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
):
146 This implementation doesn't guarantee that the polynomial
147 denominator in the coefficients is not identically zero, so
148 theoretically it could crash. The way that this is handled
149 in e.g. Faraut and Koranyi is to use a basis that guarantees
150 the denominator is non-zero. But, doing so requires knowledge
151 of at least one regular element, and we don't even know how
152 to do that. The trade-off is that, if we use the standard basis,
153 the resulting polynomial will accept the "usual" coordinates. In
154 other words, we don't have to do a change of basis before e.g.
155 computing the trace or determinant.
159 The characteristic polynomial in the spin algebra is given in
160 Alizadeh, Example 11.11::
162 sage: J = JordanSpinEJA(3)
163 sage: p = J.characteristic_polynomial(); p
164 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
165 sage: xvec = J.one().vector()
173 # The list of coefficient polynomials a_1, a_2, ..., a_n.
174 a
= [ self
._charpoly
_coeff
(i
) for i
in range(n
) ]
176 # We go to a bit of trouble here to reorder the
177 # indeterminates, so that it's easier to evaluate the
178 # characteristic polynomial at x's coordinates and get back
179 # something in terms of t, which is what we want.
181 S
= PolynomialRing(self
.base_ring(),'t')
183 S
= PolynomialRing(S
, R
.variable_names())
186 # Note: all entries past the rth should be zero. The
187 # coefficient of the highest power (x^r) is 1, but it doesn't
188 # appear in the solution vector which contains coefficients
189 # for the other powers (to make them sum to x^r).
191 a
[r
] = 1 # corresponds to x^r
193 # When the rank is equal to the dimension, trying to
194 # assign a[r] goes out-of-bounds.
195 a
.append(1) # corresponds to x^r
197 return sum( a
[k
]*(t
**k
) for k
in range(len(a
)) )
200 def inner_product(self
, x
, y
):
202 The inner product associated with this Euclidean Jordan algebra.
204 Defaults to the trace inner product, but can be overridden by
205 subclasses if they are sure that the necessary properties are
210 The inner product must satisfy its axiom for this algebra to truly
211 be a Euclidean Jordan Algebra::
213 sage: set_random_seed()
214 sage: J = random_eja()
215 sage: x = J.random_element()
216 sage: y = J.random_element()
217 sage: z = J.random_element()
218 sage: (x*y).inner_product(z) == y.inner_product(x*z)
222 if (not x
in self
) or (not y
in self
):
223 raise TypeError("arguments must live in this algebra")
224 return x
.trace_inner_product(y
)
227 def natural_basis(self
):
229 Return a more-natural representation of this algebra's basis.
231 Every finite-dimensional Euclidean Jordan Algebra is a direct
232 sum of five simple algebras, four of which comprise Hermitian
233 matrices. This method returns the original "natural" basis
234 for our underlying vector space. (Typically, the natural basis
235 is used to construct the multiplication table in the first place.)
237 Note that this will always return a matrix. The standard basis
238 in `R^n` will be returned as `n`-by-`1` column matrices.
242 sage: J = RealSymmetricEJA(2)
245 sage: J.natural_basis()
253 sage: J = JordanSpinEJA(2)
256 sage: J.natural_basis()
263 if self
._natural
_basis
is None:
264 return tuple( b
.vector().column() for b
in self
.basis() )
266 return self
._natural
_basis
271 Return the rank of this EJA.
273 if self
._rank
is None:
274 raise ValueError("no rank specified at genesis")
279 class Element(FiniteDimensionalAlgebraElement
):
281 An element of a Euclidean Jordan algebra.
284 def __init__(self
, A
, elt
=None):
288 The identity in `S^n` is converted to the identity in the EJA::
290 sage: J = RealSymmetricEJA(3)
291 sage: I = identity_matrix(QQ,3)
292 sage: J(I) == J.one()
295 This skew-symmetric matrix can't be represented in the EJA::
297 sage: J = RealSymmetricEJA(3)
298 sage: A = matrix(QQ,3, lambda i,j: i-j)
300 Traceback (most recent call last):
302 ArithmeticError: vector is not in free module
305 # Goal: if we're given a matrix, and if it lives in our
306 # parent algebra's "natural ambient space," convert it
307 # into an algebra element.
309 # The catch is, we make a recursive call after converting
310 # the given matrix into a vector that lives in the algebra.
311 # This we need to try the parent class initializer first,
312 # to avoid recursing forever if we're given something that
313 # already fits into the algebra, but also happens to live
314 # in the parent's "natural ambient space" (this happens with
317 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, elt
)
319 natural_basis
= A
.natural_basis()
320 if elt
in natural_basis
[0].matrix_space():
321 # Thanks for nothing! Matrix spaces aren't vector
322 # spaces in Sage, so we have to figure out its
323 # natural-basis coordinates ourselves.
324 V
= VectorSpace(elt
.base_ring(), elt
.nrows()**2)
325 W
= V
.span( _mat2vec(s
) for s
in natural_basis
)
326 coords
= W
.coordinates(_mat2vec(elt
))
327 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, coords
)
329 def __pow__(self
, n
):
331 Return ``self`` raised to the power ``n``.
333 Jordan algebras are always power-associative; see for
334 example Faraut and Koranyi, Proposition II.1.2 (ii).
338 We have to override this because our superclass uses row vectors
339 instead of column vectors! We, on the other hand, assume column
344 sage: set_random_seed()
345 sage: x = random_eja().random_element()
346 sage: x.operator_matrix()*x.vector() == (x^2).vector()
349 A few examples of power-associativity::
351 sage: set_random_seed()
352 sage: x = random_eja().random_element()
353 sage: x*(x*x)*(x*x) == x^5
355 sage: (x*x)*(x*x*x) == x^5
358 We also know that powers operator-commute (Koecher, Chapter
361 sage: set_random_seed()
362 sage: x = random_eja().random_element()
363 sage: m = ZZ.random_element(0,10)
364 sage: n = ZZ.random_element(0,10)
365 sage: Lxm = (x^m).operator_matrix()
366 sage: Lxn = (x^n).operator_matrix()
367 sage: Lxm*Lxn == Lxn*Lxm
377 return A( (self
.operator_matrix()**(n
-1))*self
.vector() )
380 def apply_univariate_polynomial(self
, p
):
382 Apply the univariate polynomial ``p`` to this element.
384 A priori, SageMath won't allow us to apply a univariate
385 polynomial to an element of an EJA, because we don't know
386 that EJAs are rings (they are usually not associative). Of
387 course, we know that EJAs are power-associative, so the
388 operation is ultimately kosher. This function sidesteps
389 the CAS to get the answer we want and expect.
