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         if check_axioms:
 180             # The tables are square if we're verifying that they
 181             # are commutative.
 182             mult_table = [[W.zero() for j in range(n)] for i in range(n)]
 183             ip_table = [ [ self._superalgebra.inner_product(basis[i],basis[j])
 184                            for j in range(n) ]
 185                          for i in range(n) ]
 186         else:
 187             mult_table = [[W.zero() for j in range(i+1)] for i in range(n)]
 188             ip_table = [ [ self._superalgebra.inner_product(basis[i],basis[j])
 189                            for j in range(i+1) ]
 190                          for i in range(n) ]
 191
 192         for i in range(n):
 193             for j in range(i+1):
 194                 product = basis[i]*basis[j]
 195                 # product.to_vector() might live in a vector subspace
 196                 # if our parent algebra is already a subalgebra. We
 197                 # use V.from_vector() to make it "the right size" in
 198                 # that case.
 199                 product_vector = V.from_vector(product.to_vector())
 200                 mult_table[i][j] = W.coordinate_vector(product_vector)
 201                 if check_axioms:
 202                     mult_table[j][i] = mult_table[i][j]
 203
 204         matrix_basis = tuple( b.to_matrix() for b in basis )
 205
 206
 207         self._vector_space = W
 208
 209         fdeja = super(FiniteDimensionalEuclideanJordanSubalgebra, self)
 210         fdeja.__init__(field,
 211                        mult_table,
 212                        ip_table,
 213                        prefix=prefix,
 214                        category=category,
 215                        matrix_basis=matrix_basis,
 216                        check_field=False,
 217                        check_axioms=check_axioms)
 218
 219
 220
 221     def _element_constructor_(self, elt):
 222         """
 223         Construct an element of this subalgebra from the given one.
 224         The only valid arguments are elements of the parent algebra
 225         that happen to live in this subalgebra.
 226
 227         SETUP::
 228
 229             sage: from mjo.eja.eja_algebra import RealSymmetricEJA
 230             sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
 231
 232         EXAMPLES::
 233
 234             sage: J = RealSymmetricEJA(3)
 235             sage: X = matrix(AA, [ [0,0,1],
 236             ....:                  [0,1,0],
 237             ....:                  [1,0,0] ])
 238             sage: x = J(X)
 239             sage: basis = ( x, x^2 ) # x^2 is the identity matrix
 240             sage: K = FiniteDimensionalEuclideanJordanSubalgebra(J, basis)
 241             sage: K(J.one())
 242             f1
 243             sage: K(J.one() + x)
 244             f0 + f1
 245
 246         ::
 247
 248         """
 249         if elt not in self.superalgebra():
 250             raise ValueError("not an element of this subalgebra")
 251
 252         # The extra hackery is because foo.to_vector() might not live
 253         # in foo.parent().vector_space()! Subspaces of subspaces still
 254         # have user bases in the ambient space, though, so only one
 255         # level of coordinate_vector() is needed. In other words, if V
 256         # is itself a subspace, the basis elements for W will be of
 257         # the same length as the basis elements for V -- namely
 258         # whatever the dimension of the ambient (parent of V?) space is.
 259         V = self.superalgebra().vector_space()
 260         W = self.vector_space()
 261
 262         # Multiply on the left because basis_matrix() is row-wise.
 263         ambient_coords = elt.to_vector()*V.basis_matrix()
 264         W_coords = W.coordinate_vector(ambient_coords)
 265         return self.from_vector(W_coords)
 266
 267
 268
 269     def matrix_space(self):
 270         """
 271         Return the matrix space of this algebra, which is identical to
 272         that of its superalgebra.
 273
 274         This is correct "by definition," and avoids a mismatch when
 275         the subalgebra is trivial (with no matrix basis elements to
 276         infer anything from) and the parent is not.
 277         """
 278         return self.superalgebra().matrix_space()
 279
 280
 281     def superalgebra(self):
 282         """
 283         Return the superalgebra that this algebra was generated from.
 284         """
 285         return self._superalgebra
 286
 287
 288     def vector_space(self):
 289         """
 290         SETUP::
 291
 292             sage: from mjo.eja.eja_algebra import RealSymmetricEJA
 293             sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
 294
 295         EXAMPLES::
 296
 297             sage: J = RealSymmetricEJA(3)
 298             sage: E11 = matrix(ZZ, [ [1,0,0],
 299             ....:                    [0,0,0],
 300             ....:                    [0,0,0] ])
 301             sage: E22 = matrix(ZZ, [ [0,0,0],
 302             ....:                    [0,1,0],
 303             ....:                    [0,0,0] ])
 304             sage: b1 = J(E11)
 305             sage: b2 = J(E22)
 306             sage: basis = (b1, b2)
 307             sage: K = FiniteDimensionalEuclideanJordanSubalgebra(J,basis)
 308             sage: K.vector_space()
 309             Vector space of degree 6 and dimension 2 over...
 310             User basis matrix:
 311             [1 0 0 0 0 0]
 312             [0 0 1 0 0 0]
 313             sage: b1.to_vector()
 314             (1, 0, 0, 0, 0, 0)
 315             sage: b2.to_vector()
 316             (0, 0, 1, 0, 0, 0)
 317
 318         """
 319         return self._vector_space
 320
 321
 322     Element = FiniteDimensionalEuclideanJordanSubalgebraElement