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-dimensional QuaternionHermitianAlgebra(3)) to find out why
they're so slow.
-6. We should compute whether or not the algebra is associative if it
- is unknown. I guess the "associative" argument should be ternary
- (True, False, None)? We should also figure out the correct
- True/False values for the example classes, and of course add an
- _is_associative() 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. When field=RDF, subalgebra construction is failing because the
- inner product isn't associative? Actually, it's the combination
- of field=RDF and orthonormalize=True.
+7. What the ever-loving fuck is this shit?
-8. Set check_axioms=False for element-subalgebras outside of once or
- twice in the test suite.
+ sage: O = Octonions(QQ)
+ sage: e0 = O.monomial(0)
+ sage: e0*[[[[]]]]
+ [[[[]]]]*e0
+
+8. Factor out a class for matrices with real embeddings (i.e. not the
+ octonions).
+
+9. 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...