Hey Travis,
thank you for clarification.
Dne čtvrtek 22. února 2018 2:44:44 UTC+1 Travis Scrimshaw napsal(a):
>
> Hey Vit,
> Some of these issues are probably related to bad input. Let us start
> with *.algebras_generators(). It is useful to look at the output:
>
> sage: PBW.algebra_generators()
> Finite family {-alpha[1]: PBW[-alpha[1]], alpha[1]: PBW[alpha[1]],
> alphacheck[1]: PBW[alphacheck[1]]}
>
> So it is expecting a simple root as input. Similarly, for your tensor
> product:
>
>
> sage: UC = PBW.tensor(C)
> Lazy family (Term map from Image of Cartesian product of Free abelian
> monoid indexed by {alpha[1], alphacheck[1], -alpha[1]}, Subsets of {0, 1}
> by <type 'tuple'> to Universal enveloping algebra of Lie algebra of ['A',
> 1] in the Chevalley basis in the Poincare-Birkhoff-Witt basis # The
> Clifford algebra of the Quadratic form in 2 variables over Rational Field
> with coefficients:
> [ 1 0 ]
> [ * 1 ](i))_{i in Image of Cartesian product of Free abelian monoid
> indexed by {alpha[1], alphacheck[1], -alpha[1]}, Subsets of {0, 1} by <type
> 'tuple'>}
>
> The tensor product as QQ-modules knows that it is a QQ-algebra, but it
> plays it safe and uses its basis as the generating set, which in turn, is a
> Cartesian product of the bases of its factors. Subsequently, the keys for
> the basis is the Cartesian product of the keys of the factors. So in this
> case, the (1,1) corresponds to the keys for the Clifford algebra basis
> (more of a by-product of the implementation, but the subsets are natural):
>
>
That sounds fine. But for a newcomer it's really hard to figure out what
should he input. I am still not sure myself. Consider this:
alpha = PBW.algebra_generators().keys()
PBW.basis()[alpha[0]] == PBW.algebra_generators()[alpha[0]]
I think this is a bug.
> sage: C.basis()
> Lazy family (Term map from Subsets of {0, 1} to The Clifford algebra of
> the Quadratic form in 2 variables over Rational Field with coefficients:
> [ 1 0 ]
> [ * 1 ](i))_{i in Subsets of {0, 1}}
>
> So it is expecting subsets and that the user will not input bad data.
> There is no reason for it to simplify and not a bug. Granted, we could put
> a check on the user input here, but there is a speed penalty as this can be
> a well-used code path internally. Moreover, ducktyping can also be useful.
> So we are fairly permissive here, but not without due cause IMO.
>
>
Pardon my ignorance, but If there is no simplification then what is all
this good for? It does simplify for universal enveloping algebra and so it
should do it for Clifford algebras as well. Also I have a big issue with
naming convention here. The method basis() does not produce a basis!
I understand that it might be convenient for implementation details to use
tuples of numbers (not subsets!!!) to index elements, but why can I index
with E, F and H in the universal enveloping algebra case and not with e and
f in the Clifford algebra case?
It seems that accessing Clifford algebra elements through basis just messes
up with the simplification.
x = cb[(1,1,0,1)]
print(simplify(x))
print(type(x))
print(f*f*e*f)
print(type(f*f*e*f))
print(f*f*e*f == x)
> Do not confuse a bug with a not-yet-implemented feature: (noncommutative)
> polynomial rings are not in AlgebrasWithBasis, so it is not expected that
> they work with something like tensor(). However, it would be a good feature
> to add. :)
>
I guess are referring to my complaints about not being able to take tensor
product with noncommutative ring. I see the naming convention as a BUG. If
the method has "algebra" in it's name, it should produce algebra and not a
ring. If a method has "basis" in its name, it should produce basis not a
generating set as C.basis() does or a basis and not an algebra as
pbw_basis() does.
>
> if you want to access the basis, use the .basis() method.
>
> Also, how is pbw_basis not an intuitive name for obtaining the PBW basis
> (there is also the fully spelled out version too)? If you want to change
> the ordering, RTM of "PBW?" to pass in an ordering function to
> L.pbw_basis().
>
>
For naming conventions see above. I checked out helpstring for PBW and I am
still not sure how to produce e.g. E, F, H ordering. I suggest to move this
helpstring to L.pbw_basis().
> Now the fact that you sometimes obtaining random attribute errors when
> running that code block is a definite bug. Please submit a trac ticket.
>
>
Submitted as https://trac.sagemath.org/ticket/24822
> In general, I think it is difficult to determine the tensor product of two
> (non commutative, non PBW) algebras over an arbitrary subalgebra. There is
> some code for doing smash products somewhere on #15874, but it has
> bitrotted and IDK if that will give you what you want. I did write some
> code to compute Verma modules via the PBW basis. I had to work somewhat
> hard to make it work, and it wasn't apparent to me how to extend that to
> more general framework.
>
>
Verma modules (for finite-dimensional semisimple Lie algebras) were
exactly what I had in mind. Are they already in Sage somewhere? I haven't
seen them.
> Best,
>
Travis
>
>
Best regards,
Vit
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