393 sage: R = PolynomialRing(QQ, 't')
395 sage: p = t^4 - t^3 + 5*t - 2
396 sage: J = RealCartesianProductEJA(5)
397 sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
402 We should always get back an element of the algebra::
404 sage: set_random_seed()
405 sage: p = PolynomialRing(QQ, 't').random_element()
406 sage: J = random_eja()
407 sage: x = J.random_element()
408 sage: x.apply_univariate_polynomial(p) in J
412 if len(p
.variables()) > 1:
413 raise ValueError("not a univariate polynomial")
416 # Convert the coeficcients to the parent's base ring,
417 # because a priori they might live in an (unnecessarily)
418 # larger ring for which P.sum() would fail below.
419 cs
= [ R(c
) for c
in p
.coefficients(sparse
=False) ]
420 return P
.sum( cs
[k
]*(self
**k
) for k
in range(len(cs
)) )
423 def characteristic_polynomial(self
):
425 Return the characteristic polynomial of this element.
429 The rank of `R^3` is three, and the minimal polynomial of
430 the identity element is `(t-1)` from which it follows that
431 the characteristic polynomial should be `(t-1)^3`::
433 sage: J = RealCartesianProductEJA(3)
434 sage: J.one().characteristic_polynomial()
435 t^3 - 3*t^2 + 3*t - 1
437 Likewise, the characteristic of the zero element in the
438 rank-three algebra `R^{n}` should be `t^{3}`::
440 sage: J = RealCartesianProductEJA(3)
441 sage: J.zero().characteristic_polynomial()
444 The characteristic polynomial of an element should evaluate
445 to zero on that element::
447 sage: set_random_seed()
448 sage: x = RealCartesianProductEJA(3).random_element()
449 sage: p = x.characteristic_polynomial()
450 sage: x.apply_univariate_polynomial(p)
454 p
= self
.parent().characteristic_polynomial()
455 return p(*self
.vector())
458 def inner_product(self
, other
):
460 Return the parent algebra's inner product of myself and ``other``.
464 The inner product in the Jordan spin algebra is the usual
465 inner product on `R^n` (this example only works because the
466 basis for the Jordan algebra is the standard basis in `R^n`)::
468 sage: J = JordanSpinEJA(3)
469 sage: x = vector(QQ,[1,2,3])
470 sage: y = vector(QQ,[4,5,6])
471 sage: x.inner_product(y)
473 sage: J(x).inner_product(J(y))
476 The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
477 multiplication is the usual matrix multiplication in `S^n`,
478 so the inner product of the identity matrix with itself
481 sage: J = RealSymmetricEJA(3)
482 sage: J.one().inner_product(J.one())
485 Likewise, the inner product on `C^n` is `<X,Y> =
486 Re(trace(X*Y))`, where we must necessarily take the real
487 part because the product of Hermitian matrices may not be
490 sage: J = ComplexHermitianEJA(3)
491 sage: J.one().inner_product(J.one())
494 Ditto for the quaternions::
496 sage: J = QuaternionHermitianEJA(3)
497 sage: J.one().inner_product(J.one())
502 Ensure that we can always compute an inner product, and that
503 it gives us back a real number::
505 sage: set_random_seed()
506 sage: J = random_eja()
507 sage: x = J.random_element()
508 sage: y = J.random_element()
509 sage: x.inner_product(y) in RR
515 raise TypeError("'other' must live in the same algebra")
517 return P
.inner_product(self
, other
)
520 def operator_commutes_with(self
, other
):
522 Return whether or not this element operator-commutes
527 The definition of a Jordan algebra says that any element
528 operator-commutes with its square::
530 sage: set_random_seed()
531 sage: x = random_eja().random_element()
532 sage: x.operator_commutes_with(x^2)
537 Test Lemma 1 from Chapter III of Koecher::
539 sage: set_random_seed()
540 sage: J = random_eja()
541 sage: u = J.random_element()
542 sage: v = J.random_element()
543 sage: lhs = u.operator_commutes_with(u*v)
544 sage: rhs = v.operator_commutes_with(u^2)
549 if not other
in self
.parent():
550 raise TypeError("'other' must live in the same algebra")
552 A
= self
.operator_matrix()
553 B
= other
.operator_matrix()
559 Return my determinant, the product of my eigenvalues.
563 sage: J = JordanSpinEJA(2)
564 sage: e0,e1 = J.gens()
565 sage: x = sum( J.gens() )
571 sage: J = JordanSpinEJA(3)
572 sage: e0,e1,e2 = J.gens()
573 sage: x = sum( J.gens() )
579 An element is invertible if and only if its determinant is
582 sage: set_random_seed()
583 sage: x = random_eja().random_element()
584 sage: x.is_invertible() == (x.det() != 0)
590 p
= P
._charpoly
_coeff
(0)
591 # The _charpoly_coeff function already adds the factor of
592 # -1 to ensure that _charpoly_coeff(0) is really what
593 # appears in front of t^{0} in the charpoly. However,
594 # we want (-1)^r times THAT for the determinant.
595 return ((-1)**r
)*p(*self
.vector())
600 Return the Jordan-multiplicative inverse of this element.
602 We can't use the superclass method because it relies on the
603 algebra being associative.
607 The inverse in the spin factor algebra is given in Alizadeh's
610 sage: set_random_seed()
611 sage: n = ZZ.random_element(1,10)
612 sage: J = JordanSpinEJA(n)
613 sage: x = J.random_element()
614 sage: while x.is_zero():
615 ....: x = J.random_element()
616 sage: x_vec = x.vector()
618 sage: x_bar = x_vec[1:]
619 sage: coeff = 1/(x0^2 - x_bar.inner_product(x_bar))
620 sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
621 sage: x_inverse = coeff*inv_vec
622 sage: x.inverse() == J(x_inverse)
627 The identity element is its own inverse::
629 sage: set_random_seed()
630 sage: J = random_eja()
631 sage: J.one().inverse() == J.one()
634 If an element has an inverse, it acts like one::
636 sage: set_random_seed()
637 sage: J = random_eja()
638 sage: x = J.random_element()
639 sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
643 if self
.parent().is_associative():
644 elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
647 # TODO: we can do better once the call to is_invertible()
648 # doesn't crash on irregular elements.
