Questions tagged [root-systems]
For questions on root systems (the objects classified by Dynkin diagrams).
293 questions
2
votes
0
answers
155
views
Cartan matrix and graph Laplacian
The Cartan matrix for $A_n$ is almost equal (except for the diagonal entry at the endpoints) to the graph Laplacian for its Dynkin diagram. Something similar holds for the other root systems (except ...
- 3,788
0
votes
1
answer
118
views
Find integer coefficients of polynomials from approximate irrational roots [duplicate]
Let take quadratic equations
$$x^2+ax+b=0$$
assume here $a,b$ both are integer and the roots of the equation are irrational if I give you one root in irrational form then is there any method to find $...
- 337
0
votes
0
answers
132
views
Roots and action of the Weyl group
Let $\mathfrak{g}$ be a Kac-Moody algebra a Borel pair $(\mathfrak{b},\mathfrak{t})$, let $R^{+}$ be the set of positive roots, $\alpha_1,\dots,\alpha_n$ the simple roots.
For $\alpha=\sum n_{i}\...
- 3,582
3
votes
4
answers
480
views
Fixed-point subalgebras of automorphisms of $D_4$
Let $\mathfrak{g}$ be a simple complex Lie algebra of type $D_4$, and let $\sigma$ be an automorphism of $\mathfrak{g}$.
Suppose the fixed-point subalgebra $\mathfrak{g}^\sigma$ is simple and of type ...
- 3,003
3
votes
0
answers
184
views
Contracting root data?
I am going to describe a "degeneracy functor"
$$\delta_\mathfrak{q} \ : \ \text{Par}(\text{SL}_n,B_n) \ \to \ \text{Par}(\text{SL}_{n-1},B_{n-1})$$
from parabolics of a big group to a ...
- 6,201
5
votes
0
answers
124
views
Outer automorphisms of simple real Lie algebras
For a simple complex Lie algebra $\mathfrak{g}$, it is well-known that the outer automorphisms of $\mathfrak{g}$ correspond to the automorphisms of its Dynkin diagram. Is a similar result known for a ...
- 377
6
votes
1
answer
259
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Embeddings of complex simple Lie algebras
We have an embedding of the complex simple Lie algebra $G_2$ into $\mathfrak{so}_7$. Is this embedding unique up to $\mathrm{SO}_7$-conjugacy?
Note it is easy to see the embedding of $G_2$ into $\...
- 3,003
10
votes
3
answers
423
views
A representation of $\frak{g}$ that does not decompose into weight spaces?
Let $\frak{g}$ be a complex semi simple Lie algebra. The category $\mathcal{O}$ consists of special types of $\frak{g}$-representations that have decompositions into weight spaces $M_{\lambda}$, for $\...
5
votes
1
answer
224
views
Constructing a semisimple Lie algebra from a root system without choosing a base
I'm looking for an inverse of the standard map:
$$
(\mathfrak{g}, \mathfrak{h}) \mapsto \Phi
$$
which assigns a root system, to a finite-dimensional semisimple Lie algebra with distinguished Cartan ...
- 1,556
6
votes
2
answers
308
views
Coxeter group action on type C cluster algebra?
A basic question about finite type cluster algebras. Let $A$ denote the cluster algebra of type $\mathrm C_{n-1}$, $n\ge 3$. I will view $A$ as a $\mathbb Z$-algebra with the $n^2$ generators $\Delta_{...
- 3,916
7
votes
0
answers
163
views
Is there a "Weyl Character Formula" for Jacobi Polynomials
Let $R$ be a root system, $R^+$ a choice of positive roots, $W$ the Weyl group, $\Lambda$ its weight lattice, $\Lambda^+$ the cone of dominant weights, and $\rho = \frac{1}{2} \sum_{\alpha \in R^+} \...
- 295
2
votes
0
answers
106
views
Self-contragredient representation for quasi-split Lie group
I am very new to theory of Lie algebras. I have some questions and my study requires those.
Let $\mathrm R$ be a special class of representations such that $\mathrm R$ is absolutely irreducible and $\...
- 185
6
votes
1
answer
281
views
Outer automorphisms of Lie algebras
Let $\mathfrak{g}$ be a simple complex Lie algebra. Let $\sigma$ be an automorphism of the Dynkin diagram of $\mathfrak{g}$. If we choose a pinning for $\mathfrak{g}$, we can think of $\sigma$ as an ...
- 3,003
4
votes
1
answer
256
views
Equivalent definitions of the simple $\ell$-roots $A_{i,a}$ of quantum affine algebras
Let $U_q(\widehat{\mathfrak g})$ be the quantum affine algebra over a simple Lie algebra $\mathfrak g$. I am trying to understand and compare the so called simple $\ell$-roots $A_{i,a}$ seen in both ...
- 101
6
votes
0
answers
252
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Abstract root systems in positive characteristic
Does anyone know of a reference outlining the theory of abstract root systems in positive characteristic?
In the hope of inciting useful remarks, I'll outline how I imagine such a theory might be ...
- 1,556