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 conjugate_transpose(self):
 310         r"""
 311         Return the conjugate-transpose of this matrix.
 312
 313         SETUP::
 314
 315             sage: from mjo.hurwitz import HurwitzMatrixAlgebra
 316
 317         EXAMPLES::
 318
 319             sage: A = HurwitzMatrixAlgebra(2, QQbar, ZZ)
 320             sage: M = A([ [ I,   2*I],
 321             ....:         [ 3*I, 4*I] ])
 322             +------+------+
 323             | -1*I | -3*I |
 324             +------+------+
 325             | -2*I | -4*I |
 326             +------+------+
 327
 328         """
 329         entries = [ [ self[j,i].conjugate()
 330                       for j in range(self.ncols())]
 331                     for i in range(self.nrows()) ]
 332         return self.parent()._element_constructor_(entries)
 333
 334     def is_hermitian(self):
 335         r"""
 336
 337         SETUP::
 338
 339             sage: from mjo.hurwitz import HurwitzMatrixAlgebra
 340
 341         EXAMPLES::
 342
 343             sage: A = HurwitzMatrixAlgebra(2, QQbar, ZZ)
 344             sage: M = A([ [ 0,I],
 345             ....:         [-I,0] ])
 346             sage: M.is_hermitian()
 347             True
 348
 349         """
 350         # A tiny bit faster than checking equality with the conjugate
 351         # transpose.
 352         return all( self[i,j] == self[j,i].conjugate()
 353                     for i in range(self.nrows())
 354                     for j in range(self.ncols()) )
 355
 356
 357 class HurwitzMatrixAlgebra(MatrixAlgebra):
 358     r"""
 359     A class of matrix algebras whose entries come from a Hurwitz
 360     algebra.
 361
 362     For our purposes, we consider "a Hurwitz" algebra to be the real
 363     or complex numbers, the quaternions, or the octonions. These are
 364     typically also referred to as the Euclidean Hurwitz algebras, or
 365     the normed division algebras.
 366
 367     By the Cayley-Dickson construction, each Hurwitz algebra is an
 368     algebra over the real numbers, so we restrict the scalar field in
 369     this case to be real. This also allows us to more accurately
 370     produce the generators of the matrix algebra.
 371     """
 372     Element = HurwitzMatrixAlgebraElement
 373
 374     def __init__(self, n, entry_algebra, scalars, **kwargs):
 375         from sage.rings.all import RR
 376         if not scalars.is_subring(RR):
 377             # Not perfect, but it's what we're using.
 378             raise ValueError("scalar field is not real")
 379
 380         super().__init__(n, entry_algebra, scalars, **kwargs)
 381
 382     def entry_algebra_gens(self):
 383         r"""
 384         Return a tuple of the generators of (that is, a basis for) the
 385         entries of this matrix algebra.
 386
 387         This works around the inconsistency in the ``gens()`` methods
 388         of the real/complex numbers, quaternions, and octonions.
 389
 390         SETUP::
 391
 392             sage: from mjo.hurwitz import Octonions, HurwitzMatrixAlgebra
 393
 394         EXAMPLES:
 395
 396         The inclusion of the unit element is inconsistent across
 397         (subalgebras of) Hurwitz algebras::
 398
 399             sage: AA.gens()
 400             (1,)
 401             sage: QQbar.gens()
 402             (I,)
 403             sage: QuaternionAlgebra(AA,1,-1).gens()
 404             [i, j, k]
 405             sage: Octonions().gens()
 406             (e0, e1, e2, e3, e4, e5, e6, e7)
 407
 408         The unit element is always returned by this method, so the
 409         sets of generators have cartinality 1,2,4, and 8 as you'd
 410         expect::
 411
 412             sage: HurwitzMatrixAlgebra(2, AA, AA).entry_algebra_gens()
 413             (1,)
 414             sage: HurwitzMatrixAlgebra(2, QQbar, AA).entry_algebra_gens()
 415             (1, I)
 416             sage: Q = QuaternionAlgebra(AA,-1,-1)
 417             sage: HurwitzMatrixAlgebra(2, Q, AA).entry_algebra_gens()
 418             (1, i, j, k)
 419             sage: O = Octonions()
 420             sage: HurwitzMatrixAlgebra(2, O, AA).entry_algebra_gens()
 421             (e0, e1, e2, e3, e4, e5, e6, e7)
 422
 423         """
 424         gs = self.entry_algebra().gens()
 425         one = self.entry_algebra().one()
 426         if one in gs:
 427             return gs
 428         else:
 429             return (one,) + tuple(gs)
 430
 431
 432
 433 class OctonionMatrixAlgebra(HurwitzMatrixAlgebra):
 434     r"""
 435     The algebra of ``n``-by-``n`` matrices with octonion entries over
 436     (a subfield of) the real numbers.
 437
 438     The usual matrix spaces in SageMath don't support octonion entries
 439     because they assume that the entries of the matrix come from a
 440     commutative and associative ring, and the octonions are neither.
