1
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0answers
31 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
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:= ...
1
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1answer
28 views

Any explicit pictures of root datum

Consider the root datum $(X^*, \Delta,X_*, \Delta^{\vee})$ of a reductive algebraic group, where $X^*$ is the lattice of characters of a maximal torus, $X_*$ the dual lattice (given by the 1-parameter ...
2
votes
1answer
58 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 ...
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0answers
64 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 $U(n)\simeq SU(n)\times U(1)$?
3
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1answer
58 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 ...
5
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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 ...
2
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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
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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 ...
0
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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 $P$ ...
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1answer
131 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 ...
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1answer
74 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 ...
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1answer
153 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 ...