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quaternion-valued differential forms

asked 2015-02-25 23:44:08 -0600

SL gravatar image

updated 2015-02-28 14:52:36 -0600

I'm trying to define quaternion-valued differential forms in Sage. For example, take $e_0$, $e_1$, $e_2$, $e_3$ to be the quaternion generators and $d x^0$, $d x^1$, $d x^2$, $d x^3$ be the generators of the exterior algebra. So for example I'd like to be able to compute $d X \wedge d X^\dagger$, where $d X = e_0 d x^0 + e_1 d x^1 + e_2 d x^2 + e_3 d x^3$, $d X^\dagger = e_0 d x^0 - e_1 d x^1 - e_2 d x^2 - e_3 d x^3$.

Edit: I tried taking a tensor product between an exterior algebra and a quaternion algebra

sage: ext_alg.<x0,x1,x2,x3> = ExteriorAlgebra(QQ)
sage: ext_alg in Algebras(QQ)
True
sage: quat_alg = QuaternionAlgebra(QQ,-1,-1)
sage: quat_alg in Algebras(QQ)
True
sage: tensor((ext_alg, quat_alg))  # this doesn't work!
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answered 2015-03-01 09:25:01 -0600

SL gravatar image

updated 2015-03-01 16:20:13 -0600

This seems to work fine:

sage: quat_alg = QuaternionAlgebra(QQ,-1,-1)
sage: ext_alg.<x0,x1,x2,x3> = ExteriorAlgebra(quat_alg)
sage: x = x0+x1*i+x2*j+x3*k
sage: x_dag = x0-x1*i-x2*j-x3*k
# now some computations
sage: x*x_dag
-2*i*x0^x1 - 2*j*x0^x2 - 2*k*x0^x3 - 2*k*x1^x2 + 2*j*x1^x3 - 2*i*x2^x3
sage: x*x_dag*x
-6*k*x0^x1^x2 + 6*j*x0^x1^x3 - 6*i*x0^x2^x3 + 6*x1^x2^x3
sage: x*x_dag*x*x_dag
-24*x0^x1^x2^x3
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Asked: 2015-02-25 23:44:08 -0600

Seen: 68 times

Last updated: Mar 01 '15