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
.algebras
.quatalg
.quaternion_algebra
import QuaternionAlgebra
9 from sage
.categories
.magmatic_algebras
import MagmaticAlgebras
10 from sage
.combinat
.free_module
import CombinatorialFreeModule
11 from sage
.matrix
.constructor
import matrix
12 from sage
.matrix
.matrix_space
import MatrixSpace
13 from sage
.misc
.cachefunc
import cached_method
14 from sage
.misc
.prandom
import choice
15 from sage
.misc
.table
import table
16 from sage
.modules
.free_module
import FreeModule
, VectorSpace
17 from sage
.rings
.integer_ring
import ZZ
18 from sage
.rings
.number_field
.number_field
import NumberField
, QuadraticField
19 from sage
.rings
.polynomial
.polynomial_ring_constructor
import PolynomialRing
20 from sage
.rings
.rational_field
import QQ
21 from sage
.rings
.real_lazy
import CLF
, RLF
22 from sage
.structure
.element
import is_Matrix
24 from mjo
.eja
.eja_element
import FiniteDimensionalEuclideanJordanAlgebraElement
25 from mjo
.eja
.eja_utils
import _mat2vec
27 class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule
):
28 # This is an ugly hack needed to prevent the category framework
29 # from implementing a coercion from our base ring (e.g. the
30 # rationals) into the algebra. First of all -- such a coercion is
31 # nonsense to begin with. But more importantly, it tries to do so
32 # in the category of rings, and since our algebras aren't
33 # associative they generally won't be rings.
34 _no_generic_basering_coercion
= True
46 sage: from mjo.eja.eja_algebra import random_eja
50 By definition, Jordan multiplication commutes::
52 sage: set_random_seed()
53 sage: J = random_eja()
54 sage: x = J.random_element()
55 sage: y = J.random_element()
61 self
._natural
_basis
= natural_basis
63 # TODO: HACK for the charpoly.. needs redesign badly.
64 self
._basis
_normalizers
= None
67 category
= MagmaticAlgebras(field
).FiniteDimensional()
68 category
= category
.WithBasis().Unital()
70 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
72 range(len(mult_table
)),
75 self
.print_options(bracket
='')
77 # The multiplication table we're given is necessarily in terms
78 # of vectors, because we don't have an algebra yet for
79 # anything to be an element of. However, it's faster in the
80 # long run to have the multiplication table be in terms of
81 # algebra elements. We do this after calling the superclass
82 # constructor so that from_vector() knows what to do.
83 self
._multiplication
_table
= [ map(lambda x
: self
.from_vector(x
), ls
)
84 for ls
in mult_table
]
87 def _element_constructor_(self
, elt
):
89 Construct an element of this algebra from its natural
92 This gets called only after the parent element _call_ method
93 fails to find a coercion for the argument.
97 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
98 ....: RealCartesianProductEJA,
99 ....: RealSymmetricEJA)
103 The identity in `S^n` is converted to the identity in the EJA::
105 sage: J = RealSymmetricEJA(3)
106 sage: I = matrix.identity(QQ,3)
107 sage: J(I) == J.one()
110 This skew-symmetric matrix can't be represented in the EJA::
112 sage: J = RealSymmetricEJA(3)
113 sage: A = matrix(QQ,3, lambda i,j: i-j)
115 Traceback (most recent call last):
117 ArithmeticError: vector is not in free module
121 Ensure that we can convert any element of the two non-matrix
122 simple algebras (whose natural representations are their usual
123 vector representations) back and forth faithfully::
125 sage: set_random_seed()
126 sage: J = RealCartesianProductEJA.random_instance()
127 sage: x = J.random_element()
128 sage: J(x.to_vector().column()) == x
130 sage: J = JordanSpinEJA.random_instance()
131 sage: x = J.random_element()
132 sage: J(x.to_vector().column()) == x
137 # The superclass implementation of random_element()
138 # needs to be able to coerce "0" into the algebra.
141 natural_basis
= self
.natural_basis()
142 basis_space
= natural_basis
[0].matrix_space()
143 if elt
not in basis_space
:
144 raise ValueError("not a naturally-represented algebra element")
146 # Thanks for nothing! Matrix spaces aren't vector spaces in
147 # Sage, so we have to figure out its natural-basis coordinates
148 # ourselves. We use the basis space's ring instead of the
149 # element's ring because the basis space might be an algebraic
150 # closure whereas the base ring of the 3-by-3 identity matrix
151 # could be QQ instead of QQbar.
152 V
= VectorSpace(basis_space
.base_ring(), elt
.nrows()*elt
.ncols())
153 W
= V
.span_of_basis( _mat2vec(s
) for s
in natural_basis
)
154 coords
= W
.coordinate_vector(_mat2vec(elt
))
155 return self
.from_vector(coords
)
159 def _max_test_case_size():
161 Return an integer "size" that is an upper bound on the size of
162 this algebra when it is used in a random test
163 case. Unfortunately, the term "size" is quite vague -- when
164 dealing with `R^n` under either the Hadamard or Jordan spin
165 product, the "size" refers to the dimension `n`. When dealing
166 with a matrix algebra (real symmetric or complex/quaternion
167 Hermitian), it refers to the size of the matrix, which is
168 far less than the dimension of the underlying vector space.
170 We default to five in this class, which is safe in `R^n`. The
171 matrix algebra subclasses (or any class where the "size" is
172 interpreted to be far less than the dimension) should override
173 with a smaller number.
180 Return a string representation of ``self``.
184 sage: from mjo.eja.eja_algebra import JordanSpinEJA
188 Ensure that it says what we think it says::
190 sage: JordanSpinEJA(2, field=QQ)
191 Euclidean Jordan algebra of dimension 2 over Rational Field
192 sage: JordanSpinEJA(3, field=RDF)
193 Euclidean Jordan algebra of dimension 3 over Real Double Field
196 fmt
= "Euclidean Jordan algebra of dimension {} over {}"
197 return fmt
.format(self
.dimension(), self
.base_ring())
199 def product_on_basis(self
, i
, j
):
200 return self
._multiplication
_table
[i
][j
]
202 def _a_regular_element(self
):
204 Guess a regular element. Needed to compute the basis for our
205 characteristic polynomial coefficients.
