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

Synopsis

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
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

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)
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

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

See also

Ways to use normalAutomorphism :

For the programmer

The object normalAutomorphism is a method function.