbb{C})$ have the same Lie algebra?

The Lie algebras $su(2)$ and $sl(2,\mathbb C)$ have the same Dynkin Diagram (just a blob) and therefore also have the same structure constants and isomorphic Lie algebras. Additionally, they are both, ...
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How to Calculate the Size of the Exceptional Irreducible Root Systems

There are three exceptional irreducible root systems $E_6$, $E_7$, and $E_8$ which correspond to the Dynkin diagrams $$ E_6\; \begin{aligned} &\>\bullet \\[-1ex] &\,\,\mid \\[-1ex] \...
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Young Tableaux of $SU(N)$: Group or Algebra?

I am slightly confused about the use of Young Tableaux in the context of the Lie group and Lie algebras of $SU(N)$. A Young Tableaux has an associated representation which acts on a tensor e.g. $\psi_{...
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Canonical choice for simple roots?

The Dynkin diagram of a simple Lie algebra tells me about the lengths of all simple roots and about the angles of adjacent simple roots. Since the angle between non-adjacent simple roots is not ...
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Compute Singularity from the given weighted dual graph

For a rational surface singularity, Artin's result guarantees the existence of the fundamental cycle Z. Thinking in the reverse direction, I have the following question(s): Que-1: Given a weighted ...
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Reference for reading Dynkin diagrams in Lie theory?

I have learned that given a Dynkin diagram corresponding to a Kac-Moody algebra, I should be able to use the diagram to read off the generators and relations of the Weyl group of that algebra. Each ...
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Recover Lie algebra bracket from the root system

I have coordinates of the $n$ roots in the $d$-dimensional space forming the root system of the $(n+d)$-dimensional Lie algebra. I want to implement the algorithm to recover the Lie bracket of the ...
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Simple Lie Algebra Conjugacy and Dykin Diagrams

Give two simple Lie Algebras $\mathfrak{g_1}$ and $\mathfrak{g_2}$, can we say anything about the conjugacy of $\mathfrak{g_1}$ and $\mathfrak{g_2}$ based on the properties of the corresponding ...
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An apparent contradiction in the simple Lie algebra $E_8$

The following is the Dynkin diagram for simple Lie algebra $E_8$ My question is the following: It is clear that $e_i+e_j$ for $i \neq j$ is a positive root. Let $\alpha _8$ be the fundamental ...
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How to embed $U(1)$ (or other groups) into a bigger group, using Dynkin diagrams

I am trying to find the embedding and the branching rules for some group decompositions. For example, I consider $E_7$ and its maximally compact subgroup $SU(8)$ and I want to "see" how the Dynkin ...
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At what position do we insert the new coefficient in the weights for extended Dynkin Diagrams?

Given a set of weights of a representation and the corresponding extended Dynkin diagram for some Lie algebra, we can delete a node, which yields the maximal subalgebra. I know how to draw the ...
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1-loop quiver and the classification of quivers

Gabriel's theorem states that finite type quivers are exactly the ones whose underlying graphs are ADE type Dynkin diagrams. Furthermore, the quivers whose underlying diagrams are ADE type affine ...
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How to descent to smaller groups “by chopping off a node of the Dynkin diagram”?

I read in section 2 of this paper : "There is a well-defined chain to descent from $E_8$ to smaller groups by chopping off a node of the Dynkin diagram." What exactly is here referring to here? ...
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What information can I immediately extract from a Dynkin diagram?

I have understood quite well how we construct Dynkin diagrams. My question is the following: What immediate information can I extract just by looking at a Dynkin diagram? Of course I can ...
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Definition of the Dynkin Diagram (in Humphreys)

I'm reading paragraph 11 in Humphreys' 'Introduction to Lie Algebras and Representation Theory'. The author defines Coxeter graphs and Dynkin diagrams for any rank-many distinct positive roots. He ...
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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 $\mathfrak{G}_{\mathbb{C}}=\mathfrak{sl}(n,\mathbb{C})$ (...
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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 ...