209 sage: from mjo.eja.eja_algebra import random_eja
213 Ensure that this hacky method succeeds for every algebra that we
214 know how to construct::
216 sage: set_random_seed()
217 sage: J = random_eja()
218 sage: J._a_regular_element().is_regular()
223 z
= self
.sum( (i
+1)*gs
[i
] for i
in range(len(gs
)) )
224 if not z
.is_regular():
225 raise ValueError("don't know a regular element")
230 def _charpoly_basis_space(self
):
232 Return the vector space spanned by the basis used in our
233 characteristic polynomial coefficients. This is used not only to
234 compute those coefficients, but also any time we need to
235 evaluate the coefficients (like when we compute the trace or
238 z
= self
._a
_regular
_element
()
239 # Don't use the parent vector space directly here in case this
240 # happens to be a subalgebra. In that case, we would be e.g.
241 # two-dimensional but span_of_basis() would expect three
243 V
= VectorSpace(self
.base_ring(), self
.vector_space().dimension())
244 basis
= [ (z
**k
).to_vector() for k
in range(self
.rank()) ]
245 V1
= V
.span_of_basis( basis
)
246 b
= (V1
.basis() + V1
.complement().basis())
247 return V
.span_of_basis(b
)
252 def _charpoly_coeff(self
, i
):
254 Return the coefficient polynomial "a_{i}" of this algebra's
255 general characteristic polynomial.
257 Having this be a separate cached method lets us compute and
258 store the trace/determinant (a_{r-1} and a_{0} respectively)
259 separate from the entire characteristic polynomial.
261 if self
._basis
_normalizers
is not None:
262 # Must be a matrix class?
263 # WARNING/TODO: this whole mess is mis-designed.
264 n
= self
.natural_basis_space().nrows()
265 field
= self
.base_ring().base_ring() # yeeeeaaaahhh
266 J
= self
.__class
__(n
, field
, False)
267 (_
,x
,_
,_
) = J
._charpoly
_matrix
_system
()
268 p
= J
._charpoly
_coeff
(i
)
269 # p might be missing some vars, have to substitute "optionally"
270 pairs
= zip(x
.base_ring().gens(), self
._basis
_normalizers
)
271 substitutions
= { v: v*c for (v,c) in pairs }
272 return p
.subs(substitutions
)
274 (A_of_x
, x
, xr
, detA
) = self
._charpoly
_matrix
_system
()
275 R
= A_of_x
.base_ring()
277 # Guaranteed by theory
280 # Danger: the in-place modification is done for performance
281 # reasons (reconstructing a matrix with huge polynomial
282 # entries is slow), but I don't know how cached_method works,
283 # so it's highly possible that we're modifying some global
284 # list variable by reference, here. In other words, you
285 # probably shouldn't call this method twice on the same
286 # algebra, at the same time, in two threads
287 Ai_orig
= A_of_x
.column(i
)
288 A_of_x
.set_column(i
,xr
)
289 numerator
= A_of_x
.det()
290 A_of_x
.set_column(i
,Ai_orig
)
292 # We're relying on the theory here to ensure that each a_i is
293 # indeed back in R, and the added negative signs are to make
294 # the whole charpoly expression sum to zero.
295 return R(-numerator
/detA
)
299 def _charpoly_matrix_system(self
):
301 Compute the matrix whose entries A_ij are polynomials in
302 X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
303 corresponding to `x^r` and the determinent of the matrix A =
304 [A_ij]. In other words, all of the fixed (cachable) data needed
305 to compute the coefficients of the characteristic polynomial.
310 # Turn my vector space into a module so that "vectors" can
311 # have multivatiate polynomial entries.
312 names
= tuple('X' + str(i
) for i
in range(1,n
+1))
313 R
= PolynomialRing(self
.base_ring(), names
)
315 # Using change_ring() on the parent's vector space doesn't work
316 # here because, in a subalgebra, that vector space has a basis
317 # and change_ring() tries to bring the basis along with it. And
318 # that doesn't work unless the new ring is a PID, which it usually
322 # Now let x = (X1,X2,...,Xn) be the vector whose entries are
326 # And figure out the "left multiplication by x" matrix in
329 monomial_matrices
= [ self
.monomial(i
).operator().matrix()
330 for i
in range(n
) ] # don't recompute these!
332 ek
= self
.monomial(k
).to_vector()
334 sum( x
[i
]*(monomial_matrices
[i
]*ek
)
335 for i
in range(n
) ) )
336 Lx
= matrix
.column(R
, lmbx_cols
)
338 # Now we can compute powers of x "symbolically"
339 x_powers
= [self
.one().to_vector(), x
]
340 for d
in range(2, r
+1):
341 x_powers
.append( Lx
*(x_powers
[-1]) )
343 idmat
= matrix
.identity(R
, n
)
345 W
= self
._charpoly
_basis
_space
()
346 W
= W
.change_ring(R
.fraction_field())
348 # Starting with the standard coordinates x = (X1,X2,...,Xn)
349 # and then converting the entries to W-coordinates allows us
350 # to pass in the standard coordinates to the charpoly and get
351 # back the right answer. Specifically, with x = (X1,X2,...,Xn),
354 # W.coordinates(x^2) eval'd at (standard z-coords)
358 # W-coords of (standard coords of x^2 eval'd at std-coords of z)
360 # We want the middle equivalent thing in our matrix, but use
361 # the first equivalent thing instead so that we can pass in
362 # standard coordinates.
