normalElements(NCRingMap,ZZ) -- Finds elements normalized by a ring map

Synopsis

Function: normalElements
Usage:

normalElements(f,n)
Inputs:
- f, an instance of the type NCRingMap,
- n, an integer, a homogeneous degree in which to search for normal elements
Outputs:
- a list,

Description

A normal element x in an NCRing R determines an automorphism f of R by a*x=x*f(a). Conversely, given a ring endomorphism, we may ask if any x satisfy the above equation for all a.

Given an NCRingMap f and a degree n, this method returns solutions to the equations a*x=x*f(a) for all generators a of R.

i1 : B = skewPolynomialRing(QQ,(-1)_QQ,{x,y,z,w})
--Calling Bergman for NCGB calculation.
Complete!

o1 = B

o1 : NCQuotientRing

i2 : sigma = ncMap(B,B,{y,z,w,x})

o2 = NCRingMap B <--- B

o2 : NCRingMap

i3 : C = oreExtension(B,sigma,a)
--Calling Bergman for NCGB calculation.
Complete!

o3 = C

o3 : NCQuotientRing

i4 : sigmaC = ncMap(C,C,{y,z,w,x,a})

o4 = NCRingMap C <--- C

o4 : NCRingMap

i5 : normalElements(sigmaC,1)

o5 = | a |

o5 : NCMatrix

i6 : normalElements(sigmaC,2)

o6 = 0

              1
o6 : Matrix QQ  <--- 0

i7 : normalElements(sigmaC @@ sigmaC,2)

o7 = | a^2 |

o7 : NCMatrix

Ways to use this method: