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[sage.d.git] / mjo / eja / euclidean_jordan_algebra.py
1 """
2 Euclidean Jordan Algebras. These are formally-real Jordan Algebras;
3 specifically those where u^2 + v^2 = 0 implies that u = v = 0. They
4 are used in optimization, and have some additional nice methods beyond
5 what can be supported in a general Jordan Algebra.
6 """
7
8 from sage.categories.magmatic_algebras import MagmaticAlgebras
9 from sage.structure.element import is_Matrix
10 from sage.structure.category_object import normalize_names
11
12 from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra import FiniteDimensionalAlgebra
13 from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_element import FiniteDimensionalAlgebraElement
14
15 class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
16 @staticmethod
17 def __classcall_private__(cls,
18 field,
19 mult_table,
20 names='e',
21 assume_associative=False,
22 category=None,
23 rank=None,
24 natural_basis=None):
25 n = len(mult_table)
26 mult_table = [b.base_extend(field) for b in mult_table]
27 for b in mult_table:
28 b.set_immutable()
29 if not (is_Matrix(b) and b.dimensions() == (n, n)):
30 raise ValueError("input is not a multiplication table")
31 mult_table = tuple(mult_table)
32
33 cat = MagmaticAlgebras(field).FiniteDimensional().WithBasis()
34 cat.or_subcategory(category)
35 if assume_associative:
36 cat = cat.Associative()
37
38 names = normalize_names(n, names)
39
40 fda = super(FiniteDimensionalEuclideanJordanAlgebra, cls)
41 return fda.__classcall__(cls,
42 field,
43 mult_table,
44 assume_associative=assume_associative,
45 names=names,
46 category=cat,
47 rank=rank,
48 natural_basis=natural_basis)
49
50
51 def __init__(self,
52 field,
53 mult_table,
54 names='e',
55 assume_associative=False,
56 category=None,
57 rank=None,
58 natural_basis=None):
59 """
60 EXAMPLES:
61
62 By definition, Jordan multiplication commutes::
63
64 sage: set_random_seed()
65 sage: J = random_eja()
66 sage: x = J.random_element()
67 sage: y = J.random_element()
68 sage: x*y == y*x
69 True
70
71 """
72 self._rank = rank
73 self._natural_basis = natural_basis
74 fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
75 fda.__init__(field,
76 mult_table,
77 names=names,
78 category=category)
79
80
81 def _repr_(self):
82 """
83 Return a string representation of ``self``.
84 """
85 fmt = "Euclidean Jordan algebra of degree {} over {}"
86 return fmt.format(self.degree(), self.base_ring())
87
88
89 def inner_product(self, x, y):
90 """
91 The inner product associated with this Euclidean Jordan algebra.
92
93 Defaults to the trace inner product, but can be overridden by
94 subclasses if they are sure that the necessary properties are
95 satisfied.
96
97 EXAMPLES:
98
99 The inner product must satisfy its axiom for this algebra to truly
100 be a Euclidean Jordan Algebra::
101
102 sage: set_random_seed()
103 sage: J = random_eja()
104 sage: x = J.random_element()
105 sage: y = J.random_element()
106 sage: z = J.random_element()
107 sage: (x*y).inner_product(z) == y.inner_product(x*z)
108 True
109
110 """
111 if (not x in self) or (not y in self):
112 raise TypeError("arguments must live in this algebra")
113 return x.trace_inner_product(y)
114
115
116 def natural_basis(self):
117 """
118 Return a more-natural representation of this algebra's basis.
119
120 Every finite-dimensional Euclidean Jordan Algebra is a direct
121 sum of five simple algebras, four of which comprise Hermitian
122 matrices. This method returns the original "natural" basis
123 for our underlying vector space. (Typically, the natural basis
124 is used to construct the multiplication table in the first place.)
125
126 Note that this will always return a matrix. The standard basis
127 in `R^n` will be returned as `n`-by-`1` column matrices.
128
129 EXAMPLES::
130
131 sage: J = RealSymmetricEJA(2)
132 sage: J.basis()
133 Family (e0, e1, e2)
134 sage: J.natural_basis()
135 (
136 [1 0] [0 1] [0 0]
137 [0 0], [1 0], [0 1]
138 )
139
140 ::
141
142 sage: J = JordanSpinEJA(2)
143 sage: J.basis()
144 Family (e0, e1)
145 sage: J.natural_basis()
146 (
147 [1] [0]
148 [0], [1]
149 )
150
151 """
152 if self._natural_basis is None:
153 return tuple( b.vector().column() for b in self.basis() )
154 else:
155 return self._natural_basis
156
157
158 def rank(self):
159 """
160 Return the rank of this EJA.
161 """
162 if self._rank is None:
163 raise ValueError("no rank specified at genesis")
164 else:
165 return self._rank
166
167
168 class Element(FiniteDimensionalAlgebraElement):
169 """
170 An element of a Euclidean Jordan algebra.
171 """
172
173 def __init__(self, A, elt=None):
174 """
175 EXAMPLES:
176
177 The identity in `S^n` is converted to the identity in the EJA::
178
179 sage: J = RealSymmetricEJA(3)
180 sage: I = identity_matrix(QQ,3)
181 sage: J(I) == J.one()
182 True
183
184 This skew-symmetric matrix can't be represented in the EJA::
185
186 sage: J = RealSymmetricEJA(3)
187 sage: A = matrix(QQ,3, lambda i,j: i-j)
188 sage: J(A)
189 Traceback (most recent call last):
190 ...
191 ArithmeticError: vector is not in free module
192
193 """
194 # Goal: if we're given a matrix, and if it lives in our
195 # parent algebra's "natural ambient space," convert it
196 # into an algebra element.
197 #
198 # The catch is, we make a recursive call after converting
199 # the given matrix into a vector that lives in the algebra.
200 # This we need to try the parent class initializer first,
201 # to avoid recursing forever if we're given something that
202 # already fits into the algebra, but also happens to live
203 # in the parent's "natural ambient space" (this happens with
204 # vectors in R^n).
205 try:
206 FiniteDimensionalAlgebraElement.__init__(self, A, elt)
207 except ValueError:
208 natural_basis = A.natural_basis()
209 if elt in natural_basis[0].matrix_space():
210 # Thanks for nothing! Matrix spaces aren't vector
211 # spaces in Sage, so we have to figure out its
212 # natural-basis coordinates ourselves.
213 V = VectorSpace(elt.base_ring(), elt.nrows()**2)
214 W = V.span( _mat2vec(s) for s in natural_basis )
215 coords = W.coordinates(_mat2vec(elt))
216 FiniteDimensionalAlgebraElement.__init__(self, A, coords)
217
218 def __pow__(self, n):
219 """
220 Return ``self`` raised to the power ``n``.
221
222 Jordan algebras are always power-associative; see for
223 example Faraut and Koranyi, Proposition II.1.2 (ii).
224
225 .. WARNING:
226
227 We have to override this because our superclass uses row vectors
228 instead of column vectors! We, on the other hand, assume column
229 vectors everywhere.
