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 from mjo.eja.eja_utils import gram_schmidt
   6
   7 class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement):
   8     """
   9     SETUP::
  10
  11         sage: from mjo.eja.eja_algebra import random_eja
  12
  13     TESTS::
  14
  15     The natural representation of an element in the subalgebra is
  16     the same as its natural representation in the superalgebra::
  17
  18         sage: set_random_seed()
  19         sage: A = random_eja().random_element().subalgebra_generated_by()
  20         sage: y = A.random_element()
  21         sage: actual = y.natural_representation()
  22         sage: expected = y.superalgebra_element().natural_representation()
  23         sage: actual == expected
  24         True
  25
  26     The left-multiplication-by operator for elements in the subalgebra
  27     works like it does in the superalgebra, even if we orthonormalize
  28     our basis::
  29
  30         sage: set_random_seed()
  31         sage: x = random_eja(AA).random_element()
  32         sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
  33         sage: y = A.random_element()
  34         sage: y.operator()(A.one()) == y
  35         True
  36
  37     """
  38
  39     def superalgebra_element(self):
  40         """
  41         Return the object in our algebra's superalgebra that corresponds
  42         to myself.
  43
  44         SETUP::
  45
  46             sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
  47             ....:                                  random_eja)
  48
  49         EXAMPLES::
  50
  51             sage: J = RealSymmetricEJA(3)
  52             sage: x = sum(J.gens())
  53             sage: x
  54             e0 + e1 + e2 + e3 + e4 + e5
  55             sage: A = x.subalgebra_generated_by()
  56             sage: A(x)
  57             f1
  58             sage: A(x).superalgebra_element()
  59             e0 + e1 + e2 + e3 + e4 + e5
  60
  61         TESTS:
  62
  63         We can convert back and forth faithfully::
  64
  65             sage: set_random_seed()
  66             sage: J = random_eja()
  67             sage: x = J.random_element()
  68             sage: A = x.subalgebra_generated_by()
  69             sage: A(x).superalgebra_element() == x
  70             True
  71             sage: y = A.random_element()
  72             sage: A(y.superalgebra_element()) == y
  73             True
  74
  75         """
  76         return self.parent().superalgebra().linear_combination(
  77           zip(self.parent()._superalgebra_basis, self.to_vector()) )
  78
  79
  80
  81
  82 class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanAlgebra):
  83     """
  84     The subalgebra of an EJA generated by a single element.
  85
  86     SETUP::
  87
  88         sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
  89         ....:                                  JordanSpinEJA)
  90
  91     TESTS:
  92
  93     Ensure that our generator names don't conflict with the superalgebra::
  94
  95         sage: J = JordanSpinEJA(3)
  96         sage: J.one().subalgebra_generated_by().gens()
  97         (f0,)
  98         sage: J = JordanSpinEJA(3, prefix='f')
  99         sage: J.one().subalgebra_generated_by().gens()
 100         (g0,)
 101         sage: J = JordanSpinEJA(3, prefix='b')
 102         sage: J.one().subalgebra_generated_by().gens()
 103         (c0,)
 104
 105     Ensure that we can find subalgebras of subalgebras::
 106
 107         sage: A = ComplexHermitianEJA(3).one().subalgebra_generated_by()
 108         sage: B = A.one().subalgebra_generated_by()
 109         sage: B.dimension()
 110         1
 111
 112     """
 113     def __init__(self, elt, orthonormalize_basis):
 114         self._superalgebra = elt.parent()
 115         category = self._superalgebra.category().Associative()
 116         V = self._superalgebra.vector_space()
 117         field = self._superalgebra.base_ring()
 118
 119         # A half-assed attempt to ensure that we don't collide with
 120         # the superalgebra's prefix (ignoring the fact that there
 121         # could be super-superelgrbas in scope). If possible, we
 122         # try to "increment" the parent algebra's prefix, although
 123         # this idea goes out the window fast because some prefixen
 124         # are off-limits.
 125         prefixen = [ 'f', 'g', 'h', 'a', 'b', 'c', 'd' ]
 126         try:
 127             prefix = prefixen[prefixen.index(self._superalgebra.prefix()) + 1]
 128         except ValueError:
 129             prefix = prefixen[0]
 130
 131         if elt.is_zero():
 132             # Short circuit because 0^0 == 1 is going to make us
 133             # think we have a one-dimensional algebra otherwise.
