mjo/hurwitz.py

   1 from sage.misc.cachefunc import cached_method
   2 from sage.combinat.free_module import CombinatorialFreeModule
   3 from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement
   4 from sage.rings.all import AA
   5
   6 from mjo.matrix_algebra import MatrixAlgebra, MatrixAlgebraElement
   7
   8 class Octonion(IndexedFreeModuleElement):
   9     def conjugate(self):
  10         r"""
  11         SETUP::
  12
  13             sage: from mjo.hurwitz import Octonions
  14
  15         EXAMPLES::
  16
  17             sage: O = Octonions()
  18             sage: x = sum(O.gens())
  19             sage: x.conjugate()
  20             e0 - e1 - e2 - e3 - e4 - e5 - e6 - e7
  21
  22         TESTS::
  23
  24         Conjugating twice gets you the original element::
  25
  26             sage: set_random_seed()
  27             sage: O = Octonions()
  28             sage: x = O.random_element()
  29             sage: x.conjugate().conjugate() == x
  30             True
  31
  32         """
  33         from sage.rings.all import ZZ
  34         from sage.matrix.matrix_space import MatrixSpace
  35         C = MatrixSpace(ZZ,8).diagonal_matrix((1,-1,-1,-1,-1,-1,-1,-1))
  36         return self.parent().from_vector(C*self.to_vector())
  37
  38     def real(self):
  39         r"""
  40         Return the real part of this octonion.
  41
  42         The real part of an octonion is its projection onto the span
  43         of the first generator. In other words, the "first dimension"
  44         is real and the others are imaginary.
  45
  46         SETUP::
  47
  48             sage: from mjo.hurwitz import Octonions
  49
  50         EXAMPLES::
  51
  52             sage: O = Octonions()
  53             sage: x = sum(O.gens())
  54             sage: x.real()
  55             e0
  56
  57         TESTS:
  58
  59         This method is idempotent::
  60
  61             sage: set_random_seed()
  62             sage: O = Octonions()
  63             sage: x = O.random_element()
  64             sage: x.real().real() == x.real()
  65             True
  66
  67         """
  68         return (self + self.conjugate())/2
  69
  70     def imag(self):
  71         r"""
  72         Return the imaginary part of this octonion.
  73
  74         The imaginary part of an octonion is its projection onto the
  75         orthogonal complement of the span of the first generator. In
  76         other words, the "first dimension" is real and the others are
  77         imaginary.
  78
  79         SETUP::
  80
  81             sage: from mjo.hurwitz import Octonions
  82
  83         EXAMPLES::
  84
  85             sage: O = Octonions()
  86             sage: x = sum(O.gens())
  87             sage: x.imag()
  88             e1 + e2 + e3 + e4 + e5 + e6 + e7
  89
  90         TESTS:
  91
  92         This method is idempotent::
  93
  94             sage: set_random_seed()
  95             sage: O = Octonions()
  96             sage: x = O.random_element()
  97             sage: x.imag().imag() == x.imag()
  98             True
  99
 100         """
 101         return (self - self.conjugate())/2
 102
 103     def _norm_squared(self):
 104         return (self*self.conjugate()).coefficient(0)
 105
 106     def norm(self):
 107         r"""
 108         Return the norm of this octonion.
 109
 110         SETUP::
 111
 112             sage: from mjo.hurwitz import Octonions
 113
 114         EXAMPLES::
 115
 116             sage: O = Octonions()
 117             sage: O.one().norm()
 118             1
 119
 120         TESTS:
 121
 122         The norm is nonnegative and belongs to the base field::
 123
 124             sage: set_random_seed()
 125             sage: O = Octonions()
 126             sage: n = O.random_element().norm()
 127             sage: n >= 0 and n in O.base_ring()
 128             True
 129
 130         The norm is homogeneous::
 131
 132             sage: set_random_seed()
 133             sage: O = Octonions()
 134             sage: x = O.random_element()
 135             sage: alpha = O.base_ring().random_element()
 136             sage: (alpha*x).norm() == alpha.abs()*x.norm()
 137             True
 138
 139         """
 140         return self._norm_squared().sqrt()
 141
 142     # The absolute value notation is typically used for complex numbers...
 143     # and norm() isn't supported in AA, so this lets us use abs() in all
 144     # of the division algebras we need.