649 #if not self.is_invertible():
650 # raise ValueError('element is not invertible')
652 # We do this a little different than the usual recursive
653 # call to a finite-dimensional algebra element, because we
654 # wind up with an inverse that lives in the subalgebra and
655 # we need information about the parent to convert it back.
656 V
= self
.span_of_powers()
657 assoc_subalg
= self
.subalgebra_generated_by()
658 # Mis-design warning: the basis used for span_of_powers()
659 # and subalgebra_generated_by() must be the same, and in
661 elt
= assoc_subalg(V
.coordinates(self
.vector()))
663 # This will be in the subalgebra's coordinates...
664 fda_elt
= FiniteDimensionalAlgebraElement(assoc_subalg
, elt
)
665 subalg_inverse
= fda_elt
.inverse()
667 # So we have to convert back...
668 basis
= [ self
.parent(v
) for v
in V
.basis() ]
669 pairs
= zip(subalg_inverse
.vector(), basis
)
670 return self
.parent().linear_combination(pairs
)
673 def is_invertible(self
):
675 Return whether or not this element is invertible.
677 We can't use the superclass method because it relies on
678 the algebra being associative.
682 The usual way to do this is to check if the determinant is
683 zero, but we need the characteristic polynomial for the
684 determinant. The minimal polynomial is a lot easier to get,
685 so we use Corollary 2 in Chapter V of Koecher to check
686 whether or not the paren't algebra's zero element is a root
687 of this element's minimal polynomial.
691 The identity element is always invertible::
693 sage: set_random_seed()
694 sage: J = random_eja()
695 sage: J.one().is_invertible()
698 The zero element is never invertible::
700 sage: set_random_seed()
701 sage: J = random_eja()
702 sage: J.zero().is_invertible()
706 zero
= self
.parent().zero()
707 p
= self
.minimal_polynomial()
708 return not (p(zero
) == zero
)
711 def is_nilpotent(self
):
713 Return whether or not some power of this element is zero.
715 The superclass method won't work unless we're in an
716 associative algebra, and we aren't. However, we generate
717 an assocoative subalgebra and we're nilpotent there if and
718 only if we're nilpotent here (probably).
722 The identity element is never nilpotent::
724 sage: set_random_seed()
725 sage: random_eja().one().is_nilpotent()
728 The additive identity is always nilpotent::
730 sage: set_random_seed()
731 sage: random_eja().zero().is_nilpotent()
735 # The element we're going to call "is_nilpotent()" on.
736 # Either myself, interpreted as an element of a finite-
737 # dimensional algebra, or an element of an associative
741 if self
.parent().is_associative():
742 elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
744 V
= self
.span_of_powers()
745 assoc_subalg
= self
.subalgebra_generated_by()
746 # Mis-design warning: the basis used for span_of_powers()
747 # and subalgebra_generated_by() must be the same, and in
749 elt
= assoc_subalg(V
.coordinates(self
.vector()))
751 # Recursive call, but should work since elt lives in an
752 # associative algebra.
753 return elt
.is_nilpotent()
756 def is_regular(self
):
758 Return whether or not this is a regular element.
762 The identity element always has degree one, but any element
763 linearly-independent from it is regular::
765 sage: J = JordanSpinEJA(5)
766 sage: J.one().is_regular()
768 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
769 sage: for x in J.gens():
770 ....: (J.one() + x).is_regular()
778 return self
.degree() == self
.parent().rank()
783 Compute the degree of this element the straightforward way
784 according to the definition; by appending powers to a list
785 and figuring out its dimension (that is, whether or not
786 they're linearly dependent).
790 sage: J = JordanSpinEJA(4)
791 sage: J.one().degree()
793 sage: e0,e1,e2,e3 = J.gens()
794 sage: (e0 - e1).degree()
797 In the spin factor algebra (of rank two), all elements that
798 aren't multiples of the identity are regular::
800 sage: set_random_seed()
801 sage: n = ZZ.random_element(1,10)
802 sage: J = JordanSpinEJA(n)
803 sage: x = J.random_element()
804 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
808 return self
.span_of_powers().dimension()
811 def minimal_polynomial(self
):
813 Return the minimal polynomial of this element,
814 as a function of the variable `t`.
818 We restrict ourselves to the associative subalgebra
819 generated by this element, and then return the minimal
820 polynomial of this element's operator matrix (in that
821 subalgebra). This works by Baes Proposition 2.3.16.
825 The minimal polynomial of the identity and zero elements are
828 sage: set_random_seed()
829 sage: J = random_eja()
830 sage: J.one().minimal_polynomial()
832 sage: J.zero().minimal_polynomial()
835 The degree of an element is (by one definition) the degree
836 of its minimal polynomial::
838 sage: set_random_seed()
839 sage: x = random_eja().random_element()
840 sage: x.degree() == x.minimal_polynomial().degree()
843 The minimal polynomial and the characteristic polynomial coincide
844 and are known (see Alizadeh, Example 11.11) for all elements of
845 the spin factor algebra that aren't scalar multiples of the
848 sage: set_random_seed()
849 sage: n = ZZ.random_element(2,10)
850 sage: J = JordanSpinEJA(n)
851 sage: y = J.random_element()
852 sage: while y == y.coefficient(0)*J.one():
853 ....: y = J.random_element()
854 sage: y0 = y.vector()[0]
855 sage: y_bar = y.vector()[1:]
856 sage: actual = y.minimal_polynomial()
857 sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
858 sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
859 sage: bool(actual == expected)
862 The minimal polynomial should always kill its element::
864 sage: set_random_seed()
865 sage: x = random_eja().random_element()
866 sage: p = x.minimal_polynomial()
867 sage: x.apply_univariate_polynomial(p)
871 V
= self
.span_of_powers()
872 assoc_subalg
= self
.subalgebra_generated_by()
873 # Mis-design warning: the basis used for span_of_powers()
874 # and subalgebra_generated_by() must be the same, and in
876 elt
= assoc_subalg(V
.coordinates(self
.vector()))
878 # We get back a symbolic polynomial in 'x' but want a real
880 p_of_x
= elt
.operator_matrix().minimal_polynomial()
881 return p_of_x
.change_variable_name('t')
884 def natural_representation(self
):
886 Return a more-natural representation of this element.
888 Every finite-dimensional Euclidean Jordan Algebra is a
889 direct sum of five simple algebras, four of which comprise
890 Hermitian matrices. This method returns the original
891 "natural" representation of this element as a Hermitian
892 matrix, if it has one. If not, you get the usual representation.