230
231 EXAMPLES::
232
233 sage: set_random_seed()
234 sage: x = random_eja().random_element()
235 sage: x.operator_matrix()*x.vector() == (x^2).vector()
236 True
237
238 A few examples of power-associativity::
239
240 sage: set_random_seed()
241 sage: x = random_eja().random_element()
242 sage: x*(x*x)*(x*x) == x^5
243 True
244 sage: (x*x)*(x*x*x) == x^5
245 True
246
247 We also know that powers operator-commute (Koecher, Chapter
248 III, Corollary 1)::
249
250 sage: set_random_seed()
251 sage: x = random_eja().random_element()
252 sage: m = ZZ.random_element(0,10)
253 sage: n = ZZ.random_element(0,10)
254 sage: Lxm = (x^m).operator_matrix()
255 sage: Lxn = (x^n).operator_matrix()
256 sage: Lxm*Lxn == Lxn*Lxm
257 True
258
259 """
260 A = self.parent()
261 if n == 0:
262 return A.one()
263 elif n == 1:
264 return self
265 else:
266 return A( (self.operator_matrix()**(n-1))*self.vector() )
267
268
269 def characteristic_polynomial(self):
270 """
271 Return my characteristic polynomial (if I'm a regular
272 element).
273
274 Eventually this should be implemented in terms of the parent
275 algebra's characteristic polynomial that works for ALL
276 elements.
277 """
278 if self.is_regular():
279 return self.minimal_polynomial()
280 else:
281 raise NotImplementedError('irregular element')
282
283
284 def inner_product(self, other):
285 """
286 Return the parent algebra's inner product of myself and ``other``.
287
288 EXAMPLES:
289
290 The inner product in the Jordan spin algebra is the usual
291 inner product on `R^n` (this example only works because the
292 basis for the Jordan algebra is the standard basis in `R^n`)::
293
294 sage: J = JordanSpinEJA(3)
295 sage: x = vector(QQ,[1,2,3])
296 sage: y = vector(QQ,[4,5,6])
297 sage: x.inner_product(y)
298 32
299 sage: J(x).inner_product(J(y))
300 32
301
302 The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
303 multiplication is the usual matrix multiplication in `S^n`,
304 so the inner product of the identity matrix with itself
305 should be the `n`::
306
307 sage: J = RealSymmetricEJA(3)
308 sage: J.one().inner_product(J.one())
309 3
310
311 Likewise, the inner product on `C^n` is `<X,Y> =
312 Re(trace(X*Y))`, where we must necessarily take the real
313 part because the product of Hermitian matrices may not be
314 Hermitian::
315
316 sage: J = ComplexHermitianEJA(3)
317 sage: J.one().inner_product(J.one())
318 3
319
320 Ditto for the quaternions::
321
322 sage: J = QuaternionHermitianEJA(3)
323 sage: J.one().inner_product(J.one())
324 3
325
326 TESTS:
327
328 Ensure that we can always compute an inner product, and that
329 it gives us back a real number::
330
331 sage: set_random_seed()
332 sage: J = random_eja()
333 sage: x = J.random_element()
334 sage: y = J.random_element()
335 sage: x.inner_product(y) in RR
336 True
337
338 """
339 P = self.parent()
340 if not other in P:
341 raise TypeError("'other' must live in the same algebra")
342
343 return P.inner_product(self, other)
344
345
346 def operator_commutes_with(self, other):
347 """
348 Return whether or not this element operator-commutes
349 with ``other``.
350
351 EXAMPLES:
352
353 The definition of a Jordan algebra says that any element
354 operator-commutes with its square::
355
356 sage: set_random_seed()
357 sage: x = random_eja().random_element()
358 sage: x.operator_commutes_with(x^2)
359 True
360
361 TESTS:
362
363 Test Lemma 1 from Chapter III of Koecher::
364
365 sage: set_random_seed()
366 sage: J = random_eja()
367 sage: u = J.random_element()
368 sage: v = J.random_element()
369 sage: lhs = u.operator_commutes_with(u*v)
370 sage: rhs = v.operator_commutes_with(u^2)
371 sage: lhs == rhs
372 True
373
374 """
375 if not other in self.parent():
376 raise TypeError("'other' must live in the same algebra")
377
378 A = self.operator_matrix()
379 B = other.operator_matrix()
380 return (A*B == B*A)
381
382
383 def det(self):
384 """
385 Return my determinant, the product of my eigenvalues.
386
387 EXAMPLES::
388
389 sage: J = JordanSpinEJA(2)
390 sage: e0,e1 = J.gens()
391 sage: x = e0 + e1
392 sage: x.det()
393 0
394 sage: J = JordanSpinEJA(3)
395 sage: e0,e1,e2 = J.gens()
396 sage: x = e0 + e1 + e2
397 sage: x.det()
398 -1
399
400 """
401 cs = self.characteristic_polynomial().coefficients(sparse=False)
402 r = len(cs) - 1
403 if r >= 0:
404 return cs[0] * (-1)**r
405 else:
406 raise ValueError('charpoly had no coefficients')
407
408
409 def inverse(self):
410 """
411 Return the Jordan-multiplicative inverse of this element.
412
413 We can't use the superclass method because it relies on the
414 algebra being associative.
415
416 EXAMPLES:
417
418 The inverse in the spin factor algebra is given in Alizadeh's
419 Example 11.11::
420
421 sage: set_random_seed()
422 sage: n = ZZ.random_element(1,10)
423 sage: J = JordanSpinEJA(n)
424 sage: x = J.random_element()
425 sage: while x.is_zero():
426 ....: x = J.random_element()
427 sage: x_vec = x.vector()
428 sage: x0 = x_vec[0]
429 sage: x_bar = x_vec[1:]
430 sage: coeff = 1/(x0^2 - x_bar.inner_product(x_bar))
431 sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
432 sage: x_inverse = coeff*inv_vec
433 sage: x.inverse() == J(x_inverse)
434 True
435
436 TESTS:
437
438 The identity element is its own inverse::
439
440 sage: set_random_seed()
441 sage: J = random_eja()
442 sage: J.one().inverse() == J.one()
443 True
444
445 If an element has an inverse, it acts like one. TODO: this
446 can be a lot less ugly once ``is_invertible`` doesn't crash
447 on irregular elements::
448
449 sage: set_random_seed()
450 sage: J = random_eja()
451 sage: x = J.random_element()
452 sage: try:
453 ....: x.inverse()*x == J.one()
454 ....: except:
455 ....: True
456 True
457
458 """
459 if self.parent().is_associative():
460 elt = FiniteDimensionalAlgebraElement(self.parent(), self)
461 return elt.inverse()
462
463 # TODO: we can do better once the call to is_invertible()
464 # doesn't crash on irregular elements.
465 #if not self.is_invertible():
466 # raise ValueError('element is not invertible')
467
468 # We do this a little different than the usual recursive
469 # call to a finite-dimensional algebra element, because we
470 # wind up with an inverse that lives in the subalgebra and
471 # we need information about the parent to convert it back.
