quotient of a Weyl algebra by a left ideal

[DanGrayson]

Reported by: anton (on trac)

Forming a QuotientRing as in the example below should produce an error (for noncommutative algebras)... as this makes little sense.

i1 : W = QQ[x,dx,WeylAlgebra=>{x=>dx}]

o1 = W

o1 : PolynomialRing

i2 : A := W/ideal dx

  W
o2 = --                                                                         
 dx

o2 : QuotientRing

i3 : dx_A*x_A == (dx*x)_A

o3 = false

An alternative behaviour would be to return the module W^1/I.

[DanGrayson]

Perhaps a better alternative behavior would be to declare that ideals of class Ideal should
be 2-sided. And then to leave them unimplemented until someone comes up with an
algorithm for them. Example: given generators of an ideal, find generators for it as a left ideal.

We don't need the concept of left ideal or of right ideal because we have the concept of submodule
of R^1.

Comments?

[antonleykin]

In the case of the Weyl algebra two-sided ideals are trivial...
Given that M2 does not implement many more noncommutative algebras and that for "GB-friendly algebras", in general, one-sidedness is natural there is no necessity in redefining Ideal to be two-sided at this point.

[DanGrayson]

We have to do something, though, because R/I will still try to make a ring.

On Tue, Oct 29, 2013 at 4:15 PM, antonleykin notifications@github.comwrote:

In the case of the Weyl algebra two-sided ideals are trivial...
Given that M2 does not implement many more noncommutative algebras and
that for "GB-friendly algebras", in general, one-sidedness is natural there
is no necessity in redefining Ideal to be two-sided at this point.

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[DanGrayson]

The thing to do is to have "ideal(...)" check the ring, and if it's noncommutative, to give
an error message: ideals in noncommutative rings are not implemented. And to fix
the documentation under the class Ideal to say that ideals are two-sided, so quotient
rings make sense.

[antonleykin]

... that will invalidate a lot of code in the packages. There is also no mechanism implemented to do computations in noncommutative quotient rings.

On the other hand, making a remark in documentation for Ideal that "left" is implied in the noncommutative case and giving error in Ring/Ideal does not break anything.

[DanGrayson]

Maybe your point is that we actually need left ideals for something. What
do
you use them for?

If that's the case, then we have to distinguish between left ideals and
two-sided
ideals somehow. And it would be good to design that now.

On Wed, Oct 30, 2013 at 8:04 AM, antonleykin notifications@github.comwrote:

... that will invalidate a lot of code in the packages. There is also no
mechanism implemented to do computations in noncommutative quotient rings.

On the other hand, making a remark in documentation for Ideal that "left"
is implied in the noncommutative case and giving error in Ring/Ideal does
not break anything.

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[antonleykin]

Left ideals (which are implemented in M2) have been already used in packages, two-sided ideals do not exist as a concept implemented in M2.

One of my points is that there is no pressing need to introduce two-sided ideals.

[DanGrayson]

What are left ideals currently used for in packages? Do they offer any
advantages
over submodules of R^1?

If we're going to introduce 2-sided ideals eventually, now is as good a
time as
any to plan the organizational aspects that will make that possible.

On Wed, Oct 30, 2013 at 8:46 AM, antonleykin notifications@github.comwrote:

Left ideals (which are implemented in M2) have been already used in
packages, two-sided ideals do not exist as a concept implemented in M2.

One of my points is that there is no pressing need to introduce two-sided
ideals.

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[antonleykin]

Left ideals and left submodules of the ring are, of course, the same thing. By the same token two-sides ideals are the sub-bimodules of the ring considered as a bimodule.

I think, people who would use two-sided ideals and algebras of non-polynomial type should contribute to this discussion... but suppose we do want to introduce two-sided ideals now. I see two ways:

• make a new type "TwoSidedIdeal"
• let the meaning of "Ideal" depend on the ring: e.g., for the Weyl algebra it will remain left (since two-sided ideals are useless), but for some matrix algebra of the future it would be two-sided.

[DanGrayson]

You could have a family of Weyl algebras over a polynomial ring. That would
be an algebra with additional central variables, and then it would make
sense to
form a quotient ring by an ideal generated by central elements.

It would too confusing to have the meaning of Ideal depend on the context.
If submodules
of R^1 can be used instead of left ideals, why have left ideals at all? What
are left ideals currently used for in your packages? Could you easily
transition to submodules of R^1?

I'm still leaning toward making Ideal mean 2-sided ideal, since there seems
to be no
concrete use for left ideals.

On Fri, Nov 1, 2013 at 8:54 AM, antonleykin notifications@github.comwrote:

Left ideals and left submodules of the ring are, of course, the same
thing. By the same token two-sides ideals are the sub-bimodules of the ring
considered as a bimodule.