896 sage: J = ComplexHermitianEJA(3)
899 sage: J.one().natural_representation()
909 sage: J = QuaternionHermitianEJA(3)
912 sage: J.one().natural_representation()
913 [1 0 0 0 0 0 0 0 0 0 0 0]
914 [0 1 0 0 0 0 0 0 0 0 0 0]
915 [0 0 1 0 0 0 0 0 0 0 0 0]
916 [0 0 0 1 0 0 0 0 0 0 0 0]
917 [0 0 0 0 1 0 0 0 0 0 0 0]
918 [0 0 0 0 0 1 0 0 0 0 0 0]
919 [0 0 0 0 0 0 1 0 0 0 0 0]
920 [0 0 0 0 0 0 0 1 0 0 0 0]
921 [0 0 0 0 0 0 0 0 1 0 0 0]
922 [0 0 0 0 0 0 0 0 0 1 0 0]
923 [0 0 0 0 0 0 0 0 0 0 1 0]
924 [0 0 0 0 0 0 0 0 0 0 0 1]
927 B
= self
.parent().natural_basis()
928 W
= B
[0].matrix_space()
929 return W
.linear_combination(zip(self
.vector(), B
))
932 def operator_matrix(self
):
934 Return the matrix that represents left- (or right-)
935 multiplication by this element in the parent algebra.
937 We have to override this because the superclass method
938 returns a matrix that acts on row vectors (that is, on
943 Test the first polarization identity from my notes, Koecher Chapter
944 III, or from Baes (2.3)::
946 sage: set_random_seed()
947 sage: J = random_eja()
948 sage: x = J.random_element()
949 sage: y = J.random_element()
950 sage: Lx = x.operator_matrix()
951 sage: Ly = y.operator_matrix()
952 sage: Lxx = (x*x).operator_matrix()
953 sage: Lxy = (x*y).operator_matrix()
954 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
957 Test the second polarization identity from my notes or from
960 sage: set_random_seed()
961 sage: J = random_eja()
962 sage: x = J.random_element()
963 sage: y = J.random_element()
964 sage: z = J.random_element()
965 sage: Lx = x.operator_matrix()
966 sage: Ly = y.operator_matrix()
967 sage: Lz = z.operator_matrix()
968 sage: Lzy = (z*y).operator_matrix()
969 sage: Lxy = (x*y).operator_matrix()
970 sage: Lxz = (x*z).operator_matrix()
971 sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
974 Test the third polarization identity from my notes or from
977 sage: set_random_seed()
978 sage: J = random_eja()
979 sage: u = J.random_element()
980 sage: y = J.random_element()
981 sage: z = J.random_element()
982 sage: Lu = u.operator_matrix()
983 sage: Ly = y.operator_matrix()
984 sage: Lz = z.operator_matrix()
985 sage: Lzy = (z*y).operator_matrix()
986 sage: Luy = (u*y).operator_matrix()
987 sage: Luz = (u*z).operator_matrix()
988 sage: Luyz = (u*(y*z)).operator_matrix()
989 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
990 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
991 sage: bool(lhs == rhs)
995 fda_elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
996 return fda_elt
.matrix().transpose()
999 def quadratic_representation(self
, other
=None):
1001 Return the quadratic representation of this element.
1005 The explicit form in the spin factor algebra is given by
1006 Alizadeh's Example 11.12::
1008 sage: set_random_seed()
1009 sage: n = ZZ.random_element(1,10)
1010 sage: J = JordanSpinEJA(n)
1011 sage: x = J.random_element()
1012 sage: x_vec = x.vector()
1014 sage: x_bar = x_vec[1:]
1015 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
1016 sage: B = 2*x0*x_bar.row()
1017 sage: C = 2*x0*x_bar.column()
1018 sage: D = identity_matrix(QQ, n-1)
1019 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
1020 sage: D = D + 2*x_bar.tensor_product(x_bar)
1021 sage: Q = block_matrix(2,2,[A,B,C,D])
1022 sage: Q == x.quadratic_representation()
1025 Test all of the properties from Theorem 11.2 in Alizadeh::
1027 sage: set_random_seed()
1028 sage: J = random_eja()
1029 sage: x = J.random_element()
1030 sage: y = J.random_element()
1034 sage: actual = x.quadratic_representation(y)
1035 sage: expected = ( (x+y).quadratic_representation()
1036 ....: -x.quadratic_representation()
1037 ....: -y.quadratic_representation() ) / 2
1038 sage: actual == expected
1043 sage: alpha = QQ.random_element()
1044 sage: actual = (alpha*x).quadratic_representation()
1045 sage: expected = (alpha^2)*x.quadratic_representation()
1046 sage: actual == expected
1051 sage: Qy = y.quadratic_representation()
1052 sage: actual = J(Qy*x.vector()).quadratic_representation()
1053 sage: expected = Qy*x.quadratic_representation()*Qy
1054 sage: actual == expected
1059 sage: k = ZZ.random_element(1,10)
1060 sage: actual = (x^k).quadratic_representation()
1061 sage: expected = (x.quadratic_representation())^k
1062 sage: actual == expected
1068 elif not other
in self
.parent():
1069 raise TypeError("'other' must live in the same algebra")
1071 L
= self
.operator_matrix()
1072 M
= other
.operator_matrix()
1073 return ( L
*M
+ M
*L
- (self
*other
).operator_matrix() )
1076 def span_of_powers(self
):
1078 Return the vector space spanned by successive powers of
1081 # The dimension of the subalgebra can't be greater than
1082 # the big algebra, so just put everything into a list
1083 # and let span() get rid of the excess.