 145     abs = norm
 146
 147     def inverse(self):
 148         r"""
 149         Return the inverse of this element if it exists.
 150
 151         SETUP::
 152
 153             sage: from mjo.hurwitz import Octonions
 154
 155         EXAMPLES::
 156
 157             sage: O = Octonions()
 158             sage: x = sum(O.gens())
 159             sage: x*x.inverse() == O.one()
 160             True
 161
 162         ::
 163
 164             sage: O = Octonions()
 165             sage: O.one().inverse() == O.one()
 166             True
 167
 168         TESTS::
 169
 170             sage: set_random_seed()
 171             sage: O = Octonions()
 172             sage: x = O.random_element()
 173             sage: x.is_zero() or ( x*x.inverse() == O.one() )
 174             True
 175
 176         """
 177         if self.is_zero():
 178             raise ValueError("zero is not invertible")
 179         return self.conjugate()/self._norm_squared()
 180
 181
 182
 183 class Octonions(CombinatorialFreeModule):
 184     r"""
 185     SETUP::
 186
 187         sage: from mjo.hurwitz import Octonions
 188
 189     EXAMPLES::
 190
 191         sage: Octonions()
 192         Octonion algebra with base ring Algebraic Real Field
 193         sage: Octonions(field=QQ)
 194         Octonion algebra with base ring Rational Field
 195
 196     """
 197     def __init__(self,
 198                  field=AA,
 199                  prefix="e"):
 200
 201         # Not associative, not commutative
 202         from sage.categories.magmatic_algebras import MagmaticAlgebras
 203         category = MagmaticAlgebras(field).FiniteDimensional()
 204         category = category.WithBasis().Unital()
 205
 206         super().__init__(field,
 207                          range(8),
 208                          element_class=Octonion,
 209                          category=category,
 210                          prefix=prefix,
 211                          bracket=False)
 212
 213         # The product of each basis element is plus/minus another
 214         # basis element that can simply be looked up on
 215         # https://en.wikipedia.org/wiki/Octonion
 216         e0, e1, e2, e3, e4, e5, e6, e7 = self.gens()
 217         self._multiplication_table = (
 218             (e0, e1, e2, e3, e4, e5, e6, e7),
 219             (e1,-e0, e3,-e2, e5,-e4,-e7, e6),
 220             (e2,-e3,-e0, e1, e6, e7,-e4,-e5),
 221             (e3, e2,-e1,-e0, e7,-e6, e5,-e4),
 222             (e4,-e5,-e6,-e7,-e0, e1, e2, e3),
 223             (e5, e4,-e7, e6,-e1,-e0,-e3, e2),
 224             (e6, e7, e4,-e5,-e2, e3,-e0,-e1),
 225             (e7,-e6, e5, e4,-e3,-e2, e1,-e0),
 226         )
 227
 228     def product_on_basis(self, i, j):
 229         return self._multiplication_table[i][j]
 230
 231     def one_basis(self):
 232         r"""
 233         Return the monomial index (basis element) corresponding to the
 234         octonion unit element.
 235
 236         SETUP::
 237
 238             sage: from mjo.hurwitz import Octonions
 239
 240         TESTS:
 241
 242         This gives the correct unit element::
 243
 244             sage: set_random_seed()
 245             sage: O = Octonions()
 246             sage: x = O.random_element()
 247             sage: x*O.one() == x and O.one()*x == x
 248             True
 249
 250         """
 251         return 0
 252
 253     def _repr_(self):
 254         return ("Octonion algebra with base ring %s" % self.base_ring())
 255
 256     def multiplication_table(self):
 257         """
 258         Return a visual representation of this algebra's multiplication
 259         table (on basis elements).
