# Question:Lie algebra deformation question

## Question:Lie algebra deformation question

Maple

Hi,

I am having trouble with a Lie algebra cohomology computation. Suppose I have a poset on {1,2,3,4} where 1 < 3, 1 < 4, 2 < 3, and 2 < 4. I can express this as a matrix:

* 0 * *
0 * * *
0 0 * 0
0 0 0 *

where *'s mean "any entry in my ground field," say R or C, and 0s are 0s. Basically, if there is a relation between row i and column j, there is a *. This is why there is a * in row-1 and column 3, as 1<3, but a 0 in row-1 and column 2. I can make the collection of all of these matrices into a Lie algebra using the commutator, as it is closed, and can further suppose it is of trace 0 - that is, it is Type A.

My question is this: I know this algebra has non-trivial cohology, and deforms. However, I want to make Maple do this for me, so I can try it on bigger algebras - however it always tells me that the cohomology is dead zero. What am I doing wrong? My approach is this:

Let P equal the following collection of matrices - these form my basis:

Now I can run the basic commands to get started:

L := LieAlgebraData(P, Ex1);
DGsetup(L);

I can now go straight to cohomology. If my algebra is named L, then I want to build my cochain complex C^*(L, L):

c := RelativeChains([e1,e2,e3,e4,e5,e6,e7]);

However, the answer is always that there are no non-trivial cochains: the answer is [[], []]. This will make it very difficult to have non-trivial cohomology.

I know this isn't true (see https://arxiv.org/pdf/1407.0428.pdf). I also tried the approach in the Maple documentation, where I work in the adjoint representation. This gave me non-trivial cochains, but the cohomolgy was 0.

Does anyone know what I'm doing wrong?

Thanks!

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