472 V = self.span_of_powers()
473 assoc_subalg = self.subalgebra_generated_by()
474 # Mis-design warning: the basis used for span_of_powers()
475 # and subalgebra_generated_by() must be the same, and in
476 # the same order!
477 elt = assoc_subalg(V.coordinates(self.vector()))
478
479 # This will be in the subalgebra's coordinates...
480 fda_elt = FiniteDimensionalAlgebraElement(assoc_subalg, elt)
481 subalg_inverse = fda_elt.inverse()
482
483 # So we have to convert back...
484 basis = [ self.parent(v) for v in V.basis() ]
485 pairs = zip(subalg_inverse.vector(), basis)
486 return self.parent().linear_combination(pairs)
487
488
489 def is_invertible(self):
490 """
491 Return whether or not this element is invertible.
492
493 We can't use the superclass method because it relies on
494 the algebra being associative.
495
496 ALGORITHM:
497
498 The usual way to do this is to check if the determinant is
499 zero, but we need the characteristic polynomial for the
500 determinant. The minimal polynomial is a lot easier to get,
501 so we use Corollary 2 in Chapter V of Koecher to check
502 whether or not the paren't algebra's zero element is a root
503 of this element's minimal polynomial.
504
505 TESTS:
506
507 The identity element is always invertible::
508
509 sage: set_random_seed()
510 sage: J = random_eja()
511 sage: J.one().is_invertible()
512 True
513
514 The zero element is never invertible::
515
516 sage: set_random_seed()
517 sage: J = random_eja()
518 sage: J.zero().is_invertible()
519 False
520
521 """
522 zero = self.parent().zero()
523 p = self.minimal_polynomial()
524 return not (p(zero) == zero)
525
526
527 def is_nilpotent(self):
528 """
529 Return whether or not some power of this element is zero.
530
531 The superclass method won't work unless we're in an
532 associative algebra, and we aren't. However, we generate
533 an assocoative subalgebra and we're nilpotent there if and
534 only if we're nilpotent here (probably).
535
536 TESTS:
537
538 The identity element is never nilpotent::
539
540 sage: set_random_seed()
541 sage: random_eja().one().is_nilpotent()
542 False
543
544 The additive identity is always nilpotent::
545
546 sage: set_random_seed()
547 sage: random_eja().zero().is_nilpotent()
548 True
549
550 """
551 # The element we're going to call "is_nilpotent()" on.
552 # Either myself, interpreted as an element of a finite-
553 # dimensional algebra, or an element of an associative
554 # subalgebra.
555 elt = None
556
557 if self.parent().is_associative():
558 elt = FiniteDimensionalAlgebraElement(self.parent(), self)
559 else:
560 V = self.span_of_powers()
561 assoc_subalg = self.subalgebra_generated_by()
562 # Mis-design warning: the basis used for span_of_powers()
563 # and subalgebra_generated_by() must be the same, and in
564 # the same order!
565 elt = assoc_subalg(V.coordinates(self.vector()))
566
567 # Recursive call, but should work since elt lives in an
568 # associative algebra.
569 return elt.is_nilpotent()
570
571
572 def is_regular(self):
573 """
574 Return whether or not this is a regular element.
575
576 EXAMPLES:
577
578 The identity element always has degree one, but any element
579 linearly-independent from it is regular::
580
581 sage: J = JordanSpinEJA(5)
582 sage: J.one().is_regular()
583 False
584 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
585 sage: for x in J.gens():
586 ....: (J.one() + x).is_regular()
587 False
588 True
589 True
590 True
591 True
592
593 """
594 return self.degree() == self.parent().rank()
595
596
597 def degree(self):
598 """
599 Compute the degree of this element the straightforward way
600 according to the definition; by appending powers to a list
601 and figuring out its dimension (that is, whether or not
602 they're linearly dependent).
603
604 EXAMPLES::
605
606 sage: J = JordanSpinEJA(4)
607 sage: J.one().degree()
608 1
609 sage: e0,e1,e2,e3 = J.gens()
610 sage: (e0 - e1).degree()
611 2
612
613 In the spin factor algebra (of rank two), all elements that
614 aren't multiples of the identity are regular::
615
616 sage: set_random_seed()
617 sage: n = ZZ.random_element(1,10)
618 sage: J = JordanSpinEJA(n)
619 sage: x = J.random_element()
620 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
621 True
622
623 """
624 return self.span_of_powers().dimension()
625
626
627 def minimal_polynomial(self):
628 """
629 ALGORITHM:
630
631 We restrict ourselves to the associative subalgebra
632 generated by this element, and then return the minimal
633 polynomial of this element's operator matrix (in that
634 subalgebra). This works by Baes Proposition 2.3.16.
635
636 EXAMPLES::
637
638 sage: set_random_seed()
639 sage: x = random_eja().random_element()
640 sage: x.degree() == x.minimal_polynomial().degree()
641 True
642
643 ::
644
645 sage: set_random_seed()
646 sage: x = random_eja().random_element()
647 sage: x.degree() == x.minimal_polynomial().degree()
648 True
649
650 The minimal polynomial and the characteristic polynomial coincide
651 and are known (see Alizadeh, Example 11.11) for all elements of
652 the spin factor algebra that aren't scalar multiples of the
653 identity::
654
655 sage: set_random_seed()
656 sage: n = ZZ.random_element(2,10)
657 sage: J = JordanSpinEJA(n)
658 sage: y = J.random_element()
659 sage: while y == y.coefficient(0)*J.one():
660 ....: y = J.random_element()
661 sage: y0 = y.vector()[0]
662 sage: y_bar = y.vector()[1:]
663 sage: actual = y.minimal_polynomial()
664 sage: x = SR.symbol('x', domain='real')
665 sage: expected = x^2 - 2*y0*x + (y0^2 - norm(y_bar)^2)
666 sage: bool(actual == expected)
667 True
668
669 """
670 V = self.span_of_powers()
671 assoc_subalg = self.subalgebra_generated_by()
672 # Mis-design warning: the basis used for span_of_powers()
673 # and subalgebra_generated_by() must be the same, and in
674 # the same order!
675 elt = assoc_subalg(V.coordinates(self.vector()))
676 return elt.operator_matrix().minimal_polynomial()
677
678
679 def natural_representation(self):
680 """
681 Return a more-natural representation of this element.
682
683 Every finite-dimensional Euclidean Jordan Algebra is a
684 direct sum of five simple algebras, four of which comprise
685 Hermitian matrices. This method returns the original
686 "natural" representation of this element as a Hermitian
687 matrix, if it has one. If not, you get the usual representation.