 260
 261         SETUP::
 262
 263             sage: from mjo.hurwitz import Octonions
 264
 265         EXAMPLES:
 266
 267         The multiplication table is what Wikipedia says it is::
 268
 269             sage: Octonions().multiplication_table()
 270             +----++----+-----+-----+-----+-----+-----+-----+-----+
 271             | *  || e0 | e1  | e2  | e3  | e4  | e5  | e6  | e7  |
 272             +====++====+=====+=====+=====+=====+=====+=====+=====+
 273             | e0 || e0 | e1  | e2  | e3  | e4  | e5  | e6  | e7  |
 274             +----++----+-----+-----+-----+-----+-----+-----+-----+
 275             | e1 || e1 | -e0 | e3  | -e2 | e5  | -e4 | -e7 | e6  |
 276             +----++----+-----+-----+-----+-----+-----+-----+-----+
 277             | e2 || e2 | -e3 | -e0 | e1  | e6  | e7  | -e4 | -e5 |
 278             +----++----+-----+-----+-----+-----+-----+-----+-----+
 279             | e3 || e3 | e2  | -e1 | -e0 | e7  | -e6 | e5  | -e4 |
 280             +----++----+-----+-----+-----+-----+-----+-----+-----+
 281             | e4 || e4 | -e5 | -e6 | -e7 | -e0 | e1  | e2  | e3  |
 282             +----++----+-----+-----+-----+-----+-----+-----+-----+
 283             | e5 || e5 | e4  | -e7 | e6  | -e1 | -e0 | -e3 | e2  |
 284             +----++----+-----+-----+-----+-----+-----+-----+-----+
 285             | e6 || e6 | e7  | e4  | -e5 | -e2 | e3  | -e0 | -e1 |
 286             +----++----+-----+-----+-----+-----+-----+-----+-----+
 287             | e7 || e7 | -e6 | e5  | e4  | -e3 | -e2 | e1  | -e0 |
 288             +----++----+-----+-----+-----+-----+-----+-----+-----+
 289
 290         """
 291         n = self.dimension()
 292         # Prepend the header row.
 293         M = [["*"] + list(self.gens())]
 294
 295         # And to each subsequent row, prepend an entry that belongs to
 296         # the left-side "header column."
 297         M += [ [self.monomial(i)] + [ self.monomial(i)*self.monomial(j)
 298                                     for j in range(n) ]
 299                for i in range(n) ]
 300
 301         from sage.misc.table import table
 302         return table(M, header_row=True, header_column=True, frame=True)
 303
 304
 305
 306
 307
 308 class HurwitzMatrixAlgebraElement(MatrixAlgebraElement):
 309     def is_hermitian(self):
 310         r"""
 311
 312         SETUP::
 313
 314             sage: from mjo.hurwitz import HurwitzMatrixAlgebra
 315
 316         EXAMPLES::
 317
 318             sage: A = HurwitzMatrixAlgebra(QQbar, ZZ, 2)
 319             sage: M = A([ [ 0,I],
 320             ....:         [-I,0] ])
 321             sage: M.is_hermitian()
 322             True
 323
 324         """
 325         return all( self[i,j] == self[j,i].conjugate()
 326                     for i in range(self.nrows())
 327                     for j in range(self.ncols()) )
 328
 329
 330 class HurwitzMatrixAlgebra(MatrixAlgebra):
 331     r"""
 332     A class of matrix algebras whose entries come from a Hurwitz
 333     algebra.
 334
 335     For our purposes, we consider "a Hurwitz" algebra to be the real
 336     or complex numbers, the quaternions, or the octonions. These are
 337     typically also referred to as the Euclidean Hurwitz algebras, or
 338     the normed division algebras.
 339
 340     By the Cayley-Dickson construction, each Hurwitz algebra is an
 341     algebra over the real numbers, so we restrict the scalar field in
 342     this case to be real. This also allows us to more accurately
 343     produce the generators of the matrix algebra.
 344     """
 345     Element = HurwitzMatrixAlgebraElement
 346
 347     def __init__(self, entry_algebra, scalars, n, **kwargs):
 348         from sage.rings.all import RR
 349         if not scalars.is_subring(RR):
 350             # Not perfect, but it's what we're using.
 351             raise ValueError("scalar field is not real")
 352
 353         super().__init__(entry_algebra, scalars, n, **kwargs)
 354
 355     def entry_algebra_gens(self):
 356         r"""
 357         Return a tuple of the generators of (that is, a basis for) the
 358         entries of this matrix algebra.
 359
 360         This works around the inconsistency in the ``gens()`` methods
 361         of the real/complex numbers, quaternions, and octonions.
