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

Usage:

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

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

Ways to use normalAutomorphism:

normalAutomorphism(NCRingElement)

For the programmer

The object normalAutomorphism is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NCAlgebra/NCAlgebraDoc.m2:2601:0.

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

Description

See also

Ways to use normalAutomorphism:

For the programmer