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7.3.2 bracket

bracket ( poly_expression, poly_expression )
Computes the Lie bracket [p,q]=pq-qp of the first polynomial with the second. Uses special routines, based on the Leibniz rule.
ring r=(0,Q),(x,y,z),Dp;
matrix C[3][3];  matrix D[3][3];
C[1,2]=Q2;    C[1,3]=1/Q2;  C[2,3]=Q2;
D[1,2]=-Q*z;  D[1,3]=1/Q*y; D[2,3]=-Q*x;
def R=nc_algebra(C,D); setring R; R;
==> // coefficients: QQ[Q]/(Q2-Q+1)
==> // number of vars : 3
==> //        block   1 : ordering Dp
==> //                  : names    x y z
==> //        block   2 : ordering C
==> // noncommutative relations:
==> //    yx=(Q-1)*xy+(-Q)*z
==> //    zx=(-Q)*xz+(-Q+1)*y
==> //    zy=(Q-1)*yz+(-Q)*x
// this is a quantum deformation of U(so_3),
// where Q is a 6th root of unity
poly p=Q^4*x2+y2+Q^4*z2+Q*(1-Q^4)*x*y*z;
// p is the central element of the algebra
p=p^3; // any power of a central element is central
poly q=(x+Q*y+Q^2*z)^4;
// take q to be some big noncentral element
size(q); // check how many monomials are in big polynomial q
==> 28
bracket(p,q); // check p*q=q*p
==> 0
// a more common behaviour of the bracket follows:
==> (Q+1)*xz+(Q+1)*yz+(Q-1)*x+(Q-1)*y