eja: factor out a class for real-embedded matrices.

[sage.d.git] / mjo / eja / TODO

diff --git a/mjo/eja/TODO b/mjo/eja/TODO
index 0e969cdfe4a2062c6a531b3fa84520c49e88de8f..13b00ac6a3056eaf623ac5c2905be6c7d049706c 100644 (file)
--- a/mjo/eja/TODO
+++ b/mjo/eja/TODO
@@ -1,12 +1,9 @@
-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 cartesian products to random_eja().
 
 2. Add references and start citing them.
 
-3. Implement the octonion simple EJA.
+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?
 
@@ -15,12 +12,26 @@ 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. 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
+5. Profile the construction of "large" matrix algebras (like the
    15-dimensional QuaternionHermitianAlgebra(3)) to find out why
    they're so slow.
 
-7. Drop the element-subalgebra in favor of a regular subalgebra. The
-   cached "one" can be set in the method.
+6. 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.
+
+7. 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.