eja_subalgebra.py

   1 from sage.matrix.constructor import matrix
   2 from sage.misc.cachefunc import cached_method
   3
   4 from mjo.eja.eja_algebra import FiniteDimensionalEJA
   5 from mjo.eja.eja_element import FiniteDimensionalEJAElement
   6
   7 class FiniteDimensionalEJASubalgebraElement(FiniteDimensionalEJAElement):
   8     """
   9     SETUP::
  10
  11         sage: from mjo.eja.eja_algebra import random_eja
  12
  13     TESTS::
  14
  15     The matrix representation of an element in the subalgebra is
  16     the same as its matrix representation in the superalgebra::
  17
  18         sage: x = random_eja(field=QQ,orthonormalize=False).random_element()
  19         sage: A = x.subalgebra_generated_by(orthonormalize=False)
  20         sage: y = A.random_element()
  21         sage: actual = y.to_matrix()
  22         sage: expected = y.superalgebra_element().to_matrix()
  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: x = random_eja(field=AA).random_element()           # long time
  31         sage: A = x.subalgebra_generated_by(orthonormalize=True)  # long time
  32         sage: y = A.random_element()                              # long time
  33         sage: y.operator()(A.one()) == y                          # long time
  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             b0 + b1 + b2 + b3 + b4 + b5
  54             sage: A = x.subalgebra_generated_by(orthonormalize=False)
  55             sage: A(x)
  56             c1
  57             sage: A(x).superalgebra_element()
  58             b0 + b1 + b2 + b3 + b4 + b5
  59             sage: y = sum(A.gens())
  60             sage: y
  61             c0 + c1
  62             sage: B = y.subalgebra_generated_by(orthonormalize=False)
  63             sage: B(y)
  64             d1
  65             sage: B(y).superalgebra_element()
  66             c0 + c1
  67
  68         TESTS:
  69
  70         We can convert back and forth faithfully::
  71
  72             sage: J = random_eja(field=QQ, orthonormalize=False)
  73             sage: x = J.random_element()
  74             sage: A = x.subalgebra_generated_by(orthonormalize=False)
  75             sage: A(x).superalgebra_element() == x
  76             True
  77             sage: y = A.random_element()
  78             sage: A(y.superalgebra_element()) == y
  79             True
  80             sage: B = y.subalgebra_generated_by(orthonormalize=False)
  81             sage: B(y).superalgebra_element() == y
  82             True
  83
  84         """
  85         return self.parent().superalgebra_embedding()(self)
  86
  87
  88
  89
  90 class FiniteDimensionalEJASubalgebra(FiniteDimensionalEJA):
  91     """
  92     A subalgebra of an EJA with a given basis.
  93
  94     SETUP::
  95
  96         sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
  97         ....:                                  JordanSpinEJA,
  98         ....:                                  RealSymmetricEJA)
  99         sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra
 100
 101     EXAMPLES:
 102
 103     The following Peirce subalgebras of the 2-by-2 real symmetric
 104     matrices do not contain the superalgebra's identity element::
 105
 106         sage: J = RealSymmetricEJA(2)
 107         sage: E11 = matrix(AA, [ [1,0],
 108         ....:                    [0,0] ])
 109         sage: E22 = matrix(AA, [ [0,0],
 110         ....:                    [0,1] ])
 111         sage: K1 = FiniteDimensionalEJASubalgebra(J, (J(E11),), associative=True)
 112         sage: K1.one().to_matrix()
 113         [1 0]
 114         [0 0]
 115         sage: K2 = FiniteDimensionalEJASubalgebra(J, (J(E22),), associative=True)
 116         sage: K2.one().to_matrix()
 117         [0 0]
 118         [0 1]
 119
 120     TESTS:
 121
 122     Ensure that our generator names don't conflict with the
 123     superalgebra::
 124
 125         sage: J = JordanSpinEJA(3)
 126         sage: J.one().subalgebra_generated_by().gens()
 127         (c0,)
 128         sage: J = JordanSpinEJA(3, prefix='f')
 129         sage: J.one().subalgebra_generated_by().gens()
 130         (g0,)
 131         sage: J = JordanSpinEJA(3, prefix='a')
 132         sage: J.one().subalgebra_generated_by().gens()
 133         (b0,)
 134
 135     Ensure that we can find subalgebras of subalgebras::
 136
 137         sage: A = ComplexHermitianEJA(3).one().subalgebra_generated_by()
 138         sage: B = A.one().subalgebra_generated_by()
 139         sage: B.dimension()
 140         1
 141     """
 142     def __init__(self, superalgebra, basis, **kwargs):
 143         self._superalgebra = superalgebra
 144         V = self._superalgebra.vector_space()
 145         field = self._superalgebra.base_ring()
 146
 147         # A half-assed attempt to ensure that we don't collide with
 148         # the superalgebra's prefix (ignoring the fact that there
 149         # could be super-superelgrbas in scope). If possible, we
 150         # try to "increment" the parent algebra's prefix, although
 151         # this idea goes out the window fast because some prefixen
 152         # are off-limits.
