what can be supported in a general Jordan Algebra.
"""
-from itertools import izip, repeat
+from itertools import repeat
from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra
from sage.categories.magmatic_algebras import MagmaticAlgebras
from sage.matrix.constructor import matrix
from sage.matrix.matrix_space import MatrixSpace
from sage.misc.cachefunc import cached_method
+from sage.misc.lazy_import import lazy_import
from sage.misc.prandom import choice
from sage.misc.table import table
from sage.modules.free_module import FreeModule, VectorSpace
-from sage.rings.integer_ring import ZZ
-from sage.rings.number_field.number_field import NumberField, QuadraticField
-from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
-from sage.rings.rational_field import QQ
-from sage.rings.real_lazy import CLF, RLF
-from sage.structure.element import is_Matrix
-
+from sage.rings.all import (ZZ, QQ, AA, QQbar, RR, RLF, CLF,
+ PolynomialRing,
+ QuadraticField)
from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
+lazy_import('mjo.eja.eja_subalgebra',
+ 'FiniteDimensionalEuclideanJordanSubalgebra')
from mjo.eja.eja_utils import _mat2vec
class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
- # This is an ugly hack needed to prevent the category framework
- # from implementing a coercion from our base ring (e.g. the
- # rationals) into the algebra. First of all -- such a coercion is
- # nonsense to begin with. But more importantly, it tries to do so
- # in the category of rings, and since our algebras aren't
- # associative they generally won't be rings.
- _no_generic_basering_coercion = True
+
+ def _coerce_map_from_base_ring(self):
+ """
+ Disable the map from the base ring into the algebra.
+
+ Performing a nonsense conversion like this automatically
+ is counterpedagogical. The fallback is to try the usual
+ element constructor, which should also fail.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja
+
+ TESTS::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: J(1)
+ Traceback (most recent call last):
+ ...
+ ValueError: not a naturally-represented algebra element
+
+ """
+ return None
def __init__(self,
field,
mult_table,
- rank,
prefix='e',
category=None,
- natural_basis=None):
+ natural_basis=None,
+ check_field=True,
+ check_axioms=True):
"""
SETUP::
- sage: from mjo.eja.eja_algebra import random_eja
+ sage: from mjo.eja.eja_algebra import (
+ ....: FiniteDimensionalEuclideanJordanAlgebra,
+ ....: JordanSpinEJA,
+ ....: random_eja)
EXAMPLES:
sage: x*y == y*x
True
+ TESTS:
+
+ The ``field`` we're given must be real with ``check_field=True``::
+
+ sage: JordanSpinEJA(2,QQbar)
+ Traceback (most recent call last):
+ ...
+ ValueError: scalar field is not real
+
+ The multiplication table must be square with ``check_axioms=True``::
+
+ sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,((),()))
+ Traceback (most recent call last):
+ ...
+ ValueError: multiplication table is not square
+
"""
- self._rank = rank
+ if check_field:
+ if not field.is_subring(RR):
+ # Note: this does return true for the real algebraic
+ # field, the rationals, and any quadratic field where
+ # we've specified a real embedding.
+ raise ValueError("scalar field is not real")
+
+ # The multiplication table had better be square
+ n = len(mult_table)
+ if check_axioms:
+ if not all( len(l) == n for l in mult_table ):
+ raise ValueError("multiplication table is not square")
+
self._natural_basis = natural_basis
if category is None:
fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
fda.__init__(field,
- range(len(mult_table)),
+ range(n),
prefix=prefix,
category=category)
self.print_options(bracket='')
# long run to have the multiplication table be in terms of
# algebra elements. We do this after calling the superclass
# constructor so that from_vector() knows what to do.
- self._multiplication_table = [ map(lambda x: self.from_vector(x), ls)
- for ls in mult_table ]
-
+ self._multiplication_table = [
+ list(map(lambda x: self.from_vector(x), ls))
+ for ls in mult_table
+ ]
+
+ if check_axioms:
+ if not self._is_commutative():
+ raise ValueError("algebra is not commutative")
+ if not self._is_jordanian():
+ raise ValueError("Jordan identity does not hold")
+ if not self._inner_product_is_associative():
+ raise ValueError("inner product is not associative")
def _element_constructor_(self, elt):
"""
SETUP::
sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
- ....: RealCartesianProductEJA,
+ ....: HadamardEJA,
....: RealSymmetricEJA)
EXAMPLES:
vector representations) back and forth faithfully::
sage: set_random_seed()
- sage: J = RealCartesianProductEJA.random_instance()
+ sage: J = HadamardEJA.random_instance()
sage: x = J.random_element()
sage: J(x.to_vector().column()) == x
True
True
"""
+ msg = "not a naturally-represented algebra element"
if elt == 0:
# The superclass implementation of random_element()
# needs to be able to coerce "0" into the algebra.
return self.zero()
+ elif elt in self.base_ring():
+ # Ensure that no base ring -> algebra coercion is performed
+ # by this method. There's some stupidity in sage that would
+ # otherwise propagate to this method; for example, sage thinks
+ # that the integer 3 belongs to the space of 2-by-2 matrices.
+ raise ValueError(msg)
natural_basis = self.natural_basis()
basis_space = natural_basis[0].matrix_space()
if elt not in basis_space:
- raise ValueError("not a naturally-represented algebra element")
+ raise ValueError(msg)
# Thanks for nothing! Matrix spaces aren't vector spaces in
# Sage, so we have to figure out its natural-basis coordinates
coords = W.coordinate_vector(_mat2vec(elt))
return self.from_vector(coords)
-
@staticmethod
def _max_test_case_size():
"""
"""
return 5
-
def _repr_(self):
"""
Return a string representation of ``self``.
Ensure that it says what we think it says::
- sage: JordanSpinEJA(2, field=QQ)
- Euclidean Jordan algebra of dimension 2 over Rational Field
+ sage: JordanSpinEJA(2, field=AA)
+ Euclidean Jordan algebra of dimension 2 over Algebraic Real Field
sage: JordanSpinEJA(3, field=RDF)
Euclidean Jordan algebra of dimension 3 over Real Double Field
def product_on_basis(self, i, j):
return self._multiplication_table[i][j]
- def _a_regular_element(self):
- """
- Guess a regular element. Needed to compute the basis for our
- characteristic polynomial coefficients.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import random_eja
-
- TESTS:
-
- Ensure that this hacky method succeeds for every algebra that we
- know how to construct::
-
- sage: set_random_seed()
- sage: J = random_eja()
- sage: J._a_regular_element().is_regular()
- True
-
- """
- gs = self.gens()
- z = self.sum( (i+1)*gs[i] for i in range(len(gs)) )
- if not z.is_regular():
- raise ValueError("don't know a regular element")
- return z
-
-
- @cached_method
- def _charpoly_basis_space(self):
- """
- Return the vector space spanned by the basis used in our
- characteristic polynomial coefficients. This is used not only to
- compute those coefficients, but also any time we need to
- evaluate the coefficients (like when we compute the trace or
- determinant).
- """
- z = self._a_regular_element()
- # Don't use the parent vector space directly here in case this
- # happens to be a subalgebra. In that case, we would be e.g.
- # two-dimensional but span_of_basis() would expect three
- # coordinates.
- V = VectorSpace(self.base_ring(), self.vector_space().dimension())
- basis = [ (z**k).to_vector() for k in range(self.rank()) ]
- V1 = V.span_of_basis( basis )
- b = (V1.basis() + V1.complement().basis())
- return V.span_of_basis(b)
-
-
-
- @cached_method
- def _charpoly_coeff(self, i):
- """
- Return the coefficient polynomial "a_{i}" of this algebra's
- general characteristic polynomial.
-
- Having this be a separate cached method lets us compute and
- store the trace/determinant (a_{r-1} and a_{0} respectively)
- separate from the entire characteristic polynomial.
- """
- (A_of_x, x, xr, detA) = self._charpoly_matrix_system()
- R = A_of_x.base_ring()
- if i >= self.rank():
- # Guaranteed by theory
- return R.zero()
-
- # Danger: the in-place modification is done for performance
- # reasons (reconstructing a matrix with huge polynomial
- # entries is slow), but I don't know how cached_method works,
- # so it's highly possible that we're modifying some global
- # list variable by reference, here. In other words, you
- # probably shouldn't call this method twice on the same
- # algebra, at the same time, in two threads
- Ai_orig = A_of_x.column(i)
- A_of_x.set_column(i,xr)
- numerator = A_of_x.det()
- A_of_x.set_column(i,Ai_orig)
-
- # We're relying on the theory here to ensure that each a_i is
- # indeed back in R, and the added negative signs are to make
- # the whole charpoly expression sum to zero.
- return R(-numerator/detA)
-
+ def _is_commutative(self):
+ r"""
+ Whether or not this algebra's multiplication table is commutative.
+
+ This method should of course always return ``True``, unless
+ this algebra was constructed with ``check_axioms=False`` and
+ passed an invalid multiplication table.
+ """
+ return all( self.product_on_basis(i,j) == self.product_on_basis(i,j)
+ for i in range(self.dimension())
+ for j in range(self.dimension()) )
+
+ def _is_jordanian(self):
+ r"""
+ Whether or not this algebra's multiplication table respects the
+ Jordan identity `(x^{2})(xy) = x(x^{2}y)`.
+
+ We only check one arrangement of `x` and `y`, so for a
+ ``True`` result to be truly true, you should also check
+ :meth:`_is_commutative`. This method should of course always
+ return ``True``, unless this algebra was constructed with
+ ``check_axioms=False`` and passed an invalid multiplication table.
+ """
+ return all( (self.monomial(i)**2)*(self.monomial(i)*self.monomial(j))
+ ==
+ (self.monomial(i))*((self.monomial(i)**2)*self.monomial(j))
+ for i in range(self.dimension())
+ for j in range(self.dimension()) )
+
+ def _inner_product_is_associative(self):
+ r"""
+ Return whether or not this algebra's inner product `B` is
+ associative; that is, whether or not `B(xy,z) = B(x,yz)`.
+
+ This method should of course always return ``True``, unless
+ this algebra was constructed with ``check_axioms=False`` and
+ passed an invalid multiplication table.
+ """
+
+ # Used to check whether or not something is zero in an inexact
+ # ring. This number is sufficient to allow the construction of
+ # QuaternionHermitianEJA(2, RDF) with check_axioms=True.
+ epsilon = 1e-16
+
+ for i in range(self.dimension()):
+ for j in range(self.dimension()):
+ for k in range(self.dimension()):
+ x = self.monomial(i)
+ y = self.monomial(j)
+ z = self.monomial(k)
+ diff = (x*y).inner_product(z) - x.inner_product(y*z)
+
+ if self.base_ring().is_exact():
+ if diff != 0:
+ return False
+ else:
+ if diff.abs() > epsilon:
+ return False
+
+ return True
@cached_method
- def _charpoly_matrix_system(self):
- """
- Compute the matrix whose entries A_ij are polynomials in
- X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
- corresponding to `x^r` and the determinent of the matrix A =
- [A_ij]. In other words, all of the fixed (cachable) data needed
- to compute the coefficients of the characteristic polynomial.
+ def characteristic_polynomial_of(self):
"""
- r = self.rank()
- n = self.dimension()
-
- # Turn my vector space into a module so that "vectors" can
- # have multivatiate polynomial entries.
- names = tuple('X' + str(i) for i in range(1,n+1))
- R = PolynomialRing(self.base_ring(), names)
-
- # Using change_ring() on the parent's vector space doesn't work
- # here because, in a subalgebra, that vector space has a basis
- # and change_ring() tries to bring the basis along with it. And
- # that doesn't work unless the new ring is a PID, which it usually
- # won't be.
- V = FreeModule(R,n)
-
- # Now let x = (X1,X2,...,Xn) be the vector whose entries are
- # indeterminates...
- x = V(names)
-
- # And figure out the "left multiplication by x" matrix in
- # that setting.
- lmbx_cols = []
- monomial_matrices = [ self.monomial(i).operator().matrix()
- for i in range(n) ] # don't recompute these!