363 x_powers
= [ W
.coordinate_vector(xp
) for xp
in x_powers
]
364 l2
= [idmat
.column(k
-1) for k
in range(r
+1, n
+1)]
365 A_of_x
= matrix
.column(R
, n
, (x_powers
[:r
] + l2
))
366 return (A_of_x
, x
, x_powers
[r
], A_of_x
.det())
370 def characteristic_polynomial(self
):
372 Return a characteristic polynomial that works for all elements
375 The resulting polynomial has `n+1` variables, where `n` is the
376 dimension of this algebra. The first `n` variables correspond to
377 the coordinates of an algebra element: when evaluated at the
378 coordinates of an algebra element with respect to a certain
379 basis, the result is a univariate polynomial (in the one
380 remaining variable ``t``), namely the characteristic polynomial
385 sage: from mjo.eja.eja_algebra import JordanSpinEJA
389 The characteristic polynomial in the spin algebra is given in
390 Alizadeh, Example 11.11::
392 sage: J = JordanSpinEJA(3)
393 sage: p = J.characteristic_polynomial(); p
394 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
395 sage: xvec = J.one().to_vector()
403 # The list of coefficient polynomials a_1, a_2, ..., a_n.
404 a
= [ self
._charpoly
_coeff
(i
) for i
in range(n
) ]
406 # We go to a bit of trouble here to reorder the
407 # indeterminates, so that it's easier to evaluate the
408 # characteristic polynomial at x's coordinates and get back
409 # something in terms of t, which is what we want.
411 S
= PolynomialRing(self
.base_ring(),'t')
413 S
= PolynomialRing(S
, R
.variable_names())
416 # Note: all entries past the rth should be zero. The
417 # coefficient of the highest power (x^r) is 1, but it doesn't
418 # appear in the solution vector which contains coefficients
419 # for the other powers (to make them sum to x^r).
421 a
[r
] = 1 # corresponds to x^r
423 # When the rank is equal to the dimension, trying to
424 # assign a[r] goes out-of-bounds.
425 a
.append(1) # corresponds to x^r
427 return sum( a
[k
]*(t
**k
) for k
in range(len(a
)) )
430 def inner_product(self
, x
, y
):
432 The inner product associated with this Euclidean Jordan algebra.
434 Defaults to the trace inner product, but can be overridden by
435 subclasses if they are sure that the necessary properties are
440 sage: from mjo.eja.eja_algebra import random_eja
444 The inner product must satisfy its axiom for this algebra to truly
445 be a Euclidean Jordan Algebra::
447 sage: set_random_seed()
448 sage: J = random_eja()
449 sage: x = J.random_element()
450 sage: y = J.random_element()
451 sage: z = J.random_element()
452 sage: (x*y).inner_product(z) == y.inner_product(x*z)
456 X
= x
.natural_representation()
457 Y
= y
.natural_representation()
458 return self
.natural_inner_product(X
,Y
)
461 def is_trivial(self
):
463 Return whether or not this algebra is trivial.
465 A trivial algebra contains only the zero element.
469 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
473 sage: J = ComplexHermitianEJA(3)
476 sage: A = J.zero().subalgebra_generated_by()
481 return self
.dimension() == 0
484 def multiplication_table(self
):
486 Return a visual representation of this algebra's multiplication
487 table (on basis elements).
491 sage: from mjo.eja.eja_algebra import JordanSpinEJA
495 sage: J = JordanSpinEJA(4)
496 sage: J.multiplication_table()
497 +----++----+----+----+----+
498 | * || e0 | e1 | e2 | e3 |
499 +====++====+====+====+====+
500 | e0 || e0 | e1 | e2 | e3 |
501 +----++----+----+----+----+
502 | e1 || e1 | e0 | 0 | 0 |
503 +----++----+----+----+----+
504 | e2 || e2 | 0 | e0 | 0 |
505 +----++----+----+----+----+
506 | e3 || e3 | 0 | 0 | e0 |
507 +----++----+----+----+----+
510 M
= list(self
._multiplication
_table
) # copy
511 for i
in range(len(M
)):
512 # M had better be "square"
513 M
[i
] = [self
.monomial(i
)] + M
[i
]
514 M
= [["*"] + list(self
.gens())] + M
515 return table(M
, header_row
=True, header_column
=True, frame
=True)
518 def natural_basis(self
):
520 Return a more-natural representation of this algebra's basis.
522 Every finite-dimensional Euclidean Jordan Algebra is a direct
523 sum of five simple algebras, four of which comprise Hermitian
524 matrices. This method returns the original "natural" basis
525 for our underlying vector space. (Typically, the natural basis
526 is used to construct the multiplication table in the first place.)
528 Note that this will always return a matrix. The standard basis
529 in `R^n` will be returned as `n`-by-`1` column matrices.
533 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
534 ....: RealSymmetricEJA)
538 sage: J = RealSymmetricEJA(2)
540 Finite family {0: e0, 1: e1, 2: e2}
541 sage: J.natural_basis()
543 [1 0] [ 0 1/2*sqrt2] [0 0]
544 [0 0], [1/2*sqrt2 0], [0 1]
549 sage: J = JordanSpinEJA(2)
551 Finite family {0: e0, 1: e1}
552 sage: J.natural_basis()
559 if self
._natural
_basis
is None:
560 M
= self
.natural_basis_space()
561 return tuple( M(b
.to_vector()) for b
in self
.basis() )
563 return self
._natural
_basis
566 def natural_basis_space(self
):
568 Return the matrix space in which this algebra's natural basis
571 if self
._natural
_basis
is None or len(self
._natural
_basis
) == 0:
572 return MatrixSpace(self
.base_ring(), self
.dimension(), 1)
574 return self
._natural
_basis
[0].matrix_space()
578 def natural_inner_product(X
,Y
):
580 Compute the inner product of two naturally-represented elements.
582 For example in the real symmetric matrix EJA, this will compute
583 the trace inner-product of two n-by-n symmetric matrices. The
584 default should work for the real cartesian product EJA, the
585 Jordan spin EJA, and the real symmetric matrices. The others
586 will have to be overridden.
588 return (X
.conjugate_transpose()*Y
).trace()
594 Return the unit element of this algebra.
598 sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
603 sage: J = RealCartesianProductEJA(5)
605 e0 + e1 + e2 + e3 + e4
609 The identity element acts like the identity::
611 sage: set_random_seed()
612 sage: J = random_eja()
613 sage: x = J.random_element()
614 sage: J.one()*x == x and x*J.one() == x
617 The matrix of the unit element's operator is the identity::
619 sage: set_random_seed()
620 sage: J = random_eja()
621 sage: actual = J.one().operator().matrix()
622 sage: expected = matrix.identity(J.base_ring(), J.dimension())
623 sage: actual == expected
627 # We can brute-force compute the matrices of the operators
628 # that correspond to the basis elements of this algebra.
629 # If some linear combination of those basis elements is the
630 # algebra identity, then the same linear combination of
631 # their matrices has to be the identity matrix.
633 # Of course, matrices aren't vectors in sage, so we have to
634 # appeal to the "long vectors" isometry.
635 oper_vecs
= [ _mat2vec(g
.operator().matrix()) for g
in self
.gens() ]
637 # Now we use basis linear algebra to find the coefficients,
638 # of the matrices-as-vectors-linear-combination, which should
639 # work for the original algebra basis too.
640 A
= matrix
.column(self
.base_ring(), oper_vecs
)
642 # We used the isometry on the left-hand side already, but we
643 # still need to do it for the right-hand side. Recall that we
644 # wanted something that summed to the identity matrix.
645 b
= _mat2vec( matrix
.identity(self
.base_ring(), self
.dimension()) )
647 # Now if there's an identity element in the algebra, this should work.
648 coeffs
= A
.solve_right(b
)
649 return self
.linear_combination(zip(self
.gens(), coeffs
))
652 def random_element(self
):
653 # Temporary workaround for https://trac.sagemath.org/ticket/28327
654 if self
.is_trivial():
657 s
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
658 return s
.random_element()
662 def random_instance(cls
, field
=QQ
, **kwargs
):
664 Return a random instance of this type of algebra.
666 In subclasses for algebras that we know how to construct, this
667 is a shortcut for constructing test cases and examples.
669 if cls
is FiniteDimensionalEuclideanJordanAlgebra
:
670 # Red flag! But in theory we could do this I guess. The
671 # only finite-dimensional exceptional EJA is the
672 # octononions. So, we could just create an EJA from an
673 # associative matrix algebra (generated by a subset of
674 # elements) with the symmetric product. Or, we could punt
675 # to random_eja() here, override it in our subclasses, and
676 # not worry about it.
677 raise NotImplementedError
679 n
= ZZ
.random_element(1, cls
._max
_test
_case
_size
())
680 return cls(n
, field
, **kwargs
)
685 Return the rank of this EJA.
689 The author knows of no algorithm to compute the rank of an EJA
690 where only the multiplication table is known. In lieu of one, we
691 require the rank to be specified when the algebra is created,
692 and simply pass along that number here.
696 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
697 ....: RealSymmetricEJA,
698 ....: ComplexHermitianEJA,
699 ....: QuaternionHermitianEJA,
704 The rank of the Jordan spin algebra is always two::
706 sage: JordanSpinEJA(2).rank()
708 sage: JordanSpinEJA(3).rank()
710 sage: JordanSpinEJA(4).rank()
713 The rank of the `n`-by-`n` Hermitian real, complex, or
714 quaternion matrices is `n`::
716 sage: RealSymmetricEJA(2).rank()
718 sage: ComplexHermitianEJA(2).rank()
720 sage: QuaternionHermitianEJA(2).rank()
722 sage: RealSymmetricEJA(5).rank()
724 sage: ComplexHermitianEJA(5).rank()
726 sage: QuaternionHermitianEJA(5).rank()
731 Ensure that every EJA that we know how to construct has a
732 positive integer rank::
734 sage: set_random_seed()
735 sage: r = random_eja().rank()
736 sage: r in ZZ and r > 0
743 def vector_space(self
):
745 Return the vector space that underlies this algebra.
749 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
753 sage: J = RealSymmetricEJA(2)
754 sage: J.vector_space()
755 Vector space of dimension 3 over...
758 return self
.zero().to_vector().parent().ambient_vector_space()
761 Element
= FiniteDimensionalEuclideanJordanAlgebraElement
764 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra
):
766 Return the Euclidean Jordan Algebra corresponding to the set
767 `R^n` under the Hadamard product.
769 Note: this is nothing more than the Cartesian product of ``n``
770 copies of the spin algebra. Once Cartesian product algebras
771 are implemented, this can go.
775 sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
779 This multiplication table can be verified by hand::
781 sage: J = RealCartesianProductEJA(3)
782 sage: e0,e1,e2 = J.gens()
798 We can change the generator prefix::
800 sage: RealCartesianProductEJA(3, prefix='r').gens()
803 Our inner product satisfies the Jordan axiom::
805 sage: set_random_seed()
806 sage: J = RealCartesianProductEJA.random_instance()
807 sage: x = J.random_element()
808 sage: y = J.random_element()
809 sage: z = J.random_element()
810 sage: (x*y).inner_product(z) == y.inner_product(x*z)
814 def __init__(self
, n
, field
=QQ
, **kwargs
):
815 V
= VectorSpace(field
, n
)
816 mult_table
= [ [ V
.gen(i
)*(i
== j
) for j
in range(n
) ]
819 fdeja
= super(RealCartesianProductEJA
, self
)
820 return fdeja
.__init
__(field
, mult_table
, rank
=n
, **kwargs
)
822 def inner_product(self
, x
, y
):
824 Faster to reimplement than to use natural representations.
828 sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
832 Ensure that this is the usual inner product for the algebras
835 sage: set_random_seed()
836 sage: J = RealCartesianProductEJA.random_instance()
837 sage: x = J.random_element()
838 sage: y = J.random_element()
839 sage: X = x.natural_representation()
840 sage: Y = y.natural_representation()
841 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
845 return x
.to_vector().inner_product(y
.to_vector())
850 Return a "random" finite-dimensional Euclidean Jordan Algebra.
854 For now, we choose a random natural number ``n`` (greater than zero)
855 and then give you back one of the following:
857 * The cartesian product of the rational numbers ``n`` times; this is
858 ``QQ^n`` with the Hadamard product.
860 * The Jordan spin algebra on ``QQ^n``.
862 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
865 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
866 in the space of ``2n``-by-``2n`` real symmetric matrices.
868 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
869 in the space of ``4n``-by-``4n`` real symmetric matrices.
871 Later this might be extended to return Cartesian products of the
876 sage: from mjo.eja.eja_algebra import random_eja
881 Euclidean Jordan algebra of dimension...
884 classname
= choice([RealCartesianProductEJA
,
888 QuaternionHermitianEJA
])
889 return classname
.random_instance()
896 class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra
):
898 def _max_test_case_size():
899 # Play it safe, since this will be squared and the underlying
900 # field can have dimension 4 (quaternions) too.
904 def _denormalized_basis(cls
, n
, field
):
905 raise NotImplementedError
907 def __init__(self
, n
, field
=QQ
, normalize_basis
=True, **kwargs
):
908 S
= self
._denormalized
_basis
(n
, field
)
910 if n
> 1 and normalize_basis
:
911 # We'll need sqrt(2) to normalize the basis, and this
912 # winds up in the multiplication table, so the whole
913 # algebra needs to be over the field extension.
914 R
= PolynomialRing(field
, 'z')
917 if p
.is_irreducible():
918 field
= NumberField(p
, 'sqrt2', embedding
=RLF(2).sqrt())
919 S
= [ s
.change_ring(field
) for s
in S
]
920 self
._basis
_normalizers
= tuple(
921 ~
(self
.natural_inner_product(s
,s
).sqrt()) for s
in S
)
922 S
= tuple( s
*c
for (s
,c
) in zip(S
,self
._basis
_normalizers
) )
924 Qs
= self
.multiplication_table_from_matrix_basis(S
)
926 fdeja
= super(MatrixEuclideanJordanAlgebra
, self
)
927 return fdeja
.__init
__(field
,
935 def multiplication_table_from_matrix_basis(basis
):
937 At least three of the five simple Euclidean Jordan algebras have the
938 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
939 multiplication on the right is matrix multiplication. Given a basis
940 for the underlying matrix space, this function returns a
941 multiplication table (obtained by looping through the basis
942 elements) for an algebra of those matrices.
944 # In S^2, for example, we nominally have four coordinates even
945 # though the space is of dimension three only. The vector space V
946 # is supposed to hold the entire long vector, and the subspace W
947 # of V will be spanned by the vectors that arise from symmetric
948 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
949 field
= basis
[0].base_ring()
950 dimension
= basis
[0].nrows()
952 V
= VectorSpace(field
, dimension
**2)
953 W
= V
.span_of_basis( _mat2vec(s
) for s
in basis
)
955 mult_table
= [[W
.zero() for j
in range(n
)] for i
in range(n
)]
958 mat_entry
= (basis
[i
]*basis
[j
] + basis
[j
]*basis
[i
])/2
959 mult_table
[i
][j
] = W
.coordinate_vector(_mat2vec(mat_entry
))
967 Embed the matrix ``M`` into a space of real matrices.
969 The matrix ``M`` can have entries in any field at the moment:
970 the real numbers, complex numbers, or quaternions. And although
971 they are not a field, we can probably support octonions at some
972 point, too. This function returns a real matrix that "acts like"
973 the original with respect to matrix multiplication; i.e.
975 real_embed(M*N) = real_embed(M)*real_embed(N)
978 raise NotImplementedError
984 The inverse of :meth:`real_embed`.
986 raise NotImplementedError
990 def natural_inner_product(cls
,X
,Y
):
991 Xu
= cls
.real_unembed(X
)
992 Yu
= cls
.real_unembed(Y
)
998 # Otherwise, try the thing that works for complex numbers; and
999 # if that doesn't work, the thing that works for quaternions.
1001 return tr
.vector()[0] # real part, imag part is index 1
1002 except AttributeError:
1003 # A quaternions doesn't have a vector() method, but does
1004 # have coefficient_tuple() method that returns the
1005 # coefficients of 1, i, j, and k -- in that order.
1006 return tr
.coefficient_tuple()[0]
1009 class RealSymmetricEJA(MatrixEuclideanJordanAlgebra
):
1011 The rank-n simple EJA consisting of real symmetric n-by-n
1012 matrices, the usual symmetric Jordan product, and the trace inner
1013 product. It has dimension `(n^2 + n)/2` over the reals.
1017 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1021 sage: J = RealSymmetricEJA(2)
1022 sage: e0, e1, e2 = J.gens()
1032 The dimension of this algebra is `(n^2 + n) / 2`::
1034 sage: set_random_seed()
1035 sage: n_max = RealSymmetricEJA._max_test_case_size()
1036 sage: n = ZZ.random_element(1, n_max)
1037 sage: J = RealSymmetricEJA(n)
1038 sage: J.dimension() == (n^2 + n)/2
1041 The Jordan multiplication is what we think it is::
1043 sage: set_random_seed()
1044 sage: J = RealSymmetricEJA.random_instance()
1045 sage: x = J.random_element()
1046 sage: y = J.random_element()
1047 sage: actual = (x*y).natural_representation()
1048 sage: X = x.natural_representation()
1049 sage: Y = y.natural_representation()
1050 sage: expected = (X*Y + Y*X)/2
1051 sage: actual == expected
1053 sage: J(expected) == x*y
1056 We can change the generator prefix::
1058 sage: RealSymmetricEJA(3, prefix='q').gens()
1059 (q0, q1, q2, q3, q4, q5)
1061 Our inner product satisfies the Jordan axiom::
1063 sage: set_random_seed()
1064 sage: J = RealSymmetricEJA.random_instance()
1065 sage: x = J.random_element()
1066 sage: y = J.random_element()
1067 sage: z = J.random_element()
1068 sage: (x*y).inner_product(z) == y.inner_product(x*z)
1071 Our natural basis is normalized with respect to the natural inner
1072 product unless we specify otherwise::
1074 sage: set_random_seed()
1075 sage: J = RealSymmetricEJA.random_instance()
1076 sage: all( b.norm() == 1 for b in J.gens() )
1079 Since our natural basis is normalized with respect to the natural
1080 inner product, and since we know that this algebra is an EJA, any
1081 left-multiplication operator's matrix will be symmetric because
1082 natural->EJA basis representation is an isometry and within the EJA
1083 the operator is self-adjoint by the Jordan axiom::
1085 sage: set_random_seed()
1086 sage: x = RealSymmetricEJA.random_instance().random_element()
1087 sage: x.operator().matrix().is_symmetric()
1092 def _denormalized_basis(cls
, n
, field
):
1094 Return a basis for the space of real symmetric n-by-n matrices.
1098 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1102 sage: set_random_seed()
1103 sage: n = ZZ.random_element(1,5)
1104 sage: B = RealSymmetricEJA._denormalized_basis(n,QQ)
1105 sage: all( M.is_symmetric() for M in B)
1109 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1113 for j
in xrange(i
+1):
1114 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1118 Sij
= Eij
+ Eij
.transpose()
1124 def _max_test_case_size():
1125 return 5 # Dimension 10
1130 Embed the matrix ``M`` into a space of real matrices.
1132 The matrix ``M`` can have entries in any field at the moment:
1133 the real numbers, complex numbers, or quaternions. And although
1134 they are not a field, we can probably support octonions at some
1135 point, too. This function returns a real matrix that "acts like"
1136 the original with respect to matrix multiplication; i.e.
1138 real_embed(M*N) = real_embed(M)*real_embed(N)
1145 def real_unembed(M
):
1147 The inverse of :meth:`real_embed`.
1153 class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1157 Embed the n-by-n complex matrix ``M`` into the space of real
1158 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1159 bi` to the block matrix ``[[a,b],[-b,a]]``.
1163 sage: from mjo.eja.eja_algebra import \
1164 ....: ComplexMatrixEuclideanJordanAlgebra
1168 sage: F = QuadraticField(-1, 'i')
1169 sage: x1 = F(4 - 2*i)
1170 sage: x2 = F(1 + 2*i)
1173 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1174 sage: ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1183 Embedding is a homomorphism (isomorphism, in fact)::
1185 sage: set_random_seed()
1186 sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size()
1187 sage: n = ZZ.random_element(n_max)
1188 sage: F = QuadraticField(-1, 'i')
1189 sage: X = random_matrix(F, n)
1190 sage: Y = random_matrix(F, n)
1191 sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X)
1192 sage: Ye = ComplexMatrixEuclideanJordanAlgebra.real_embed(Y)
1193 sage: XYe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1200 raise ValueError("the matrix 'M' must be square")
1201 field
= M
.base_ring()
1204 a
= z
.vector()[0] # real part, I guess
1205 b
= z
.vector()[1] # imag part, I guess
1206 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1208 # We can drop the imaginaries here.
1209 return matrix
.block(field
.base_ring(), n
, blocks
)
1213 def real_unembed(M
):
1215 The inverse of _embed_complex_matrix().
1219 sage: from mjo.eja.eja_algebra import \
1220 ....: ComplexMatrixEuclideanJordanAlgebra
1224 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1225 ....: [-2, 1, -4, 3],
1226 ....: [ 9, 10, 11, 12],
1227 ....: [-10, 9, -12, 11] ])
1228 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A)
1230 [ 10*i + 9 12*i + 11]
1234 Unembedding is the inverse of embedding::
1236 sage: set_random_seed()
1237 sage: F = QuadraticField(-1, 'i')
1238 sage: M = random_matrix(F, 3)
1239 sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1240 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1246 raise ValueError("the matrix 'M' must be square")
1247 if not n
.mod(2).is_zero():
1248 raise ValueError("the matrix 'M' must be a complex embedding")
1250 field
= M
.base_ring() # This should already have sqrt2
1251 R
= PolynomialRing(field
, 'z')
1253 F
= NumberField(z
**2 + 1,'i', embedding
=CLF(-1).sqrt())
1256 # Go top-left to bottom-right (reading order), converting every
1257 # 2-by-2 block we see to a single complex element.
1259 for k
in xrange(n
/2):
1260 for j
in xrange(n
/2):
1261 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1262 if submat
[0,0] != submat
[1,1]:
1263 raise ValueError('bad on-diagonal submatrix')
1264 if submat
[0,1] != -submat
[1,0]:
1265 raise ValueError('bad off-diagonal submatrix')
1266 z
= submat
[0,0] + submat
[0,1]*i
1269 return matrix(F
, n
/2, elements
)
1272 class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra
):
1274 The rank-n simple EJA consisting of complex Hermitian n-by-n
1275 matrices over the real numbers, the usual symmetric Jordan product,
1276 and the real-part-of-trace inner product. It has dimension `n^2` over
1281 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1285 The dimension of this algebra is `n^2`::
1287 sage: set_random_seed()
1288 sage: n_max = ComplexHermitianEJA._max_test_case_size()
1289 sage: n = ZZ.random_element(1, n_max)
1290 sage: J = ComplexHermitianEJA(n)
1291 sage: J.dimension() == n^2
1294 The Jordan multiplication is what we think it is::
1296 sage: set_random_seed()
1297 sage: J = ComplexHermitianEJA.random_instance()
1298 sage: x = J.random_element()
1299 sage: y = J.random_element()
1300 sage: actual = (x*y).natural_representation()
1301 sage: X = x.natural_representation()
1302 sage: Y = y.natural_representation()
1303 sage: expected = (X*Y + Y*X)/2
1304 sage: actual == expected
1306 sage: J(expected) == x*y
1309 We can change the generator prefix::
1311 sage: ComplexHermitianEJA(2, prefix='z').gens()
1314 Our inner product satisfies the Jordan axiom::
1316 sage: set_random_seed()
1317 sage: J = ComplexHermitianEJA.random_instance()
1318 sage: x = J.random_element()
1319 sage: y = J.random_element()
1320 sage: z = J.random_element()
1321 sage: (x*y).inner_product(z) == y.inner_product(x*z)
1324 Our natural basis is normalized with respect to the natural inner
1325 product unless we specify otherwise::
1327 sage: set_random_seed()
1328 sage: J = ComplexHermitianEJA.random_instance()
1329 sage: all( b.norm() == 1 for b in J.gens() )
1332 Since our natural basis is normalized with respect to the natural
1333 inner product, and since we know that this algebra is an EJA, any
1334 left-multiplication operator's matrix will be symmetric because
1335 natural->EJA basis representation is an isometry and within the EJA
1336 the operator is self-adjoint by the Jordan axiom::
1338 sage: set_random_seed()
1339 sage: x = ComplexHermitianEJA.random_instance().random_element()
1340 sage: x.operator().matrix().is_symmetric()
1345 def _denormalized_basis(cls
, n
, field
):
