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.
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
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
15 class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra
):
17 def __classcall_private__(cls
,
21 assume_associative
=False,
26 mult_table
= [b
.base_extend(field
) for b
in mult_table
]
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
)
33 cat
= MagmaticAlgebras(field
).FiniteDimensional().WithBasis()
34 cat
.or_subcategory(category
)
35 if assume_associative
:
36 cat
= cat
.Associative()
38 names
= normalize_names(n
, names
)
40 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, cls
)
41 return fda
.__classcall
__(cls
,
44 assume_associative
=assume_associative
,
48 natural_basis
=natural_basis
)
55 assume_associative
=False,
62 By definition, Jordan multiplication commutes::
64 sage: set_random_seed()
65 sage: J = random_eja()
66 sage: x = J.random_element()
67 sage: y = J.random_element()
73 self
._natural
_basis
= natural_basis
74 self
._multiplication
_table
= mult_table
75 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
84 Return a string representation of ``self``.
86 fmt
= "Euclidean Jordan algebra of degree {} over {}"
87 return fmt
.format(self
.degree(), self
.base_ring())
92 def _charpoly_coeff(self
, i
):
94 Return the coefficient polynomial "a_{i}" of this algebra's
95 general characteristic polynomial.
97 Having this be a separate cached method lets us compute and
98 store the trace/determinant (a_{r-1} and a_{0} respectively)
99 separate from the entire characteristic polynomial.
101 (A_of_x
, x
) = self
._charpoly
_matrix
()
102 R
= A_of_x
.base_ring()
103 A_cols
= A_of_x
.columns()
104 A_cols
[i
] = (x
**self
.rank()).vector()
105 numerator
= column_matrix(A_of_x
.base_ring(), A_cols
).det()
106 denominator
= A_of_x
.det()
108 # We're relying on the theory here to ensure that each a_i is
109 # indeed back in R, and the added negative signs are to make
110 # the whole charpoly expression sum to zero.
111 return R(-numerator
/denominator
)
115 def _charpoly_matrix(self
):
117 Compute the matrix whose entries A_ij are polynomials in
118 X1,...,XN. This same matrix is used in more than one method and
119 it's not so fast to construct.
124 # Construct a new algebra over a multivariate polynomial ring...
125 names
= ['X' + str(i
) for i
in range(1,n
+1)]
126 R
= PolynomialRing(self
.base_ring(), names
)
127 J
= FiniteDimensionalEuclideanJordanAlgebra(R
,
128 self
._multiplication
_table
,
131 idmat
= identity_matrix(J
.base_ring(), n
)
133 x
= J(vector(R
, R
.gens()))
134 l1
= [column_matrix((x
**k
).vector()) for k
in range(r
)]
135 l2
= [idmat
.column(k
-1).column() for k
in range(r
+1, n
+1)]
136 A_of_x
= block_matrix(R
, 1, n
, (l1
+ l2
))
141 def characteristic_polynomial(self
):
145 The characteristic polynomial in the spin algebra is given in
146 Alizadeh, Example 11.11::
148 sage: J = JordanSpinEJA(3)
149 sage: p = J.characteristic_polynomial(); p
150 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
151 sage: xvec = J.one().vector()
159 # The list of coefficient polynomials a_1, a_2, ..., a_n.
160 a
= [ self
._charpoly
_coeff
(i
) for i
in range(n
) ]
162 # We go to a bit of trouble here to reorder the
163 # indeterminates, so that it's easier to evaluate the
164 # characteristic polynomial at x's coordinates and get back
165 # something in terms of t, which is what we want.
167 S
= PolynomialRing(self
.base_ring(),'t')
169 S
= PolynomialRing(S
, R
.variable_names())
172 # Note: all entries past the rth should be zero. The
173 # coefficient of the highest power (x^r) is 1, but it doesn't
174 # appear in the solution vector which contains coefficients
175 # for the other powers (to make them sum to x^r).
177 a
[r
] = 1 # corresponds to x^r
179 # When the rank is equal to the dimension, trying to
180 # assign a[r] goes out-of-bounds.
181 a
.append(1) # corresponds to x^r
183 return sum( a
[k
]*(t
**k
) for k
in range(len(a
)) )
186 def inner_product(self
, x
, y
):
188 The inner product associated with this Euclidean Jordan algebra.
190 Defaults to the trace inner product, but can be overridden by
191 subclasses if they are sure that the necessary properties are
196 The inner product must satisfy its axiom for this algebra to truly
197 be a Euclidean Jordan Algebra::
199 sage: set_random_seed()
200 sage: J = random_eja()
201 sage: x = J.random_element()
202 sage: y = J.random_element()
203 sage: z = J.random_element()
204 sage: (x*y).inner_product(z) == y.inner_product(x*z)
208 if (not x
in self
) or (not y
in self
):
209 raise TypeError("arguments must live in this algebra")
210 return x
.trace_inner_product(y
)
213 def natural_basis(self
):
215 Return a more-natural representation of this algebra's basis.
217 Every finite-dimensional Euclidean Jordan Algebra is a direct
218 sum of five simple algebras, four of which comprise Hermitian
219 matrices. This method returns the original "natural" basis
220 for our underlying vector space. (Typically, the natural basis
221 is used to construct the multiplication table in the first place.)
223 Note that this will always return a matrix. The standard basis
224 in `R^n` will be returned as `n`-by-`1` column matrices.
228 sage: J = RealSymmetricEJA(2)
231 sage: J.natural_basis()
239 sage: J = JordanSpinEJA(2)
242 sage: J.natural_basis()
249 if self
._natural
_basis
is None:
250 return tuple( b
.vector().column() for b
in self
.basis() )
252 return self
._natural
_basis
257 Return the rank of this EJA.
259 if self
._rank
is None:
260 raise ValueError("no rank specified at genesis")
265 class Element(FiniteDimensionalAlgebraElement
):
267 An element of a Euclidean Jordan algebra.
270 def __init__(self
, A
, elt
=None):
274 The identity in `S^n` is converted to the identity in the EJA::
276 sage: J = RealSymmetricEJA(3)
277 sage: I = identity_matrix(QQ,3)
278 sage: J(I) == J.one()
281 This skew-symmetric matrix can't be represented in the EJA::
283 sage: J = RealSymmetricEJA(3)
284 sage: A = matrix(QQ,3, lambda i,j: i-j)
286 Traceback (most recent call last):
288 ArithmeticError: vector is not in free module
291 # Goal: if we're given a matrix, and if it lives in our
292 # parent algebra's "natural ambient space," convert it
293 # into an algebra element.