- for k in range(n):
- ek = self.monomial(k).to_vector()
- lmbx_cols.append(
- sum( x[i]*(monomial_matrices[i]*ek)
- for i in range(n) ) )
- Lx = matrix.column(R, lmbx_cols)
-
- # Now we can compute powers of x "symbolically"
- x_powers = [self.one().to_vector(), x]
- for d in range(2, r+1):
- x_powers.append( Lx*(x_powers[-1]) )
-
- idmat = matrix.identity(R, n)
-
- W = self._charpoly_basis_space()
- W = W.change_ring(R.fraction_field())
-
- # Starting with the standard coordinates x = (X1,X2,...,Xn)
- # and then converting the entries to W-coordinates allows us
- # to pass in the standard coordinates to the charpoly and get
- # back the right answer. Specifically, with x = (X1,X2,...,Xn),
- # we have
- #
- # W.coordinates(x^2) eval'd at (standard z-coords)
- # =
- # W-coords of (z^2)
- # =
- # W-coords of (standard coords of x^2 eval'd at std-coords of z)
- #
- # We want the middle equivalent thing in our matrix, but use
- # the first equivalent thing instead so that we can pass in
- # standard coordinates.
- x_powers = [ W.coordinate_vector(xp) for xp in x_powers ]
- l2 = [idmat.column(k-1) for k in range(r+1, n+1)]
- A_of_x = matrix.column(R, n, (x_powers[:r] + l2))
- return (A_of_x, x, x_powers[r], A_of_x.det())
-
-
- @cached_method
- def characteristic_polynomial(self):
- """
- Return a characteristic polynomial that works for all elements
- of this algebra.
+ Return the algebra's "characteristic polynomial of" function,
+ which is itself a multivariate polynomial that, when evaluated
+ at the coordinates of some algebra element, returns that
+ element's characteristic polynomial.
The resulting polynomial has `n+1` variables, where `n` is the
dimension of this algebra. The first `n` variables correspond to
SETUP::
- sage: from mjo.eja.eja_algebra import JordanSpinEJA
+ sage: from mjo.eja.eja_algebra import JordanSpinEJA, TrivialEJA
EXAMPLES:
Alizadeh, Example 11.11::
sage: J = JordanSpinEJA(3)
- sage: p = J.characteristic_polynomial(); p
+ sage: p = J.characteristic_polynomial_of(); p
X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
sage: xvec = J.one().to_vector()
sage: p(*xvec)
t^2 - 2*t + 1
+ By definition, the characteristic polynomial is a monic
+ degree-zero polynomial in a rank-zero algebra. Note that
+ Cayley-Hamilton is indeed satisfied since the polynomial
+ ``1`` evaluates to the identity element of the algebra on
+ any argument::
+
+ sage: J = TrivialEJA()
+ sage: J.characteristic_polynomial_of()
+ 1
+
"""
r = self.rank()
n = self.dimension()
- # The list of coefficient polynomials a_1, a_2, ..., a_n.
- a = [ self._charpoly_coeff(i) for i in range(n) ]
+ # The list of coefficient polynomials a_0, a_1, a_2, ..., a_(r-1).
+ a = self._charpoly_coefficients()
# We go to a bit of trouble here to reorder the
# indeterminates, so that it's easier to evaluate the
# characteristic polynomial at x's coordinates and get back
# something in terms of t, which is what we want.
- R = a[0].parent()
S = PolynomialRing(self.base_ring(),'t')
t = S.gen(0)
- S = PolynomialRing(S, R.variable_names())
- t = S(t)
-
- # Note: all entries past the rth should be zero. The
- # coefficient of the highest power (x^r) is 1, but it doesn't
- # appear in the solution vector which contains coefficients
- # for the other powers (to make them sum to x^r).
- if (r < n):
- a[r] = 1 # corresponds to x^r
- else:
- # When the rank is equal to the dimension, trying to
- # assign a[r] goes out-of-bounds.
