mjo/eja/eja_subalgebra.py

   1 from sage.matrix.constructor import matrix
   2
   3 from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra
   4 from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
   5
   6 class FiniteDimensionalEuclideanJordanSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement):
   7     """
   8     SETUP::
   9
  10         sage: from mjo.eja.eja_algebra import random_eja
  11
  12     TESTS::
  13
  14     The natural representation of an element in the subalgebra is
  15     the same as its natural representation in the superalgebra::
  16
  17         sage: set_random_seed()
  18         sage: A = random_eja().random_element().subalgebra_generated_by()
  19         sage: y = A.random_element()
  20         sage: actual = y.natural_representation()
  21         sage: expected = y.superalgebra_element().natural_representation()
  22         sage: actual == expected
  23         True
  24
  25     The left-multiplication-by operator for elements in the subalgebra
  26     works like it does in the superalgebra, even if we orthonormalize
  27     our basis::
  28
  29         sage: set_random_seed()
  30         sage: x = random_eja(AA).random_element()
  31         sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
  32         sage: y = A.random_element()
  33         sage: y.operator()(A.one()) == y
  34         True
  35
  36     """
  37
  38     def superalgebra_element(self):
  39         """
  40         Return the object in our algebra's superalgebra that corresponds
  41         to myself.
  42
  43         SETUP::
  44
  45             sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
  46             ....:                                  random_eja)
  47
  48         EXAMPLES::
  49
  50             sage: J = RealSymmetricEJA(3)
  51             sage: x = sum(J.gens())
  52             sage: x
  53             e0 + e1 + e2 + e3 + e4 + e5
  54             sage: A = x.subalgebra_generated_by()
  55             sage: A(x)
  56             f1
  57             sage: A(x).superalgebra_element()
  58             e0 + e1 + e2 + e3 + e4 + e5
  59             sage: y = sum(A.gens())
  60             sage: y
  61             f0 + f1
  62             sage: B = y.subalgebra_generated_by()
  63             sage: B(y)
  64             g1
  65             sage: B(y).superalgebra_element()
  66             f0 + f1
  67
  68         TESTS:
  69
  70         We can convert back and forth faithfully::
  71
  72             sage: set_random_seed()
  73             sage: J = random_eja()
  74             sage: x = J.random_element()
  75             sage: A = x.subalgebra_generated_by()
  76             sage: A(x).superalgebra_element() == x
  77             True
  78             sage: y = A.random_element()
  79             sage: A(y.superalgebra_element()) == y
  80             True
  81             sage: B = y.subalgebra_generated_by()
  82             sage: B(y).superalgebra_element() == y
  83             True
  84
  85         """
  86         # As with the _element_constructor_() method on the
  87         # algebra... even in a subspace of a subspace, the basis
  88         # elements belong to the ambient space. As a result, only one
  89         # level of coordinate_vector() is needed, regardless of how
  90         # deeply we're nested.
  91         W = self.parent().vector_space()
  92         V = self.parent().superalgebra().vector_space()
  93
  94         # Multiply on the left because basis_matrix() is row-wise.
  95         ambient_coords = self.to_vector()*W.basis_matrix()
  96         V_coords = V.coordinate_vector(ambient_coords)
  97         return self.parent().superalgebra().from_vector(V_coords)
  98
  99
 100
 101
 102 class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJordanAlgebra):
 103     """
 104     A subalgebra of an EJA with a given basis.
 105
 106     SETUP::
 107
 108         sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
 109         ....:                                  JordanSpinEJA,
 110         ....:                                  RealSymmetricEJA)
 111         sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
 112
 113     EXAMPLES:
 114
 115     The following Peirce subalgebras of the 2-by-2 real symmetric
 116     matrices do not contain the superalgebra's identity element::
 117
 118         sage: J = RealSymmetricEJA(2)
 119         sage: E11 = matrix(AA, [ [1,0],
 120         ....:                    [0,0] ])
 121         sage: E22 = matrix(AA, [ [0,0],
 122         ....:                    [0,1] ])
 123         sage: K1 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E11),))
 124         sage: K1.one().natural_representation()
 125         [1 0]
 126         [0 0]
 127         sage: K2 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E22),))
 128         sage: K2.one().natural_representation()
 129         [0 0]
 130         [0 1]
 131
 132     TESTS:
 133
 134     Ensure that our generator names don't conflict with the superalgebra::
 135
 136         sage: J = JordanSpinEJA(3)
 137         sage: J.one().subalgebra_generated_by().gens()
 138         (f0,)
 139         sage: J = JordanSpinEJA(3, prefix='f')
 140         sage: J.one().subalgebra_generated_by().gens()
 141         (g0,)
 142         sage: J = JordanSpinEJA(3, prefix='b')
 143         sage: J.one().subalgebra_generated_by().gens()
 144         (c0,)
 145
 146     Ensure that we can find subalgebras of subalgebras::
 147
 148         sage: A = ComplexHermitianEJA(3).one().subalgebra_generated_by()
 149         sage: B = A.one().subalgebra_generated_by()
 150         sage: B.dimension()
 151         1
 152
 153     """
 154     def __init__(self, superalgebra, basis, category=None, check_axioms=True):
 155         self._superalgebra = superalgebra
 156         V = self._superalgebra.vector_space()
 157         field = self._superalgebra.base_ring()
 158         if category is None:
 159             category = self._superalgebra.category()
 160
 161         # A half-assed attempt to ensure that we don't collide with
 162         # the superalgebra's prefix (ignoring the fact that there
 163         # could be super-superelgrbas in scope). If possible, we
 164         # try to "increment" the parent algebra's prefix, although
 165         # this idea goes out the window fast because some prefixen
 166         # are off-limits.