688
689 EXAMPLES::
690
691 sage: J = ComplexHermitianEJA(3)
692 sage: J.one()
693 e0 + e5 + e8
694 sage: J.one().natural_representation()
695 [1 0 0 0 0 0]
696 [0 1 0 0 0 0]
697 [0 0 1 0 0 0]
698 [0 0 0 1 0 0]
699 [0 0 0 0 1 0]
700 [0 0 0 0 0 1]
701
702 ::
703
704 sage: J = QuaternionHermitianEJA(3)
705 sage: J.one()
706 e0 + e9 + e14
707 sage: J.one().natural_representation()
708 [1 0 0 0 0 0 0 0 0 0 0 0]
709 [0 1 0 0 0 0 0 0 0 0 0 0]
710 [0 0 1 0 0 0 0 0 0 0 0 0]
711 [0 0 0 1 0 0 0 0 0 0 0 0]
712 [0 0 0 0 1 0 0 0 0 0 0 0]
713 [0 0 0 0 0 1 0 0 0 0 0 0]
714 [0 0 0 0 0 0 1 0 0 0 0 0]
715 [0 0 0 0 0 0 0 1 0 0 0 0]
716 [0 0 0 0 0 0 0 0 1 0 0 0]
717 [0 0 0 0 0 0 0 0 0 1 0 0]
718 [0 0 0 0 0 0 0 0 0 0 1 0]
719 [0 0 0 0 0 0 0 0 0 0 0 1]
720
721 """
722 B = self.parent().natural_basis()
723 W = B[0].matrix_space()
724 return W.linear_combination(zip(self.vector(), B))
725
726
727 def operator_matrix(self):
728 """
729 Return the matrix that represents left- (or right-)
730 multiplication by this element in the parent algebra.
731
732 We have to override this because the superclass method
733 returns a matrix that acts on row vectors (that is, on
734 the right).
735
736 EXAMPLES:
737
738 Test the first polarization identity from my notes, Koecher Chapter
739 III, or from Baes (2.3)::
740
741 sage: set_random_seed()
742 sage: J = random_eja()
743 sage: x = J.random_element()
744 sage: y = J.random_element()
745 sage: Lx = x.operator_matrix()
746 sage: Ly = y.operator_matrix()
747 sage: Lxx = (x*x).operator_matrix()
748 sage: Lxy = (x*y).operator_matrix()
749 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
750 True
751
752 Test the second polarization identity from my notes or from
753 Baes (2.4)::
754
755 sage: set_random_seed()
756 sage: J = random_eja()
757 sage: x = J.random_element()
758 sage: y = J.random_element()
759 sage: z = J.random_element()
760 sage: Lx = x.operator_matrix()
761 sage: Ly = y.operator_matrix()
762 sage: Lz = z.operator_matrix()
763 sage: Lzy = (z*y).operator_matrix()
764 sage: Lxy = (x*y).operator_matrix()
765 sage: Lxz = (x*z).operator_matrix()
766 sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
767 True
768
769 Test the third polarization identity from my notes or from
770 Baes (2.5)::
771
772 sage: set_random_seed()
773 sage: J = random_eja()
774 sage: u = J.random_element()
775 sage: y = J.random_element()
776 sage: z = J.random_element()
777 sage: Lu = u.operator_matrix()
778 sage: Ly = y.operator_matrix()
779 sage: Lz = z.operator_matrix()
780 sage: Lzy = (z*y).operator_matrix()
781 sage: Luy = (u*y).operator_matrix()
782 sage: Luz = (u*z).operator_matrix()
783 sage: Luyz = (u*(y*z)).operator_matrix()
784 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
785 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
786 sage: bool(lhs == rhs)
787 True
788
789 """
790 fda_elt = FiniteDimensionalAlgebraElement(self.parent(), self)
791 return fda_elt.matrix().transpose()
792
793
794 def quadratic_representation(self, other=None):
795 """
796 Return the quadratic representation of this element.
797
798 EXAMPLES:
799
800 The explicit form in the spin factor algebra is given by
801 Alizadeh's Example 11.12::
802
803 sage: set_random_seed()
804 sage: n = ZZ.random_element(1,10)
805 sage: J = JordanSpinEJA(n)
806 sage: x = J.random_element()
807 sage: x_vec = x.vector()
808 sage: x0 = x_vec[0]
809 sage: x_bar = x_vec[1:]
810 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
811 sage: B = 2*x0*x_bar.row()
812 sage: C = 2*x0*x_bar.column()
813 sage: D = identity_matrix(QQ, n-1)
814 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
815 sage: D = D + 2*x_bar.tensor_product(x_bar)
816 sage: Q = block_matrix(2,2,[A,B,C,D])
817 sage: Q == x.quadratic_representation()
818 True
819
820 Test all of the properties from Theorem 11.2 in Alizadeh::
821
822 sage: set_random_seed()
823 sage: J = random_eja()
824 sage: x = J.random_element()
825 sage: y = J.random_element()
826
827 Property 1:
828
829 sage: actual = x.quadratic_representation(y)
830 sage: expected = ( (x+y).quadratic_representation()
831 ....: -x.quadratic_representation()
832 ....: -y.quadratic_representation() ) / 2
833 sage: actual == expected
834 True
835
836 Property 2:
837
838 sage: alpha = QQ.random_element()
839 sage: actual = (alpha*x).quadratic_representation()
840 sage: expected = (alpha^2)*x.quadratic_representation()
841 sage: actual == expected
842 True
843
844 Property 5:
845
846 sage: Qy = y.quadratic_representation()
847 sage: actual = J(Qy*x.vector()).quadratic_representation()
848 sage: expected = Qy*x.quadratic_representation()*Qy
849 sage: actual == expected
850 True
851
852 Property 6:
853
854 sage: k = ZZ.random_element(1,10)
855 sage: actual = (x^k).quadratic_representation()
856 sage: expected = (x.quadratic_representation())^k
857 sage: actual == expected
858 True
859
860 """
861 if other is None:
862 other=self
863 elif not other in self.parent():
864 raise TypeError("'other' must live in the same algebra")
865
866 L = self.operator_matrix()
867 M = other.operator_matrix()
868 return ( L*M + M*L - (self*other).operator_matrix() )
869
870
871 def span_of_powers(self):
872 """
873 Return the vector space spanned by successive powers of
874 this element.
875 """
876 # The dimension of the subalgebra can't be greater than
877 # the big algebra, so just put everything into a list
878 # and let span() get rid of the excess.
879 V = self.vector().parent()
880 return V.span( (self**d).vector() for d in xrange(V.dimension()) )
881
882
883 def subalgebra_generated_by(self):
884 """
885 Return the associative subalgebra of the parent EJA generated
886 by this element.
887
888 TESTS::
889
890 sage: set_random_seed()
891 sage: x = random_eja().random_element()
892 sage: x.subalgebra_generated_by().is_associative()
893 True
894
895 Squaring in the subalgebra should be the same thing as
896 squaring in the superalgebra::
897
898 sage: set_random_seed()
899 sage: x = random_eja().random_element()
900 sage: u = x.subalgebra_generated_by().random_element()
901 sage: u.operator_matrix()*u.vector() == (u**2).vector()
902 True
903
904 """
905 # First get the subspace spanned by the powers of myself...
906 V = self.span_of_powers()
907 F = self.base_ring()
908
909 # Now figure out the entries of the right-multiplication
910 # matrix for the successive basis elements b0, b1,... of
911 # that subspace.
