#15300: Weyl and Clifford Algebras
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Reporter: tscrim | Owner: tscrim
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-6.4
Component: algebra | Resolution:
Keywords: days54 | Merged in:
Authors: Travis Scrimshaw | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
public/algebras/weyl_clifford-15300| 130d1498f68ae297e2298abc13265133f39eca8f
Dependencies: #16037 | Stopgaps:
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Comment (by jhpalmieri):
A few more comments: the documentation for `expand_derivative` could use a
little work. The sum in the displayed math is over which monomials? What
is `c_{\alpha}(X)`?
I think you could also add more to the documentation for the
`DifferentialWeylAlgebra` class: all you really have now is a link to a
wikipedia article. At least mention the generators and relations. Your
documentation for `CliffordAlgebra` is very nice, in contrast.
I'm wondering about the generators for a Weyl algebra. I think I expect
`gen` (and `gens` and `ngens`) to return the algebra generators, so if we
start with a polynomial algebra on one generator `x`, we should get both
`x` and `dx`, not just `dx`. I can see that it would be useful to have a
method returning just `dx` as a generator, but maybe that method should be
called something else. Alternatively, the `gen` method could just return
`dx`, but it should be clearly documented, explaining that it doesn't
return all of the algebra generators; to get those, you should do
`algebra_generators` instead (which at the moment is available via tab-
completion but is not implemented). The method `ngens` should be changed
accordingly.
I guess you could be viewing the Weyl algebra as an algebra over the
polynomial ring, but isn't it usual, if you have an algebra `A` over a
ring `R`, to assume that `R` is central in `A`? The Weyl algebra is an
iterated Ore extension over the polynomial ring on `n` generators, adding
in one `dx` at a time, but even so, the first two places I look
(Wikipedia, and !McConnell & Robson's ''Noncommutative Noetherian Rings'')
describe such a Weyl algebra as having `2n` generators, the original
polynomial generators and the `dx`'s. So I would discourage your current
use of `gen`, `gens`, and `ngens`.
--
Ticket URL: <http://trac.sagemath.org/ticket/15300#comment:132>
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