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2023-12-29 02:12:29 +0100 commented answer Exterior Powers of A Cohomology

Hi Professor Palmieri, I just want to say hello and thank you for your help on this thread. This allowed me to advanced

2023-12-29 01:48:25 +0100 commented answer Exterior Powers of A Cohomology

Hi Professor Palmieri, I just want to say hello and thank you for your help on this thread. This allowed me to advanced

2023-07-12 06:31:17 +0100 marked best answer Exterior Powers of A Cohomology

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!

2023-07-12 06:31:14 +0100 commented answer Exterior Powers of A Cohomology

Thank you so much!

2023-07-12 03:38:52 +0100 commented answer Exterior Powers of A Cohomology

Thank you for the answer! This is very useful for extracting basis information. I have another quick question: How do I

2023-07-11 06:57:42 +0100 asked a question Exterior Powers of A Cohomology

Exterior Powers of A Cohomology Hi guys, I am working on computing basis of the exterior powers of a CDGA's cohomology.

2023-06-16 01:38:47 +0100 received badge  Supporter (source)
2023-06-15 05:08:02 +0100 marked best answer Commutative Graded Algebra basis shows as Expression

I am trying to create a CDGA using an existing Lie Algebra basis as follows:

sage: L = LieAlgebra(QQ, 3, step=3)
sage: gen_list = L.basis().list()
sage: gen_list
[X_1, X_2, X_3, X_12, X_13, X_23, X_112, X_113,
 X_122, X_123, X_132, X_133, X_223, X_233]

To use these generators as the ones for my CGA, after poking around a bit I tried:

sage: gen_list = str(gen_list)[1:-1] # turn list to string
sage: A = GradedCommutativeAlgebra(QQ, names=var(gen_list))
sage: A
Graded Commutative Algebra with generators
('X_1', 'X_2', 'X_3', 'X_12', 'X_13', 'X_23', 'X_112', 'X_113',
'X_122', 'X_123', 'X_132', 'X_133', 'X_223', 'X_233')
in degrees (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) over Rational Field

But when I try to access a specific generator, e.g. X_1, its type is actually just an expression:

sage: type(X_1)
<class 'sage.symbolic.expression.Expression'>

A correct generator should have the following type:

sage: type(A.gen(0))
<class 'sage.algebras.commutative_dga.GCAlgebra_with_category.element_class'>

I have to generalize this method so I cannot define the CGA as in the library guide using something like A.<x, y, z> = ....

How do I define the generators correctly to get the type that I want?

2023-06-15 05:08:02 +0100 received badge  Scholar (source)
2023-06-15 05:07:54 +0100 commented answer Commutative Graded Algebra basis shows as Expression

Thank you so much for such a descriptive explanation! That worked perfectly!

2023-06-14 09:19:55 +0100 received badge  Student (source)
2023-06-14 09:11:28 +0100 asked a question Commutative Graded Algebra basis shows as Expression

Commutative Graded Algebra bases showed as Expression Hi guys, I am trying to create a CDGA using an existed Lie Algebra