912 mats = []
913 for b_right in V.basis():
914 eja_b_right = self.parent()(b_right)
915 b_right_rows = []
916 # The first row of the right-multiplication matrix by
917 # b1 is what we get if we apply that matrix to b1. The
918 # second row of the right multiplication matrix by b1
919 # is what we get when we apply that matrix to b2...
920 #
921 # IMPORTANT: this assumes that all vectors are COLUMN
922 # vectors, unlike our superclass (which uses row vectors).
923 for b_left in V.basis():
924 eja_b_left = self.parent()(b_left)
925 # Multiply in the original EJA, but then get the
926 # coordinates from the subalgebra in terms of its
927 # basis.
928 this_row = V.coordinates((eja_b_left*eja_b_right).vector())
929 b_right_rows.append(this_row)
930 b_right_matrix = matrix(F, b_right_rows)
931 mats.append(b_right_matrix)
932
933 # It's an algebra of polynomials in one element, and EJAs
934 # are power-associative.
935 #
936 # TODO: choose generator names intelligently.
937 return FiniteDimensionalEuclideanJordanAlgebra(F, mats, assume_associative=True, names='f')
938
939
940 def subalgebra_idempotent(self):
941 """
942 Find an idempotent in the associative subalgebra I generate
943 using Proposition 2.3.5 in Baes.
944
945 TESTS::
946
947 sage: set_random_seed()
948 sage: J = RealCartesianProductEJA(5)
949 sage: c = J.random_element().subalgebra_idempotent()
950 sage: c^2 == c
951 True
952 sage: J = JordanSpinEJA(5)
953 sage: c = J.random_element().subalgebra_idempotent()
954 sage: c^2 == c
955 True
956
957 """
958 if self.is_nilpotent():
959 raise ValueError("this only works with non-nilpotent elements!")
960
961 V = self.span_of_powers()
962 J = self.subalgebra_generated_by()
963 # Mis-design warning: the basis used for span_of_powers()
964 # and subalgebra_generated_by() must be the same, and in
965 # the same order!
966 u = J(V.coordinates(self.vector()))
967
968 # The image of the matrix of left-u^m-multiplication
969 # will be minimal for some natural number s...
970 s = 0
971 minimal_dim = V.dimension()
972 for i in xrange(1, V.dimension()):
973 this_dim = (u**i).operator_matrix().image().dimension()
974 if this_dim < minimal_dim:
975 minimal_dim = this_dim
976 s = i
977
978 # Now minimal_matrix should correspond to the smallest
979 # non-zero subspace in Baes's (or really, Koecher's)
980 # proposition.
981 #
982 # However, we need to restrict the matrix to work on the
983 # subspace... or do we? Can't we just solve, knowing that
984 # A(c) = u^(s+1) should have a solution in the big space,
985 # too?
986 #
987 # Beware, solve_right() means that we're using COLUMN vectors.
988 # Our FiniteDimensionalAlgebraElement superclass uses rows.
989 u_next = u**(s+1)
990 A = u_next.operator_matrix()
991 c_coordinates = A.solve_right(u_next.vector())
992
993 # Now c_coordinates is the idempotent we want, but it's in
994 # the coordinate system of the subalgebra.
995 #
996 # We need the basis for J, but as elements of the parent algebra.
997 #
998 basis = [self.parent(v) for v in V.basis()]
999 return self.parent().linear_combination(zip(c_coordinates, basis))
1000
1001
1002 def trace(self):
1003 """
1004 Return my trace, the sum of my eigenvalues.
1005
1006 EXAMPLES::
1007
1008 sage: J = JordanSpinEJA(3)
1009 sage: e0,e1,e2 = J.gens()
1010 sage: x = e0 + e1 + e2
1011 sage: x.trace()
1012 2
1013
1014 """
1015 cs = self.characteristic_polynomial().coefficients(sparse=False)
1016 if len(cs) >= 2:
1017 return -1*cs[-2]
1018 else:
1019 raise ValueError('charpoly had fewer than 2 coefficients')
1020
1021
1022 def trace_inner_product(self, other):
1023 """
1024 Return the trace inner product of myself and ``other``.
1025 """
1026 if not other in self.parent():
1027 raise TypeError("'other' must live in the same algebra")
1028
1029 return (self*other).trace()
1030
1031
1032 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra):
1033 """
1034 Return the Euclidean Jordan Algebra corresponding to the set
1035 `R^n` under the Hadamard product.
1036
1037 Note: this is nothing more than the Cartesian product of ``n``
1038 copies of the spin algebra. Once Cartesian product algebras
1039 are implemented, this can go.
1040
1041 EXAMPLES:
1042
1043 This multiplication table can be verified by hand::
1044
1045 sage: J = RealCartesianProductEJA(3)
1046 sage: e0,e1,e2 = J.gens()
1047 sage: e0*e0
1048 e0
1049 sage: e0*e1
1050 0
1051 sage: e0*e2
1052 0
1053 sage: e1*e1
1054 e1
1055 sage: e1*e2
1056 0
1057 sage: e2*e2
1058 e2
1059
1060 """
1061 @staticmethod
1062 def __classcall_private__(cls, n, field=QQ):
1063 # The FiniteDimensionalAlgebra constructor takes a list of
1064 # matrices, the ith representing right multiplication by the ith
1065 # basis element in the vector space. So if e_1 = (1,0,0), then
1066 # right (Hadamard) multiplication of x by e_1 picks out the first
1067 # component of x; and likewise for the ith basis element e_i.
1068 Qs = [ matrix(field, n, n, lambda k,j: 1*(k == j == i))
1069 for i in xrange(n) ]
1070
1071 fdeja = super(RealCartesianProductEJA, cls)
1072 return fdeja.__classcall_private__(cls, field, Qs, rank=n)
1073
1074 def inner_product(self, x, y):
1075 return _usual_ip(x,y)
1076
1077
1078 def random_eja():
1079 """
1080 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1081
1082 ALGORITHM:
1083
1084 For now, we choose a random natural number ``n`` (greater than zero)
1085 and then give you back one of the following:
1086
1087 * The cartesian product of the rational numbers ``n`` times; this is
1088 ``QQ^n`` with the Hadamard product.
1089
1090 * The Jordan spin algebra on ``QQ^n``.
1091
1092 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
1093 product.
1094
1095 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
1096 in the space of ``2n``-by-``2n`` real symmetric matrices.
1097
1098 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
1099 in the space of ``4n``-by-``4n`` real symmetric matrices.
1100
1101 Later this might be extended to return Cartesian products of the
1102 EJAs above.
1103
1104 TESTS::
1105
1106 sage: random_eja()
1107 Euclidean Jordan algebra of degree...
1108
1109 """
1110 n = ZZ.random_element(1,5)
1111 constructor = choice([RealCartesianProductEJA,
1112 JordanSpinEJA,
1113 RealSymmetricEJA,
1114 ComplexHermitianEJA,
1115 QuaternionHermitianEJA])
1116 return constructor(n, field=QQ)
1117
1118
1119
1120 def _real_symmetric_basis(n, field=QQ):
1121 """
1122 Return a basis for the space of real symmetric n-by-n matrices.
1123 """
1124 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1125 # coordinates.