295 # The catch is, we make a recursive call after converting
296 # the given matrix into a vector that lives in the algebra.
297 # This we need to try the parent class initializer first,
298 # to avoid recursing forever if we're given something that
299 # already fits into the algebra, but also happens to live
300 # in the parent's "natural ambient space" (this happens with
303 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, elt
)
305 natural_basis
= A
.natural_basis()
306 if elt
in natural_basis
[0].matrix_space():
307 # Thanks for nothing! Matrix spaces aren't vector
308 # spaces in Sage, so we have to figure out its
309 # natural-basis coordinates ourselves.
310 V
= VectorSpace(elt
.base_ring(), elt
.nrows()**2)
311 W
= V
.span( _mat2vec(s
) for s
in natural_basis
)
312 coords
= W
.coordinates(_mat2vec(elt
))
313 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, coords
)
315 def __pow__(self
, n
):
317 Return ``self`` raised to the power ``n``.
319 Jordan algebras are always power-associative; see for
320 example Faraut and Koranyi, Proposition II.1.2 (ii).
324 We have to override this because our superclass uses row vectors
325 instead of column vectors! We, on the other hand, assume column
330 sage: set_random_seed()
331 sage: x = random_eja().random_element()
332 sage: x.operator_matrix()*x.vector() == (x^2).vector()
335 A few examples of power-associativity::
337 sage: set_random_seed()
338 sage: x = random_eja().random_element()
339 sage: x*(x*x)*(x*x) == x^5
341 sage: (x*x)*(x*x*x) == x^5
344 We also know that powers operator-commute (Koecher, Chapter
347 sage: set_random_seed()
348 sage: x = random_eja().random_element()
349 sage: m = ZZ.random_element(0,10)
350 sage: n = ZZ.random_element(0,10)
351 sage: Lxm = (x^m).operator_matrix()
352 sage: Lxn = (x^n).operator_matrix()
353 sage: Lxm*Lxn == Lxn*Lxm
363 return A( (self
.operator_matrix()**(n
-1))*self
.vector() )
366 def apply_univariate_polynomial(self
, p
):
368 Apply the univariate polynomial ``p`` to this element.
370 A priori, SageMath won't allow us to apply a univariate
371 polynomial to an element of an EJA, because we don't know
372 that EJAs are rings (they are usually not associative). Of
373 course, we know that EJAs are power-associative, so the
374 operation is ultimately kosher. This function sidesteps
375 the CAS to get the answer we want and expect.
379 sage: R = PolynomialRing(QQ, 't')
381 sage: p = t^4 - t^3 + 5*t - 2
382 sage: J = RealCartesianProductEJA(5)
383 sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
388 We should always get back an element of the algebra::
390 sage: set_random_seed()
391 sage: p = PolynomialRing(QQ, 't').random_element()
392 sage: J = random_eja()
393 sage: x = J.random_element()
394 sage: x.apply_univariate_polynomial(p) in J
398 if len(p
.variables()) > 1:
399 raise ValueError("not a univariate polynomial")
402 # Convert the coeficcients to the parent's base ring,
403 # because a priori they might live in an (unnecessarily)
404 # larger ring for which P.sum() would fail below.
405 cs
= [ R(c
) for c
in p
.coefficients(sparse
=False) ]
406 return P
.sum( cs
[k
]*(self
**k
) for k
in range(len(cs
)) )
409 def characteristic_polynomial(self
):
411 Return the characteristic polynomial of this element.
415 The rank of `R^3` is three, and the minimal polynomial of
416 the identity element is `(t-1)` from which it follows that
417 the characteristic polynomial should be `(t-1)^3`::
419 sage: J = RealCartesianProductEJA(3)
420 sage: J.one().characteristic_polynomial()
421 t^3 - 3*t^2 + 3*t - 1
423 Likewise, the characteristic of the zero element in the
424 rank-three algebra `R^{n}` should be `t^{3}`::
426 sage: J = RealCartesianProductEJA(3)
427 sage: J.zero().characteristic_polynomial()
430 The characteristic polynomial of an element should evaluate
431 to zero on that element::
433 sage: set_random_seed()
434 sage: x = RealCartesianProductEJA(3).random_element()
435 sage: p = x.characteristic_polynomial()
436 sage: x.apply_univariate_polynomial(p)
440 p
= self
.parent().characteristic_polynomial()
441 return p(*self
.vector())
444 def inner_product(self
, other
):
446 Return the parent algebra's inner product of myself and ``other``.
450 The inner product in the Jordan spin algebra is the usual
451 inner product on `R^n` (this example only works because the
452 basis for the Jordan algebra is the standard basis in `R^n`)::
454 sage: J = JordanSpinEJA(3)
455 sage: x = vector(QQ,[1,2,3])
456 sage: y = vector(QQ,[4,5,6])
457 sage: x.inner_product(y)
459 sage: J(x).inner_product(J(y))
462 The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
463 multiplication is the usual matrix multiplication in `S^n`,
464 so the inner product of the identity matrix with itself
467 sage: J = RealSymmetricEJA(3)
468 sage: J.one().inner_product(J.one())
471 Likewise, the inner product on `C^n` is `<X,Y> =
472 Re(trace(X*Y))`, where we must necessarily take the real
473 part because the product of Hermitian matrices may not be
476 sage: J = ComplexHermitianEJA(3)
477 sage: J.one().inner_product(J.one())
480 Ditto for the quaternions::
482 sage: J = QuaternionHermitianEJA(3)
483 sage: J.one().inner_product(J.one())
488 Ensure that we can always compute an inner product, and that
489 it gives us back a real number::
491 sage: set_random_seed()
492 sage: J = random_eja()
493 sage: x = J.random_element()
494 sage: y = J.random_element()
495 sage: x.inner_product(y) in RR
501 raise TypeError("'other' must live in the same algebra")
503 return P
.inner_product(self
, other
)
506 def operator_commutes_with(self
, other
):
508 Return whether or not this element operator-commutes
513 The definition of a Jordan algebra says that any element
514 operator-commutes with its square::
516 sage: set_random_seed()
517 sage: x = random_eja().random_element()
518 sage: x.operator_commutes_with(x^2)
523 Test Lemma 1 from Chapter III of Koecher::
525 sage: set_random_seed()
526 sage: J = random_eja()
527 sage: u = J.random_element()
528 sage: v = J.random_element()
529 sage: lhs = u.operator_commutes_with(u*v)
530 sage: rhs = v.operator_commutes_with(u^2)
535 if not other
in self
.parent():
536 raise TypeError("'other' must live in the same algebra")
538 A
= self
.operator_matrix()
539 B
= other
.operator_matrix()