 167         prefixen = [ 'f', 'g', 'h', 'a', 'b', 'c', 'd' ]
 168         try:
 169             prefix = prefixen[prefixen.index(self._superalgebra.prefix()) + 1]
 170         except ValueError:
 171             prefix = prefixen[0]
 172
 173         # If our superalgebra is a subalgebra of something else, then
 174         # these vectors won't have the right coordinates for
 175         # V.span_of_basis() unless we use V.from_vector() on them.
 176         W = V.span_of_basis( V.from_vector(b.to_vector()) for b in basis )
 177
 178         n = len(basis)
 179         mult_table = [[W.zero() for i in range(n)] for j in range(n)]
 180         for i in range(n):
 181             for j in range(n):
 182                 product = basis[i]*basis[j]
 183                 # product.to_vector() might live in a vector subspace
 184                 # if our parent algebra is already a subalgebra. We
 185                 # use V.from_vector() to make it "the right size" in
 186                 # that case.
 187                 product_vector = V.from_vector(product.to_vector())
 188                 mult_table[i][j] = W.coordinate_vector(product_vector)
 189
 190         natural_basis = tuple( b.natural_representation() for b in basis )
 191
 192
 193         self._vector_space = W
 194
 195         fdeja = super(FiniteDimensionalEuclideanJordanSubalgebra, self)
 196         fdeja.__init__(field,
 197                        mult_table,
 198                        prefix=prefix,
 199                        category=category,
 200                        natural_basis=natural_basis,
 201                        check_field=False,
 202                        check_axioms=check_axioms)
 203
 204
 205
 206     def _element_constructor_(self, elt):
 207         """
 208         Construct an element of this subalgebra from the given one.
 209         The only valid arguments are elements of the parent algebra
 210         that happen to live in this subalgebra.
 211
 212         SETUP::
 213
 214             sage: from mjo.eja.eja_algebra import RealSymmetricEJA
 215             sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
 216
 217         EXAMPLES::
 218
 219             sage: J = RealSymmetricEJA(3)
 220             sage: X = matrix(AA, [ [0,0,1],
 221             ....:                  [0,1,0],
 222             ....:                  [1,0,0] ])
 223             sage: x = J(X)
 224             sage: basis = ( x, x^2 ) # x^2 is the identity matrix
 225             sage: K = FiniteDimensionalEuclideanJordanSubalgebra(J, basis)
 226             sage: K(J.one())
 227             f1
 228             sage: K(J.one() + x)
 229             f0 + f1
 230
 231         ::
 232
 233         """
 234         if elt not in self.superalgebra():
 235             raise ValueError("not an element of this subalgebra")
 236
 237         # The extra hackery is because foo.to_vector() might not live
 238         # in foo.parent().vector_space()! Subspaces of subspaces still
 239         # have user bases in the ambient space, though, so only one
 240         # level of coordinate_vector() is needed. In other words, if V
 241         # is itself a subspace, the basis elements for W will be of
 242         # the same length as the basis elements for V -- namely
 243         # whatever the dimension of the ambient (parent of V?) space is.
 244         V = self.superalgebra().vector_space()
 245         W = self.vector_space()
 246
 247         # Multiply on the left because basis_matrix() is row-wise.
 248         ambient_coords = elt.to_vector()*V.basis_matrix()
 249         W_coords = W.coordinate_vector(ambient_coords)
 250         return self.from_vector(W_coords)
 251
 252
 253
 254     def natural_basis_space(self):
 255         """
 256         Return the natural basis space of this algebra, which is identical
 257         to that of its superalgebra.
 258
 259         This is correct "by definition," and avoids a mismatch when the
 260         subalgebra is trivial (with no natural basis to infer anything
 261         from) and the parent is not.
 262         """
 263         return self.superalgebra().natural_basis_space()
 264
 265
 266     def superalgebra(self):
 267         """
 268         Return the superalgebra that this algebra was generated from.
 269         """
 270         return self._superalgebra
 271
 272
 273     def vector_space(self):
 274         """
 275         SETUP::
 276
 277             sage: from mjo.eja.eja_algebra import RealSymmetricEJA
 278             sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
 279
 280         EXAMPLES::
 281
 282             sage: J = RealSymmetricEJA(3)
 283             sage: E11 = matrix(ZZ, [ [1,0,0],
 284             ....:                    [0,0,0],
 285             ....:                    [0,0,0] ])
 286             sage: E22 = matrix(ZZ, [ [0,0,0],
 287             ....:                    [0,1,0],
 288             ....:                    [0,0,0] ])
 289             sage: b1 = J(E11)
 290             sage: b2 = J(E22)
 291             sage: basis = (b1, b2)
 292             sage: K = FiniteDimensionalEuclideanJordanSubalgebra(J,basis)
 293             sage: K.vector_space()
 294             Vector space of degree 6 and dimension 2 over...
 295             User basis matrix:
 296             [1 0 0 0 0 0]
 297             [0 0 1 0 0 0]
 298             sage: b1.to_vector()
 299             (1, 0, 0, 0, 0, 0)
 300             sage: b2.to_vector()
 301             (0, 0, 1, 0, 0, 0)
 302
 303         """
 304         return self._vector_space
 305
 306
 307     Element = FiniteDimensionalEuclideanJordanSubalgebraElement