# qTensorProduct -- Define the (q-)commuting tensor product

## Synopsis

• Usage:
C = qTensorProduct(A,B,q)
• Inputs:
• Outputs:

## Description

This function returns the algebra that contains A and B as a subalgebra, with the commutation law on the images of A and B given by a*b = q*b*a for all a in A and b in B. In the case of A ** B, q = 1.

 i1 : A = QQ<|x,y|> o1 = A o1 : FreeAlgebra i2 : B = skewPolynomialRing(QQ,(-1)_QQ, {a,b}) o2 = B o2 : FreeAlgebraQuotient i3 : C = qTensorProduct(A,B,-1_QQ) o3 = C o3 : FreeAlgebraQuotient i4 : ideal C o4 = ideal (a*b + b*a, x*a + a*x, x*b + b*x, y*a + a*y, y*b + b*y) o4 : Ideal of QQ <|x, y, a, b|> i5 : D = A ** B o5 = D o5 : FreeAlgebraQuotient i6 : ideal D o6 = ideal (a*b + b*a, - x*a + a*x, - x*b + b*x, - y*a + a*y, - y*b + b*y) o6 : Ideal of QQ <|x, y, a, b|>

## Ways to use qTensorProduct :

• "qTensorProduct(Ring,Ring,QQ)"
• "qTensorProduct(Ring,Ring,RingElement)"
• "qTensorProduct(Ring,Ring,ZZ)"

## For the programmer

The object qTensorProduct is .