# Tagged Questions

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### 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 ...

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**1**answer

67 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 ...

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**1**answer

217 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 ...

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**1**answer

222 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 ...

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**2**answers

333 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 ...

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**2**answers

182 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 ...

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**0**answers

121 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 = ...

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645 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}} ...

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**3**answers

385 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 ...

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vote

**2**answers

332 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 ...

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460 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 ...

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403 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 ...

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**0**answers

533 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 ...

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**3**answers

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: ...

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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 ...

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744 views

### Complex root systems

This question is twofold.
1) What is the best reference on root systems?
2) Do complex root systems exist?

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**4**answers

866 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 + ...