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skewPolynomialRing -- Defines a skew polynomial ring via a skewing matrix

Synopsis

Description

This method constructs a skew polynomial ring with coefficients in the ring R and generators from the list L. A valid input matrix is a square matrix over R with at least #L rows such that M_{ij} = M_{ji}^{(-1)} and M_{ii}=1. The relations of the resulting ring have the form g_i*g_j - M_{ij}*g_j*g_i.

i1 : R = QQ[q]/ideal{q^4+q^3+q^2+q+1}

o1 = R

o1 : QuotientRing
 
i2 : M = matrix{{1,q,q},{q^4,1,1},{q^4,1,1}}

o2 = | 1          q q |
     | -q3-q2-q-1 1 1 |
     | -q3-q2-q-1 1 1 |

             3      3
o2 : Matrix R  <-- R
 
i3 : B = skewPolynomialRing(R,M,{x,y,z})

o3 = B

o3 : FreeAlgebraQuotient
 
i4 : x*y == q^4*y*x

o4 = true
 
i5 : N = matrix{{1,1,1,1},{1,1,1,1},{1,1,1,1},{1,1,1,1}}

o5 = | 1 1 1 1 |
     | 1 1 1 1 |
     | 1 1 1 1 |
     | 1 1 1 1 |

              4       4
o5 : Matrix ZZ  <-- ZZ
 
i6 : C = skewPolynomialRing(QQ,promote(N,QQ), {a,b,c,d})

o6 = C

o6 : FreeAlgebraQuotient
 
i7 : isCommutative C

o7 = false
 
i8 : isCommutative B

o8 = false
 
i9 : Bop = oppositeRing B
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

o9 = Bop

o9 : FreeAlgebraQuotient
 
i10 : y*x == q^4*x*y

o10 = true

See also

Ways to use skewPolynomialRing :

For the programmer

The object skewPolynomialRing is a method function.