Singular

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 ( ,#) is an algebra ( ,*) with the same vector space but with the opposite multiplication, i.e.
, a new multiplication on is defined to be .

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