#15300: Clifford algebras and differential Weyl 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: Darij Grinberg, John
Report Upstream: N/A | Palmieri
Branch: | Work issues:
public/algebras/weyl_clifford-15300| Commit:
Dependencies: #16037 | 86982750ec7fcf5a9a531702ee038dc26cadb401
| Stopgaps:
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Comment (by darij):
Thanks from me, too, John and Travis! Care to review one final piece of
code? I have added the lifted bilinear form on the exterior algebra.
TODO list for further patches:
WEYL:
- Weyl algebras in general. This will be mostly copypasting the structure
of the Clifford algebra class.
- Differential Weyl algebra as a particular case of a Weyl algebra (I
think the matrix is the block matrix \\ (0 -I \\ I 0) \\ here). Is this
best achieved by having DifferentialWeylAlgebra inherit from WeylAlgebra,
or better by coercions from one to the other? (I hope it's the former.)
CLIFFORD:
- The canonical isomorphism between a Clifford algebra and an exterior
algebra induced by a bilinear (! not quadratic !) form. This isomorphism,
and its inverse, are particular cases of a common construction (Bourbaki's
"Algèbre IX", §9, no. 2-3; Ricardo Baeza's "Quadratic Forms over Semilocal
Rings", Lecture Notes in Mathematics 655, Springer 1978): If f and g are
two bilinear forms on a module V, and if F and G are their corresponding
quadratic forms (so F(x) = f(x, x) and G(x) = g(x, x)), then there is a
canonical isomorphism from Cl(V, F) to Cl(V, F + G), which however depends
on g and not just on G.
- Use this Clifford-exterior isomorphism to define constant coefficients
and scalar products on Clifford algebras. This will depend on a bilinear
form.
- Check if the scalar product on the Clifford algebra allows a faster way
to compute the lifted bilinear form on the exterior algebra.
--
Ticket URL: <http://trac.sagemath.org/ticket/15300#comment:158>
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