# normalAutomorphism -- Computes the automorphism determined by a normal homogeneous element

## Synopsis

• Usage:
normalAutomorphism x
• Inputs:
• x, an instance of the type NCRingElement, a homogeneous normal element
• Outputs:

## Description

Let x be a homogeneous element in an NCRing R. If x is normal then x determines a graded ring automorphism f of R by x*a = f(x)*a. This method returns this automorphism as an NCRingMap.

 i1 : A = QQ{a,b,c} o1 = A o1 : NCPolynomialRing i2 : I = ncIdeal {a*b+b*a,a*c+c*a,b*c+c*b} o2 = Two-sided ideal {ba+ab, ca+ac, cb+bc} o2 : NCIdeal i3 : B = A/I --Calling Bergman for NCGB calculation. Complete! o3 = B o3 : NCQuotientRing i4 : sigma = ncMap(B,B,{b,c,a}) o4 = NCRingMap B <--- B o4 : NCRingMap i5 : isWellDefined sigma o5 = true i6 : C = oreExtension(B,sigma,w) --Calling Bergman for NCGB calculation. Complete! o6 = C o6 : NCQuotientRing i7 : isNormal w^2 o7 = true i8 : phi = normalAutomorphism w^2 o8 = NCRingMap C <--- C o8 : NCRingMap i9 : matrix phi o9 = | c a b w | o9 : NCMatrix i10 : (matrix sigma @@ sigma) o10 = | c a b | o10 : NCMatrix