mjo/eja/TODO

   1 1. Add cartesian products to random_eja().
   2
   3 2. Add references and start citing them.
   4
   5 3. Implement the octonion simple EJA. We don't actually need octonions
   6    for this to work, only their real embedding (some 8x8 monstrosity).
   7
   8 4. Pre-cache charpoly for some small algebras?
   9
  10 RealSymmetricEJA(4):
  11
  12 sage: F = J.base_ring()
  13 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]
  14
  15 5. Profile the construction of "large" matrix algebras (like the
  16    15-dimensional QuaternionHermitianAlgebra(3)) to find out why
  17    they're so slow.
  18
  19 6. Instead of storing a basis multiplication matrix, just make
  20    product_on_basis() a cached method and manually cache its
  21    entries. The cython cached method lookup should be faster than a
  22    python-based matrix lookup anyway. NOTE: we should still be able
  23    to recompute the table somehow. Is this worth it?
  24
  25 7. What the ever-loving fuck is this shit?
  26
  27        sage: O = Octonions(QQ)
  28        sage: e0 = O.monomial(0)
  29        sage: e0*[[[[]]]]
  30        [[[[]]]]*e0
  31
  32 8. In fact, could my octonion matrix algebra be generalized for any
  33    algebra of matrices over the reals whose entries are not real? Then
  34    we wouldn't need real embeddings at all. They might even be fricking
  35    vector spaces if I did that...
  36
  37 9. Every once in a long while, the test
  38
  39        sage: set_random_seed()
  40        sage: x = random_eja().random_element()
  41        sage: x.is_invertible() == (x.det() != 0)
  42
  43    in eja_element.py returns False.