Hi!
I would like to do some calculations but I hit the wall with in the very
beginning. The following was tested on 8.2-beta6
L = lie_algebras.sl(QQ, 2)
U = L.universal_enveloping_algebra() # Noncommutative Multivariate
Polynomial Ring => can't do tensor product with algebra. BUG?
PBW = L.pbw_basis() # universal enveloping algebra => can do tensor
product with algebra but very nonintuitive name BUG? Also, how do I define
ordering on the basis?
E, F, H = PBW.algebra_generators()
Q = QuadraticForm(QQ, 2, [1,0,1])
C = CliffordAlgebra(Q)
e, f = C.algebra_generators()
UC = PBW.tensor(C) # but how to access its basis?
x = UC.algebra_generators()[(E, e)] # this actually defines some element
but it has broken _repr_ and _latex_ so I can't see it.
UC.algebra_generators()[E*F - F*E, (1, 1)] # works but what's with (1, 1)?
I found it by accident! Wait, it doesn't simplify e1*e1 to 1! BUG!
# So let's try tensor product in other order to see if it always simplifies
the first term in the product
CU = C.tensor(PBW) # AttributeError: 'Rings_with_category' object has no
attribute 'TensorProducts'
### Strange. I tried running it in a restarted kernel for the second time
to be sure and it worked.
# Hmmm. But this tensor product of C with itself works.
CC = C.tensor(C)
CC.algebra_generators()[(0,0), (1,1)] # But it does not simplify on either
side!
Acutally, you might encounter random AttributeErrors when tensoring
algebras. The trace is attached and it's appearance was completely random.
This was produced by running
set_random_seed(2)
L = lie_algebras.sl(QQ, 2)
U = L.universal_enveloping_algebra()
PBW = L.pbw_basis()
E, F, H = PBW.algebra_generators()
Q = QuadraticForm(QQ, 2, [1,0,1])
C = CliffordAlgebra(Q)
e, f = C.algebra_generators()
UC = PBW.tensor(C)
x = UC.algebra_generators()[(E, e)]
UC.algebra_generators()[E*F - F*E, (1, 1)]
In several fresh Sage shells. Once in ten times it gives you an error.
Best regards,
Vit Tucek
P.S.: How would I define tensor product of two algebras over a common
subalgebra?
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