mjo/eja/TODO

   1 1. Finish DirectSumEJA: add to_matrix(), random_instance(),
   2    one()... methods. Make it subclass RationalBasisEuclideanJordanAlgebra.
   3    This is not a general direct sum / cartesian product implementation,
   4    it's used only with the other rationalbasis algebras (to make non-
   5    simple EJAs out of the simple ones).
   6
   7 2. Add references and start citing them.
   8
   9 3. Implement the octonion simple EJA.
  10
  11 4. Pre-cache charpoly for some small algebras?
  12
  13 RealSymmetricEJA(4):
  14
  15 sage: F = J.base_ring()
  16 sage: a0 = (1/4)*X[4]**2*X[6]**2 - (1/2)*X[2]*X[5]*X[6]**2 - (1/2)*X[3]*X[4]*X[6]*X[7] + (F(2).sqrt()/2)*X[1]*X[5]*X[6]*X[7] + (1/4)*X[3]**2*X[7]**2 - (1/2)*X[0]*X[5]*X[7]**2 + (F(2).sqrt()/2)*X[2]*X[3]*X[6]*X[8] - (1/2)*X[1]*X[4]*X[6*X[8] - (1/2)*X[1]*X[3]*X[7]*X[8] + (F(2).sqrt()/2)*X[0]*X[4]*X[7]*X[8] + (1/4)*X[1]**2*X[8]**2 - (1/2)*X[0]*X[2]*X[8]**2 - (1/2)*X[2]*X[3]**2*X[9] + (F(2).sqrt()/2)*X[1]*X[3]*X[4]*X[9] - (1/2)*X[0]*X[4]**2*X[9] - (1/2)*X[1]**2*X[5]*X[9] + X[0]*X[2]*X[5]*X[9]
  17
  18 5. Compute the scalar in the general natural_inner_product() for
  19    matrices, so no overrides are necessary.
  20
  21 6. The main EJA element constructor is happy to convert between
  22    e.g. HadamardEJA(3) and JordanSpinEJA(3).