1085 # We do the extra ambient_vector_space() in case we're messing
1086 # with polynomials and the direct parent is a module.
1087 V
= self
.vector().parent().ambient_vector_space()
1088 return V
.span( (self
**d
).vector() for d
in xrange(V
.dimension()) )
1091 def subalgebra_generated_by(self
):
1093 Return the associative subalgebra of the parent EJA generated
1098 sage: set_random_seed()
1099 sage: x = random_eja().random_element()
1100 sage: x.subalgebra_generated_by().is_associative()
1103 Squaring in the subalgebra should be the same thing as
1104 squaring in the superalgebra::
1106 sage: set_random_seed()
1107 sage: x = random_eja().random_element()
1108 sage: u = x.subalgebra_generated_by().random_element()
1109 sage: u.operator_matrix()*u.vector() == (u**2).vector()
1113 # First get the subspace spanned by the powers of myself...
1114 V
= self
.span_of_powers()
1115 F
= self
.base_ring()
1117 # Now figure out the entries of the right-multiplication
1118 # matrix for the successive basis elements b0, b1,... of
1121 for b_right
in V
.basis():
1122 eja_b_right
= self
.parent()(b_right
)
1124 # The first row of the right-multiplication matrix by
1125 # b1 is what we get if we apply that matrix to b1. The
1126 # second row of the right multiplication matrix by b1
1127 # is what we get when we apply that matrix to b2...
1129 # IMPORTANT: this assumes that all vectors are COLUMN
1130 # vectors, unlike our superclass (which uses row vectors).
1131 for b_left
in V
.basis():
1132 eja_b_left
= self
.parent()(b_left
)
1133 # Multiply in the original EJA, but then get the
1134 # coordinates from the subalgebra in terms of its
1136 this_row
= V
.coordinates((eja_b_left
*eja_b_right
).vector())
1137 b_right_rows
.append(this_row
)
1138 b_right_matrix
= matrix(F
, b_right_rows
)
1139 mats
.append(b_right_matrix
)
1141 # It's an algebra of polynomials in one element, and EJAs
1142 # are power-associative.
1144 # TODO: choose generator names intelligently.
1145 return FiniteDimensionalEuclideanJordanAlgebra(F
, mats
, assume_associative
=True, names
='f')
1148 def subalgebra_idempotent(self
):
1150 Find an idempotent in the associative subalgebra I generate
1151 using Proposition 2.3.5 in Baes.
1155 sage: set_random_seed()
1156 sage: J = RealCartesianProductEJA(5)
1157 sage: c = J.random_element().subalgebra_idempotent()
1160 sage: J = JordanSpinEJA(5)
1161 sage: c = J.random_element().subalgebra_idempotent()
1166 if self
.is_nilpotent():
1167 raise ValueError("this only works with non-nilpotent elements!")
1169 V
= self
.span_of_powers()
1170 J
= self
.subalgebra_generated_by()
1171 # Mis-design warning: the basis used for span_of_powers()
1172 # and subalgebra_generated_by() must be the same, and in
1174 u
= J(V
.coordinates(self
.vector()))
1176 # The image of the matrix of left-u^m-multiplication
1177 # will be minimal for some natural number s...
1179 minimal_dim
= V
.dimension()
1180 for i
in xrange(1, V
.dimension()):
1181 this_dim
= (u
**i
).operator_matrix().image().dimension()
1182 if this_dim
< minimal_dim
:
1183 minimal_dim
= this_dim
1186 # Now minimal_matrix should correspond to the smallest
1187 # non-zero subspace in Baes's (or really, Koecher's)
1190 # However, we need to restrict the matrix to work on the
1191 # subspace... or do we? Can't we just solve, knowing that
1192 # A(c) = u^(s+1) should have a solution in the big space,
1195 # Beware, solve_right() means that we're using COLUMN vectors.
1196 # Our FiniteDimensionalAlgebraElement superclass uses rows.
1198 A
= u_next
.operator_matrix()
1199 c_coordinates
= A
.solve_right(u_next
.vector())
1201 # Now c_coordinates is the idempotent we want, but it's in
1202 # the coordinate system of the subalgebra.
1204 # We need the basis for J, but as elements of the parent algebra.
1206 basis
= [self
.parent(v
) for v
in V
.basis()]
1207 return self
.parent().linear_combination(zip(c_coordinates
, basis
))
1212 Return my trace, the sum of my eigenvalues.
1216 sage: J = JordanSpinEJA(3)
1217 sage: x = sum(J.gens())
1223 sage: J = RealCartesianProductEJA(5)
1224 sage: J.one().trace()
1229 The trace of an element is a real number::
1231 sage: set_random_seed()
1232 sage: J = random_eja()
1233 sage: J.random_element().trace() in J.base_ring()
1239 p
= P
._charpoly
_coeff
(r
-1)
1240 # The _charpoly_coeff function already adds the factor of
1241 # -1 to ensure that _charpoly_coeff(r-1) is really what
1242 # appears in front of t^{r-1} in the charpoly. However,
1243 # we want the negative of THAT for the trace.
1244 return -p(*self
.vector())
1247 def trace_inner_product(self
, other
):
1249 Return the trace inner product of myself and ``other``.
1253 The trace inner product is commutative::
1255 sage: set_random_seed()
1256 sage: J = random_eja()
1257 sage: x = J.random_element(); y = J.random_element()
1258 sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1261 The trace inner product is bilinear::
1263 sage: set_random_seed()
1264 sage: J = random_eja()
1265 sage: x = J.random_element()
1266 sage: y = J.random_element()
1267 sage: z = J.random_element()
1268 sage: a = QQ.random_element();
1269 sage: actual = (a*(x+z)).trace_inner_product(y)
1270 sage: expected = ( a*x.trace_inner_product(y) +
1271 ....: a*z.trace_inner_product(y) )
1272 sage: actual == expected
1274 sage: actual = x.trace_inner_product(a*(y+z))
1275 sage: expected = ( a*x.trace_inner_product(y) +
1276 ....: a*x.trace_inner_product(z) )
1277 sage: actual == expected
1280 The trace inner product satisfies the compatibility
1281 condition in the definition of a Euclidean Jordan algebra::
1283 sage: set_random_seed()
1284 sage: J = random_eja()
1285 sage: x = J.random_element()
1286 sage: y = J.random_element()
1287 sage: z = J.random_element()
1288 sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1292 if not other
in self
.parent():
1293 raise TypeError("'other' must live in the same algebra")
1295 return (self
*other
).trace()
1298 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1300 Return the Euclidean Jordan Algebra corresponding to the set
1301 `R^n` under the Hadamard product.
