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3.18 opposite

Syntax:
opposite ( ring_name )
Type:
ring
Purpose:
creates an opposite algebra of a given algebra.
Note:
You have to activate the ring with the setring command.
An opposite algebra of a given algebra is an algebra with the same vectorspace but with the opposite multiplication, i.e. $\forall \; f,g \in A^{opp}$, a new multiplication $*$ on $A^{opp}$ is defined to be $f*g := g \cdot f$.

Remark:
Starting from the variables x_1,...,x_N and the ordering < of the given algebra, an opposite algebra will have variables X_N,...,X_1 (where the case and the position are reverted). Moreover, it is equipped with an opposed ordering <_opp (it is given by the matrix, obtained from the matrix ordering of < with the reverse order of columns).
 
LIB "ncalg.lib";
def B = Qso3(3);
setring B; B;
==> //   characteristic : 0
==> //   1 parameter    : Q 
==> //   minpoly        : (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
def Bopp = opposite(B);
setring Bopp; 
Bopp;
==> //   characteristic : 0
==> //   1 parameter    : Q 
==> //   minpoly        : (Q2-Q+1)
==> //   number of vars : 3
==> //        block   1 : ordering a
==> //                  : names    Z Y X 
==> //                  : weights  1 1 1 
==> //        block   2 : ordering ls
==> //                  : names    Z Y X 
==> //        block   3 : ordering C
==> //   noncommutative relations:
==> //    YZ=(Q-1)*ZY+(-Q)*X
==> //    XZ=(-Q)*ZX+(-Q+1)*Y
==> //    XY=(Q-1)*YX+(-Q)*Z
See envelope; oppose.


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