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 matrix representation of an element in the subalgebra is
  15     the same as its matrix 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.to_matrix()
  21         sage: expected = y.superalgebra_element().to_matrix()
  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().to_matrix()
 125         [1 0]
 126         [0 0]
 127         sage: K2 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E22),))
 128         sage: K2.one().to_matrix()
 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         ip_table = [ [ self._superalgebra.inner_product(basis[i],basis[j])
 181                        for i in range(n) ]
 182                      for j in range(n) ]
 183
 184         for i in range(n):
 185             for j in range(n):
 186                 product = basis[i]*basis[j]
 187                 # product.to_vector() might live in a vector subspace
 188                 # if our parent algebra is already a subalgebra. We
 189                 # use V.from_vector() to make it "the right size" in
 190                 # that case.
 191                 product_vector = V.from_vector(product.to_vector())
 192                 mult_table[i][j] = W.coordinate_vector(product_vector)
 193
 194         self._inner_product_matrix = matrix(field, ip_table)
 195         matrix_basis = tuple( b.to_matrix() for b in basis )
 196
 197
 198         self._vector_space = W
 199
 200         fdeja = super(FiniteDimensionalEuclideanJordanSubalgebra, self)
 201         fdeja.__init__(field,
 202                        mult_table,
 203                        prefix=prefix,
 204                        category=category,
 205                        matrix_basis=matrix_basis,
 206                        check_field=False,
 207                        check_axioms=check_axioms)
 208
 209
 210
 211     def _element_constructor_(self, elt):
 212         """
 213         Construct an element of this subalgebra from the given one.
 214         The only valid arguments are elements of the parent algebra
 215         that happen to live in this subalgebra.
 216
 217         SETUP::
 218
 219             sage: from mjo.eja.eja_algebra import RealSymmetricEJA
 220             sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
 221
 222         EXAMPLES::
 223
 224             sage: J = RealSymmetricEJA(3)
 225             sage: X = matrix(AA, [ [0,0,1],
 226             ....:                  [0,1,0],
 227             ....:                  [1,0,0] ])
 228             sage: x = J(X)
 229             sage: basis = ( x, x^2 ) # x^2 is the identity matrix
 230             sage: K = FiniteDimensionalEuclideanJordanSubalgebra(J, basis)
 231             sage: K(J.one())
 232             f1
 233             sage: K(J.one() + x)
 234             f0 + f1
 235
 236         ::
 237
 238         """
 239         if elt not in self.superalgebra():
 240             raise ValueError("not an element of this subalgebra")
 241
 242         # The extra hackery is because foo.to_vector() might not live
 243         # in foo.parent().vector_space()! Subspaces of subspaces still
 244         # have user bases in the ambient space, though, so only one
 245         # level of coordinate_vector() is needed. In other words, if V
 246         # is itself a subspace, the basis elements for W will be of
 247         # the same length as the basis elements for V -- namely
 248         # whatever the dimension of the ambient (parent of V?) space is.
 249         V = self.superalgebra().vector_space()
 250         W = self.vector_space()
 251
 252         # Multiply on the left because basis_matrix() is row-wise.
 253         ambient_coords = elt.to_vector()*V.basis_matrix()
 254         W_coords = W.coordinate_vector(ambient_coords)
 255         return self.from_vector(W_coords)
 256
 257
 258
 259     def matrix_space(self):
 260         """
 261         Return the matrix space of this algebra, which is identical to
 262         that of its superalgebra.
 263
 264         This is correct "by definition," and avoids a mismatch when
 265         the subalgebra is trivial (with no matrix basis elements to
 266         infer anything from) and the parent is not.
 267         """
 268         return self.superalgebra().matrix_space()
 269
 270
 271     def superalgebra(self):
 272         """
 273         Return the superalgebra that this algebra was generated from.
 274         """
 275         return self._superalgebra
 276
 277
 278     def vector_space(self):
 279         """
 280         SETUP::
 281
 282             sage: from mjo.eja.eja_algebra import RealSymmetricEJA
 283             sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
 284
 285         EXAMPLES::
 286
 287             sage: J = RealSymmetricEJA(3)
 288             sage: E11 = matrix(ZZ, [ [1,0,0],
 289             ....:                    [0,0,0],
 290             ....:                    [0,0,0] ])
 291             sage: E22 = matrix(ZZ, [ [0,0,0],
 292             ....:                    [0,1,0],
 293             ....:                    [0,0,0] ])
 294             sage: b1 = J(E11)
 295             sage: b2 = J(E22)
 296             sage: basis = (b1, b2)
 297             sage: K = FiniteDimensionalEuclideanJordanSubalgebra(J,basis)
 298             sage: K.vector_space()
 299             Vector space of degree 6 and dimension 2 over...
 300             User basis matrix:
 301             [1 0 0 0 0 0]
 302             [0 0 1 0 0 0]
 303             sage: b1.to_vector()
 304             (1, 0, 0, 0, 0, 0)
 305             sage: b2.to_vector()
 306             (0, 0, 1, 0, 0, 0)
 307
 308         """
 309         return self._vector_space
 310
 311
 312     Element = FiniteDimensionalEuclideanJordanSubalgebraElement