1303 Note: this is nothing more than the Cartesian product of ``n``
1304 copies of the spin algebra. Once Cartesian product algebras
1305 are implemented, this can go.
1309 This multiplication table can be verified by hand::
1311 sage: J = RealCartesianProductEJA(3)
1312 sage: e0,e1,e2 = J.gens()
1328 def __classcall_private__(cls
, n
, field
=QQ
):
1329 # The FiniteDimensionalAlgebra constructor takes a list of
1330 # matrices, the ith representing right multiplication by the ith
1331 # basis element in the vector space. So if e_1 = (1,0,0), then
1332 # right (Hadamard) multiplication of x by e_1 picks out the first
1333 # component of x; and likewise for the ith basis element e_i.
1334 Qs
= [ matrix(field
, n
, n
, lambda k
,j
: 1*(k
== j
== i
))
1335 for i
in xrange(n
) ]
1337 fdeja
= super(RealCartesianProductEJA
, cls
)
1338 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=n
)
1340 def inner_product(self
, x
, y
):
1341 return _usual_ip(x
,y
)
1346 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1350 For now, we choose a random natural number ``n`` (greater than zero)
1351 and then give you back one of the following:
1353 * The cartesian product of the rational numbers ``n`` times; this is
1354 ``QQ^n`` with the Hadamard product.
1356 * The Jordan spin algebra on ``QQ^n``.
1358 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
1361 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
1362 in the space of ``2n``-by-``2n`` real symmetric matrices.
1364 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
1365 in the space of ``4n``-by-``4n`` real symmetric matrices.
1367 Later this might be extended to return Cartesian products of the
1373 Euclidean Jordan algebra of degree...
1377 # The max_n component lets us choose different upper bounds on the
1378 # value "n" that gets passed to the constructor. This is needed
1379 # because e.g. R^{10} is reasonable to test, while the Hermitian
1380 # 10-by-10 quaternion matrices are not.
1381 (constructor
, max_n
) = choice([(RealCartesianProductEJA
, 6),
1383 (RealSymmetricEJA
, 5),
1384 (ComplexHermitianEJA
, 4),
1385 (QuaternionHermitianEJA
, 3)])
1386 n
= ZZ
.random_element(1, max_n
)
1387 return constructor(n
, field
=QQ
)
1391 def _real_symmetric_basis(n
, field
=QQ
):
1393 Return a basis for the space of real symmetric n-by-n matrices.
1395 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1399 for j
in xrange(i
+1):
1400 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1404 # Beware, orthogonal but not normalized!
1405 Sij
= Eij
+ Eij
.transpose()
1410 def _complex_hermitian_basis(n
, field
=QQ
):
1412 Returns a basis for the space of complex Hermitian n-by-n matrices.
1416 sage: set_random_seed()
1417 sage: n = ZZ.random_element(1,5)
1418 sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
1422 F
= QuadraticField(-1, 'I')
1425 # This is like the symmetric case, but we need to be careful:
1427 # * We want conjugate-symmetry, not just symmetry.
1428 # * The diagonal will (as a result) be real.
1432 for j
in xrange(i
+1):
1433 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1435 Sij
= _embed_complex_matrix(Eij
)
1438 # Beware, orthogonal but not normalized! The second one
1439 # has a minus because it's conjugated.
1440 Sij_real
= _embed_complex_matrix(Eij
+ Eij
.transpose())
1442 Sij_imag
= _embed_complex_matrix(I
*Eij
- I
*Eij
.transpose())
1447 def _quaternion_hermitian_basis(n
, field
=QQ
):
1449 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1453 sage: set_random_seed()
1454 sage: n = ZZ.random_element(1,5)
1455 sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
1459 Q
= QuaternionAlgebra(QQ
,-1,-1)
1462 # This is like the symmetric case, but we need to be careful:
1464 # * We want conjugate-symmetry, not just symmetry.
1465 # * The diagonal will (as a result) be real.
1469 for j
in xrange(i
+1):
1470 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1472 Sij
= _embed_quaternion_matrix(Eij
)
1475 # Beware, orthogonal but not normalized! The second,
1476 # third, and fourth ones have a minus because they're
1478 Sij_real
= _embed_quaternion_matrix(Eij
+ Eij
.transpose())
1480 Sij_I
= _embed_quaternion_matrix(I
*Eij
- I
*Eij
.transpose())
1482 Sij_J
= _embed_quaternion_matrix(J
*Eij
- J
*Eij
.transpose())
1484 Sij_K
= _embed_quaternion_matrix(K
*Eij
- K
*Eij
.transpose())
1490 return vector(m
.base_ring(), m
.list())
1493 return matrix(v
.base_ring(), sqrt(v
.degree()), v
.list())
1495 def _multiplication_table_from_matrix_basis(basis
):
1497 At least three of the five simple Euclidean Jordan algebras have the
1498 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1499 multiplication on the right is matrix multiplication. Given a basis
1500 for the underlying matrix space, this function returns a
1501 multiplication table (obtained by looping through the basis
1502 elements) for an algebra of those matrices. A reordered copy
1503 of the basis is also returned to work around the fact that
1504 the ``span()`` in this function will change the order of the basis
1505 from what we think it is, to... something else.
1507 # In S^2, for example, we nominally have four coordinates even
1508 # though the space is of dimension three only. The vector space V
1509 # is supposed to hold the entire long vector, and the subspace W
1510 # of V will be spanned by the vectors that arise from symmetric
1511 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1512 field
= basis
[0].base_ring()
1513 dimension
= basis
[0].nrows()
1515 V
= VectorSpace(field
, dimension
**2)
1516 W
= V
.span( _mat2vec(s
) for s
in basis
)
1518 # Taking the span above reorders our basis (thanks, jerk!) so we
1519 # need to put our "matrix basis" in the same order as the
1520 # (reordered) vector basis.
