ASKSAGE: Sage Q&A Forum - Latest question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Thu, 29 Aug 2019 05:30:20 -0500Virasoro Verma module Basishttps://ask.sagemath.org/question/47657/virasoro-verma-module-basis/Hi I am starting to look at the implementation of the Virasoro algebra and some of its modules in
https://doc.sagemath.org/html/en/reference/algebras/sage/algebras/lie_algebras/virasoro.html
I have a question regarding the method basis() that returns a basis of the module. As the following example shows this particular linear combination of basis elements is zero so I wander in which sense are we getting a basis or a generating set, or even better how to actually get sage to recognize that the combination is actually zero?
<pre><code>sage: g = lie_algebras.VirasoroAlgebra(QQ)
sage: V = g.verma_module(1/2,0)
sage: d = g.basis()
sage: B = V.basis()
sage: v = V.highest_weight_vector()
sage: B[-2,-1] - B[-1,-2] + B[-3] == V.zero()
False
sage: B[-2,-1] - B[-1,-2] + B[-3]
d[-3]*v + d[-2]*d[-1]*v - d[-1]*d[-2]*v
sage: d[-3]*v + d[-2]*(d[-1]*v) - d[-1]*(d[-2]*v) == V.zero()
True</pre>heluaniThu, 29 Aug 2019 05:30:20 -0500https://ask.sagemath.org/question/47657/A Sage implementation of the Virasoro algebra and its representationshttps://ask.sagemath.org/question/10809/a-sage-implementation-of-the-virasoro-algebra-and-its-representations/Hello everyone,
A few years ago I tried to implement the Virasoro algebra on Sage, but due to having a lot of other things to do, I didn't really get anywhere. Now I'd like to try again in earnest. But due to not having a lot of experience with programming, I'm having a hard time reading the Sage documentation. So I was hoping people could give me some pointers to get me going.
First some math: The Virasoro algebra is a complex infinite dimensional Lie algebra with generators $L_n, n\in \mathbb{Z}$ and bracket
$[L_m,L_n]=(m-n)L_{m+n}+\delta_{m,-n}(m^3-m)\frac{C}{12}$, where $\delta$ is the Kronecker delta and $c$ is the central element of the Virasoro algebra (aka central charge).
A Verma module $M(h,c)$ of the Virasoro algebra is generated by a highest weight vector $v(h,c)$ such that
$L_0v(h,c)=hv(h,c)$, $Cv(h,c)=cv(h,c)$ and $L_nv(h,c)=0,\ n>0$.
The remaining generators act freely up to the relations coming from the Virasoro algebra itself. Therefore, a basis of $M(h,c)$ is given by
$L_{-n_1}L_{-n_2}\cdots L_{-n_m}v(h,c)$,
where $n_1\geq n_2\geq \cdots \geq n_n\geq1,\ m\geq0$, i.e. it is parametrised by all partitions of integers.
Here is what I've learned from talking to people on this forum a few years ago:
Since the Virasoro algebra and its Verma modules are parametrised by integers and partitions of integers, it seems natural to define them using CombinatorialFreeModule, e.g.
`Vir=CombinatorialFreeModule(QQbar,Integer(), prefix='L')`
(this does not include the central element $C$). Unfortunately I'm clueless as to how to define the Lie bracket, orderings of generators and an action on the Verma module.
The list below is a detailed list of things I wish to implement. If someone could help me with one or two of those, I think I should be able to figure out the rest by example.
1. Define a bracket on the generators that extends linearly to the whole Virasoro algebra.
2. Define an ordering of generators such that a multiplication of generators can be defined, i.e. the universal enveloping algebra. The ordering is: If $m < n$ then $L_m \cdot L_n$ just stays $L_m\cdot L_n$ but $L_n\cdot L_m=[L_n,L_M]+L_m\cdot L_n$. All the better if this product could overload the multiplication symbol *.
3. Define an action of Virasoro generators on basis elements of $M(h,c)$ and extend this action linearly to all of $M(h,c)$.
4. Extend the action of Virasoro generators on $M(h,c)$ to products of Virasoro generators acting on $M(h,c)$.
Thanks in advance for any advice.GonnemanThu, 05 Dec 2013 11:31:19 -0600https://ask.sagemath.org/question/10809/