1347 Returns a basis for the space of complex Hermitian n-by-n matrices.
1349 Why do we embed these? Basically, because all of numerical linear
1350 algebra assumes that you're working with vectors consisting of `n`
1351 entries from a field and scalars from the same field. There's no way
1352 to tell SageMath that (for example) the vectors contain complex
1353 numbers, while the scalar field is real.
1357 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1361 sage: set_random_seed()
1362 sage: n = ZZ.random_element(1,5)
1363 sage: field = QuadraticField(2, 'sqrt2')
1364 sage: B = ComplexHermitianEJA._denormalized_basis(n, field)
1365 sage: all( M.is_symmetric() for M in B)
1369 R
= PolynomialRing(field
, 'z')
1371 F
= NumberField(z
**2 + 1, 'I', embedding
=CLF(-1).sqrt())
1374 # This is like the symmetric case, but we need to be careful:
1376 # * We want conjugate-symmetry, not just symmetry.
1377 # * The diagonal will (as a result) be real.
1381 for j
in xrange(i
+1):
1382 Eij
= matrix(F
, n
, lambda k
,l
: k
==i
and l
==j
)
1384 Sij
= cls
.real_embed(Eij
)
1387 # The second one has a minus because it's conjugated.
1388 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1390 Sij_imag
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1393 # Since we embedded these, we can drop back to the "field" that we
1394 # started with instead of the complex extension "F".
1395 return tuple( s
.change_ring(field
) for s
in S
)
1399 class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1403 Embed the n-by-n quaternion matrix ``M`` into the space of real
1404 matrices of size 4n-by-4n by first sending each quaternion entry `z
1405 = a + bi + cj + dk` to the block-complex matrix ``[[a + bi,
1406 c+di],[-c + di, a-bi]]`, and then embedding those into a real
1411 sage: from mjo.eja.eja_algebra import \
1412 ....: QuaternionMatrixEuclideanJordanAlgebra
1416 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1417 sage: i,j,k = Q.gens()
1418 sage: x = 1 + 2*i + 3*j + 4*k
1419 sage: M = matrix(Q, 1, [[x]])
1420 sage: QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1426 Embedding is a homomorphism (isomorphism, in fact)::
1428 sage: set_random_seed()
1429 sage: n_max = QuaternionMatrixEuclideanJordanAlgebra._max_test_case_size()
1430 sage: n = ZZ.random_element(n_max)
1431 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1432 sage: X = random_matrix(Q, n)
1433 sage: Y = random_matrix(Q, n)
1434 sage: Xe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X)
1435 sage: Ye = QuaternionMatrixEuclideanJordanAlgebra.real_embed(Y)
1436 sage: XYe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1441 quaternions
= M
.base_ring()
1444 raise ValueError("the matrix 'M' must be square")
1446 F
= QuadraticField(-1, 'i')
1451 t
= z
.coefficient_tuple()
1456 cplxM
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1457 [-c
+ d
*i
, a
- b
*i
]])
1458 realM
= ComplexMatrixEuclideanJordanAlgebra
.real_embed(cplxM
)
1459 blocks
.append(realM
)
1461 # We should have real entries by now, so use the realest field
1462 # we've got for the return value.
1463 return matrix
.block(quaternions
.base_ring(), n
, blocks
)
1468 def real_unembed(M
):
1470 The inverse of _embed_quaternion_matrix().
1474 sage: from mjo.eja.eja_algebra import \
1475 ....: QuaternionMatrixEuclideanJordanAlgebra
1479 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1480 ....: [-2, 1, -4, 3],
1481 ....: [-3, 4, 1, -2],
1482 ....: [-4, -3, 2, 1]])
1483 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(M)
1484 [1 + 2*i + 3*j + 4*k]
1488 Unembedding is the inverse of embedding::
1490 sage: set_random_seed()
1491 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1492 sage: M = random_matrix(Q, 3)
1493 sage: Me = QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1494 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1500 raise ValueError("the matrix 'M' must be square")
1501 if not n
.mod(4).is_zero():
1502 raise ValueError("the matrix 'M' must be a complex embedding")
1504 # Use the base ring of the matrix to ensure that its entries can be
1505 # multiplied by elements of the quaternion algebra.
