Defining Clifford Algebras

asked 2013-01-11 13:29:58 +0100

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Dear all.

I'm a high energy physicist, interested in using SAGE for manipulations of Clifford algebras.

Being accustomed to the architecture of SAGE, I believe that the fundamental structures of these algebras is defined somewhere (probably as an algebra with basis).

Question(s)

How could I define the Clifford algebra structure from the basis elements?
Is it possible to manipulate the order of terms like $v w u \to u v w$ ?

Thank you.

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answered 2013-01-12 04:24:28 +0100

benjaminfjones
2775 ●8 ●44 ●76 http://bfj7.com/

It should certainly be possible. One place you might start looking is the implementation of Quaternion algebras in Sage. This is in the sage.algebras.quatalgpackage: https://github.com/sagemath/sagelib/t...

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@benjaminfjones : I have a doubt with the QuaternionAlgebra. The basis is form by `i,j,k` and the definition of what `i` and `j` are, is related with the values of `arg1` and `arg2`. But I still don't get how and when the base `k` is defined. Also, where in the code is imposed that `i` and `j` anticommute? Thank you!

Dox ( 2013-01-17 10:32:34 +0100 )edit

For a basis over a field K you also need a vector representing 1. I think the reason `i` and `j` are there but not `k` is that you only need 2 of the 3 to generate the quaternions as an algebra (recall `k = i * j`). To see where the multiplicative structure on the algebra is defined look at the ring multiplication for generic elements (see the code starting with `cpdef RingElement _mul_(self, RingElement _right):` in `quaternion_algebra_element.pyx`.

benjaminfjones ( 2013-01-21 16:15:40 +0100 )edit

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answered 2013-01-12 18:41:21 +0100

burcin
1208 ●16 ●36 http://erocal.org/burcin

GiNaC, the C++ library used as the backend for symbolic expressions in Sage, supports Clifford algebras. See this article for more details. Using this implementation might be easier than starting from scratch in Python/Cython. I expect the performance of GiNaC to be quite competitive, thanks to Pynac we can also use arbitrary coefficient domains from Sage.

After a brief look at the article mentioned above, I can say that it would be fairly straightforward to wrap those C++ functions and provide access to these data structures from Sage. I would be happy to do this and create a prototype implementation if somebody is willing to take over the polish. :)

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That sounds interesting. I'd certainly be willing to help polish.

benjaminfjones ( 2013-01-14 12:48:04 +0100 )edit

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