- a.append(1) # corresponds to x^r
+ if r > 0:
+ R = a[0].parent()
+ S = PolynomialRing(S, R.variable_names())
+ t = S(t)
- return sum( a[k]*(t**k) for k in xrange(len(a)) )
+ return (t**r + sum( a[k]*(t**k) for k in range(r) ))
def inner_product(self, x, y):
EXAMPLES:
- Our inner product satisfies the Jordan axiom, which is also
- referred to as "associativity" for a symmetric bilinear form::
+ Our inner product is "associative," which means the following for
+ a symmetric bilinear form::
sage: set_random_seed()
sage: J = random_eja()
SETUP::
- sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
+ sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+ ....: TrivialEJA)
EXAMPLES::
sage: J = ComplexHermitianEJA(3)
sage: J.is_trivial()
False
- sage: A = J.zero().subalgebra_generated_by()
- sage: A.is_trivial()
+
+ ::
+
+ sage: J = TrivialEJA()
+ sage: J.is_trivial()
True
"""
"""
M = list(self._multiplication_table) # copy
- for i in xrange(len(M)):
+ for i in range(len(M)):
# M had better be "square"
M[i] = [self.monomial(i)] + M[i]
M = [["*"] + list(self.gens())] + M
Finite family {0: e0, 1: e1, 2: e2}
sage: J.natural_basis()
(
- [1 0] [ 0 1/2*sqrt2] [0 0]
- [0 0], [1/2*sqrt2 0], [0 1]
+ [1 0] [ 0 0.7071067811865475?] [0 0]
+ [0 0], [0.7071067811865475? 0], [0 1]
)
::
"""
Return the matrix space in which this algebra's natural basis
elements live.
+
+ Generally this will be an `n`-by-`1` column-vector space,
+ except when the algebra is trivial. There it's `n`-by-`n`
+ (where `n` is zero), to ensure that two elements of the
+ natural basis space (empty matrices) can be multiplied.
"""
- if self._natural_basis is None or len(self._natural_basis) == 0:
+ if self.is_trivial():
+ return MatrixSpace(self.base_ring(), 0)
+ elif self._natural_basis is None or len(self._natural_basis) == 0:
return MatrixSpace(self.base_ring(), self.dimension(), 1)
else:
return self._natural_basis[0].matrix_space()
SETUP::
- sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
....: random_eja)
EXAMPLES::
- sage: J = RealCartesianProductEJA(5)
+ sage: J = HadamardEJA(5)
sage: J.one()
e0 + e1 + e2 + e3 + e4
return self.linear_combination(zip(self.gens(), coeffs))
- def random_element(self):
- # Temporary workaround for https://trac.sagemath.org/ticket/28327
- if self.is_trivial():
- return self.zero()
- else:
- s = super(FiniteDimensionalEuclideanJordanAlgebra, self)
- return s.random_element()
+ def peirce_decomposition(self, c):
+ """
+ The Peirce decomposition of this algebra relative to the
+ idempotent ``c``.
+
+ In the future, this can be extended to a complete system of
+ orthogonal idempotents.
+
+ INPUT:
+
+ - ``c`` -- an idempotent of this algebra.
+
+ OUTPUT:
+
+ A triple (J0, J5, J1) containing two subalgebras and one subspace
+ of this algebra,
+
+ - ``J0`` -- the algebra on the eigenspace of ``c.operator()``
+ corresponding to the eigenvalue zero.
+
+ - ``J5`` -- the eigenspace (NOT a subalgebra) of ``c.operator()``
+ corresponding to the eigenvalue one-half.
+
+ - ``J1`` -- the algebra on the eigenspace of ``c.operator()``
+ corresponding to the eigenvalue one.
+
+ These are the only possible eigenspaces for that operator, and this
+ algebra is a direct sum of them. The spaces ``J0`` and ``J1`` are
+ orthogonal, and are subalgebras of this algebra with the appropriate
+ restrictions.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja, RealSymmetricEJA
+
+ EXAMPLES:
+
+ The canonical example comes from the symmetric matrices, which
+ decompose into diagonal and off-diagonal parts::
+
+ sage: J = RealSymmetricEJA(3)
+ sage: C = matrix(QQ, [ [1,0,0],
+ ....: [0,1,0],
+ ....: [0,0,0] ])
+ sage: c = J(C)
+ sage: J0,J5,J1 = J.peirce_decomposition(c)
+ sage: J0
+ Euclidean Jordan algebra of dimension 1...
+ sage: J5
+ Vector space of degree 6 and dimension 2...
+ sage: J1
+ Euclidean Jordan algebra of dimension 3...