545 Return my determinant, the product of my eigenvalues.
549 sage: J = JordanSpinEJA(2)
550 sage: e0,e1 = J.gens()
554 sage: J = JordanSpinEJA(3)
555 sage: e0,e1,e2 = J.gens()
556 sage: x = e0 + e1 + e2
561 cs
= self
.characteristic_polynomial().coefficients(sparse
=False)
564 return cs
[0] * (-1)**r
566 raise ValueError('charpoly had no coefficients')
571 Return the Jordan-multiplicative inverse of this element.
573 We can't use the superclass method because it relies on the
574 algebra being associative.
578 The inverse in the spin factor algebra is given in Alizadeh's
581 sage: set_random_seed()
582 sage: n = ZZ.random_element(1,10)
583 sage: J = JordanSpinEJA(n)
584 sage: x = J.random_element()
585 sage: while x.is_zero():
586 ....: x = J.random_element()
587 sage: x_vec = x.vector()
589 sage: x_bar = x_vec[1:]
590 sage: coeff = 1/(x0^2 - x_bar.inner_product(x_bar))
591 sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
592 sage: x_inverse = coeff*inv_vec
593 sage: x.inverse() == J(x_inverse)
598 The identity element is its own inverse::
600 sage: set_random_seed()
601 sage: J = random_eja()
602 sage: J.one().inverse() == J.one()
605 If an element has an inverse, it acts like one. TODO: this
606 can be a lot less ugly once ``is_invertible`` doesn't crash
607 on irregular elements::
609 sage: set_random_seed()
610 sage: J = random_eja()
611 sage: x = J.random_element()
613 ....: x.inverse()*x == J.one()
619 if self
.parent().is_associative():
620 elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
623 # TODO: we can do better once the call to is_invertible()
624 # doesn't crash on irregular elements.
625 #if not self.is_invertible():
626 # raise ValueError('element is not invertible')
628 # We do this a little different than the usual recursive
629 # call to a finite-dimensional algebra element, because we
630 # wind up with an inverse that lives in the subalgebra and
631 # we need information about the parent to convert it back.
632 V
= self
.span_of_powers()
633 assoc_subalg
= self
.subalgebra_generated_by()
634 # Mis-design warning: the basis used for span_of_powers()
635 # and subalgebra_generated_by() must be the same, and in
637 elt
= assoc_subalg(V
.coordinates(self
.vector()))
639 # This will be in the subalgebra's coordinates...
640 fda_elt
= FiniteDimensionalAlgebraElement(assoc_subalg
, elt
)
641 subalg_inverse
= fda_elt
.inverse()
643 # So we have to convert back...
644 basis
= [ self
.parent(v
) for v
in V
.basis() ]
645 pairs
= zip(subalg_inverse
.vector(), basis
)
646 return self
.parent().linear_combination(pairs
)
649 def is_invertible(self
):
651 Return whether or not this element is invertible.
653 We can't use the superclass method because it relies on
654 the algebra being associative.
658 The usual way to do this is to check if the determinant is
659 zero, but we need the characteristic polynomial for the
660 determinant. The minimal polynomial is a lot easier to get,
661 so we use Corollary 2 in Chapter V of Koecher to check
662 whether or not the paren't algebra's zero element is a root
663 of this element's minimal polynomial.
667 The identity element is always invertible::
669 sage: set_random_seed()
670 sage: J = random_eja()
671 sage: J.one().is_invertible()
674 The zero element is never invertible::
676 sage: set_random_seed()
677 sage: J = random_eja()
678 sage: J.zero().is_invertible()
682 zero
= self
.parent().zero()
683 p
= self
.minimal_polynomial()
684 return not (p(zero
) == zero
)
687 def is_nilpotent(self
):
689 Return whether or not some power of this element is zero.
691 The superclass method won't work unless we're in an
692 associative algebra, and we aren't. However, we generate
693 an assocoative subalgebra and we're nilpotent there if and
694 only if we're nilpotent here (probably).
698 The identity element is never nilpotent::
700 sage: set_random_seed()
701 sage: random_eja().one().is_nilpotent()
704 The additive identity is always nilpotent::
706 sage: set_random_seed()
707 sage: random_eja().zero().is_nilpotent()
711 # The element we're going to call "is_nilpotent()" on.
712 # Either myself, interpreted as an element of a finite-
713 # dimensional algebra, or an element of an associative
717 if self
.parent().is_associative():
718 elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
720 V
= self
.span_of_powers()
721 assoc_subalg
= self
.subalgebra_generated_by()
722 # Mis-design warning: the basis used for span_of_powers()
723 # and subalgebra_generated_by() must be the same, and in
725 elt
= assoc_subalg(V
.coordinates(self
.vector()))
727 # Recursive call, but should work since elt lives in an
728 # associative algebra.
729 return elt
.is_nilpotent()
732 def is_regular(self
):
734 Return whether or not this is a regular element.
738 The identity element always has degree one, but any element
739 linearly-independent from it is regular::
741 sage: J = JordanSpinEJA(5)
742 sage: J.one().is_regular()
744 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
745 sage: for x in J.gens():
746 ....: (J.one() + x).is_regular()
754 return self
.degree() == self
.parent().rank()
759 Compute the degree of this element the straightforward way
760 according to the definition; by appending powers to a list
761 and figuring out its dimension (that is, whether or not
762 they're linearly dependent).
766 sage: J = JordanSpinEJA(4)
767 sage: J.one().degree()
769 sage: e0,e1,e2,e3 = J.gens()
770 sage: (e0 - e1).degree()
773 In the spin factor algebra (of rank two), all elements that
774 aren't multiples of the identity are regular::
776 sage: set_random_seed()
777 sage: n = ZZ.random_element(1,10)
778 sage: J = JordanSpinEJA(n)
779 sage: x = J.random_element()
780 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
784 return self
.span_of_powers().dimension()
787 def minimal_polynomial(self
):
789 Return the minimal polynomial of this element,
790 as a function of the variable `t`.
