-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. Add references and start citing them.
-2. Add references and start citing them.
+2. Pre-cache charpoly for some small algebras?
-3. Implement the octonion simple EJA.
+RealSymmetricEJA(4):
-4. Pass already_echelonized (default: False) and echelon_basis
- (default: None) into the subalgebra constructor. The value of
- already_echelonized can be passed to V.span_of_basis() to save
- some time, and using e.g. FreeModule_submodule_with_basis_field
- we may somehow be able to pass the echelon basis straight in to
- save time.
+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]
- This may require supporting "basis" as a list of basis vectors
- (as opposed to superalgebra elements) in the subalgebra constructor.
+3. Profile the construction of "large" matrix algebras (like the
+ 15-dimensional QuaternionHermitianAlgebra(3)) to find out why
+ they're so slow.
-5. Pre-cache charpoly for some small algebras?
+4. Instead of storing a basis multiplication matrix, just make
+ product_on_basis() a cached method and manually cache its
+ entries. The cython cached method lookup should be faster than a
+ python-based matrix lookup anyway. NOTE: we should still be able
+ to recompute the table somehow. Is this worth it?
-RealSymmetricEJA(4):
+5. What the ever-loving fuck is this shit?
-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]
+ sage: O = Octonions(QQ)
+ sage: e0 = O.monomial(0)
+ sage: e0*[[[[]]]]
+ [[[[]]]]*e0
-6. Compute the scalar in the general natural_inner_product() for
- matrices, so no overrides are necessary.
+6. Can we convert the complex/quaternion algebras to avoid real-
+ (un)embeddings? Quaternions would need their own
+ QuaternionMatrixAlgebra, since Sage matrices have to have entries
+ in a commutative ring.
-7. The main EJA element constructor is happy to convert between
- e.g. HadamardEJA(3) and JordanSpinEJA(3).
+7. Every once in a long while, the test
-8. Figure out if CombinatorialFreeModule's use of IndexedGenerators
- can be used to replace the matrix_basis().
+ sage: set_random_seed()
+ sage: x = random_eja().random_element()
+ sage: x.is_invertible() == (x.det() != 0)
-9. Move the "field" argument to a keyword after basis, jp, and ip.
+ in eja_element.py returns False.