1126 S = []
1127 for i in xrange(n):
1128 for j in xrange(i+1):
1129 Eij = matrix(field, n, lambda k,l: k==i and l==j)
1130 if i == j:
1131 Sij = Eij
1132 else:
1133 # Beware, orthogonal but not normalized!
1134 Sij = Eij + Eij.transpose()
1135 S.append(Sij)
1136 return tuple(S)
1137
1138
1139 def _complex_hermitian_basis(n, field=QQ):
1140 """
1141 Returns a basis for the space of complex Hermitian n-by-n matrices.
1142
1143 TESTS::
1144
1145 sage: set_random_seed()
1146 sage: n = ZZ.random_element(1,5)
1147 sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
1148 True
1149
1150 """
1151 F = QuadraticField(-1, 'I')
1152 I = F.gen()
1153
1154 # This is like the symmetric case, but we need to be careful:
1155 #
1156 # * We want conjugate-symmetry, not just symmetry.
1157 # * The diagonal will (as a result) be real.
1158 #
1159 S = []
1160 for i in xrange(n):
1161 for j in xrange(i+1):
1162 Eij = matrix(field, n, lambda k,l: k==i and l==j)
1163 if i == j:
1164 Sij = _embed_complex_matrix(Eij)
1165 S.append(Sij)
1166 else:
1167 # Beware, orthogonal but not normalized! The second one
1168 # has a minus because it's conjugated.
1169 Sij_real = _embed_complex_matrix(Eij + Eij.transpose())
1170 S.append(Sij_real)
1171 Sij_imag = _embed_complex_matrix(I*Eij - I*Eij.transpose())
1172 S.append(Sij_imag)
1173 return tuple(S)
1174
1175
1176 def _quaternion_hermitian_basis(n, field=QQ):
1177 """
1178 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1179
1180 TESTS::
1181
1182 sage: set_random_seed()
1183 sage: n = ZZ.random_element(1,5)
1184 sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
1185 True
1186
1187 """
1188 Q = QuaternionAlgebra(QQ,-1,-1)
1189 I,J,K = Q.gens()
1190
1191 # This is like the symmetric case, but we need to be careful:
1192 #
1193 # * We want conjugate-symmetry, not just symmetry.
1194 # * The diagonal will (as a result) be real.
1195 #
1196 S = []
1197 for i in xrange(n):
1198 for j in xrange(i+1):
1199 Eij = matrix(Q, n, lambda k,l: k==i and l==j)
1200 if i == j:
1201 Sij = _embed_quaternion_matrix(Eij)
1202 S.append(Sij)
1203 else:
1204 # Beware, orthogonal but not normalized! The second,
1205 # third, and fourth ones have a minus because they're
1206 # conjugated.
1207 Sij_real = _embed_quaternion_matrix(Eij + Eij.transpose())
1208 S.append(Sij_real)
1209 Sij_I = _embed_quaternion_matrix(I*Eij - I*Eij.transpose())
1210 S.append(Sij_I)
1211 Sij_J = _embed_quaternion_matrix(J*Eij - J*Eij.transpose())
1212 S.append(Sij_J)
1213 Sij_K = _embed_quaternion_matrix(K*Eij - K*Eij.transpose())
1214 S.append(Sij_K)
1215 return tuple(S)
1216
1217
1218 def _mat2vec(m):
1219 return vector(m.base_ring(), m.list())
1220
1221 def _vec2mat(v):
1222 return matrix(v.base_ring(), sqrt(v.degree()), v.list())
1223
1224 def _multiplication_table_from_matrix_basis(basis):
1225 """
1226 At least three of the five simple Euclidean Jordan algebras have the
1227 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1228 multiplication on the right is matrix multiplication. Given a basis
1229 for the underlying matrix space, this function returns a
1230 multiplication table (obtained by looping through the basis
1231 elements) for an algebra of those matrices. A reordered copy
1232 of the basis is also returned to work around the fact that
1233 the ``span()`` in this function will change the order of the basis
1234 from what we think it is, to... something else.
1235 """
1236 # In S^2, for example, we nominally have four coordinates even
1237 # though the space is of dimension three only. The vector space V
1238 # is supposed to hold the entire long vector, and the subspace W
1239 # of V will be spanned by the vectors that arise from symmetric
1240 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1241 field = basis[0].base_ring()
1242 dimension = basis[0].nrows()
1243
1244 V = VectorSpace(field, dimension**2)
1245 W = V.span( _mat2vec(s) for s in basis )
1246
1247 # Taking the span above reorders our basis (thanks, jerk!) so we
1248 # need to put our "matrix basis" in the same order as the
1249 # (reordered) vector basis.
1250 S = tuple( _vec2mat(b) for b in W.basis() )
1251
1252 Qs = []
1253 for s in S:
1254 # Brute force the multiplication-by-s matrix by looping
1255 # through all elements of the basis and doing the computation
1256 # to find out what the corresponding row should be. BEWARE:
1257 # these multiplication tables won't be symmetric! It therefore
1258 # becomes REALLY IMPORTANT that the underlying algebra
1259 # constructor uses ROW vectors and not COLUMN vectors. That's
1260 # why we're computing rows here and not columns.
1261 Q_rows = []
1262 for t in S:
1263 this_row = _mat2vec((s*t + t*s)/2)
1264 Q_rows.append(W.coordinates(this_row))
1265 Q = matrix(field, W.dimension(), Q_rows)
1266 Qs.append(Q)
1267
1268 return (Qs, S)
1269
1270
1271 def _embed_complex_matrix(M):
1272 """
1273 Embed the n-by-n complex matrix ``M`` into the space of real
1274 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1275 bi` to the block matrix ``[[a,b],[-b,a]]``.
1276
1277 EXAMPLES::
1278
1279 sage: F = QuadraticField(-1,'i')
1280 sage: x1 = F(4 - 2*i)
1281 sage: x2 = F(1 + 2*i)
1282 sage: x3 = F(-i)
1283 sage: x4 = F(6)
1284 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1285 sage: _embed_complex_matrix(M)
1286 [ 4 -2| 1 2]
1287 [ 2 4|-2 1]
1288 [-----+-----]
1289 [ 0 -1| 6 0]
1290 [ 1 0| 0 6]
1291
1292 TESTS:
1293
1294 Embedding is a homomorphism (isomorphism, in fact)::
1295
1296 sage: set_random_seed()
1297 sage: n = ZZ.random_element(5)
1298 sage: F = QuadraticField(-1, 'i')
1299 sage: X = random_matrix(F, n)
1300 sage: Y = random_matrix(F, n)
1301 sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
1302 sage: expected = _embed_complex_matrix(X*Y)
1303 sage: actual == expected
1304 True
1305
1306 """
1307 n = M.nrows()
1308 if M.ncols() != n:
1309 raise ValueError("the matrix 'M' must be square")
1310 field = M.base_ring()
1311 blocks = []
1312 for z in M.list():
1313 a = z.real()
1314 b = z.imag()
1315 blocks.append(matrix(field, 2, [[a,b],[-b,a]]))
1316
1317 # We can drop the imaginaries here.
