Signs and graded algebras

[John H Palmieri]

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

[Travis Scrimshaw]

Hey John,
   I'm not quite sure. There are two approaches that come to mind. The first is to implement an analog of CombinatorialFreeModule_Tensor for this setting. Then it is a matter of overriding the Tensor attribute on the classes you want to use for your special tensor multiplication. However, it is a more general construction, i.e., should happen for all superalgebras (with basis), then you should do something in the corresponding TensorCategory by having a default product_on_basis method.

Best,
Travis