ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Thu, 25 Jul 2024 12:27:25 +0200Where does this minus come from in the squared Clifford algebra, Cl$(\mathbb Z,\mathrm{diag}[1,1])^{\otimes 2}$ ?https://ask.sagemath.org/question/78451/where-does-this-minus-come-from-in-the-squared-clifford-algebra-clmathbb-zmathrmdiag11otimes-2/In the Clifford algebra tensor-squared, I find a minus I cannot explain:
C.<a0,a1>=CliffordAlgebra(DiagonalQuadraticForm(ZZ,[1,1])) ; C.rename('Cl(Z,2)')
a0.tensor(a0) in C.tensor(C)
a0.tensor(a1) * a0.tensor(a1)
# result: -1#1 ( why? )
In mathematics $a_0\otimes a_1$ squared equals its components squared, so $a_0^2\otimes a_1^2$ which should be the unit of the $\mathrm{Cl}(\mathbb Z,2) \otimes \mathrm{Cl}(\mathbb Z,2) $ above. Am I missing something?
I supposed first that Sage interprets the wandering of the, say, left $a_1$ to the 'right of the second' $a_0$
in
$ (a_0\otimes a_1) \cdot (a_0\otimes a_1) = a_0^2\otimes a_1^2 = 1\otimes 1$ (trivial quad. form)
as some anti-commutation rule ($a_0a_1 + a_1 a_0 =0$, hence the minus), but such products never happened
in the computation (i.e. they must not occur for this example, if I'm not mistaken). I tried renaming creating a copy of the Clifford algebra with new-names for generators, but that doesn't work either.Thu, 25 Jul 2024 12:27:25 +0200https://ask.sagemath.org/question/78451/where-does-this-minus-come-from-in-the-squared-clifford-algebra-clmathbb-zmathrmdiag11otimes-2/