Singular Manual: quantMat

| ##### 7.5.24.0. quantMat Procedure from library `qmatrix.lib` (see qmatrix_lib). Usage: quantMat(n [, p]); n integer (n>1), p an optional integer Return: ring (of quantum matrices). If p is specified, the quantum parameter q will be specialized at the p-th root of unity Purpose: compute the quantum matrix ring of order n Note: activate this ring with the "setring" command. The usual representation of the variables in this quantum algebra is not used because double indexes are not allowed in the variables. Instead the variables are listed by reading the rows of the usual matrix representation, that is, there will be n*n variables (one for each entry an n*N generic matrix), listed row-wise Example:
    LIB "qmatrix.lib";
    def r = quantMat(2); // generate O_q(M_2) at q generic
    setring r;   r;
    ==> // coefficients: QQ(q)
    ==> // number of vars : 4
    ==> //        block   1 : ordering Dp
    ==> //                  : names    y(1) y(2) y(3) y(4)
    ==> //        block   2 : ordering C
    ==> // noncommutative relations:
    ==> //    y(2)y(1)=1/(q)*y(1)*y(2)
    ==> //    y(3)y(1)=1/(q)*y(1)*y(3)
    ==> //    y(4)y(1)=y(1)*y(4)+(-q^2+1)/(q)*y(2)*y(3)
    ==> //    y(4)y(2)=1/(q)*y(2)*y(4)
    ==> //    y(4)y(3)=1/(q)*y(3)*y(4)
    kill r;
    def r = quantMat(2,5); // generate O_q(M_2) at q^5=1
    setring r;   r;
    ==> // coefficients: QQ[q]/(q^4+q^3+q^2+q+1)
    ==> // number of vars : 4
    ==> //        block   1 : ordering Dp
    ==> //                  : names    y(1) y(2) y(3) y(4)
    ==> //        block   2 : ordering C
    ==> // noncommutative relations:
    ==> //    y(2)y(1)=(-q^3-q^2-q-1)*y(1)*y(2)
    ==> //    y(3)y(1)=(-q^3-q^2-q-1)*y(1)*y(3)
    ==> //    y(4)y(1)=y(1)*y(4)+(-q^3-q^2-2*q-1)*y(2)*y(3)
    ==> //    y(4)y(2)=(-q^3-q^2-q-1)*y(2)*y(4)
    ==> //    y(4)y(3)=(-q^3-q^2-q-1)*y(3)*y(4)
See also: qminor.

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