All Questions
Tagged with lie-groups root-systems
60 questions
4
votes
1
answer
381
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The existence of a finite dimensional Lie algebra with a given symmetric invariant metric
The question is motivated by a more broad perspective in another MO post and here, but here we would like to understand a specific case (our question potentially connects to / is motivated b Quantum ...
- 755
6
votes
1
answer
1k
views
Finite dimensional Lie algebra with non-degenerate invariant bilinear forms $\Omega_{ab}$
Firstly, my apology to MO experts that I am in a more science/physics background (a PhD). So please feel free refine/modify/comment my language if I have different math accents than yours. From ...
- 755
5
votes
3
answers
2k
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Complete classification of six dimensional non-semi simple Lie algebra
I would aim to know the complete classification of 6 dimensional non-semi simple Lie algebra (here the dimension stands for the generators; or the dimension $\leq 6$).
In this paper, in page 7, it ...
- 81
5
votes
1
answer
453
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A subgroup of the Weyl group
Let $D$ be a connected Dynkin diagram with an automorphism $\nu$ of order 2.
Let $Q=Q(D)$ denote the root lattice of $D$.
Let $W=W(D)$ denote the Weyl group, it acts effectively on $Q$ and it is ...
- 14.2k
9
votes
2
answers
634
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Extension of the Weyl dimension formula
Let $G$ be a compact semisimple group and let $\Gamma$ be a finite subgroup of $G$. I am interested, for $(\pi,V)\in \widehat G$ (irred rep of $G$), in a formula for $\mathrm{dim} V^\Gamma$, the ...
- 2,446
4
votes
1
answer
282
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Name for a class of parabolic subgroups
This is a reference request for a (the) name of the following class of parabolic subgroups of a complex simple Lie group $G$:
Recall that parabolic subgroups of $G$, containing fixed Borel subgroup, ...
- 31.2k
3
votes
0
answers
289
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Conjugation of faces in root systems / of parabolic subgroups having same Levi in split reductive groups
If $(V,\Phi)$ is a root system of rank $n$, one knows that its Weyl group $W$ acts simply and transitively on Weyl chambers. But in general, if $d\lt n$, the action of $W$ on faces of dimension $d$ is ...
- 31
7
votes
1
answer
743
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schur weyl duality for real orthogonal groups and relation to hyperoctahedral groups
I am wondering whether the Lie groups $SO(n)$ and the hyperoctahedral groups $H_n$ form some sort of duality. I am mainly interested in how to parametrize the conjugacy classes of $H_n$ in terms of ...
- 4,466
1
vote
2
answers
465
views
subgroups with the same number of roots that the group.
When thinking in terms of Dynkin diagrams, I am naively used to see that the diagram for a subgroup can be extracted from the diagram of the group by removing some roots. Now, I noticed that for SO(10)...
- 437
5
votes
2
answers
1k
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Complex root systems
This question is twofold.
1) What is the best reference on root systems?
2) Do complex root systems exist?
- 475