1318 return block_matrix(field.base_ring(), n, blocks)
1319
1320
1321 def _unembed_complex_matrix(M):
1322 """
1323 The inverse of _embed_complex_matrix().
1324
1325 EXAMPLES::
1326
1327 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1328 ....: [-2, 1, -4, 3],
1329 ....: [ 9, 10, 11, 12],
1330 ....: [-10, 9, -12, 11] ])
1331 sage: _unembed_complex_matrix(A)
1332 [ 2*i + 1 4*i + 3]
1333 [ 10*i + 9 12*i + 11]
1334
1335 TESTS:
1336
1337 Unembedding is the inverse of embedding::
1338
1339 sage: set_random_seed()
1340 sage: F = QuadraticField(-1, 'i')
1341 sage: M = random_matrix(F, 3)
1342 sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
1343 True
1344
1345 """
1346 n = ZZ(M.nrows())
1347 if M.ncols() != n:
1348 raise ValueError("the matrix 'M' must be square")
1349 if not n.mod(2).is_zero():
1350 raise ValueError("the matrix 'M' must be a complex embedding")
1351
1352 F = QuadraticField(-1, 'i')
1353 i = F.gen()
1354
1355 # Go top-left to bottom-right (reading order), converting every
1356 # 2-by-2 block we see to a single complex element.
1357 elements = []
1358 for k in xrange(n/2):
1359 for j in xrange(n/2):
1360 submat = M[2*k:2*k+2,2*j:2*j+2]
1361 if submat[0,0] != submat[1,1]:
1362 raise ValueError('bad on-diagonal submatrix')
1363 if submat[0,1] != -submat[1,0]:
1364 raise ValueError('bad off-diagonal submatrix')
1365 z = submat[0,0] + submat[0,1]*i
1366 elements.append(z)
1367
1368 return matrix(F, n/2, elements)
1369
1370
1371 def _embed_quaternion_matrix(M):
1372 """
1373 Embed the n-by-n quaternion matrix ``M`` into the space of real
1374 matrices of size 4n-by-4n by first sending each quaternion entry
1375 `z = a + bi + cj + dk` to the block-complex matrix
1376 ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
1377 a real matrix.
1378
1379 EXAMPLES::
1380
1381 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1382 sage: i,j,k = Q.gens()
1383 sage: x = 1 + 2*i + 3*j + 4*k
1384 sage: M = matrix(Q, 1, [[x]])
1385 sage: _embed_quaternion_matrix(M)
1386 [ 1 2 3 4]
1387 [-2 1 -4 3]
1388 [-3 4 1 -2]
1389 [-4 -3 2 1]
1390
1391 Embedding is a homomorphism (isomorphism, in fact)::
1392
1393 sage: set_random_seed()
1394 sage: n = ZZ.random_element(5)
1395 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1396 sage: X = random_matrix(Q, n)
1397 sage: Y = random_matrix(Q, n)
1398 sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
1399 sage: expected = _embed_quaternion_matrix(X*Y)
1400 sage: actual == expected
1401 True
1402
1403 """
1404 quaternions = M.base_ring()
1405 n = M.nrows()
1406 if M.ncols() != n:
1407 raise ValueError("the matrix 'M' must be square")
1408
1409 F = QuadraticField(-1, 'i')
1410 i = F.gen()
1411
1412 blocks = []
1413 for z in M.list():
1414 t = z.coefficient_tuple()
1415 a = t[0]
1416 b = t[1]
1417 c = t[2]
1418 d = t[3]
1419 cplx_matrix = matrix(F, 2, [[ a + b*i, c + d*i],
1420 [-c + d*i, a - b*i]])
1421 blocks.append(_embed_complex_matrix(cplx_matrix))
1422
1423 # We should have real entries by now, so use the realest field
1424 # we've got for the return value.
1425 return block_matrix(quaternions.base_ring(), n, blocks)
1426
1427
1428 def _unembed_quaternion_matrix(M):
1429 """
1430 The inverse of _embed_quaternion_matrix().
1431
1432 EXAMPLES::
1433
1434 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1435 ....: [-2, 1, -4, 3],
1436 ....: [-3, 4, 1, -2],
1437 ....: [-4, -3, 2, 1]])
1438 sage: _unembed_quaternion_matrix(M)
1439 [1 + 2*i + 3*j + 4*k]
1440
1441 TESTS:
1442
1443 Unembedding is the inverse of embedding::
1444
1445 sage: set_random_seed()
1446 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1447 sage: M = random_matrix(Q, 3)
1448 sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
1449 True
1450
1451 """
1452 n = ZZ(M.nrows())
1453 if M.ncols() != n:
1454 raise ValueError("the matrix 'M' must be square")
1455 if not n.mod(4).is_zero():
1456 raise ValueError("the matrix 'M' must be a complex embedding")
1457
1458 Q = QuaternionAlgebra(QQ,-1,-1)
1459 i,j,k = Q.gens()
1460
1461 # Go top-left to bottom-right (reading order), converting every
1462 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1463 # quaternion block.
1464 elements = []
1465 for l in xrange(n/4):
1466 for m in xrange(n/4):
1467 submat = _unembed_complex_matrix(M[4*l:4*l+4,4*m:4*m+4])
1468 if submat[0,0] != submat[1,1].conjugate():
1469 raise ValueError('bad on-diagonal submatrix')
1470 if submat[0,1] != -submat[1,0].conjugate():
1471 raise ValueError('bad off-diagonal submatrix')
1472 z = submat[0,0].real() + submat[0,0].imag()*i
1473 z += submat[0,1].real()*j + submat[0,1].imag()*k
1474 elements.append(z)
1475
1476 return matrix(Q, n/4, elements)
1477
1478
1479 # The usual inner product on R^n.
1480 def _usual_ip(x,y):
1481 return x.vector().inner_product(y.vector())
1482
1483 # The inner product used for the real symmetric simple EJA.
1484 # We keep it as a separate function because e.g. the complex
1485 # algebra uses the same inner product, except divided by 2.
1486 def _matrix_ip(X,Y):
1487 X_mat = X.natural_representation()
1488 Y_mat = Y.natural_representation()
1489 return (X_mat*Y_mat).trace()
1490
1491
1492 class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra):
1493 """
1494 The rank-n simple EJA consisting of real symmetric n-by-n
1495 matrices, the usual symmetric Jordan product, and the trace inner
1496 product. It has dimension `(n^2 + n)/2` over the reals.
