FUNCTION: LieTools/UgC2
USAGE: UgC2(upoly);
SYNOPSIS: UgC2(upoly) computes the action of the 2nd order Casimir element in U(g)
  on a polynomial upoly in U[g]
CAVEATS: requires a previously initialized global Lie algebra environment.
  (See LieTools[gSetup].)
EXAMPLE: 
gSetup(A,2);Setting up Gtype = A   rnk = 2
> upoly := u(g[1],g[2]);

                        upoly := u(g[1], g[2])

> UgC2(upoly);

  -1/9 u(g[4], g[1], g[2], g[5]) - 1/3 u(g[6], g[1], g[2], g[3])

         + 1/6 u(g[1], g[2], g[3], g[6])

         + 1/18 u(g[4], g[5], g[1], g[2])

         + 1/9 u(g[1], g[2], g[4], g[4])

         + 1/6 u(g[8], g[1], g[1], g[2])

         - 1/3 u(g[1], g[1], g[2], g[8])

         - 1/3 u(g[8], g[1], g[2], g[1])

         + 1/6 u(g[1], g[2], g[1], g[8])

         + 1/6 u(g[7], g[2], g[1], g[2])

         - 1/3 u(g[2], g[1], g[2], g[7])

         - 1/3 u(g[7], g[1], g[2], g[2])

         + 1/6 u(g[1], g[2], g[2], g[7])

         + 1/6 u(g[6], g[3], g[1], g[2])

         - 1/3 u(g[3], g[1], g[2], g[6])

         + 1/9 u(g[4], g[4], g[1], g[2])

         - 2/9 u(g[4], g[1], g[2], g[4])

         + 1/18 u(g[5], g[4], g[1], g[2])

         - 1/9 u(g[5], g[1], g[2], g[4])

         + 1/18 u(g[1], g[2], g[4], g[5])

         + 1/18 u(g[1], g[2], g[5], g[4])

         + 1/9 u(g[5], g[5], g[1], g[2])

         - 2/9 u(g[5], g[1], g[2], g[5])

         + 1/9 u(g[1], g[2], g[5], g[5])

         + 1/6 u(g[3], g[6], g[1], g[2])

         + 1/6 u(g[1], g[2], g[6], g[3])

         + 1/6 u(g[2], g[7], g[1], g[2])

         + 1/6 u(g[1], g[2], g[7], g[2])

         + 1/6 u(g[1], g[8], g[1], g[2])

         + 1/6 u(g[1], g[2], g[8], g[1])

> PBWexpand(%);

                          8/3 u(g[1], g[2])

SEE ALSO: LieTools[UgC2Reduce]
