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.Sun, 02 Jan 2022 17:22:35 +0100Constructing a non-commutative algebra over Z[q, q^-1] given some relationshttps://ask.sagemath.org/question/60516/constructing-a-non-commutative-algebra-over-zq-q-1-given-some-relations/
I would like to construct a non-commutative alegebra over Z[q, q^-1] generated by the variables u1, u2, u3 with the relations:
u2*u1 = q*u1*u2
u3*u2 = (q^2)*u2*u3
u3*u1 = u1*u3
but I am having some trouble getting this to work. I have primarily been trying to do this using the FreeAlgebra structure. Here is what I have tried:
----------
Zq.<q> = LaurentPolynomialRing(ZZ)
A.<u1,u2, u3> = FreeAlgebra(Zq, 3)
G = A.g_algebra({u2*u1: q*u1*u2, u3*u2: (q**2)*u2*u3})
G
but I get the errors:
AttributeError: 'FreeAlgebra_generic_with_category.element_class' object has no attribute 'lift'
TypeError: unable to coerce <class 'sage.algebras.free_algebra.FreeAlgebra_generic_with_category.element_class'> to `an integer`
----------
Zqring.<q, qinv> = ZZ[]
qideal = Zqring.ideal(q*qinv - 1)
Zq.<q, qinv> = Zqring.quotient(qideal)
A.<u1,u2,u3> = FreeAlgebra(Zq, 3)
I = A.ideal(u2*u1-q*u1*u2, u3*u2-(q**2)*u2*u3, side = "twosided")
W.<u1, u2, u3> = quotient(A,I)
u2*u1-q*u1*u2
but this outputs "(-q)*u1*u2 + u2*u1" and not "0"
----------
Zqring.<q, qinv> = ZZ[]
qideal = Zqring.ideal(q*qinv - 1)
Zq.<q, qinv> = Zqring.quotient(qideal)
A.<u1,u2,u3> = FreeAlgebra(Zq, 3)
I = A*[u2*u1-q*u1*u2]*A
W.<u1,u2,u3> = A.quo(I)
W(u2*u1-q*u1*u2)
and, again, the output is not "0".
Is there a better way I can construct such a non-commutative algebra?
Thanks!kAllenMathSun, 02 Jan 2022 17:22:35 +0100https://ask.sagemath.org/question/60516/Calculations in quotient of a free algebrahttps://ask.sagemath.org/question/41219/calculations-in-quotient-of-a-free-algebra/I want to define (the algebra part of) Sweedler's four-dimensional Hopf algebra, which is freely generated by $x,y$ and subject to the relations
$$
x^2 = 1, \qquad y^2 = 0, \qquad x\cdot y = - y\cdot x~ ,
$$
but I don't see how to do it.
I have tried the following:
sage: A.<x,y> = FreeAlgebra(QQbar)
sage: I = A*[x*x - 1, y*y, x*y + y*x]*A
sage: H.<x,y> = A.quo(I)
sage: H
Quotient of Free Algebra on 2 generators (x, y) over Algebraic Field by the ideal (-1 + x^2, y^2, x*y + y*x)
But then I get
sage: H.one() == H(x*x)
False
So is this currently possibly using a different method?
ThanksJo BeWed, 21 Feb 2018 14:44:18 +0100https://ask.sagemath.org/question/41219/