 134             natural_basis = tuple()
 135             mult_table = tuple()
 136             rank = 0
 137             self._vector_space = V.zero_subspace()
 138             self._superalgebra_basis = []
 139             fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra,
 140                           self)
 141             return fdeja.__init__(field,
 142                                   mult_table,
 143                                   rank,
 144                                   prefix=prefix,
 145                                   category=category,
 146                                   natural_basis=natural_basis)
 147
 148
 149         # This list is guaranteed to contain all independent powers,
 150         # because it's the maximal set of powers that could possibly
 151         # be independent (by a dimension argument).
 152         powers = [ elt**k for k in range(V.dimension()) ]
 153
 154         if orthonormalize_basis == False:
 155             # In this case, we just need to figure out which elements
 156             # of the "powers" list are redundant... First compute the
 157             # vector subspace spanned by the powers of the given
 158             # element.
 159             power_vectors = [ p.to_vector() for p in powers ]
 160
 161             # Figure out which powers form a linearly-independent set.
 162             ind_rows = matrix(field, power_vectors).pivot_rows()
 163
 164             # Pick those out of the list of all powers.
 165             superalgebra_basis = tuple(map(powers.__getitem__, ind_rows))
 166
 167             # If our superalgebra is a subalgebra of something else, then
 168             # these vectors won't have the right coordinates for
 169             # V.span_of_basis() unless we use V.from_vector() on them.
 170             basis_vectors = map(power_vectors.__getitem__, ind_rows)
 171         else:
 172             # If we're going to orthonormalize the basis anyway, we
 173             # might as well just do Gram-Schmidt on the whole list of
 174             # powers. The redundant ones will get zero'd out.
 175             superalgebra_basis = gram_schmidt(powers)
 176             basis_vectors = [ b.to_vector() for b in superalgebra_basis ]
 177
 178         W = V.span_of_basis( V.from_vector(v) for v in basis_vectors )
 179         n = len(superalgebra_basis)
 180         mult_table = [[W.zero() for i in range(n)] for j in range(n)]
 181         for i in range(n):
 182             for j in range(n):
 183                 product = superalgebra_basis[i]*superalgebra_basis[j]
 184                 # product.to_vector() might live in a vector subspace
 185                 # if our parent algebra is already a subalgebra. We
 186                 # use V.from_vector() to make it "the right size" in
 187                 # that case.
 188                 product_vector = V.from_vector(product.to_vector())
 189                 mult_table[i][j] = W.coordinate_vector(product_vector)
 190
 191         # The rank is the highest possible degree of a minimal
 192         # polynomial, and is bounded above by the dimension. We know
 193         # in this case that there's an element whose minimal
 194         # polynomial has the same degree as the space's dimension
 195         # (remember how we constructed the space?), so that must be
 196         # its rank too.
 197         rank = W.dimension()
 198
 199         natural_basis = tuple( b.natural_representation()
 200                                for b in superalgebra_basis )
 201
 202
 203         self._vector_space = W
 204         self._superalgebra_basis = superalgebra_basis
 205
 206
 207         fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self)
 208         return fdeja.__init__(field,
 209                               mult_table,
 210                               rank,
 211                               prefix=prefix,
 212                               category=category,
 213                               natural_basis=natural_basis)
 214
 215
 216     def _a_regular_element(self):
 217         """
 218         Override the superalgebra method to return the one
 219         regular element that is sure to exist in this
 220         subalgebra, namely the element that generated it.
 221
 222         SETUP::
 223
 224             sage: from mjo.eja.eja_algebra import random_eja
 225
 226         TESTS::
 227
 228             sage: set_random_seed()
 229             sage: J = random_eja().random_element().subalgebra_generated_by()
 230             sage: J._a_regular_element().is_regular()
 231             True
 232
 233         """
 234         if self.dimension() == 0:
 235             return self.zero()
 236         else:
 237             return self.monomial(1)
 238
 239
 240     def _element_constructor_(self, elt):
 241         """
 242         Construct an element of this subalgebra from the given one.
 243         The only valid arguments are elements of the parent algebra
 244         that happen to live in this subalgebra.
 245
 246         SETUP::
 247
 248             sage: from mjo.eja.eja_algebra import RealSymmetricEJA
 249             sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra
 250
 251         EXAMPLES::
 252
 253             sage: J = RealSymmetricEJA(3)
 254             sage: x = sum( i*J.gens()[i] for i in range(6) )
 255             sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x,False)
 256             sage: [ K(x^k) for k in range(J.rank()) ]
 257             [f0, f1, f2]
 258
 259         ::
 260
 261         """
 262         if elt == 0:
 263             # Just as in the superalgebra class, we need to hack
 264             # this special case to ensure that random_element() can
 265             # coerce a ring zero into the algebra.