1497
1498 EXAMPLES::
1499
1500 sage: J = RealSymmetricEJA(2)
1501 sage: e0, e1, e2 = J.gens()
1502 sage: e0*e0
1503 e0
1504 sage: e1*e1
1505 e0 + e2
1506 sage: e2*e2
1507 e2
1508
1509 TESTS:
1510
1511 The degree of this algebra is `(n^2 + n) / 2`::
1512
1513 sage: set_random_seed()
1514 sage: n = ZZ.random_element(1,5)
1515 sage: J = RealSymmetricEJA(n)
1516 sage: J.degree() == (n^2 + n)/2
1517 True
1518
1519 The Jordan multiplication is what we think it is::
1520
1521 sage: set_random_seed()
1522 sage: n = ZZ.random_element(1,5)
1523 sage: J = RealSymmetricEJA(n)
1524 sage: x = J.random_element()
1525 sage: y = J.random_element()
1526 sage: actual = (x*y).natural_representation()
1527 sage: X = x.natural_representation()
1528 sage: Y = y.natural_representation()
1529 sage: expected = (X*Y + Y*X)/2
1530 sage: actual == expected
1531 True
1532 sage: J(expected) == x*y
1533 True
1534
1535 """
1536 @staticmethod
1537 def __classcall_private__(cls, n, field=QQ):
1538 S = _real_symmetric_basis(n, field=field)
1539 (Qs, T) = _multiplication_table_from_matrix_basis(S)
1540
1541 fdeja = super(RealSymmetricEJA, cls)
1542 return fdeja.__classcall_private__(cls,
1543 field,
1544 Qs,
1545 rank=n,
1546 natural_basis=T)
1547
1548 def inner_product(self, x, y):
1549 return _matrix_ip(x,y)
1550
1551
1552 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
1553 """
1554 The rank-n simple EJA consisting of complex Hermitian n-by-n
1555 matrices over the real numbers, the usual symmetric Jordan product,
1556 and the real-part-of-trace inner product. It has dimension `n^2` over
1557 the reals.
1558
1559 TESTS:
1560
1561 The degree of this algebra is `n^2`::
1562
1563 sage: set_random_seed()
1564 sage: n = ZZ.random_element(1,5)
1565 sage: J = ComplexHermitianEJA(n)
1566 sage: J.degree() == n^2
1567 True
1568
1569 The Jordan multiplication is what we think it is::
1570
1571 sage: set_random_seed()
1572 sage: n = ZZ.random_element(1,5)
1573 sage: J = ComplexHermitianEJA(n)
1574 sage: x = J.random_element()
1575 sage: y = J.random_element()
1576 sage: actual = (x*y).natural_representation()
1577 sage: X = x.natural_representation()
1578 sage: Y = y.natural_representation()
1579 sage: expected = (X*Y + Y*X)/2
1580 sage: actual == expected
1581 True
1582 sage: J(expected) == x*y
1583 True
1584
1585 """
1586 @staticmethod
1587 def __classcall_private__(cls, n, field=QQ):
1588 S = _complex_hermitian_basis(n)
1589 (Qs, T) = _multiplication_table_from_matrix_basis(S)
1590
1591 fdeja = super(ComplexHermitianEJA, cls)
1592 return fdeja.__classcall_private__(cls,
1593 field,
1594 Qs,
1595 rank=n,
1596 natural_basis=T)
1597
1598 def inner_product(self, x, y):
1599 # Since a+bi on the diagonal is represented as
1600 #
1601 # a + bi = [ a b ]
1602 # [ -b a ],
1603 #
1604 # we'll double-count the "a" entries if we take the trace of
1605 # the embedding.
1606 return _matrix_ip(x,y)/2
1607
1608
1609 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
1610 """
1611 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1612 matrices, the usual symmetric Jordan product, and the
1613 real-part-of-trace inner product. It has dimension `2n^2 - n` over
1614 the reals.
1615
1616 TESTS:
1617
1618 The degree of this algebra is `n^2`::
1619
1620 sage: set_random_seed()
1621 sage: n = ZZ.random_element(1,5)
1622 sage: J = QuaternionHermitianEJA(n)
1623 sage: J.degree() == 2*(n^2) - n
1624 True
1625
1626 The Jordan multiplication is what we think it is::
1627
1628 sage: set_random_seed()
1629 sage: n = ZZ.random_element(1,5)
1630 sage: J = QuaternionHermitianEJA(n)
1631 sage: x = J.random_element()
1632 sage: y = J.random_element()
1633 sage: actual = (x*y).natural_representation()
1634 sage: X = x.natural_representation()
1635 sage: Y = y.natural_representation()
1636 sage: expected = (X*Y + Y*X)/2
1637 sage: actual == expected
1638 True
1639 sage: J(expected) == x*y
1640 True
1641
1642 """
1643 @staticmethod
1644 def __classcall_private__(cls, n, field=QQ):
1645 S = _quaternion_hermitian_basis(n)
1646 (Qs, T) = _multiplication_table_from_matrix_basis(S)
1647
1648 fdeja = super(QuaternionHermitianEJA, cls)
1649 return fdeja.__classcall_private__(cls,
1650 field,
1651 Qs,
1652 rank=n,
1653 natural_basis=T)
1654
1655 def inner_product(self, x, y):
1656 # Since a+bi+cj+dk on the diagonal is represented as
1657 #
1658 # a + bi +cj + dk = [ a b c d]
1659 # [ -b a -d c]
1660 # [ -c d a -b]
1661 # [ -d -c b a],
1662 #
1663 # we'll quadruple-count the "a" entries if we take the trace of
1664 # the embedding.
1665 return _matrix_ip(x,y)/4
1666
1667
1668 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra):
1669 """
1670 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
1671 with the usual inner product and jordan product ``x*y =
1672 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
1673 the reals.
1674
1675 EXAMPLES:
1676
1677 This multiplication table can be verified by hand::
1678
1679 sage: J = JordanSpinEJA(4)
1680 sage: e0,e1,e2,e3 = J.gens()
1681 sage: e0*e0
1682 e0
1683 sage: e0*e1
1684 e1
1685 sage: e0*e2
1686 e2
1687 sage: e0*e3
1688 e3
1689 sage: e1*e2
1690 0
1691 sage: e1*e3
1692 0
1693 sage: e2*e3
1694 0
1695
1696 """
1697 @staticmethod
1698 def __classcall_private__(cls, n, field=QQ):
1699 Qs = []
1700 id_matrix = identity_matrix(field, n)
1701 for i in xrange(n):
1702 ei = id_matrix.column(i)
1703 Qi = zero_matrix(field, n)
1704 Qi.set_row(0, ei)
1705 Qi.set_column(0, ei)
1706 Qi += diagonal_matrix(n, [ei[0]]*n)
1707 # The addition of the diagonal matrix adds an extra ei[0] in the
1708 # upper-left corner of the matrix.
1709 Qi[0,0] = Qi[0,0] * ~field(2)
1710 Qs.append(Qi)
1711
1712 # The rank of the spin algebra is two, unless we're in a
1713 # one-dimensional ambient space (because the rank is bounded by
1714 # the ambient dimension).
1715 fdeja = super(JordanSpinEJA, cls)
1716 return fdeja.__classcall_private__(cls, field, Qs, rank=min(n,2))
1717
1718 def inner_product(self, x, y):
1719 return _usual_ip(x,y)
My personal library of Sage code.