 441
 442     SETUP::
 443
 444         sage: from mjo.hurwitz import Octonions, OctonionMatrixAlgebra
 445
 446     EXAMPLES::
 447
 448         sage: OctonionMatrixAlgebra(3)
 449         Module of 3 by 3 matrices with entries in Octonion algebra with base
 450         ring Algebraic Real Field over the scalar ring Algebraic Real Field
 451
 452     ::
 453
 454         sage: OctonionMatrixAlgebra(3,scalars=QQ)
 455         Module of 3 by 3 matrices with entries in Octonion algebra with
 456         base ring Rational Field over the scalar ring Rational Field
 457
 458     ::
 459
 460         sage: O = Octonions(RR)
 461         sage: A = OctonionMatrixAlgebra(1,O)
 462         sage: A
 463         Module of 1 by 1 matrices with entries in Octonion algebra with
 464         base ring Real Field with 53 bits of precision over the scalar
 465         ring Algebraic Real Field
 466         sage: A.one()
 467         +---------------------+
 468         | 1.00000000000000*e0 |
 469         +---------------------+
 470         sage: A.gens()
 471         (+---------------------+
 472         | 1.00000000000000*e0 |
 473         +---------------------+,
 474         +---------------------+
 475         | 1.00000000000000*e1 |
 476         +---------------------+,
 477         +---------------------+
 478         | 1.00000000000000*e2 |
 479         +---------------------+,
 480         +---------------------+
 481         | 1.00000000000000*e3 |
 482         +---------------------+,
 483         +---------------------+
 484         | 1.00000000000000*e4 |
 485         +---------------------+,
 486         +---------------------+
 487         | 1.00000000000000*e5 |
 488         +---------------------+,
 489         +---------------------+
 490         | 1.00000000000000*e6 |
 491         +---------------------+,
 492         +---------------------+
 493         | 1.00000000000000*e7 |
 494         +---------------------+)
 495
 496     ::
 497
 498         sage: A = OctonionMatrixAlgebra(2)
 499         sage: e0,e1,e2,e3,e4,e5,e6,e7 = A.entry_algebra().gens()
 500         sage: A([ [e0+e4, e1+e5],
 501         ....:     [e2-e6, e3-e7] ])
 502         +---------+---------+
 503         | e0 + e4 | e1 + e5 |
 504         +---------+---------+
 505         | e2 - e6 | e3 - e7 |
 506         +---------+---------+
 507
 508     ::
 509
 510         sage: A1 = OctonionMatrixAlgebra(1,scalars=QQ)
 511         sage: A2 = OctonionMatrixAlgebra(1,scalars=QQ)
 512         sage: cartesian_product([A1,A2])
 513         Module of 1 by 1 matrices with entries in Octonion algebra with
 514         base ring Rational Field over the scalar ring Rational Field (+)
 515         Module of 1 by 1 matrices with entries in Octonion algebra with
 516         base ring Rational Field over the scalar ring Rational Field
 517
 518     TESTS::
 519
 520         sage: set_random_seed()
 521         sage: A = OctonionMatrixAlgebra(ZZ.random_element(10))
 522         sage: x = A.random_element()
 523         sage: x*A.one() == x and A.one()*x == x
 524         True
 525
 526     """
 527     def __init__(self, n, entry_algebra=None, scalars=AA, **kwargs):
 528         if entry_algebra is None:
 529             entry_algebra = Octonions(field=scalars)
 530         super().__init__(n,
 531                          entry_algebra,
 532                          scalars,
 533                          **kwargs)
 534
 535 class QuaternionMatrixAlgebra(HurwitzMatrixAlgebra):
 536     r"""
 537     The algebra of ``n``-by-``n`` matrices with quaternion entries over
 538     (a subfield of) the real numbers.
 539
 540     The usual matrix spaces in SageMath don't support quaternion entries
 541     because they assume that the entries of the matrix come from a
 542     commutative ring, and the quaternions are not commutative.
