agh314

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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:

 [Matrix(4, 4, [[1, 0, 0, 0], [0, -1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]), Matrix(4, 4, [[0, 0, 1, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]), Matrix(4, 4, [[0, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]), Matrix(4, 4, [[0, 0, 0, 0], [0, 1, 0, 0], [0, 0, -1, 0], [0, 0, 0, 0]]), Matrix(4, 4, [[0, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 0], [0, 0, 0, 0]]), Matrix(4, 4, [[0, 0, 0, 0], [0, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0]]), Matrix(4, 4, [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, -1]])]

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!

Howdy,

I am trying to understand various approaches to threading in Maple. I can naturally parallelize some computations, but only if I can figure out threading. The punchline is this - running something on a single core takes significantly longer when I break it into threads.

I picked a command that takes some time - suppose I have a Lie algebra G, and I want to know if H is a subalgebra. If I do this 16 times in a row, it takes about 3 seconds consistently. Here I made an array of 16 copies of H. Then I can simply ask which element is a subalgebra:

is_subalgebra:=proc(list_of_algs)
local i;
for i in list_of algs do
Query(i, "Subalgebra");
end do;
end proc;


Then running the following gives me an answer of about 3:

st := time();
is_subalgebra(my_list);
time() - st;

Suppose I want to run this on cores = 4 or 5 or...

are_subalgebras_t := proc(list, cores)

local ...., mutex, threadlist;
threadlist:= [];
mutex := Threads[Mutex][Create]();
<create cores chunks of list, call it sublist>
for i in sublist do
 Threads[Mutex][Lock](mutex);
 threadlist:= [op(ts), Create(are_subalgebra(i))];
 Threads[Mutex][Unlock](mutex);
end do;
Wait(op(threadlist));
return;
end proc:

The unecessary code is removed, but I:

1: Create a mutex

2: Chunk up my list into as many pieces as I want

3: Lock my mutex

4: Spawn a thread on the current chunk, save the thread id in a list

5: Unlock the mutex (I just want to add thread ids to my list safely)

6: Repeat 3-5 for each chunk

7: Wait until all threads are done

My machine has 16 cores, and if I run this on say 8 cores it takes almost 10 seconds. I can see that it starts executing the threads immediately - an example run shows I might spawn two threads, then the first thread starts, the third spawns, and then they complete in some order. Predictably, every run through is different. This tells me that I am getting parallelization, but at the detriment of runtime.

Can anyone tell me where my approach is going wrong? I was unable to get any Map functions to work at all.

Thanks!

Hey everyone,

I see I can use SimpleLieAlgebraData to create Lie algebras of types A, B, C, D, and also G2 and F4. Is there a built-in way to generate E6, E7, and E8? If not, is there any plan to add these?

Thanks!

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