2
votes
2answers
118 views

Which linear combinations of simple roots are roots

Let $\Delta$ be the root system of a complex simple Lie algebra, $\Delta^+$ be positive roots and $\Pi$ be simple roots. We view $\Pi$ as nodes of the Dynkin diagram. Then for any two simple roots ...
0
votes
1answer
63 views

A property of compact involutions of semi-simple Lie algebras?

I need to prove the statement below. Since my background on Lie theory is rather weak, I post it here. Let $\frak{g}$ be a complex semi-simple Lie algebra. Fix a Cartan subalgebra $\frak{h}$ with ...
4
votes
1answer
216 views

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 ...
4
votes
1answer
218 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 ...
3
votes
2answers
326 views

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 ...
3
votes
2answers
170 views

How to find faces of polytope defined by a Weyl orbit

A few days ago I asked the following question at MSE and received no answer. I thought I would try here. Let $\xi$ be an integral dominant weight of an irreducible root system $\Delta$, and let ...
2
votes
0answers
117 views

simple roots of a reflection subgroup

Consider a Hermitian symmetric pair of complex Lie algebras $(\mathfrak{g},\mathfrak{k})$ and split the set of roots into compact roots (i.e. roots of $\mathfrak{k}$) and noncomapt roots $\Delta = ...
3
votes
1answer
642 views

A possible mistake in Kac's “Infinite Dimensional Lie Algebras”

I have a paperback 3rd edition and on page 65 you can find Proposition 5.8. My question is about part (c): If $A$ is of indefinite type, then $$ \overline{X} = \{ h \in \{ \frak h_{\mathbb{R}} ...
7
votes
3answers
379 views

Can one easily pick out a basis of a simple Lie algebra after picking a convex order?

One of the basic results of Lie theory is that if one picks a Cartan subalgebra of a simple Lie algebra, there there is a canonical decomposition of $\mathfrak{g}$ into the Cartan and a bunch of ...
1
vote
2answers
331 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 ...
4
votes
1answer
453 views

Cartan Matrices of type B and C.

I was using the built-in functions for Root Systems in SAGE, and I noticed that the Cartan Matrices for Type $B_n$ and type $C_n$ are interchanged from what I thought they would be, i.e. following the ...
6
votes
1answer
396 views

Motivation behind Kac's notation for affine root systems

I'm reading Kac's Infinite Dimensional Lie Algebras. In Chapter 4, he classifies the affine root systems. Bourbaki classified the affine Coxeter groups, but multiple root systems can give the same ...
2
votes
0answers
525 views

dual Coxeter number, affine algebra, invariants under twisting

Sometime ago we came across invariant quantities under twisting of all affine algebra. (See the appendix E of http://arxiv.org/abs/hep-th/0403076 .) Choose the convention so that the longest root has ...
17
votes
3answers
1k views

Number of triples of roots (of a simply-laced root system) which sum to zero

In a paper 1105.5073, the authors took a simply-laced root system $\Delta$ of type $G=A,D,E$, and then counted the number of unordered triples $(\alpha,\beta,\gamma)$ of roots which sum to zero: ...
4
votes
3answers
1k views

Modular Forms and Root Systems

In the study of semisimple Lie groups, lattices appear all over the place. In the theory of elliptic functions and modular forms, (equivalence classes of) lattices correspond to elliptic curves and to ...
2
votes
2answers
744 views

Complex root systems

This question is twofold. 1) What is the best reference on root systems? 2) Do complex root systems exist?
9
votes
4answers
863 views

Root systems and sums of squares

It is easy to see that the quadratic form for the root system $A_n$ is a sum of $n+1$ squares of integral linear forms: $$q_{A_n} = 2 \sum_{i=1}^n x_i^2 - 2 \sum_{i=1}^{n-1} x_i x_{i+1} = x_1^2 + ...