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
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i3 : B = A/I o3 = B o3 : FreeAlgebraQuotient |
i4 : sigma = map(B,B,{b,c,a})
o4 = map (B, B, {b, c, a})
o4 : RingMap B <--- B
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i5 : C = oreExtension(B,sigma,w) 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
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i9 : matrix (sigma * sigma)
o9 = | c a b |
1 3
o9 : Matrix B <--- B
|
The object normalAutomorphism is a method function.