I have a graded algebra A over a field k, and I would like the following
behavior: when I multiply homogeneous elements of the tensor square (A
tensor A), I want signs to appear, as in:
(a tensor b) (c tensor d) = (-1)^(deg b deg c) (ac tensor bd)
You could ask for the same when multiplying elements of (A tensor B) for
two graded algebras A and B. This may not be desirable for all graded
algebras in Sage, but it might be useful in more than one case. How should
this be implemented?
I'm guessing and/or hoping that modifying something in the category code
would help, and that one could appropriately initialize the categories of A
and B to turn this feature on, but I'm confused enough by the category code
that I don't know where to start. Any suggestions? (Or is the category
approach not viable, so something else (and what?) should be done?)
To illustrate my confusion, if A is the mod 3 Steenrod algebra and if y is
an element in (A tensor A), I don't even know how the multiplication y*y is
defined. Is this category code, coercion, something else? Note that this
example leads to a bug:
sage: A = SteenrodAlgebra(3)
sage: x = A.Q(0)
sage: x**2
0
sage: y = x.coproduct()
sage: y**2
2*Q_0 # Q_0
The coproduct is an algebra map, so if x**2=0, then (x.coproduct())**2
should also be zero, but it's not. If the signs were dealt with
appropriately, this would be okay, but as it is, we have a bug.
--
John
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