I think, people who would use two-sided ideals and algebras of
non-polynomial type should contribute to this discussion... but suppose we
do want to introduce two-sided ideals now. I see two ways:

• make a new type "TwoSidedIdeal"
• let the meaning of "Ideal" depend on the ring: e.g., for the Weyl
  algebra it will remain left (since two-sided ideals are useless), but for
  some matrix algebra of the future it would be two-sided.

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[mikestillman]

I think that Anton is correct though, in that people use Ideal currently to mean left ideal, and that to change it would break significant amount of code. (At least, I suspect that it is significant). I would like to hear from Frank Moore about what he thinks, as he is programming non-commutative algebras.

[moorewf]

In my NCAlgebra package, I have been distinguishing between left, right and two-sided ideals (though the package would not handle the Weyl algebra very well (if at all), as it is not graded). Left and right ideal types are not as fully featured as the two-sided ideal types at this time.

I think allowing 'Ideal' to be dependent on the ring is indeed dangerous, since left/right ideals and two-sided ideals are so fundamentally different. I think Dan's example of a family of Weyl algebras also shows the utility of this approach. It does seem that to make this change would break a lot of code, but a new type called LeftIdeal can be made, with all the relevant utility functions that Ideal has, so that the current code would not need any 'substantial' changes.

It seems that if one implemented functions with sided modules (or bi-modules), then one would by extension also have operations on sided ideals (or ideals) as well. The NCAlgebra package does not yet have functionality for modules, but will soon I think.

Hope this helps.

[DanGrayson]

Frank, do you see any use for left ideals or right ideals that wouldn't be
served adequately by submodules
of R^1? And do you really need both left and right, or will one suffice?
If not, do you need also left and right
free modules, etc?

On Fri, Nov 1, 2013 at 3:43 PM, Frank Moore notifications@github.comwrote:

In my NCAlgebra package, I have been distinguishing between left, right
and two-sided ideals (though the package would not handle the Weyl algebra
very well (if at all), as it is not graded). Left and right ideal types are
not as fully featured as the two-sided ideal types at this time.

I think allowing 'Ideal' to be dependent on the ring is indeed dangerous,
since left/right ideals and two-sided ideals are so fundamentally
different. I think Dan's example of a family of Weyl algebras also shows
the utility of this approach. It does seem that to make this change would
break a lot of code, but a new type called LeftIdeal can be made, with all
the relevant utility functions that Ideal has, so that the current code
would not need any 'substantial' changes.

It seems that if one implemented functions with sided modules (or
bi-modules), then one would by extension also have operations on sided
ideals (or ideals) as well. The NCAlgebra package does not yet have
functionality for modules, but will soon I think.

Hope this helps.

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[moorewf]

We haven't decided whether we need both left and right, since of course left modules over R are right modules over R^op, and that transition from R to R^op is aided by some functionality already.

If you want to take a look at what we have done thus far, you can find the package under "workshops/WFU-2012/NCAlgebra" on subversion. (I know, shame on me for not moving to git. Is there a 'friendly' introduction that you could point me to?)

While I don't necessarily see any theoretical benefit of having left ideals vs. left submodules of the left free module R^1, I do think it is beneficial to the user, much in the same way that M2 has facility for both Ideals as well as submodules of free modules, has them typed differently, etc.

[DanGrayson]

So maybe Ideal should be augmented with a new class called LeftIdeal. (We
provide
only left modules, so providing right ideals but not right modules would be
overkill.)
And then "ideal", the function, could be augmented with a function
"leftIdeal". Fixing
any broken code would then be trivial, and I could do it at the same
moment, by running
all of the packages' tests.

I see that in your package you have created three classes for the three
types of ideals,
bypassing the issue with Ideal that we're discussing. In particular, your
code would
probably not break if the functionality of "ideal" changes slightly.

Does anyone get confused that scalars and matrices both act on the left?
That's why
matrix multiplication is implemented with the opposite multiplication, so
it will give
a homomorphism of left R-modules.

On Fri, Nov 1, 2013 at 4:18 PM, Frank Moore notifications@github.comwrote:

We haven't decided whether we need both left and right, since of course
left modules over R are right modules over R^op, and that transition from R
to R^op is aided by some functionality already.

If you want to take a look at what we have done thus far, you can find the
package under "workshops/WFU-2012/NCAlgebra" on subversion. (I know, shame
on me for not moving to git. Is there a 'friendly' introduction that you
could point me to?)

While I don't necessarily see any theoretical benefit of having left
ideals vs. left submodules of the left free module R^1, I do think it is
beneficial to the user, much in the same way that M2 has facility for both
Ideals as well as submodules of free modules, has them typed differently,
etc.

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