 543
 544     SETUP::
 545
 546         sage: from mjo.hurwitz import QuaternionMatrixAlgebra
 547
 548     EXAMPLES::
 549
 550         sage: QuaternionMatrixAlgebra(3)
 551         Module of 3 by 3 matrices with entries in Quaternion
 552         Algebra (-1, -1) with base ring Algebraic Real Field
 553         over the scalar ring Algebraic Real Field
 554
 555     ::
 556
 557         sage: QuaternionMatrixAlgebra(3,scalars=QQ)
 558         Module of 3 by 3 matrices with entries in Quaternion
 559         Algebra (-1, -1) with base ring Rational Field over
 560         the scalar ring Rational Field
 561
 562     ::
 563
 564         sage: Q = QuaternionAlgebra(RDF, -1, -1)
 565         sage: A = QuaternionMatrixAlgebra(1,Q)
 566         sage: A
 567         Module of 1 by 1 matrices with entries in Quaternion Algebra
 568         (-1.0, -1.0) with base ring Real Double Field over the scalar
 569         ring Algebraic Real Field
 570         sage: A.one()
 571         +-----+
 572         | 1.0 |
 573         +-----+
 574         sage: A.gens()
 575         (+-----+
 576         | 1.0 |
 577         +-----+,
 578         +---+
 579         | i |
 580         +---+,
 581         +---+
 582         | j |
 583         +---+,
 584         +---+
 585         | k |
 586         +---+)
 587
 588     ::
 589
 590         sage: A = QuaternionMatrixAlgebra(2)
 591         sage: i,j,k = A.entry_algebra().gens()
 592         sage: A([ [1+i, j-2],
 593         ....:     [k,   k+j] ])
 594         +-------+--------+
 595         | 1 + i | -2 + j |
 596         +-------+--------+
 597         | k     | j + k  |
 598         +-------+--------+
 599
 600     ::
 601
 602         sage: A1 = QuaternionMatrixAlgebra(1,scalars=QQ)
 603         sage: A2 = QuaternionMatrixAlgebra(2,scalars=QQ)
 604         sage: cartesian_product([A1,A2])
 605         Module of 1 by 1 matrices with entries in Quaternion Algebra
 606         (-1, -1) with base ring Rational Field over the scalar ring
 607         Rational Field (+) Module of 2 by 2 matrices with entries in
 608         Quaternion Algebra (-1, -1) with base ring Rational Field over
 609         the scalar ring Rational Field
 610
 611     TESTS::
 612
 613         sage: set_random_seed()
 614         sage: A = QuaternionMatrixAlgebra(ZZ.random_element(10))
 615         sage: x = A.random_element()
 616         sage: x*A.one() == x and A.one()*x == x
 617         True
 618
 619     """
 620     def __init__(self, n, entry_algebra=None, scalars=AA, **kwargs):
 621         if entry_algebra is None:
 622             # The -1,-1 gives us the "usual" definition of quaternion
 623             from sage.algebras.quatalg.quaternion_algebra import (
 624                 QuaternionAlgebra
 625             )
 626             entry_algebra = QuaternionAlgebra(scalars,-1,-1)
 627         super().__init__(n, entry_algebra, scalars, **kwargs)
 628
 629
 630 class ComplexMatrixAlgebra(HurwitzMatrixAlgebra):
 631     r"""
 632     The algebra of ``n``-by-``n`` matrices with complex entries over
 633     (a subfield of) the real numbers.
 634
 635     These differ from the usual complex matrix spaces in SageMath
 636     because the scalar field is real (and not assumed to be the same
 637     as the space from which the entries are drawn). The space of
 638     `1`-by-`1` complex matrices will have dimension two, for example.
 639
 640     SETUP::
 641
 642         sage: from mjo.hurwitz import ComplexMatrixAlgebra
 643
 644     EXAMPLES::
 645
 646         sage: ComplexMatrixAlgebra(3)
 647         Module of 3 by 3 matrices with entries in Algebraic Field
 648         over the scalar ring Algebraic Real Field
 649
 650     ::
 651
 652         sage: ComplexMatrixAlgebra(3,scalars=QQ)
 653         Module of 3 by 3 matrices with entries in Algebraic Field
 654         over the scalar ring Rational Field
 655
 656     ::
 657
 658         sage: A = ComplexMatrixAlgebra(1,CC)
 659         sage: A
 660         Module of 1 by 1 matrices with entries in Complex Field with
 661         53 bits of precision over the scalar ring Algebraic Real Field
 662         sage: A.one()
 663         +------------------+
 664         | 1.00000000000000 |
 665         +------------------+
 666         sage: A.gens()
 667         (+------------------+
 668         | 1.00000000000000 |
 669         +------------------+,
 670         +--------------------+
 671         | 1.00000000000000*I |
 672         +--------------------+)
 673
 674     ::
 675
 676         sage: A = ComplexMatrixAlgebra(2)
 677         sage: (I,) = A.entry_algebra().gens()
 678         sage: A([ [1+I, 1],
 679         ....:     [-1, -I] ])
 680         +-------+----+
 681         | I + 1 | 1  |
 682         +-------+----+
 683         | -1    | -I |
 684         +-------+----+
 685
 686     ::
 687
 688         sage: A1 = ComplexMatrixAlgebra(1,scalars=QQ)
 689         sage: A2 = ComplexMatrixAlgebra(2,scalars=QQ)
 690         sage: cartesian_product([A1,A2])
 691         Module of 1 by 1 matrices with entries in Algebraic Field over
 692         the scalar ring Rational Field (+) Module of 2 by 2 matrices with
 693         entries in Algebraic Field over the scalar ring Rational Field
 694
 695     TESTS::
 696
 697         sage: set_random_seed()
 698         sage: A = ComplexMatrixAlgebra(ZZ.random_element(10))
 699         sage: x = A.random_element()
 700         sage: x*A.one() == x and A.one()*x == x
 701         True
 702
 703     """
 704     def __init__(self, n, entry_algebra=None, scalars=AA, **kwargs):
 705         if entry_algebra is None:
 706             from sage.rings.all import QQbar
 707             entry_algebra = QQbar
 708         super().__init__(n, entry_algebra, scalars, **kwargs)