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2019-10-24 04:57:07 -0600 asked a question Vector subspaces of Verma Modules

I know how to construct ordinary vector spaces over fields. For example a vector space of dimension 4 over rationals:

sage: V=VectorSpace(QQ, 4)

Given a set of linearly independent elements, we can construct the subspace of V formed by that set. For example we can construct the subspace whose basis is {$(1,0,0,0), (0,1,0,0)$}:

sage: U=V.subspace_with_basis([(1,0,0,0), (0,1,0,0)])

Now we given $u\in U$, we can also find its coordinates with respect to the basis {$(1,0,0,0), (0,1,0,0)$}. For example, if we take the vector $(2,2,0,0)\in U$:

sage: U.coordinates((2,2,0,0))
sage: [2, 2]

I want to do something similar with Verma Modules. So given a linearly independent set and a vector, I want to get its coordinates w.r.t this basis.

So first let's construct a Verma Module over $\frak{sp}(4)$:

sage: L = lie_algebras.sp(QQ, 4)
sage: La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
sage: M = L.verma_module(La[1] - 3*La[2])

Let $v$ be the highest weight vector.

sage: v=M.highest_weight_vector()
sage: pbw = M.pbw_basis()
sage: x1,x2,y1,y2,h1,h2 = [pbw(g) for g in L.gens()]

I want to construct the $\mathbb{Q}$-subspace of $M$ spanned by a given basis. For example I tried to construct a subspace with basis {$y_1^2\cdot v, y_2^3\cdot v$}. So quite naively I tired:

sage: M.subspace_with_basis([y1^2*v, y2^3*v])

which gave a big error message.

Is there a way to solve this problem?

On a related note, is there any way to check if a given set of elements in a Verma module are linearly independent?

2019-10-14 20:43:47 -0600 answered a question sage-8.9 fails compilation in Ubuntu 18.04

I recently installed SAGE on ubuntu 18.04. This command worked for me

sudo apt-get install sagemath

Check http://www.sagemath.org/download-linu... (go to the distributions section)

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2019-10-14 19:33:14 -0600 asked a question Verma modules and accessing constants of proportionality

The Math Part: Let me first describe the math without going into the programming. Start with two vectors $v$ and $w$ in a vector space (just a regular vector space with no additional structure). Let's say we know that $w=\lambda\cdot v$ for some scalar $\lambda$. Given $w$ and $v$, can we figure out what $\lambda$ is?

The Programming Part: Now let me describe specifics of my calculation. I am working with a Verma Module over $\frak{sp}(4)$.

sage: L = lie_algebras.sp(QQ, 4)
sage: La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
sage: M = L.verma_module(La[1] - 3*La[2])
sage: pbw = M.pbw_basis()
sage: x1,x2,y1,y2,h1,h2 = [pbw(g) for g in L.gens()]

We will call the highest weight vector $v$. In code,

sage: v = M.highest_weight_vector()
sage: v
sage: v[Lambda[1] - 3*Lambda[2]]

Now we have $x_2y_2\cdot v=-3\cdot v$ and $x_2^2y_2^2\cdot v= 24\cdot v$. So in code,

sage: x2*y2*v
sage: -3*v[Lambda[1] - 3*Lambda[2]]
sage: x2^2*y2^2*v
sage: 24*v[Lambda[1] - 3*Lambda[2]]

In general, we will have $$x_2^ny_2^n\cdot v=c_n\cdot v$$ for some constant $c_n$ (with $c_1=-3$ and $c_2=24$).

My questions is the following.

How to access this constant $c_n$, given that we know $v$ and $c_n\cdot v$?

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2019-09-30 18:17:44 -0600 commented answer Questions about Lie algebra

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 $i$.

2019-09-30 03:44:33 -0600 asked a question 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?

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2019-09-24 02:49:33 -0600 asked a question Computations on Verma Modules

I want to do some computation on Verma Modules. Consider the Verma Module $W_\lambda$ of weight $\lambda$. We know that $W_\lambda$ has a unique maximal submodule $N_\lambda$ and a corresponding irreducible quotient $L(\lambda)=W_\lambda/N_\lambda$.

  1. I found some documentation about Verma Modules on the SAGE website. But it does not tell how to find the irreducible quotient. I want to figure out this quotient $L(\lambda)$.
  2. Secondly, I want to see know the dimensions of weight spaces of $W_\lambda$ and $L(\lambda)$ and what they look like explicitly.

Could you please help me with the syntax? Thanks is advance.