Questions about Lie algebra

asked 2019-09-30 10:44:33 +0200

I am trying to do some calculations and I don't understand what the output is.

L = lie_algebras.sp(QQ, 4)

L.gens()

The output is the following

(E[alpha[1]], E[alpha[2]], E[-alpha[1]], E[-alpha[2]], h1, h2)

I know that the command L.gens() gives a set of genearators of the Lie algebra. So I understand that this is giving us an element from $e_\alpha\in L_\alpha$ for each $\alpha\in \Delta$ and the corresponding elements $h_\alpha \in H$, (where $\Delta$ is a base of the root system and $H$ is a Cartan Subalgebra). But I don't understand what these elements exactly are. Are these elements of a Chevalley basis?
Let's say I want to figure out $\alpha_1(h_1)$. So I thought maybe alpha1(h1) will give me the answer. But I am getting an error. I also tried L.alpha[1](h1) which results in an error as well. How can I fix this?

answered 2019-09-30 12:47:36 +0200

heluani
304 ●8 ●19

1) Yes,

sage: L = lie_algebras.sp(QQ,4)
sage: L
Lie algebra of ['C', 2] in the Chevalley basis

If you want a basis try L.basis()

2) you can get this directly from the Cartan Matrix which by definition will get you 2 since h(1)=alphacheck[1] is the coroot associated to alpha[1]. In this particular case you could try


sage: L = lie_algebras.sp(QQ,4, representation='matrix')
sage: L.simple_root(1,L.h(1))
2

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Comments

Thanks for the answer. I have one further question, when you are writing L.h(1) what does it mean? Is this some generic element? In general, what does it mean if we write L.h(i) for some integer $i$.

slartibartfast ( 2019-10-01 01:17:44 +0200 )edit

L.h(i) is the element h_i in the Cartan subalgebra for a simple root alpha_i. Take a look in https://doc.sagemath.org/html/en/refe...

heluani ( 2019-10-06 03:04:50 +0200 )edit

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