794 We restrict ourselves to the associative subalgebra
795 generated by this element, and then return the minimal
796 polynomial of this element's operator matrix (in that
797 subalgebra). This works by Baes Proposition 2.3.16.
801 The minimal polynomial of the identity and zero elements are
804 sage: set_random_seed()
805 sage: J = random_eja()
806 sage: J.one().minimal_polynomial()
808 sage: J.zero().minimal_polynomial()
811 The degree of an element is (by one definition) the degree
812 of its minimal polynomial::
814 sage: set_random_seed()
815 sage: x = random_eja().random_element()
816 sage: x.degree() == x.minimal_polynomial().degree()
819 The minimal polynomial and the characteristic polynomial coincide
820 and are known (see Alizadeh, Example 11.11) for all elements of
821 the spin factor algebra that aren't scalar multiples of the
824 sage: set_random_seed()
825 sage: n = ZZ.random_element(2,10)
826 sage: J = JordanSpinEJA(n)
827 sage: y = J.random_element()
828 sage: while y == y.coefficient(0)*J.one():
829 ....: y = J.random_element()
830 sage: y0 = y.vector()[0]
831 sage: y_bar = y.vector()[1:]
832 sage: actual = y.minimal_polynomial()
833 sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
834 sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
835 sage: bool(actual == expected)
838 The minimal polynomial should always kill its element::
840 sage: set_random_seed()
841 sage: x = random_eja().random_element()
842 sage: p = x.minimal_polynomial()
843 sage: x.apply_univariate_polynomial(p)
847 V
= self
.span_of_powers()
848 assoc_subalg
= self
.subalgebra_generated_by()
849 # Mis-design warning: the basis used for span_of_powers()
850 # and subalgebra_generated_by() must be the same, and in
852 elt
= assoc_subalg(V
.coordinates(self
.vector()))
854 # We get back a symbolic polynomial in 'x' but want a real
856 p_of_x
= elt
.operator_matrix().minimal_polynomial()
857 return p_of_x
.change_variable_name('t')
860 def natural_representation(self
):
862 Return a more-natural representation of this element.
864 Every finite-dimensional Euclidean Jordan Algebra is a
865 direct sum of five simple algebras, four of which comprise
866 Hermitian matrices. This method returns the original
867 "natural" representation of this element as a Hermitian
868 matrix, if it has one. If not, you get the usual representation.
872 sage: J = ComplexHermitianEJA(3)
875 sage: J.one().natural_representation()
885 sage: J = QuaternionHermitianEJA(3)
888 sage: J.one().natural_representation()
889 [1 0 0 0 0 0 0 0 0 0 0 0]
890 [0 1 0 0 0 0 0 0 0 0 0 0]
891 [0 0 1 0 0 0 0 0 0 0 0 0]
892 [0 0 0 1 0 0 0 0 0 0 0 0]
893 [0 0 0 0 1 0 0 0 0 0 0 0]
894 [0 0 0 0 0 1 0 0 0 0 0 0]
895 [0 0 0 0 0 0 1 0 0 0 0 0]
896 [0 0 0 0 0 0 0 1 0 0 0 0]
897 [0 0 0 0 0 0 0 0 1 0 0 0]
898 [0 0 0 0 0 0 0 0 0 1 0 0]
899 [0 0 0 0 0 0 0 0 0 0 1 0]
900 [0 0 0 0 0 0 0 0 0 0 0 1]
903 B
= self
.parent().natural_basis()
904 W
= B
[0].matrix_space()
905 return W
.linear_combination(zip(self
.vector(), B
))
908 def operator_matrix(self
):
910 Return the matrix that represents left- (or right-)
911 multiplication by this element in the parent algebra.
913 We have to override this because the superclass method
914 returns a matrix that acts on row vectors (that is, on
919 Test the first polarization identity from my notes, Koecher Chapter
920 III, or from Baes (2.3)::
922 sage: set_random_seed()
923 sage: J = random_eja()
924 sage: x = J.random_element()
925 sage: y = J.random_element()
926 sage: Lx = x.operator_matrix()
927 sage: Ly = y.operator_matrix()
928 sage: Lxx = (x*x).operator_matrix()
929 sage: Lxy = (x*y).operator_matrix()
930 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
933 Test the second polarization identity from my notes or from
936 sage: set_random_seed()
937 sage: J = random_eja()
938 sage: x = J.random_element()
939 sage: y = J.random_element()
940 sage: z = J.random_element()
941 sage: Lx = x.operator_matrix()
942 sage: Ly = y.operator_matrix()
943 sage: Lz = z.operator_matrix()
944 sage: Lzy = (z*y).operator_matrix()
945 sage: Lxy = (x*y).operator_matrix()
946 sage: Lxz = (x*z).operator_matrix()
947 sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
950 Test the third polarization identity from my notes or from
953 sage: set_random_seed()
954 sage: J = random_eja()
955 sage: u = J.random_element()
956 sage: y = J.random_element()
957 sage: z = J.random_element()
958 sage: Lu = u.operator_matrix()
959 sage: Ly = y.operator_matrix()
960 sage: Lz = z.operator_matrix()
961 sage: Lzy = (z*y).operator_matrix()
962 sage: Luy = (u*y).operator_matrix()
963 sage: Luz = (u*z).operator_matrix()
964 sage: Luyz = (u*(y*z)).operator_matrix()
965 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
966 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
967 sage: bool(lhs == rhs)
971 fda_elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
972 return fda_elt
.matrix().transpose()
975 def quadratic_representation(self
, other
=None):
