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

## Synopsis

• Usage:
C = qTensorProduct(A,B,q)
• Inputs:
• A, an instance of the type NCRing,
• B, an instance of the type NCRing,
• q, ,
• Outputs:
• C, an instance of the type NCRing,

## 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 : NCPolynomialRing i2 : B = skewPolynomialRing(QQ,(-1)_QQ, {a,b}) --Calling Bergman for NCGB calculation. Complete! o2 = B o2 : NCQuotientRing i3 : C = qTensorProduct(A,B,-1_QQ) --Calling Bergman for NCGB calculation. Complete! --Calling Bergman for NCGB calculation. Complete! o3 = C o3 : NCQuotientRing i4 : ideal C o4 = Two-sided ideal {ba+ab, ax+xa, bx+xb, ay+ya, by+yb} o4 : NCIdeal i5 : D = A ** B --Calling Bergman for NCGB calculation. Complete! --Calling Bergman for NCGB calculation. Complete! o5 = D o5 : NCQuotientRing i6 : ideal D o6 = Two-sided ideal {ba+ab, ax-xa, bx-xb, ay-ya, by-yb} o6 : NCIdeal

## Ways to use qTensorProduct :

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

## For the programmer

The object qTensorProduct is .