1521 S
= tuple( _vec2mat(b
) for b
in W
.basis() )
1525 # Brute force the multiplication-by-s matrix by looping
1526 # through all elements of the basis and doing the computation
1527 # to find out what the corresponding row should be. BEWARE:
1528 # these multiplication tables won't be symmetric! It therefore
1529 # becomes REALLY IMPORTANT that the underlying algebra
1530 # constructor uses ROW vectors and not COLUMN vectors. That's
1531 # why we're computing rows here and not columns.
1534 this_row
= _mat2vec((s
*t
+ t
*s
)/2)
1535 Q_rows
.append(W
.coordinates(this_row
))
1536 Q
= matrix(field
, W
.dimension(), Q_rows
)
1542 def _embed_complex_matrix(M
):
1544 Embed the n-by-n complex matrix ``M`` into the space of real
1545 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1546 bi` to the block matrix ``[[a,b],[-b,a]]``.
1550 sage: F = QuadraticField(-1,'i')
1551 sage: x1 = F(4 - 2*i)
1552 sage: x2 = F(1 + 2*i)
1555 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1556 sage: _embed_complex_matrix(M)
1565 Embedding is a homomorphism (isomorphism, in fact)::
1567 sage: set_random_seed()
1568 sage: n = ZZ.random_element(5)
1569 sage: F = QuadraticField(-1, 'i')
1570 sage: X = random_matrix(F, n)
1571 sage: Y = random_matrix(F, n)
1572 sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
1573 sage: expected = _embed_complex_matrix(X*Y)
1574 sage: actual == expected
1580 raise ValueError("the matrix 'M' must be square")
1581 field
= M
.base_ring()
1586 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1588 # We can drop the imaginaries here.
1589 return block_matrix(field
.base_ring(), n
, blocks
)
1592 def _unembed_complex_matrix(M
):
1594 The inverse of _embed_complex_matrix().
1598 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1599 ....: [-2, 1, -4, 3],
1600 ....: [ 9, 10, 11, 12],
1601 ....: [-10, 9, -12, 11] ])
1602 sage: _unembed_complex_matrix(A)
1604 [ 10*i + 9 12*i + 11]
1608 Unembedding is the inverse of embedding::
1610 sage: set_random_seed()
1611 sage: F = QuadraticField(-1, 'i')
1612 sage: M = random_matrix(F, 3)
1613 sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
1619 raise ValueError("the matrix 'M' must be square")
1620 if not n
.mod(2).is_zero():
1621 raise ValueError("the matrix 'M' must be a complex embedding")
1623 F
= QuadraticField(-1, 'i')
1626 # Go top-left to bottom-right (reading order), converting every
1627 # 2-by-2 block we see to a single complex element.
1629 for k
in xrange(n
/2):
1630 for j
in xrange(n
/2):
1631 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1632 if submat
[0,0] != submat
[1,1]:
1633 raise ValueError('bad on-diagonal submatrix')
1634 if submat
[0,1] != -submat
[1,0]:
1635 raise ValueError('bad off-diagonal submatrix')
1636 z
= submat
[0,0] + submat
[0,1]*i
1639 return matrix(F
, n
/2, elements
)
1642 def _embed_quaternion_matrix(M
):
1644 Embed the n-by-n quaternion matrix ``M`` into the space of real
1645 matrices of size 4n-by-4n by first sending each quaternion entry
1646 `z = a + bi + cj + dk` to the block-complex matrix
1647 ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
1652 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1653 sage: i,j,k = Q.gens()
1654 sage: x = 1 + 2*i + 3*j + 4*k
1655 sage: M = matrix(Q, 1, [[x]])
1656 sage: _embed_quaternion_matrix(M)
1662 Embedding is a homomorphism (isomorphism, in fact)::
1664 sage: set_random_seed()
1665 sage: n = ZZ.random_element(5)
1666 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1667 sage: X = random_matrix(Q, n)
1668 sage: Y = random_matrix(Q, n)
1669 sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
1670 sage: expected = _embed_quaternion_matrix(X*Y)
1671 sage: actual == expected
1675 quaternions
= M
.base_ring()
1678 raise ValueError("the matrix 'M' must be square")
1680 F
= QuadraticField(-1, 'i')
1685 t
= z
.coefficient_tuple()
1690 cplx_matrix
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1691 [-c
+ d
*i
, a
- b
*i
]])
1692 blocks
.append(_embed_complex_matrix(cplx_matrix
))
1694 # We should have real entries by now, so use the realest field
1695 # we've got for the return value.
1696 return block_matrix(quaternions
.base_ring(), n
, blocks
)
1699 def _unembed_quaternion_matrix(M
):
1701 The inverse of _embed_quaternion_matrix().
1705 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1706 ....: [-2, 1, -4, 3],
1707 ....: [-3, 4, 1, -2],
1708 ....: [-4, -3, 2, 1]])
1709 sage: _unembed_quaternion_matrix(M)
1710 [1 + 2*i + 3*j + 4*k]
1714 Unembedding is the inverse of embedding::
1716 sage: set_random_seed()
1717 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1718 sage: M = random_matrix(Q, 3)
1719 sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
1725 raise ValueError("the matrix 'M' must be square")
1726 if not n
.mod(4).is_zero():
1727 raise ValueError("the matrix 'M' must be a complex embedding")
1729 Q
= QuaternionAlgebra(QQ
,-1,-1)
1732 # Go top-left to bottom-right (reading order), converting every
1733 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1736 for l
in xrange(n
/4):
1737 for m
in xrange(n
/4):
1738 submat
= _unembed_complex_matrix(M
[4*l
:4*l
+4,4*m
:4*m
+4])
1739 if submat
[0,0] != submat
[1,1].conjugate():
1740 raise ValueError('bad on-diagonal submatrix')
1741 if submat
[0,1] != -submat
[1,0].conjugate():
1742 raise ValueError('bad off-diagonal submatrix')
1743 z
= submat
[0,0].real() + submat
[0,0].imag()*i
1744 z
+= submat
[0,1].real()*j
+ submat
[0,1].imag()*k
1747 return matrix(Q
, n
/4, elements
)
1750 # The usual inner product on R^n.