977 Return the quadratic representation of this element.
981 The explicit form in the spin factor algebra is given by
982 Alizadeh's Example 11.12::
984 sage: set_random_seed()
985 sage: n = ZZ.random_element(1,10)
986 sage: J = JordanSpinEJA(n)
987 sage: x = J.random_element()
988 sage: x_vec = x.vector()
990 sage: x_bar = x_vec[1:]
991 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
992 sage: B = 2*x0*x_bar.row()
993 sage: C = 2*x0*x_bar.column()
994 sage: D = identity_matrix(QQ, n-1)
995 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
996 sage: D = D + 2*x_bar.tensor_product(x_bar)
997 sage: Q = block_matrix(2,2,[A,B,C,D])
998 sage: Q == x.quadratic_representation()
1001 Test all of the properties from Theorem 11.2 in Alizadeh::
1003 sage: set_random_seed()
1004 sage: J = random_eja()
1005 sage: x = J.random_element()
1006 sage: y = J.random_element()
1010 sage: actual = x.quadratic_representation(y)
1011 sage: expected = ( (x+y).quadratic_representation()
1012 ....: -x.quadratic_representation()
1013 ....: -y.quadratic_representation() ) / 2
1014 sage: actual == expected
1019 sage: alpha = QQ.random_element()
1020 sage: actual = (alpha*x).quadratic_representation()
1021 sage: expected = (alpha^2)*x.quadratic_representation()
1022 sage: actual == expected
1027 sage: Qy = y.quadratic_representation()
1028 sage: actual = J(Qy*x.vector()).quadratic_representation()
1029 sage: expected = Qy*x.quadratic_representation()*Qy
1030 sage: actual == expected
1035 sage: k = ZZ.random_element(1,10)
1036 sage: actual = (x^k).quadratic_representation()
1037 sage: expected = (x.quadratic_representation())^k
1038 sage: actual == expected
1044 elif not other
in self
.parent():
1045 raise TypeError("'other' must live in the same algebra")
1047 L
= self
.operator_matrix()
1048 M
= other
.operator_matrix()
1049 return ( L
*M
+ M
*L
- (self
*other
).operator_matrix() )
1052 def span_of_powers(self
):
1054 Return the vector space spanned by successive powers of
1057 # The dimension of the subalgebra can't be greater than
1058 # the big algebra, so just put everything into a list
1059 # and let span() get rid of the excess.
1061 # We do the extra ambient_vector_space() in case we're messing
1062 # with polynomials and the direct parent is a module.
1063 V
= self
.vector().parent().ambient_vector_space()
1064 return V
.span( (self
**d
).vector() for d
in xrange(V
.dimension()) )
1067 def subalgebra_generated_by(self
):
1069 Return the associative subalgebra of the parent EJA generated
1074 sage: set_random_seed()
1075 sage: x = random_eja().random_element()
1076 sage: x.subalgebra_generated_by().is_associative()
1079 Squaring in the subalgebra should be the same thing as
1080 squaring in the superalgebra::
1082 sage: set_random_seed()
1083 sage: x = random_eja().random_element()
1084 sage: u = x.subalgebra_generated_by().random_element()
1085 sage: u.operator_matrix()*u.vector() == (u**2).vector()
1089 # First get the subspace spanned by the powers of myself...
1090 V
= self
.span_of_powers()
1091 F
= self
.base_ring()
1093 # Now figure out the entries of the right-multiplication
1094 # matrix for the successive basis elements b0, b1,... of
1097 for b_right
in V
.basis():
1098 eja_b_right
= self
.parent()(b_right
)
1100 # The first row of the right-multiplication matrix by
1101 # b1 is what we get if we apply that matrix to b1. The
1102 # second row of the right multiplication matrix by b1
1103 # is what we get when we apply that matrix to b2...
1105 # IMPORTANT: this assumes that all vectors are COLUMN
1106 # vectors, unlike our superclass (which uses row vectors).
1107 for b_left
in V
.basis():
1108 eja_b_left
= self
.parent()(b_left
)
1109 # Multiply in the original EJA, but then get the
1110 # coordinates from the subalgebra in terms of its
1112 this_row
= V
.coordinates((eja_b_left
*eja_b_right
).vector())
1113 b_right_rows
.append(this_row
)
1114 b_right_matrix
= matrix(F
, b_right_rows
)
1115 mats
.append(b_right_matrix
)
1117 # It's an algebra of polynomials in one element, and EJAs
1118 # are power-associative.
1120 # TODO: choose generator names intelligently.
1121 return FiniteDimensionalEuclideanJordanAlgebra(F
, mats
, assume_associative
=True, names
='f')
1124 def subalgebra_idempotent(self
):
1126 Find an idempotent in the associative subalgebra I generate
1127 using Proposition 2.3.5 in Baes.
1131 sage: set_random_seed()
1132 sage: J = RealCartesianProductEJA(5)
1133 sage: c = J.random_element().subalgebra_idempotent()
1136 sage: J = JordanSpinEJA(5)
1137 sage: c = J.random_element().subalgebra_idempotent()
1142 if self
.is_nilpotent():
1143 raise ValueError("this only works with non-nilpotent elements!")
1145 V
= self
.span_of_powers()
1146 J
= self
.subalgebra_generated_by()
1147 # Mis-design warning: the basis used for span_of_powers()
1148 # and subalgebra_generated_by() must be the same, and in
1150 u
= J(V
.coordinates(self
.vector()))
1152 # The image of the matrix of left-u^m-multiplication
1153 # will be minimal for some natural number s...
1155 minimal_dim
= V
.dimension()
1156 for i
in xrange(1, V
.dimension()):
1157 this_dim
= (u
**i
).operator_matrix().image().dimension()
1158 if this_dim
< minimal_dim
:
1159 minimal_dim
= this_dim
1162 # Now minimal_matrix should correspond to the smallest
1163 # non-zero subspace in Baes's (or really, Koecher's)
1166 # However, we need to restrict the matrix to work on the
1167 # subspace... or do we? Can't we just solve, knowing that
1168 # A(c) = u^(s+1) should have a solution in the big space,
1171 # Beware, solve_right() means that we're using COLUMN vectors.
1172 # Our FiniteDimensionalAlgebraElement superclass uses rows.
1174 A
= u_next
.operator_matrix()
1175 c_coordinates
= A
.solve_right(u_next
.vector())
1177 # Now c_coordinates is the idempotent we want, but it's in
1178 # the coordinate system of the subalgebra.
1180 # We need the basis for J, but as elements of the parent algebra.
