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)  A correct generator should have the following type: sage: type(A.gen(0))  I have to generalize this method so I cannot define the CGA as in the library guide using something like A. = .... 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