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7.3.20 opposite

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

This is an identity functor on commutative algebras.

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).
Currently not implemented for non-global orderings.
 
LIB "ncalg.lib";
def B = makeQso3(3);
// this algebra is a quantum deformation of U(so_3),
// where the quantum parameter is a 6th root of unity
setring B; B;
==> // 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
def Bopp = opposite(B);
setring Bopp;
Bopp;
==> // coefficients: QQ[Q]/(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
def Bcheck = opposite(Bopp);
setring Bcheck; Bcheck;  // check that (B-opp)-opp = B
==> // coefficients: QQ[Q]/(Q2-Q+1)
==> // number of vars : 3
==> //        block   1 : ordering wp
==> //                  : names    x y z
==> //                  : weights  1 1 1
==> //        block   2 : ordering C
==> //        block   3 : ordering C
==> // noncommutative relations:
==> //    yx=(Q-1)*xy+(-Q)*z
==> //    zx=(-Q)*xz+(-Q+1)*y
==> //    zy=(Q-1)*yz+(-Q)*x
See Matrix orderings; envelope; oppose.


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