Let us say I have the following commutation relations for an algebra:
[J0,J1]=J2
[J0,J2]=−J1
[J1,J2]=2J0
I would like to define these commutation relations and do manipulations with them. For example, I would like to calculate
[J0,[J0,J1]]+[[J1,J2],J1].
How do (or, can) we do this in SageMath?
Bonus question: Can we find the Casimir operator for this algebra in SageMath?
Lie algebra approach:
sage: d = {('J0', 'J1'): {'J2': 1}, ('J0', 'J2'): {'J1': -1}, ('J1', 'J2'): {'J0': 2}}
sage: F.<J0, J1, J2> = LieAlgebra(QQ, d)
sage: F
Lie algebra on 3 generators (J0, J1, J2) over Rational Field
sage: F.bracket(J0, J1)
J2
sage: F.bracket(J0, F.bracket(J0, J1))
-J1
sage: F.bracket(J0, F.bracket(J0, J1)) + F.bracket(F.bracket(J1, J2), J1)
-J1 + 2*J2Flawed approach below; feel free to ignore (or work on Sage to improve things!).
The following naïve approach doesn't seem to work well in Sage:
sage: F.<J0,J1,J2> = FreeAlgebra(QQ)
sage: def comm(x,y): return x*y-y*x
sage: I = F.ideal(comm(J0, J1) - J2, comm(J0, J2) + J1, comm(J1, J2) - 2*J0)
sage: A = F.quotient(I)
sage: comm(J0, comm(J0, J1)) + comm(comm(J1, J2), J1)
J0^2*J1 - 2*J0*J1*J0 + J1*J0^2 - J1^2*J2 + 2*J1*J2*J1 - J2*J1^2
sage: A(comm(J0, comm(J0, J1)) + comm(comm(J1, J2), J1))
J0bar^2*J1bar - 2*J0bar*J1bar*J0bar + J1bar*J0bar^2 - J1bar^2*J2bar + 2*J1bar*J2bar*J1bar - J2bar*J1bar^2This is clearly flawed. It can't even tell that the generators of the ideal are sent to zero in the quotient:
sage: [A(x)==0 for x in I.gens()]
[False, False, False]Another approach would be to construct the algebra from scratch. You should be able to describe a vector space basis for it, so you can then construct it using CombinatorialFreeModule, the way Clifford algebras (among many other examples) are constructed in Sage.
Thank you! This should do the job for me.
Asked: 2 years ago
Seen: 611 times
Last updated: Apr 23 '22
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Searching the source code for 'Casimir' comes up almost empty.
Thank you! I was expecting that as it is not a straightforward calculation.
Maybe I am pushing my luck but can we define calculations like
[J1,J0J2]=[J1,J0]J2+J0[J1,J2]
?
You could define
Fas in my answer and then useA = F.enveloping_algebra(), and take a quotient ofAbyA.ideal( <your relation> , side='twosided').Thank you, I changed
.enveloping_algebra()to.universal_enveloping_algebra()as in the followingand it worked. However, if I change QQ to CC or SR, this yields the error "AttributeError: 'LieAlgebraWithStructureCoefficients_with_category' object has no attribute 'lift'". I may need to use imaginary numbers or symbolic variables in the future.