1182 basis
= [self
.parent(v
) for v
in V
.basis()]
1183 return self
.parent().linear_combination(zip(c_coordinates
, basis
))
1188 Return my trace, the sum of my eigenvalues.
1192 sage: J = JordanSpinEJA(3)
1193 sage: x = sum(J.gens())
1199 sage: J = RealCartesianProductEJA(5)
1200 sage: J.one().trace()
1205 The trace of an element is a real number::
1207 sage: set_random_seed()
1208 sage: J = random_eja()
1209 sage: J.random_element().trace() in J.base_ring()
1215 p
= P
._charpoly
_coeff
(r
-1)
1216 # The _charpoly_coeff function already adds the factor of
1217 # -1 to ensure that _charpoly_coeff(r-1) is really what
1218 # appears in front of t^{r-1} in the charpoly. However,
1219 # we want the negative of THAT for the trace.
1220 return -p(*self
.vector())
1223 def trace_inner_product(self
, other
):
1225 Return the trace inner product of myself and ``other``.
1227 if not other
in self
.parent():
1228 raise TypeError("'other' must live in the same algebra")
1230 return (self
*other
).trace()
1233 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1235 Return the Euclidean Jordan Algebra corresponding to the set
1236 `R^n` under the Hadamard product.
1238 Note: this is nothing more than the Cartesian product of ``n``
1239 copies of the spin algebra. Once Cartesian product algebras
1240 are implemented, this can go.
1244 This multiplication table can be verified by hand::
1246 sage: J = RealCartesianProductEJA(3)
1247 sage: e0,e1,e2 = J.gens()
1263 def __classcall_private__(cls
, n
, field
=QQ
):
1264 # The FiniteDimensionalAlgebra constructor takes a list of
1265 # matrices, the ith representing right multiplication by the ith
1266 # basis element in the vector space. So if e_1 = (1,0,0), then
1267 # right (Hadamard) multiplication of x by e_1 picks out the first
1268 # component of x; and likewise for the ith basis element e_i.
1269 Qs
= [ matrix(field
, n
, n
, lambda k
,j
: 1*(k
== j
== i
))
1270 for i
in xrange(n
) ]
1272 fdeja
= super(RealCartesianProductEJA
, cls
)
1273 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=n
)
1275 def inner_product(self
, x
, y
):
1276 return _usual_ip(x
,y
)
1281 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1285 For now, we choose a random natural number ``n`` (greater than zero)
1286 and then give you back one of the following:
1288 * The cartesian product of the rational numbers ``n`` times; this is
1289 ``QQ^n`` with the Hadamard product.
1291 * The Jordan spin algebra on ``QQ^n``.
1293 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
1296 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
1297 in the space of ``2n``-by-``2n`` real symmetric matrices.
1299 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
1300 in the space of ``4n``-by-``4n`` real symmetric matrices.
1302 Later this might be extended to return Cartesian products of the
1308 Euclidean Jordan algebra of degree...
1312 # The max_n component lets us choose different upper bounds on the
1313 # value "n" that gets passed to the constructor. This is needed
1314 # because e.g. R^{10} is reasonable to test, while the Hermitian
1315 # 10-by-10 quaternion matrices are not.
1316 (constructor
, max_n
) = choice([(RealCartesianProductEJA
, 6),
1318 (RealSymmetricEJA
, 5),
1319 (ComplexHermitianEJA
, 4),
1320 (QuaternionHermitianEJA
, 3)])
1321 n
= ZZ
.random_element(1, max_n
)
1322 return constructor(n
, field
=QQ
)
1326 def _real_symmetric_basis(n
, field
=QQ
):
1328 Return a basis for the space of real symmetric n-by-n matrices.
1330 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1334 for j
in xrange(i
+1):
1335 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1339 # Beware, orthogonal but not normalized!
1340 Sij
= Eij
+ Eij
.transpose()
1345 def _complex_hermitian_basis(n
, field
=QQ
):
1347 Returns a basis for the space of complex Hermitian n-by-n matrices.
1351 sage: set_random_seed()
1352 sage: n = ZZ.random_element(1,5)
1353 sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
1357 F
= QuadraticField(-1, 'I')
1360 # This is like the symmetric case, but we need to be careful:
1362 # * We want conjugate-symmetry, not just symmetry.
1363 # * The diagonal will (as a result) be real.
1367 for j
in xrange(i
+1):
1368 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1370 Sij
= _embed_complex_matrix(Eij
)
1373 # Beware, orthogonal but not normalized! The second one
1374 # has a minus because it's conjugated.
1375 Sij_real
= _embed_complex_matrix(Eij
+ Eij
.transpose())
1377 Sij_imag
= _embed_complex_matrix(I
*Eij
- I
*Eij
.transpose())
1382 def _quaternion_hermitian_basis(n
, field
=QQ
):
1384 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1388 sage: set_random_seed()
1389 sage: n = ZZ.random_element(1,5)
1390 sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
1394 Q
= QuaternionAlgebra(QQ
,-1,-1)
1397 # This is like the symmetric case, but we need to be careful:
1399 # * We want conjugate-symmetry, not just symmetry.
1400 # * The diagonal will (as a result) be real.
1404 for j
in xrange(i
+1):
1405 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1407 Sij
= _embed_quaternion_matrix(Eij
)
1410 # Beware, orthogonal but not normalized! The second,
1411 # third, and fourth ones have a minus because they're
1413 Sij_real
= _embed_quaternion_matrix(Eij
+ Eij
.transpose())
1415 Sij_I
= _embed_quaternion_matrix(I
*Eij
- I
*Eij
.transpose())
1417 Sij_J
= _embed_quaternion_matrix(J
*Eij
- J
*Eij
.transpose())
1419 Sij_K
= _embed_quaternion_matrix(K
*Eij
- K
*Eij
.transpose())
1425 return vector(m
.base_ring(), m
.list())
1428 return matrix(v
.base_ring(), sqrt(v
.degree()), v
.list())
1430 def _multiplication_table_from_matrix_basis(basis
):
1432 At least three of the five simple Euclidean Jordan algebras have the
1433 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1434 multiplication on the right is matrix multiplication. Given a basis
1435 for the underlying matrix space, this function returns a
1436 multiplication table (obtained by looping through the basis
1437 elements) for an algebra of those matrices. A reordered copy
1438 of the basis is also returned to work around the fact that
1439 the ``span()`` in this function will change the order of the basis
1440 from what we think it is, to... something else.
