mjo/eja/DESIGN

   1 Overview
   2 --------
   3 This is a collection of design notes that should eventually wind up in
   4 the documentation. I'm just not sure where they go yet.
   5
   6 Matrix representations
   7 ----------------------
   8
   9 All algebras have a "matrix representation" of their elements. This is
  10 the original, ambient representation of the elements as either column
  11 (n-by-1) or square (n-by-n) matrices. For example, the elements of the
  12 Jordan spin algebra are column vectors, and the elements of real
  13 symmetric matrix algebra are... real symmetric matrices.
  14
  15 The CombinatorialFreeModule class actually supports such an
  16 alternative representation of its generators since it subclasses
  17 IndexedGenerators. However, using matrices as the index set turns out
  18 to be ugly: printing the generators, and especially printing an
  19 algebra element as a sum of said generators comes out wonky, since
  20 the matrices require more than one line. For example,
  21
  22     sage: A = matrix(QQ,[[1,2],[3,4]])
  23     sage: B = matrix(QQ,[[5,6],[7,8]])
  24     sage: A.set_immutable()
  25     sage: B.set_immutable()
  26     sage: M = CombinatorialFreeModule(QQ,[A,B],bracket=False,prefix="")
  27     sage: 2*M(A) + 3*M(B)
  28     2*[1 2]
  29     [3 4] + 3*[5 6]
  30     [7 8]
  31
  32 And things only get worse if you leave the prefix in there to
  33 distinguish between e.g. the super- and sub-algebra elements
  34 corresponding to the same matrix. Thus, we store out own copy
  35 of the matrix generators, and have our own set of methods for
  36 accessing them.
  37
  38 Why allow matrix representations for all algebras, rather than just
  39 the matrix algebras?
  40
  41 1. We already have a to_vector() operation that turns an algebra
  42    element into a vector whose coordinates live in the algebra's
  43    base_ring(). Adding a to_matrix() operation is a natural
  44    generalization of that.
  45
  46 2. Having access to the ambient coordinates in a general way is useful
  47    when converting between other coordinate systems. If we have two
  48    subalgebras B and C of A, we can use to_matrix() to go from, say,
  49    C -> A -> B rather than having to convert from C to B directly.
  50
  51
  52 Fielding questions
  53 ------------------
  54
  55 All Euclidean Jordan algebras are over the real scalar field. This
  56 presents a problem: in SageMath, the matrix and vector classes don't
  57 support scalar fields that are different than their entries. And three
  58 of the simple families of Euclidean Jordan algebras are matrices with
  59 non-real entries: the Hermitian comples, quaternion, and octonion
  60 algebras.
  61
  62 At least in the complex and quaternion case, we can "embed" the
  63 complex numbers and quaternions into the space of 2-by-2 or 4-by-4
  64 matrices. But the octonions are not associative, so they can't be
  65 embedded (via a homomorphism) into any real matrix space. So what
  66 do we do? Write it ourselves, obviously.
  67
  68 In contrast to the algebra of real symmetric matrices, the complex,
  69 quaternion, and octonion matrix algebras are implemented separately,
  70 as a subclasses of CombinatorialFreeModule, to work around that
  71 issue. The custom class supports a scalar field that is different than
  72 the entries of the matrices. However, this means that we actually have
  73 FOUR different types of "matrices" to support:
  74
  75   (1) Sage vectors
  76   (2) Sage matrices
  77   (3) Our custom matrices
  78   (4) Cartesian products of the (1) through (3)
  79
  80 The real symmetric matrices could of course be implemented in the same
  81 manner as the others; but for the sake of the user interface, we must
  82 also support at least the usual SageMath vectors and matrices. Having
  83 the real symmetric matrices actually be (SageMath) matrices ensures
  84 that we don't accidentally break support for such things.
  85
  86 Note: this has one less-than-obvious consequence: we have to assume
  87 that the user has supplied an entirely-correct basis (with entries in
  88 the correct structure). We generally cannot mess witht the entries of
  89 his basis, or use them to figure out what (for example) the ambient
  90 scalar ring is. None of these are insurmountable obstacles; we just
  91 have to be a little careful distinguishing between what's inside the
  92 algebra elements and what's outside them.
  93
  94 Basis normalization
  95 -------------------
  96 For performance reasons, we prefer the algebra constructor to
  97 orthonormalize its own basis. We _could_ ask the user to do that,
  98 but there's a good reason to do it ourselves: if _we_ run
  99 Gram-Schmidt, then we can compute/store the matrix that undoes the
 100 process. Undoing the change-of-coordinates allows us to perform some
 101 computations in the original basis (like the "characteristic
 102 polynomial of"), which can be much faster when the original basis
 103 contains only rational entries.
 104
 105 Debugging
 106 ---------
 107 There are several places in the code where we set check_field=False
 108 and check_axioms=False because the theory guarantees that they will be
 109 satisfied. Well, you know what they say about theory and practice. The
 110 first thing you should do when a problem is discovered it replace all
 111 of those with check_field=True and check_axioms=True, and then re-run
 112 the test suite. The Cartesian product class bypasses its superclass
 113 constructor, so any extra axiom/field checks on product algebras must
 114 be inserted at debug-time.