 362
 363         SETUP::
 364
 365             sage: from mjo.hurwitz import Octonions, HurwitzMatrixAlgebra
 366
 367         EXAMPLES:
 368
 369         The inclusion of the unit element is inconsistent across
 370         (subalgebras of) Hurwitz algebras::
 371
 372             sage: AA.gens()
 373             (1,)
 374             sage: QQbar.gens()
 375             (I,)
 376             sage: QuaternionAlgebra(AA,1,-1).gens()
 377             [i, j, k]
 378             sage: Octonions().gens()
 379             (e0, e1, e2, e3, e4, e5, e6, e7)
 380
 381         The unit element is always returned by this method, so the
 382         sets of generators have cartinality 1,2,4, and 8 as you'd
 383         expect::
 384
 385             sage: HurwitzMatrixAlgebra(AA, AA, 2).entry_algebra_gens()
 386             (1,)
 387             sage: HurwitzMatrixAlgebra(QQbar, AA, 2).entry_algebra_gens()
 388             (1, I)
 389             sage: Q = QuaternionAlgebra(AA,-1,-1)
 390             sage: HurwitzMatrixAlgebra(Q, AA, 2).entry_algebra_gens()
 391             (1, i, j, k)
 392             sage: O = Octonions()
 393             sage: HurwitzMatrixAlgebra(O, AA, 2).entry_algebra_gens()
 394             (e0, e1, e2, e3, e4, e5, e6, e7)
 395
 396         """
 397         gs = self.entry_algebra().gens()
 398         one = self.entry_algebra().one()
 399         if one in gs:
 400             return gs
 401         else:
 402             return (one,) + tuple(gs)
 403
 404
 405
 406 class OctonionMatrixAlgebra(HurwitzMatrixAlgebra):
 407     r"""
 408     The algebra of ``n``-by-``n`` matrices with octonion entries over
 409     (a subfield of) the real numbers.
 410
 411     The usual matrix spaces in SageMath don't support octonion entries
 412     because they assume that the entries of the matrix come from a
 413     commutative and associative ring, and the octonions are neither.
 414
 415     SETUP::
 416
 417         sage: from mjo.hurwitz import OctonionMatrixAlgebra
 418
 419     EXAMPLES::
 420
 421         sage: OctonionMatrixAlgebra(3)
 422         Module of 3 by 3 matrices with entries in Octonion algebra with base
 423         ring Algebraic Real Field over the scalar ring Algebraic Real Field
 424         sage: OctonionMatrixAlgebra(3,QQ)
 425         Module of 3 by 3 matrices with entries in Octonion algebra with base
 426         ring Rational Field over the scalar ring Rational Field
 427
 428     ::
 429
 430         sage: A = OctonionMatrixAlgebra(2)
 431         sage: e0,e1,e2,e3,e4,e5,e6,e7 = A.entry_algebra().gens()
 432         sage: A([ [e0+e4, e1+e5],
 433         ....:     [e2-e6, e3-e7] ])
 434         +---------+---------+
 435         | e0 + e4 | e1 + e5 |
 436         +---------+---------+
 437         | e2 - e6 | e3 - e7 |
 438         +---------+---------+
 439
 440     ::
 441
 442         sage: A1 = OctonionMatrixAlgebra(1,QQ)
 443         sage: A2 = OctonionMatrixAlgebra(1,QQ)
 444         sage: cartesian_product([A1,A2])
 445         Module of 1 by 1 matrices with entries in Octonion algebra with
 446         base ring Rational Field over the scalar ring Rational Field (+)
 447         Module of 1 by 1 matrices with entries in Octonion algebra with
 448         base ring Rational Field over the scalar ring Rational Field
 449
 450     TESTS::
 451
 452         sage: set_random_seed()
 453         sage: A = OctonionMatrixAlgebra(ZZ.random_element(10))
 454         sage: x = A.random_element()
 455         sage: x*A.one() == x and A.one()*x == x
 456         True
 457
 458     """
 459     def __init__(self, n, scalars=AA, prefix="E", **kwargs):
 460         super().__init__(Octonions(field=scalars),
 461                          scalars,
 462                          n,
 463                          prefix=prefix,
 464                          **kwargs)
 465
 466 class QuaternionMatrixAlgebra(HurwitzMatrixAlgebra):
 467     r"""
 468     The algebra of ``n``-by-``n`` matrices with quaternion entries over
 469     (a subfield of) the real numbers.
 470
 471     The usual matrix spaces in SageMath don't support quaternion entries
 472     because they assume that the entries of the matrix come from a
 473     commutative ring, and the quaternions are not commutative.