 153         prefixen = ["b","c","d","e","f","g","h","l","m"]
 154         try:
 155             prefix = prefixen[prefixen.index(self._superalgebra.prefix()) + 1]
 156         except ValueError:
 157             prefix = prefixen[0]
 158
 159         # The superalgebra constructor expects these to be in original matrix
 160         # form, not algebra-element form.
 161         matrix_basis = tuple( b.to_matrix() for b in basis )
 162         def jordan_product(x,y):
 163             return (self._superalgebra(x)*self._superalgebra(y)).to_matrix()
 164
 165         def inner_product(x,y):
 166             return self._superalgebra(x).inner_product(self._superalgebra(y))
 167
 168         super().__init__(matrix_basis,
 169                          jordan_product,
 170                          inner_product,
 171                          field=field,
 172                          matrix_space=superalgebra.matrix_space(),
 173                          prefix=prefix,
 174                          **kwargs)
 175
 176
 177
 178     def _element_constructor_(self, elt):
 179         """
 180         Construct an element of this subalgebra from the given one.
 181         The only valid arguments are elements of the parent algebra
 182         that happen to live in this subalgebra.
 183
 184         SETUP::
 185
 186             sage: from mjo.eja.eja_algebra import RealSymmetricEJA
 187             sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra
 188
 189         EXAMPLES::
 190
 191             sage: J = RealSymmetricEJA(3)
 192             sage: X = matrix(AA, [ [0,0,1],
 193             ....:                  [0,1,0],
 194             ....:                  [1,0,0] ])
 195             sage: x = J(X)
 196             sage: basis = ( x, x^2 ) # x^2 is the identity matrix
 197             sage: K = FiniteDimensionalEJASubalgebra(J,
 198             ....:                                    basis,
 199             ....:                                    associative=True,
 200             ....:                                    orthonormalize=False)
 201             sage: K(J.one())
 202             c1
 203             sage: K(J.one() + x)
 204             c0 + c1
 205
 206         ::
 207
 208         """
 209         if elt in self.superalgebra():
 210             # If the subalgebra is trivial, its _matrix_span will be empty
 211             # but we still want to be able convert the superalgebra's zero()
 212             # element into the subalgebra's zero() element. There's no great
 213             # workaround for this because sage checks that your basis is
 214             # linearly-independent everywhere, so we can't just give it a
 215             # basis consisting of the zero element.
 216             m = elt.to_matrix()
 217             if self.is_trivial() and m.is_zero():
 218                 return self.zero()
 219             else:
 220                 return super()._element_constructor_(m)
 221         else:
 222             return super()._element_constructor_(elt)
 223
 224
 225     def superalgebra(self):
 226         """
 227         Return the superalgebra that this algebra was generated from.
 228         """
 229         return self._superalgebra
 230
 231
 232     @cached_method
 233     def superalgebra_embedding(self):
 234         r"""
 235         Return the embedding from this subalgebra into the superalgebra.
 236
 237         SETUP::
 238
 239             sage: from mjo.eja.eja_algebra import HadamardEJA
 240
 241         EXAMPLES::
 242
 243             sage: J = HadamardEJA(4)
 244             sage: A = J.one().subalgebra_generated_by()
 245             sage: iota = A.superalgebra_embedding()
 246             sage: iota
 247             Linear operator between finite-dimensional Euclidean Jordan algebras represented by the matrix:
 248             [1/2]
 249             [1/2]
 250             [1/2]
 251             [1/2]
 252             Domain: Euclidean Jordan algebra of dimension 1 over Algebraic Real Field
 253             Codomain: Euclidean Jordan algebra of dimension 4 over Algebraic Real Field
 254             sage: iota(A.one()) == J.one()
 255             True
 256
 257         """
 258         from mjo.eja.eja_operator import FiniteDimensionalEJAOperator
 259         mm = self._module_morphism(lambda j: self.superalgebra()(self.monomial(j).to_matrix()),
 260                                    codomain=self.superalgebra())
 261         return FiniteDimensionalEJAOperator(self,
 262                                             self.superalgebra(),
 263                                             mm.matrix())
 264
 265
 266
 267     Element = FiniteDimensionalEJASubalgebraElement