Incorrect ring map evaluation over Weyl algebras

[mahrud]

Ring maps are still not working correctly for ring maps between Weyl algebras and polynomial rings:

needsPackage "WeylAlgebras"

D = QQ[x,y,dx,dy, WeylAlgebra => {x => dx, y => dy}]
-- the associated graded ring is commutative
R = (extractVarsAlgebra D) (monoid[D.dpairVars#1])

x_D * dx_D
sub(x_D * dx_D, R) -- good

x_R * dx_R
sub(x_R * dx_R, D) -- bad: x*dx + 1

cc: @mikestillman @antonleykin

(this doesn't have anything to do with non-associative algebras, but also cc @moorewf just in case you have any ideas)

[antonleykin]

I think both subs and e.g.

f = map (R,D)

should result in error, since

f(dx_D*x_D) == f(dx_D)*f(x_D)

is false.
(Alternatively, class f should not be RingMap.)

[moorewf]

But we use RingMaps even when the maps are not well-defined all the time. For example, when R --> S is a surjection and you need a quick way to map linear forms from S to linear forms in R, one often considers map(R,S,gens R).

[antonleykin]

I understand it's hacked at the moment, so perhaps a "good" approach would be to have a type named e.g. "SubstitutionMap" for which we can say what behavior is "good" (e.g. for @mahrud 's scenario, this is a map of vector spaces with distinguished monomial bases.)

[mahrud]

@antonleykin note that in my example I didn't use a ring map, I used substitute. I don't care what's the internal mechanism of substitute. I don't need a new type either, just for substitute to work correctly for sending elements to and from the associated graded ring of the Weyl algebra. How would you suggest this should happen?

[antonleykin]

@antonleykin note that in my example I didn't use a ring map, I used substitute.
Well, you didn't use a map but sub did. The behavior of sub is described (in the "caveats") of it's doc. (Also, the subj of this issue seems to imply this 😄 )
While creating this map it probably flattened the ring:

(last flattenRing R) (x_R*dx_R)

gives you dx*x.

I don't care what's the internal mechanism of substitute. I don't need a new type either, just for substitute to work correctly for sending elements to and from the associated graded ring of the Weyl algebra.

As an M2 user I agree, M2 should do what we want "correctly" and with ease.

How would you suggest this should happen?

That depends on what "correctly" means (for @mahrud, @moorewf, @antonleykin, etc.). Note: "to/from" for (D and gr D) happen to be not available as maps. Perhaps the most common meaning (for an average user) is "substitute, literally". Should one wrap something like this

use D
value toString (x_R*dx_R)

in a method? Should it still be called sub?

[mahrud]

My point was that I'm fine with ring maps involving Weyl algebras giving an error in certain circumstances, and changing sub to not use ring maps at all.

Incidentally, sub is a frequent source of inefficiency for many people, I think because it constructs a ring map, uses it, then discards it. I think a vector space based sub could solve this issue.