1752 return x
.vector().inner_product(y
.vector())
1754 # The inner product used for the real symmetric simple EJA.
1755 # We keep it as a separate function because e.g. the complex
1756 # algebra uses the same inner product, except divided by 2.
1757 def _matrix_ip(X
,Y
):
1758 X_mat
= X
.natural_representation()
1759 Y_mat
= Y
.natural_representation()
1760 return (X_mat
*Y_mat
).trace()
1763 class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1765 The rank-n simple EJA consisting of real symmetric n-by-n
1766 matrices, the usual symmetric Jordan product, and the trace inner
1767 product. It has dimension `(n^2 + n)/2` over the reals.
1771 sage: J = RealSymmetricEJA(2)
1772 sage: e0, e1, e2 = J.gens()
1782 The degree of this algebra is `(n^2 + n) / 2`::
1784 sage: set_random_seed()
1785 sage: n = ZZ.random_element(1,5)
1786 sage: J = RealSymmetricEJA(n)
1787 sage: J.degree() == (n^2 + n)/2
1790 The Jordan multiplication is what we think it is::
1792 sage: set_random_seed()
1793 sage: n = ZZ.random_element(1,5)
1794 sage: J = RealSymmetricEJA(n)
1795 sage: x = J.random_element()
1796 sage: y = J.random_element()
1797 sage: actual = (x*y).natural_representation()
1798 sage: X = x.natural_representation()
1799 sage: Y = y.natural_representation()
1800 sage: expected = (X*Y + Y*X)/2
1801 sage: actual == expected
1803 sage: J(expected) == x*y
1808 def __classcall_private__(cls
, n
, field
=QQ
):
1809 S
= _real_symmetric_basis(n
, field
=field
)
1810 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
1812 fdeja
= super(RealSymmetricEJA
, cls
)
1813 return fdeja
.__classcall
_private
__(cls
,
1819 def inner_product(self
, x
, y
):
1820 return _matrix_ip(x
,y
)
1823 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1825 The rank-n simple EJA consisting of complex Hermitian n-by-n
1826 matrices over the real numbers, the usual symmetric Jordan product,
1827 and the real-part-of-trace inner product. It has dimension `n^2` over
1832 The degree of this algebra is `n^2`::
1834 sage: set_random_seed()
1835 sage: n = ZZ.random_element(1,5)
1836 sage: J = ComplexHermitianEJA(n)
1837 sage: J.degree() == n^2
1840 The Jordan multiplication is what we think it is::
1842 sage: set_random_seed()
1843 sage: n = ZZ.random_element(1,5)
1844 sage: J = ComplexHermitianEJA(n)
1845 sage: x = J.random_element()
1846 sage: y = J.random_element()
1847 sage: actual = (x*y).natural_representation()
1848 sage: X = x.natural_representation()
1849 sage: Y = y.natural_representation()
1850 sage: expected = (X*Y + Y*X)/2
1851 sage: actual == expected
1853 sage: J(expected) == x*y
1858 def __classcall_private__(cls
, n
, field
=QQ
):
1859 S
= _complex_hermitian_basis(n
)
1860 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
1862 fdeja
= super(ComplexHermitianEJA
, cls
)
1863 return fdeja
.__classcall
_private
__(cls
,
1869 def inner_product(self
, x
, y
):
1870 # Since a+bi on the diagonal is represented as
1875 # we'll double-count the "a" entries if we take the trace of
1877 return _matrix_ip(x
,y
)/2
1880 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1882 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1883 matrices, the usual symmetric Jordan product, and the
1884 real-part-of-trace inner product. It has dimension `2n^2 - n` over
1889 The degree of this algebra is `n^2`::
1891 sage: set_random_seed()
1892 sage: n = ZZ.random_element(1,5)
1893 sage: J = QuaternionHermitianEJA(n)
1894 sage: J.degree() == 2*(n^2) - n
1897 The Jordan multiplication is what we think it is::
1899 sage: set_random_seed()
1900 sage: n = ZZ.random_element(1,5)
1901 sage: J = QuaternionHermitianEJA(n)
1902 sage: x = J.random_element()
1903 sage: y = J.random_element()
1904 sage: actual = (x*y).natural_representation()
1905 sage: X = x.natural_representation()
1906 sage: Y = y.natural_representation()
1907 sage: expected = (X*Y + Y*X)/2
1908 sage: actual == expected
1910 sage: J(expected) == x*y
1915 def __classcall_private__(cls
, n
, field
=QQ
):
1916 S
= _quaternion_hermitian_basis(n
)
1917 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
1919 fdeja
= super(QuaternionHermitianEJA
, cls
)
1920 return fdeja
.__classcall
_private
__(cls
,
1926 def inner_product(self
, x
, y
):
1927 # Since a+bi+cj+dk on the diagonal is represented as
1929 # a + bi +cj + dk = [ a b c d]
1934 # we'll quadruple-count the "a" entries if we take the trace of
1936 return _matrix_ip(x
,y
)/4
1939 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1941 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
1942 with the usual inner product and jordan product ``x*y =
1943 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
1948 This multiplication table can be verified by hand::
1950 sage: J = JordanSpinEJA(4)
1951 sage: e0,e1,e2,e3 = J.gens()
1969 def __classcall_private__(cls
, n
, field
=QQ
):
1971 id_matrix
= identity_matrix(field
, n
)
1973 ei
= id_matrix
.column(i
)
1974 Qi
= zero_matrix(field
, n
)
1976 Qi
.set_column(0, ei
)
1977 Qi
+= diagonal_matrix(n
, [ei
[0]]*n
)
1978 # The addition of the diagonal matrix adds an extra ei[0] in the
1979 # upper-left corner of the matrix.
1980 Qi
[0,0] = Qi
[0,0] * ~
field(2)
1983 # The rank of the spin algebra is two, unless we're in a
1984 # one-dimensional ambient space (because the rank is bounded by
1985 # the ambient dimension).
1986 fdeja
= super(JordanSpinEJA
, cls
)
1987 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=min(n
,2))
1989 def inner_product(self
, x
, y
):
1990 return _usual_ip(x
,y
)