-1. Finish DirectSumEJA: add to_matrix(), random_instance(),
- one()... methods. Make it subclass RationalBasisEuclideanJordanAlgebra.
- This is not a general direct sum / cartesian product implementation,
- it's used only with the other rationalbasis algebras (to make non-
- simple EJAs out of the simple ones).
+1. Finish CartesisnProductEJA: add to_matrix(), random_instance(),
+ one()... methods. This will require rethinking what a "matrix
+ representation" and "matrix space" means for a cartesian product
+ algebra. Do we want our matrix basis to consist of ordered pairs
+ (or triples, or...)? Should the matrix_space() of the algebra
+ be the cartesian product of the factors' matrix spaces? Can
+ the FDEJA initializer be made to work on tuples, or will it
+ need to be overridden?
2. Add references and start citing them.
sage: F = J.base_ring()
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]
-5. Compute the scalar in the general natural_inner_product() for
- matrices, so no overrides are necessary. Actually, this is
- probably better implemented as a dimension_over_reals() method
- that returns 1, 2, or 4.
-
-6. The main EJA element constructor is happy to convert between
+5. The main EJA element constructor is happy to convert between
e.g. HadamardEJA(3) and JordanSpinEJA(3).
+
+6. Profile the construction of "large" matrix algebras (like the
+ 15-dimensional QuaternionHermitianAlgebra(3)) to find out why
+ they're so slow.