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

## Synopsis

• Usage:
normalAutomorphism x
• Inputs:
• x, , a homogeneous normal element
• Outputs:
• ,

## Description

Let x be a homogeneous element in a noncommutative ring 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 a RingMap.

 i1 : A = QQ<|a,b,c|> o1 = A o1 : FreeAlgebra i2 : I = ideal {a*b+b*a,a*c+c*a,b*c+c*b} o2 = ideal (a*b + b*a, a*c + c*a, b*c + c*b) o2 : Ideal of A i3 : B = A/I Using numthreads = 0 o3 = B o3 : FreeAlgebraQuotient i4 : sigma = map(B,B,{b,c,a}) o4 = map (B, B, {b, c, a}) o4 : RingMap B <--- B i5 : C = oreExtension(B,sigma,w) Using numthreads = 0 o5 = C o5 : FreeAlgebraQuotient

By construction, w is normal, and the normalizing automorphism is sigma extended to C sending w to itself. It follows that therefore w^2 is also normal whose automorphism is the square of sigma extended to C in a similar way. We verify these facts with the following commands:

 i6 : isNormal w^2 o6 = true i7 : phi = normalAutomorphism w^2 o7 = map (C, C, {c, a, b, w}) o7 : RingMap C <--- C i8 : matrix phi o8 = | c a b w | 1 4 o8 : Matrix C <--- C i9 : matrix (sigma * sigma) o9 = | c a b | 1 3 o9 : Matrix B <--- B