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how to calcualte the basis of quotient module over the steenrod algebra in Sage

asked 2021-05-04 14:29:04 +0100

wittenhe gravatar image

A(2) is steenrod algebra generated by sq1, sq2,sq4 with Adem relations, we knew A(2) has dimension 64, how to calculate the basis like quotient algebra A(2)/A(2){sq1} and A(2)/A(2){sq1,sq2sq3} and A(2)/A(2){sq1, sq7,sq3sq7+sq4sq6} in Sage? Thanks !

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answered 2021-05-04 18:38:22 +0100

Quotients like this are not well implemented in Sage, and in particular, there is no simple way to compute a basis. Examples of ways in which the implementation is not great:

sage: A2 = SteenrodAlgebra(profile=(3,2,1))
sage: s1 = A2.Sq(1)
sage: I = A2.ideal(s1)
sage: I
Twosided Ideal (Sq(1)) of sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [3, 2, 1]
sage: s1 in I
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)

....

NotImplementedError: 
sage: R = A2.quotient(I)
sage: R(s1) == 0
False
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Asked: 2021-05-04 14:28:08 +0100

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Last updated: May 04