1442 # In S^2, for example, we nominally have four coordinates even
1443 # though the space is of dimension three only. The vector space V
1444 # is supposed to hold the entire long vector, and the subspace W
1445 # of V will be spanned by the vectors that arise from symmetric
1446 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1447 field
= basis
[0].base_ring()
1448 dimension
= basis
[0].nrows()
1450 V
= VectorSpace(field
, dimension
**2)
1451 W
= V
.span( _mat2vec(s
) for s
in basis
)
1453 # Taking the span above reorders our basis (thanks, jerk!) so we
1454 # need to put our "matrix basis" in the same order as the
1455 # (reordered) vector basis.
1456 S
= tuple( _vec2mat(b
) for b
in W
.basis() )
1460 # Brute force the multiplication-by-s matrix by looping
1461 # through all elements of the basis and doing the computation
1462 # to find out what the corresponding row should be. BEWARE:
1463 # these multiplication tables won't be symmetric! It therefore
1464 # becomes REALLY IMPORTANT that the underlying algebra
1465 # constructor uses ROW vectors and not COLUMN vectors. That's
1466 # why we're computing rows here and not columns.
1469 this_row
= _mat2vec((s
*t
+ t
*s
)/2)
1470 Q_rows
.append(W
.coordinates(this_row
))
1471 Q
= matrix(field
, W
.dimension(), Q_rows
)
1477 def _embed_complex_matrix(M
):
1479 Embed the n-by-n complex matrix ``M`` into the space of real
1480 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1481 bi` to the block matrix ``[[a,b],[-b,a]]``.
1485 sage: F = QuadraticField(-1,'i')
1486 sage: x1 = F(4 - 2*i)
1487 sage: x2 = F(1 + 2*i)
1490 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1491 sage: _embed_complex_matrix(M)
1500 Embedding is a homomorphism (isomorphism, in fact)::
1502 sage: set_random_seed()
1503 sage: n = ZZ.random_element(5)
1504 sage: F = QuadraticField(-1, 'i')
1505 sage: X = random_matrix(F, n)
1506 sage: Y = random_matrix(F, n)
1507 sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
1508 sage: expected = _embed_complex_matrix(X*Y)
1509 sage: actual == expected
1515 raise ValueError("the matrix 'M' must be square")
1516 field
= M
.base_ring()
1521 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1523 # We can drop the imaginaries here.
1524 return block_matrix(field
.base_ring(), n
, blocks
)
1527 def _unembed_complex_matrix(M
):
1529 The inverse of _embed_complex_matrix().
1533 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1534 ....: [-2, 1, -4, 3],
1535 ....: [ 9, 10, 11, 12],
1536 ....: [-10, 9, -12, 11] ])
1537 sage: _unembed_complex_matrix(A)
1539 [ 10*i + 9 12*i + 11]
1543 Unembedding is the inverse of embedding::
1545 sage: set_random_seed()
1546 sage: F = QuadraticField(-1, 'i')
1547 sage: M = random_matrix(F, 3)
1548 sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
1554 raise ValueError("the matrix 'M' must be square")
1555 if not n
.mod(2).is_zero():
1556 raise ValueError("the matrix 'M' must be a complex embedding")
1558 F
= QuadraticField(-1, 'i')
1561 # Go top-left to bottom-right (reading order), converting every
1562 # 2-by-2 block we see to a single complex element.
1564 for k
in xrange(n
/2):
1565 for j
in xrange(n
/2):
1566 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1567 if submat
[0,0] != submat
[1,1]:
1568 raise ValueError('bad on-diagonal submatrix')
1569 if submat
[0,1] != -submat
[1,0]:
1570 raise ValueError('bad off-diagonal submatrix')
1571 z
= submat
[0,0] + submat
[0,1]*i
1574 return matrix(F
, n
/2, elements
)
1577 def _embed_quaternion_matrix(M
):
1579 Embed the n-by-n quaternion matrix ``M`` into the space of real
1580 matrices of size 4n-by-4n by first sending each quaternion entry
1581 `z = a + bi + cj + dk` to the block-complex matrix
1582 ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
1587 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1588 sage: i,j,k = Q.gens()
1589 sage: x = 1 + 2*i + 3*j + 4*k
1590 sage: M = matrix(Q, 1, [[x]])
1591 sage: _embed_quaternion_matrix(M)
1597 Embedding is a homomorphism (isomorphism, in fact)::
1599 sage: set_random_seed()
1600 sage: n = ZZ.random_element(5)
1601 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1602 sage: X = random_matrix(Q, n)
1603 sage: Y = random_matrix(Q, n)
1604 sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
1605 sage: expected = _embed_quaternion_matrix(X*Y)
1606 sage: actual == expected
1610 quaternions
= M
.base_ring()
1613 raise ValueError("the matrix 'M' must be square")
1615 F
= QuadraticField(-1, 'i')
1620 t
= z
.coefficient_tuple()
1625 cplx_matrix
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1626 [-c
+ d
*i
, a
- b
*i
]])
1627 blocks
.append(_embed_complex_matrix(cplx_matrix
))
1629 # We should have real entries by now, so use the realest field
1630 # we've got for the return value.
1631 return block_matrix(quaternions
.base_ring(), n
, blocks
)
1634 def _unembed_quaternion_matrix(M
):
1636 The inverse of _embed_quaternion_matrix().
1640 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1641 ....: [-2, 1, -4, 3],
1642 ....: [-3, 4, 1, -2],
1643 ....: [-4, -3, 2, 1]])
1644 sage: _unembed_quaternion_matrix(M)
1645 [1 + 2*i + 3*j + 4*k]
1649 Unembedding is the inverse of embedding::
1651 sage: set_random_seed()
1652 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1653 sage: M = random_matrix(Q, 3)
1654 sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
1660 raise ValueError("the matrix 'M' must be square")
1661 if not n
.mod(4).is_zero():
1662 raise ValueError("the matrix 'M' must be a complex embedding")
1664 Q
= QuaternionAlgebra(QQ
,-1,-1)
1667 # Go top-left to bottom-right (reading order), converting every
1668 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1671 for l
in xrange(n
/4):
1672 for m
in xrange(n
/4):
1673 submat
= _unembed_complex_matrix(M
[4*l
:4*l
+4,4*m
:4*m
+4])
1674 if submat
[0,0] != submat
[1,1].conjugate():
1675 raise ValueError('bad on-diagonal submatrix')
1676 if submat
[0,1] != -submat
[1,0].conjugate():
1677 raise ValueError('bad off-diagonal submatrix')
1678 z
= submat
[0,0].real() + submat
[0,0].imag()*i
1679 z
+= submat
[0,1].real()*j
+ submat
[0,1].imag()*k
1682 return matrix(Q
, n
/4, elements
)
1685 # The usual inner product on R^n.