1506 field
= M
.base_ring()
1507 Q
= QuaternionAlgebra(field
,-1,-1)
1510 # Go top-left to bottom-right (reading order), converting every
1511 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1514 for l
in xrange(n
/4):
1515 for m
in xrange(n
/4):
1516 submat
= ComplexMatrixEuclideanJordanAlgebra
.real_unembed(
1517 M
[4*l
:4*l
+4,4*m
:4*m
+4] )
1518 if submat
[0,0] != submat
[1,1].conjugate():
1519 raise ValueError('bad on-diagonal submatrix')
1520 if submat
[0,1] != -submat
[1,0].conjugate():
1521 raise ValueError('bad off-diagonal submatrix')
1522 z
= submat
[0,0].vector()[0] # real part
1523 z
+= submat
[0,0].vector()[1]*i
# imag part
1524 z
+= submat
[0,1].vector()[0]*j
# real part
1525 z
+= submat
[0,1].vector()[1]*k
# imag part
1528 return matrix(Q
, n
/4, elements
)
1532 class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra
):
1534 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1535 matrices, the usual symmetric Jordan product, and the
1536 real-part-of-trace inner product. It has dimension `2n^2 - n` over
1541 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1545 The dimension of this algebra is `2*n^2 - n`::
1547 sage: set_random_seed()
1548 sage: n_max = QuaternionHermitianEJA._max_test_case_size()
1549 sage: n = ZZ.random_element(1, n_max)
1550 sage: J = QuaternionHermitianEJA(n)
1551 sage: J.dimension() == 2*(n^2) - n
1554 The Jordan multiplication is what we think it is::
1556 sage: set_random_seed()
1557 sage: J = QuaternionHermitianEJA.random_instance()
1558 sage: x = J.random_element()
1559 sage: y = J.random_element()
1560 sage: actual = (x*y).natural_representation()
1561 sage: X = x.natural_representation()
1562 sage: Y = y.natural_representation()
1563 sage: expected = (X*Y + Y*X)/2
1564 sage: actual == expected
1566 sage: J(expected) == x*y
1569 We can change the generator prefix::
1571 sage: QuaternionHermitianEJA(2, prefix='a').gens()
1572 (a0, a1, a2, a3, a4, a5)
1574 Our inner product satisfies the Jordan axiom::
1576 sage: set_random_seed()
1577 sage: J = QuaternionHermitianEJA.random_instance()
1578 sage: x = J.random_element()
1579 sage: y = J.random_element()
1580 sage: z = J.random_element()
1581 sage: (x*y).inner_product(z) == y.inner_product(x*z)
1584 Our natural basis is normalized with respect to the natural inner
1585 product unless we specify otherwise::
1587 sage: set_random_seed()
1588 sage: J = QuaternionHermitianEJA.random_instance()
1589 sage: all( b.norm() == 1 for b in J.gens() )
1592 Since our natural basis is normalized with respect to the natural
1593 inner product, and since we know that this algebra is an EJA, any
1594 left-multiplication operator's matrix will be symmetric because
1595 natural->EJA basis representation is an isometry and within the EJA
1596 the operator is self-adjoint by the Jordan axiom::
1598 sage: set_random_seed()
1599 sage: x = QuaternionHermitianEJA.random_instance().random_element()
1600 sage: x.operator().matrix().is_symmetric()
1605 def _denormalized_basis(cls
, n
, field
):
1607 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1609 Why do we embed these? Basically, because all of numerical
1610 linear algebra assumes that you're working with vectors consisting
1611 of `n` entries from a field and scalars from the same field. There's
1612 no way to tell SageMath that (for example) the vectors contain
1613 complex numbers, while the scalar field is real.
1617 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1621 sage: set_random_seed()
1622 sage: n = ZZ.random_element(1,5)
1623 sage: B = QuaternionHermitianEJA._denormalized_basis(n,QQ)
1624 sage: all( M.is_symmetric() for M in B )
1628 Q
= QuaternionAlgebra(QQ
,-1,-1)
1631 # This is like the symmetric case, but we need to be careful:
1633 # * We want conjugate-symmetry, not just symmetry.
1634 # * The diagonal will (as a result) be real.
1638 for j
in xrange(i
+1):
1639 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1641 Sij
= cls
.real_embed(Eij
)
1644 # The second, third, and fourth ones have a minus
1645 # because they're conjugated.
1646 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1648 Sij_I
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1650 Sij_J
= cls
.real_embed(J
*Eij
- J
*Eij
.transpose())
1652 Sij_K
= cls
.real_embed(K
*Eij
- K
*Eij
.transpose())
1658 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1660 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
1661 with the usual inner product and jordan product ``x*y =
1662 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
1667 sage: from mjo.eja.eja_algebra import JordanSpinEJA
1671 This multiplication table can be verified by hand::
1673 sage: J = JordanSpinEJA(4)
1674 sage: e0,e1,e2,e3 = J.gens()
1690 We can change the generator prefix::
1692 sage: JordanSpinEJA(2, prefix='B').gens()
1695 Our inner product satisfies the Jordan axiom::
1697 sage: set_random_seed()
1698 sage: J = JordanSpinEJA.random_instance()
1699 sage: x = J.random_element()
1700 sage: y = J.random_element()
1701 sage: z = J.random_element()
1702 sage: (x*y).inner_product(z) == y.inner_product(x*z)
1706 def __init__(self
, n
, field
=QQ
, **kwargs
):
1707 V
= VectorSpace(field
, n
)
1708 mult_table
= [[V
.zero() for j
in range(n
)] for i
in range(n
)]
1718 z0
= x
.inner_product(y
)
1719 zbar
= y0
*xbar
+ x0
*ybar
1720 z
= V([z0
] + zbar
.list())
1721 mult_table
[i
][j
] = z
1723 # The rank of the spin algebra is two, unless we're in a
1724 # one-dimensional ambient space (because the rank is bounded by
1725 # the ambient dimension).
1726 fdeja
= super(JordanSpinEJA
, self
)
1727 return fdeja
.__init
__(field
, mult_table
, rank
=min(n
,2), **kwargs
)
1729 def inner_product(self
, x
, y
):
1731 Faster to reimplement than to use natural representations.
1735 sage: from mjo.eja.eja_algebra import JordanSpinEJA
1739 Ensure that this is the usual inner product for the algebras
1742 sage: set_random_seed()
1743 sage: J = JordanSpinEJA.random_instance()
1744 sage: x = J.random_element()
1745 sage: y = J.random_element()
1746 sage: X = x.natural_representation()
1747 sage: Y = y.natural_representation()
1748 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
1752 return x
.to_vector().inner_product(y
.to_vector())