mjo/eja/eja_subalgebra.py

   1 from sage.matrix.constructor import matrix
   2 from sage.structure.category_object import normalize_names
   3
   4 from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra
   5 from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
   6
   7
   8 class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanAlgebra):
   9     """
  10     The subalgebra of an EJA generated by a single element.
  11     """
  12     @staticmethod
  13     def __classcall_private__(cls, elt):
  14         superalgebra = elt.parent()
  15
  16         # First compute the vector subspace spanned by the powers of
  17         # the given element.
  18         V = superalgebra.vector_space()
  19         superalgebra_basis = [superalgebra.one()]
  20         basis_vectors = [superalgebra.one().vector()]
  21         W = V.span_of_basis(basis_vectors)
  22         for exponent in range(1, V.dimension()):
  23             new_power = elt**exponent
  24             basis_vectors.append( new_power.vector() )
  25             try:
  26                 W = V.span_of_basis(basis_vectors)
  27                 superalgebra_basis.append( new_power )
  28             except ValueError:
  29                 # Vectors weren't independent; bail and keep the
  30                 # last subspace that worked.
  31                 break
  32
  33         # Make the basis hashable for UniqueRepresentation.
  34         superalgebra_basis = tuple(superalgebra_basis)
  35
  36         # Now figure out the entries of the right-multiplication
  37         # matrix for the successive basis elements b0, b1,... of
  38         # that subspace.
  39         F = superalgebra.base_ring()
  40         mult_table = []
  41         for b_right in superalgebra_basis:
  42                 b_right_rows = []
  43                 # The first row of the right-multiplication matrix by
  44                 # b1 is what we get if we apply that matrix to b1. The
  45                 # second row of the right multiplication matrix by b1
  46                 # is what we get when we apply that matrix to b2...
  47                 #
  48                 # IMPORTANT: this assumes that all vectors are COLUMN
  49                 # vectors, unlike our superclass (which uses row vectors).
  50                 for b_left in superalgebra_basis:
  51                     # Multiply in the original EJA, but then get the
  52                     # coordinates from the subalgebra in terms of its
  53                     # basis.
  54                     this_row = W.coordinates((b_left*b_right).vector())
  55                     b_right_rows.append(this_row)
  56                 b_right_matrix = matrix(F, b_right_rows)
  57                 mult_table.append(b_right_matrix)
  58
  59         for m in mult_table:
  60             m.set_immutable()
  61         mult_table = tuple(mult_table)
  62
  63         # The rank is the highest possible degree of a minimal
  64         # polynomial, and is bounded above by the dimension. We know
  65         # in this case that there's an element whose minimal
  66         # polynomial has the same degree as the space's dimension
  67         # (remember how we constructed the space?), so that must be
  68         # its rank too.
  69         rank = W.dimension()
  70
  71         # EJAs are power-associative, and this algebra is nothin but
  72         # powers.
  73         assume_associative=True
  74
  75         # TODO: Un-hard-code this. It should be possible to get the "next"
  76         # name based on the parent's generator names.
  77         names = 'f'
  78         names = normalize_names(W.dimension(), names)
  79
  80         cat = superalgebra.category().Associative()
  81
  82         # TODO: compute this and actually specify it.
  83         natural_basis = None
  84
  85         fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, cls)
  86         return fdeja.__classcall__(cls,
  87                                    F,
  88                                    mult_table,
  89                                    rank,
  90                                    superalgebra_basis,
  91                                    W,
  92                                    assume_associative=assume_associative,
  93                                    names=names,
  94                                    category=cat,
  95                                    natural_basis=natural_basis)
  96
  97     def __init__(self,
  98                  field,
  99                  mult_table,
 100                  rank,
 101                  superalgebra_basis,
 102                  vector_space,
 103                  assume_associative=True,
 104                  names='f',
 105                  category=None,
 106                  natural_basis=None):
 107
 108         self._superalgebra = superalgebra_basis[0].parent()
 109         self._vector_space = vector_space
 110         self._superalgebra_basis = superalgebra_basis
 111
 112         fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self)
 113         fdeja.__init__(field,
 114                        mult_table,
 115                        rank,
 116                        assume_associative=assume_associative,
 117                        names=names,
 118                        category=category,
 119                        natural_basis=natural_basis)
 120
 121
 122     def superalgebra(self):
 123         """
