mjo/eja/TODO

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