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Exterior Powers of A Cohomology

asked 2023-07-11 06:57:42 +0200

Hieu Nguyen gravatar image

Hi guys, I am working on computing basis of the exterior powers of a CDGA's cohomology. The following code is what I currently have:

L = lie_algebras.Heisenberg(QQ, 2)
A = GradedCommutativeAlgebra(QQ, names=L.basis())
A.inject_variables()
Output: Defining p1, p2, q1, q2, z
B = A.cdg_algebra({p1: 0, p2: 0, q1: 0, q2: 0, z: p1*q1 + p2*q2})
C = B.cohomology(1)
C.basis()
Output: Finite family {[p1]: B[[p1]], [p2]: B[[p2]], [q1]: B[[q1]], [q2]: B[[q2]]}

First, I tried to manipulate the basis to possibly calculate all the possible wedge products but I found out that type(C) is sage.combinat.free_module.CombinatorialFreeModule_with_category; thus, I cannot do wedges. Then, I tried to compute the dual exterior powers of C using preexisted method. I imported ExtPowerDualFreeModule and run E = ExtPowerDualFreeModule(C, 2) but did not succeed. Hence, is there a way to compute the dual exterior powers and see their bases? If not, can the free module C be converted to another object so that I can perform wedges on its basis? It is very important for me to be able to see the basis of the exterior powers of the cohomology.
I apologize for my lack of mathematical knowledge. If I am missing any information, please let me know. I really appreciate your help, thank you!

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answered 2023-07-11 18:57:36 +0200

You can first compute the cohomology algebra in a range of dimensions:

HB = B.cohomology_algebra(10)

Then you can form an exterior algebra using either a basis for HB in a particular dimension or all of its algebra generators:

ExteriorAlgebra(QQ, HB.basis(1))

or

ExteriorAlgebra(QQ, HB.gens())

One potential difficulty is that the generators have been renamed when computing cohomology, so it may be hard to tell where they came from in B. You can try to recover that information using B.cohomology_generators(10), which lists the generators in each degree. You can then match them up to HB.gens().

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Thank you for the answer! This is very useful for extracting basis information. I have another quick question: How do I compute exterior powers on such algebra? I am trying to find a linear map (matrix) between such k-th exterior power to the k-th cohomology of the original Lie algebra.

Hieu Nguyen gravatar imageHieu Nguyen ( 2023-07-12 03:38:52 +0200 )edit

If E = ExteriorAlgebra(...), then E.basis(k) will return the basis for the kth exterior power.

John Palmieri gravatar imageJohn Palmieri ( 2023-07-12 03:42:19 +0200 )edit

Thank you so much!

Hieu Nguyen gravatar imageHieu Nguyen ( 2023-07-12 06:31:14 +0200 )edit

Hi Professor Palmieri, I just want to say hello and thank you for your help on this thread. This allowed me to advanced on my undergraduate research and gained a quite good result. Your work inspired me to applied for the Mathematics Ph.D. at UW for Fall 2024. If my application ever comes to you, I hope you would give it a careful consideration and that our interests align. There are so many things I want to learn from you. May we meet again!

Hieu Nguyen gravatar imageHieu Nguyen ( 2023-12-29 01:48:25 +0200 )edit

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Asked: 2023-07-11 06:57:42 +0200

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Last updated: Jul 11 '23