 124         Return the superalgebra that this algebra was generated from.
 125         """
 126         return self._superalgebra
 127
 128
 129     def vector_space(self):
 130         """
 131         SETUP::
 132
 133             sage: from mjo.eja.eja_algebra import RealSymmetricEJA
 134             sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra
 135
 136         EXAMPLES::
 137
 138             sage: J = RealSymmetricEJA(3)
 139             sage: x = sum( i*J.gens()[i] for i in range(6) )
 140             sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x)
 141             sage: K.vector_space()
 142             Vector space of degree 6 and dimension 3 over Rational Field
 143             User basis matrix:
 144             [ 1  0  0  1  0  1]
 145             [ 0  1  2  3  4  5]
 146             [ 5 11 14 26 34 45]
 147             sage: (x^0).vector()
 148             (1, 0, 0, 1, 0, 1)
 149             sage: (x^1).vector()
 150             (0, 1, 2, 3, 4, 5)
 151             sage: (x^2).vector()
 152             (5, 11, 14, 26, 34, 45)
 153
 154         """
 155         return self._vector_space
 156
 157
 158     class Element(FiniteDimensionalEuclideanJordanAlgebraElement):
 159         def __init__(self, A, elt=None):
 160             """
 161             SETUP::
 162
 163                 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
 164                 sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra
 165
 166             EXAMPLES::
 167
 168                 sage: J = RealSymmetricEJA(3)
 169                 sage: x = sum( i*J.gens()[i] for i in range(6) )
 170                 sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x)
 171                 sage: [ K(x^k) for k in range(J.rank()) ]
 172                 [f0, f1, f2]
 173
 174             ::
 175
 176             """
 177             if elt in A.superalgebra():
 178                     # Try to convert a parent algebra element into a
 179                     # subalgebra element...
 180                 try:
 181                     coords = A.vector_space().coordinates(elt.vector())
 182                     elt = A(coords)
 183                 except AttributeError:
 184                     # Catches a missing method in elt.vector()
 185                     pass
 186
 187             FiniteDimensionalEuclideanJordanAlgebraElement.__init__(self,
 188                                                                     A,
 189                                                                     elt)
 190
 191         def superalgebra_element(self):
 192             """
 193             Return the object in our algebra's superalgebra that corresponds
 194             to myself.
 195
 196             SETUP::
 197
 198                 sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
 199                 ....:                                  random_eja)
 200
 201             EXAMPLES::
 202
 203                 sage: J = RealSymmetricEJA(3)
 204                 sage: x = sum(J.gens())
 205                 sage: x
 206                 e0 + e1 + e2 + e3 + e4 + e5
 207                 sage: A = x.subalgebra_generated_by()
 208                 sage: A(x)
 209                 f1
 210                 sage: A(x).superalgebra_element()
 211                 e0 + e1 + e2 + e3 + e4 + e5
 212
 213             TESTS:
 214
 215             We can convert back and forth faithfully::
 216
 217                 sage: set_random_seed()
 218                 sage: J = random_eja()
 219                 sage: x = J.random_element()
 220                 sage: A = x.subalgebra_generated_by()
 221                 sage: A(x).superalgebra_element() == x
 222                 True
 223                 sage: y = A.random_element()
 224                 sage: A(y.superalgebra_element()) == y
 225                 True
 226
 227             """
 228             return self.parent().superalgebra().linear_combination(
 229               zip(self.vector(), self.parent()._superalgebra_basis) )