-1. Add cartesian products to random_eja().
+1. Add references and start citing them.
-2. Add references and start citing them.
-
-3. Implement the octonion simple EJA. We don't actually need octonions
- for this to work, only their real embedding (some 8x8 monstrosity).
-
-4. Pre-cache charpoly for some small algebras?
+2. Pre-cache charpoly for some small algebras?
RealSymmetricEJA(4):
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. Profile the construction of "large" matrix algebras (like the
+3. Profile the construction of "large" matrix algebras (like the
15-dimensional QuaternionHermitianAlgebra(3)) to find out why
they're so slow.
-6. Instead of storing a basis multiplication matrix, just make
+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?
-7. What the ever-loving fuck is this shit?
+5. What the ever-loving fuck is this shit?
sage: O = Octonions(QQ)
sage: e0 = O.monomial(0)
sage: e0*[[[[]]]]
[[[[]]]]*e0
-8. In fact, could my octonion matrix algebra be generalized for any
- algebra of matrices over the reals whose entries are not real? Then
- we wouldn't need real embeddings at all. They might even be fricking
- vector spaces if I did that...
-
-9. Add HurwitzMatrixAlgebra subclass between MatrixAlgebra and
- OctonionMatrixAlgebra.
+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. Those and the octonion stuff could be moved
+ to hurwitz.py along with the HurwitzMatrixAlgebra.
-10. Every once in a long while, the test
+7. Every once in a long while, the test
- sage: set_random_seed()
- sage: x = random_eja().random_element()
- sage: x.is_invertible() == (x.det() != 0)
+ sage: set_random_seed()
+ sage: x = random_eja().random_element()
+ sage: x.is_invertible() == (x.det() != 0)
- in eja_element.py returns False.
+ in eja_element.py returns False.