1687 return x
.vector().inner_product(y
.vector())
1689 # The inner product used for the real symmetric simple EJA.
1690 # We keep it as a separate function because e.g. the complex
1691 # algebra uses the same inner product, except divided by 2.
1692 def _matrix_ip(X
,Y
):
1693 X_mat
= X
.natural_representation()
1694 Y_mat
= Y
.natural_representation()
1695 return (X_mat
*Y_mat
).trace()
1698 class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1700 The rank-n simple EJA consisting of real symmetric n-by-n
1701 matrices, the usual symmetric Jordan product, and the trace inner
1702 product. It has dimension `(n^2 + n)/2` over the reals.
1706 sage: J = RealSymmetricEJA(2)
1707 sage: e0, e1, e2 = J.gens()
1717 The degree of this algebra is `(n^2 + n) / 2`::
1719 sage: set_random_seed()
1720 sage: n = ZZ.random_element(1,5)
1721 sage: J = RealSymmetricEJA(n)
1722 sage: J.degree() == (n^2 + n)/2
1725 The Jordan multiplication is what we think it is::
1727 sage: set_random_seed()
1728 sage: n = ZZ.random_element(1,5)
1729 sage: J = RealSymmetricEJA(n)
1730 sage: x = J.random_element()
1731 sage: y = J.random_element()
1732 sage: actual = (x*y).natural_representation()
1733 sage: X = x.natural_representation()
1734 sage: Y = y.natural_representation()
1735 sage: expected = (X*Y + Y*X)/2
1736 sage: actual == expected
1738 sage: J(expected) == x*y
1743 def __classcall_private__(cls
, n
, field
=QQ
):
1744 S
= _real_symmetric_basis(n
, field
=field
)
1745 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
1747 fdeja
= super(RealSymmetricEJA
, cls
)
1748 return fdeja
.__classcall
_private
__(cls
,
1754 def inner_product(self
, x
, y
):
1755 return _matrix_ip(x
,y
)
1758 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1760 The rank-n simple EJA consisting of complex Hermitian n-by-n
1761 matrices over the real numbers, the usual symmetric Jordan product,
1762 and the real-part-of-trace inner product. It has dimension `n^2` over
1767 The degree of this algebra is `n^2`::
1769 sage: set_random_seed()
1770 sage: n = ZZ.random_element(1,5)
1771 sage: J = ComplexHermitianEJA(n)
1772 sage: J.degree() == n^2
1775 The Jordan multiplication is what we think it is::
1777 sage: set_random_seed()
1778 sage: n = ZZ.random_element(1,5)
1779 sage: J = ComplexHermitianEJA(n)
1780 sage: x = J.random_element()
1781 sage: y = J.random_element()
1782 sage: actual = (x*y).natural_representation()
1783 sage: X = x.natural_representation()
1784 sage: Y = y.natural_representation()
1785 sage: expected = (X*Y + Y*X)/2
1786 sage: actual == expected
1788 sage: J(expected) == x*y
1793 def __classcall_private__(cls
, n
, field
=QQ
):
1794 S
= _complex_hermitian_basis(n
)
1795 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
1797 fdeja
= super(ComplexHermitianEJA
, cls
)
1798 return fdeja
.__classcall
_private
__(cls
,
1804 def inner_product(self
, x
, y
):
1805 # Since a+bi on the diagonal is represented as
1810 # we'll double-count the "a" entries if we take the trace of
1812 return _matrix_ip(x
,y
)/2
1815 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1817 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1818 matrices, the usual symmetric Jordan product, and the
1819 real-part-of-trace inner product. It has dimension `2n^2 - n` over
1824 The degree of this algebra is `n^2`::
1826 sage: set_random_seed()
1827 sage: n = ZZ.random_element(1,5)
1828 sage: J = QuaternionHermitianEJA(n)
1829 sage: J.degree() == 2*(n^2) - n
1832 The Jordan multiplication is what we think it is::
1834 sage: set_random_seed()
1835 sage: n = ZZ.random_element(1,5)
1836 sage: J = QuaternionHermitianEJA(n)
1837 sage: x = J.random_element()
1838 sage: y = J.random_element()
1839 sage: actual = (x*y).natural_representation()
1840 sage: X = x.natural_representation()
1841 sage: Y = y.natural_representation()
1842 sage: expected = (X*Y + Y*X)/2
1843 sage: actual == expected
1845 sage: J(expected) == x*y
1850 def __classcall_private__(cls
, n
, field
=QQ
):
1851 S
= _quaternion_hermitian_basis(n
)
1852 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
1854 fdeja
= super(QuaternionHermitianEJA
, cls
)
1855 return fdeja
.__classcall
_private
__(cls
,
1861 def inner_product(self
, x
, y
):
1862 # Since a+bi+cj+dk on the diagonal is represented as
1864 # a + bi +cj + dk = [ a b c d]
1869 # we'll quadruple-count the "a" entries if we take the trace of
1871 return _matrix_ip(x
,y
)/4
1874 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1876 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
1877 with the usual inner product and jordan product ``x*y =
1878 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
1883 This multiplication table can be verified by hand::
1885 sage: J = JordanSpinEJA(4)
1886 sage: e0,e1,e2,e3 = J.gens()
1904 def __classcall_private__(cls
, n
, field
=QQ
):
1906 id_matrix
= identity_matrix(field
, n
)
1908 ei
= id_matrix
.column(i
)
1909 Qi
= zero_matrix(field
, n
)
1911 Qi
.set_column(0, ei
)
1912 Qi
+= diagonal_matrix(n
, [ei
[0]]*n
)
1913 # The addition of the diagonal matrix adds an extra ei[0] in the
1914 # upper-left corner of the matrix.
1915 Qi
[0,0] = Qi
[0,0] * ~
field(2)
1918 # The rank of the spin algebra is two, unless we're in a
1919 # one-dimensional ambient space (because the rank is bounded by
1920 # the ambient dimension).
1921 fdeja
= super(JordanSpinEJA
, cls
)
1922 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=min(n
,2))
1924 def inner_product(self
, x
, y
):
1925 return _usual_ip(x
,y
)