 266             return self.zero()
 267
 268         if elt in self.superalgebra():
 269             coords = self.vector_space().coordinate_vector(elt.to_vector())
 270             return self.from_vector(coords)
 271
 272
 273
 274     def one(self):
 275         """
 276         Return the multiplicative identity element of this algebra.
 277
 278         The superclass method computes the identity element, which is
 279         beyond overkill in this case: the superalgebra identity
 280         restricted to this algebra is its identity. Note that we can't
 281         count on the first basis element being the identity -- it migth
 282         have been scaled if we orthonormalized the basis.
 283
 284         SETUP::
 285
 286             sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
 287             ....:                                  random_eja)
 288
 289         EXAMPLES::
 290
 291             sage: J = RealCartesianProductEJA(5)
 292             sage: J.one()
 293             e0 + e1 + e2 + e3 + e4
 294             sage: x = sum(J.gens())
 295             sage: A = x.subalgebra_generated_by()
 296             sage: A.one()
 297             f0
 298             sage: A.one().superalgebra_element()
 299             e0 + e1 + e2 + e3 + e4
 300
 301         TESTS:
 302
 303         The identity element acts like the identity over the rationals::
 304
 305             sage: set_random_seed()
 306             sage: x = random_eja().random_element()
 307             sage: A = x.subalgebra_generated_by()
 308             sage: x = A.random_element()
 309             sage: A.one()*x == x and x*A.one() == x
 310             True
 311
 312         The identity element acts like the identity over the algebraic
 313         reals with an orthonormal basis::
 314
 315             sage: set_random_seed()
 316             sage: x = random_eja(AA).random_element()
 317             sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
 318             sage: x = A.random_element()
 319             sage: A.one()*x == x and x*A.one() == x
 320             True
 321
 322         The matrix of the unit element's operator is the identity over
 323         the rationals::
 324
 325             sage: set_random_seed()
 326             sage: x = random_eja().random_element()
 327             sage: A = x.subalgebra_generated_by()
 328             sage: actual = A.one().operator().matrix()
 329             sage: expected = matrix.identity(A.base_ring(), A.dimension())
 330             sage: actual == expected
 331             True
 332
 333         The matrix of the unit element's operator is the identity over
 334         the algebraic reals with an orthonormal basis::
 335
 336             sage: set_random_seed()
 337             sage: x = random_eja(AA).random_element()
 338             sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
 339             sage: actual = A.one().operator().matrix()
 340             sage: expected = matrix.identity(A.base_ring(), A.dimension())
 341             sage: actual == expected
 342             True
 343
 344         """
 345         if self.dimension() == 0:
 346             return self.zero()
 347         else:
 348             sa_one = self.superalgebra().one().to_vector()
 349             sa_coords = self.vector_space().coordinate_vector(sa_one)
 350             return self.from_vector(sa_coords)
 351
 352
 353     def natural_basis_space(self):
 354         """
 355         Return the natural basis space of this algebra, which is identical
 356         to that of its superalgebra.
 357
 358         This is correct "by definition," and avoids a mismatch when the
 359         subalgebra is trivial (with no natural basis to infer anything
 360         from) and the parent is not.
 361         """
 362         return self.superalgebra().natural_basis_space()
 363
 364
 365     def superalgebra(self):
 366         """
 367         Return the superalgebra that this algebra was generated from.
 368         """
 369         return self._superalgebra
 370
 371
 372     def vector_space(self):
 373         """
 374         SETUP::
 375
 376             sage: from mjo.eja.eja_algebra import RealSymmetricEJA
 377             sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra
 378
 379         EXAMPLES::
 380
 381             sage: J = RealSymmetricEJA(3)
 382             sage: x = J.monomial(0) + 2*J.monomial(2) + 5*J.monomial(5)
 383             sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x,False)
 384             sage: K.vector_space()
 385             Vector space of degree 6 and dimension 3 over...
 386             User basis matrix:
 387             [ 1  0  1  0  0  1]
 388             [ 1  0  2  0  0  5]
 389             [ 1  0  4  0  0 25]
 390             sage: (x^0).to_vector()
 391             (1, 0, 1, 0, 0, 1)
 392             sage: (x^1).to_vector()
 393             (1, 0, 2, 0, 0, 5)
 394             sage: (x^2).to_vector()
 395             (1, 0, 4, 0, 0, 25)
 396
 397         """
 398         return self._vector_space
 399
 400
 401     Element = FiniteDimensionalEuclideanJordanElementSubalgebraElement