 474
 475     SETUP::
 476
 477         sage: from mjo.hurwitz import QuaternionMatrixAlgebra
 478
 479     EXAMPLES::
 480
 481         sage: QuaternionMatrixAlgebra(3)
 482         Module of 3 by 3 matrices with entries in Quaternion
 483         Algebra (-1, -1) with base ring Algebraic Real Field
 484         over the scalar ring Algebraic Real Field
 485         sage: QuaternionMatrixAlgebra(3,QQ)
 486         Module of 3 by 3 matrices with entries in Quaternion
 487         Algebra (-1, -1) with base ring Rational Field over
 488         the scalar ring Rational Field
 489
 490     ::
 491
 492         sage: A = QuaternionMatrixAlgebra(2)
 493         sage: i,j,k = A.entry_algebra().gens()
 494         sage: A([ [1+i, j-2],
 495         ....:     [k,   k+j] ])
 496         +-------+--------+
 497         | 1 + i | -2 + j |
 498         +-------+--------+
 499         | k     | j + k  |
 500         +-------+--------+
 501
 502     ::
 503
 504         sage: A1 = QuaternionMatrixAlgebra(1,QQ)
 505         sage: A2 = QuaternionMatrixAlgebra(2,QQ)
 506         sage: cartesian_product([A1,A2])
 507         Module of 1 by 1 matrices with entries in Quaternion Algebra
 508         (-1, -1) with base ring Rational Field over the scalar ring
 509         Rational Field (+) Module of 2 by 2 matrices with entries in
 510         Quaternion Algebra (-1, -1) with base ring Rational Field over
 511         the scalar ring Rational Field
 512
 513     TESTS::
 514
 515         sage: set_random_seed()
 516         sage: A = QuaternionMatrixAlgebra(ZZ.random_element(10))
 517         sage: x = A.random_element()
 518         sage: x*A.one() == x and A.one()*x == x
 519         True
 520
 521     """
 522     def __init__(self, n, scalars=AA, **kwargs):
 523         # The -1,-1 gives us the "usual" definition of quaternion
 524         from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra
 525         Q = QuaternionAlgebra(scalars,-1,-1)
 526         super().__init__(Q, scalars, n, **kwargs)
 527
 528
 529 class ComplexMatrixAlgebra(HurwitzMatrixAlgebra):
 530     r"""
 531     The algebra of ``n``-by-``n`` matrices with complex entries over
 532     (a subfield of) the real numbers.
 533
 534     These differ from the usual complex matrix spaces in SageMath
 535     because the scalar field is real (and not assumed to be the same
 536     as the space from which the entries are drawn). The space of
 537     `1`-by-`1` complex matrices will have dimension two, for example.
 538
 539     SETUP::
 540
 541         sage: from mjo.hurwitz import ComplexMatrixAlgebra
 542
 543     EXAMPLES::
 544
 545         sage: ComplexMatrixAlgebra(3)
 546         Module of 3 by 3 matrices with entries in Algebraic Field
 547         over the scalar ring Algebraic Real Field
 548         sage: ComplexMatrixAlgebra(3,QQ)
 549         Module of 3 by 3 matrices with entries in Algebraic Field
 550         over the scalar ring Rational Field
 551
 552     ::
 553
 554         sage: A = ComplexMatrixAlgebra(2)
 555         sage: (I,) = A.entry_algebra().gens()
 556         sage: A([ [1+I, 1],
 557         ....:     [-1, -I] ])
 558         +-------+----+
 559         | I + 1 | 1  |
 560         +-------+----+
 561         | -1    | -I |
 562         +-------+----+
 563
 564     ::
 565
 566         sage: A1 = ComplexMatrixAlgebra(1,QQ)
 567         sage: A2 = ComplexMatrixAlgebra(2,QQ)
 568         sage: cartesian_product([A1,A2])
 569         Module of 1 by 1 matrices with entries in Algebraic Field over
 570         the scalar ring Rational Field (+) Module of 2 by 2 matrices with
 571         entries in Algebraic Field over the scalar ring Rational Field
 572
 573     TESTS::
 574
 575         sage: set_random_seed()
 576         sage: A = ComplexMatrixAlgebra(ZZ.random_element(10))
 577         sage: x = A.random_element()
 578         sage: x*A.one() == x and A.one()*x == x
 579         True
 580
 581     """
 582     def __init__(self, n, scalars=AA, **kwargs):
 583         from sage.rings.all import QQbar
 584         super().__init__(QQbar, scalars, n, **kwargs)