Singular Manual: opposite

  | ### 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 ( 190#190,#) is an algebra ( 190#190,*) with the same vector space but with the opposite multiplication, i.e. 216#216, a new multiplication 217#217 on 206#206 is defined to be 218#218. 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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