mjo/eja/TODO

   1 1. Finish CartesisnProductEJA: add to_matrix(), random_instance(),
   2    one()... methods. This will require rethinking what a "matrix
   3    representation" and "matrix space" means for a cartesian product
   4    algebra. Do we want our matrix basis to consist of ordered pairs
   5    (or triples, or...)? Should the matrix_space() of the algebra
   6    be the cartesian product of the factors' matrix spaces? Can
   7    the FDEJA initializer be made to work on tuples, or will it
   8    need to be overridden?
   9
  10 2. Add references and start citing them.
  11
  12 3. Implement the octonion simple EJA.
  13
  14 4. Pre-cache charpoly for some small algebras?
  15
  16 RealSymmetricEJA(4):
  17
  18 sage: F = J.base_ring()
  19 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]
  20
  21 5. The main EJA element constructor is happy to convert between
  22    e.g. HadamardEJA(3) and JordanSpinEJA(3).
  23
  24 6. Profile the construction of "large" matrix algebras (like the
  25    15-dimensional QuaternionHermitianAlgebra(3)) to find out why
  26    they're so slow.