ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sun, 06 Oct 2019 03:04:50 +0200Questions about Lie algebrahttps://ask.sagemath.org/question/48103/questions-about-lie-algebra/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)
1. 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?
2. 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?Mon, 30 Sep 2019 10:44:33 +0200https://ask.sagemath.org/question/48103/questions-about-lie-algebra/Answer by heluani for <p>I am trying to do some calculations and I don't understand what the output is. </p>
<pre><code>L = lie_algebras.sp(QQ, 4)
L.gens()
</code></pre>
<p>The output is the following</p>
<pre><code>(E[alpha[1]], E[alpha[2]], E[-alpha[1]], E[-alpha[2]], h1, h2)
</code></pre>
<ol>
<li><p>I know that the command <code>L.gens()</code> 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? </p></li>
<li><p>Let's say I want to figure out $\alpha_1(h_1)$. So I thought maybe <code>alpha1(h1)</code> will give me the answer. But I am getting an error. I also tried <code>L.alpha[1](h1)</code> which results in an error as well. How can I fix this?</p></li>
</ol>
https://ask.sagemath.org/question/48103/questions-about-lie-algebra/?answer=48104#post-id-481041) Yes,
<pre><code>sage: L = lie_algebras.sp(QQ,4)
sage: L
Lie algebra of ['C', 2] in the Chevalley basis
</code></pre>
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
<pre><code>
sage: L = lie_algebras.sp(QQ,4, representation='matrix')
sage: L.simple_root(1,L.h(1))
2
</code></pre>
Mon, 30 Sep 2019 12:47:36 +0200https://ask.sagemath.org/question/48103/questions-about-lie-algebra/?answer=48104#post-id-48104Comment by heluani for <p>1) Yes, </p>
<pre><code>sage: L = lie_algebras.sp(QQ,4)
sage: L
Lie algebra of ['C', 2] in the Chevalley basis
</code></pre>
<p>If you want a basis try <code>L.basis()</code></p>
<p>2) you can get this directly from the Cartan Matrix which by definition will get you <code>2</code> since <code>h(1)=alphacheck[1]</code> is the coroot associated to <code>alpha[1]</code>. In this particular case you could try</p>
<pre><code>
sage: L = lie_algebras.sp(QQ,4, representation='matrix')
sage: L.simple_root(1,L.h(1))
2
</code></pre>
https://ask.sagemath.org/question/48103/questions-about-lie-algebra/?comment=48199#post-id-48199L.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/reference/algebras/sage/algebras/lie_algebras/classical_lie_algebra.htmlSun, 06 Oct 2019 03:04:50 +0200https://ask.sagemath.org/question/48103/questions-about-lie-algebra/?comment=48199#post-id-48199Comment by slartibartfast for <p>1) Yes, </p>
<pre><code>sage: L = lie_algebras.sp(QQ,4)
sage: L
Lie algebra of ['C', 2] in the Chevalley basis
</code></pre>
<p>If you want a basis try <code>L.basis()</code></p>
<p>2) you can get this directly from the Cartan Matrix which by definition will get you <code>2</code> since <code>h(1)=alphacheck[1]</code> is the coroot associated to <code>alpha[1]</code>. In this particular case you could try</p>
<pre><code>
sage: L = lie_algebras.sp(QQ,4, representation='matrix')
sage: L.simple_root(1,L.h(1))
2
</code></pre>
https://ask.sagemath.org/question/48103/questions-about-lie-algebra/?comment=48109#post-id-48109Thanks 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 $i$.Tue, 01 Oct 2019 01:17:44 +0200https://ask.sagemath.org/question/48103/questions-about-lie-algebra/?comment=48109#post-id-48109