+ sage: J0.one().natural_representation()
+ [0 0 0]
+ [0 0 0]
+ [0 0 1]
+ sage: orig_df = AA.options.display_format
+ sage: AA.options.display_format = 'radical'
+ sage: J.from_vector(J5.basis()[0]).natural_representation()
+ [ 0 0 1/2*sqrt(2)]
+ [ 0 0 0]
+ [1/2*sqrt(2) 0 0]
+ sage: J.from_vector(J5.basis()[1]).natural_representation()
+ [ 0 0 0]
+ [ 0 0 1/2*sqrt(2)]
+ [ 0 1/2*sqrt(2) 0]
+ sage: AA.options.display_format = orig_df
+ sage: J1.one().natural_representation()
+ [1 0 0]
+ [0 1 0]
+ [0 0 0]
+
+ TESTS:
+
+ Every algebra decomposes trivially with respect to its identity
+ element::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: J0,J5,J1 = J.peirce_decomposition(J.one())
+ sage: J0.dimension() == 0 and J5.dimension() == 0
+ True
+ sage: J1.superalgebra() == J and J1.dimension() == J.dimension()
+ True
+
+ The decomposition is into eigenspaces, and its components are
+ therefore necessarily orthogonal. Moreover, the identity
+ elements in the two subalgebras are the projections onto their
+ respective subspaces of the superalgebra's identity element::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: if not J.is_trivial():
+ ....: while x.is_nilpotent():
+ ....: x = J.random_element()
+ sage: c = x.subalgebra_idempotent()
+ sage: J0,J5,J1 = J.peirce_decomposition(c)
+ sage: ipsum = 0
+ sage: for (w,y,z) in zip(J0.basis(), J5.basis(), J1.basis()):
+ ....: w = w.superalgebra_element()
+ ....: y = J.from_vector(y)
+ ....: z = z.superalgebra_element()
+ ....: ipsum += w.inner_product(y).abs()
+ ....: ipsum += w.inner_product(z).abs()
+ ....: ipsum += y.inner_product(z).abs()
+ sage: ipsum
+ 0
+ sage: J1(c) == J1.one()
+ True
+ sage: J0(J.one() - c) == J0.one()
+ True
- def random_elements(self, count):
+ """
+ if not c.is_idempotent():
+ raise ValueError("element is not idempotent: %s" % c)
+
+ # Default these to what they should be if they turn out to be
+ # trivial, because eigenspaces_left() won't return eigenvalues
+ # corresponding to trivial spaces (e.g. it returns only the
+ # eigenspace corresponding to lambda=1 if you take the
+ # decomposition relative to the identity element).
+ trivial = FiniteDimensionalEuclideanJordanSubalgebra(self, ())
+ J0 = trivial # eigenvalue zero
+ J5 = VectorSpace(self.base_ring(), 0) # eigenvalue one-half
+ J1 = trivial # eigenvalue one
+
+ for (eigval, eigspace) in c.operator().matrix().right_eigenspaces():
+ if eigval == ~(self.base_ring()(2)):
+ J5 = eigspace
+ else:
+ gens = tuple( self.from_vector(b) for b in eigspace.basis() )
+ subalg = FiniteDimensionalEuclideanJordanSubalgebra(self,
+ gens,
+ check_axioms=False)
+ if eigval == 0:
+ J0 = subalg
+ elif eigval == 1:
+ J1 = subalg
+ else:
+ raise ValueError("unexpected eigenvalue: %s" % eigval)
+
+ return (J0, J5, J1)
+
+
+ def random_element(self, thorough=False):
+ r"""
+ Return a random element of this algebra.
+
+ Our algebra superclass method only returns a linear
+ combination of at most two basis elements. We instead
+ want the vector space "random element" method that
+ returns a more diverse selection.
+
+ INPUT:
+
+ - ``thorough`` -- (boolean; default False) whether or not we
+ should generate irrational coefficients for the random
+ element when our base ring is irrational; this slows the
+ algebra operations to a crawl, but any truly random method
+ should include them
+
+ """
+ # For a general base ring... maybe we can trust this to do the
+ # right thing? Unlikely, but.
+ V = self.vector_space()
+ v = V.random_element()
+
+ if self.base_ring() is AA:
+ # The "random element" method of the algebraic reals is
+ # stupid at the moment, and only returns integers between
+ # -2 and 2, inclusive:
+ #
+ # https://trac.sagemath.org/ticket/30875
+ #
+ # Instead, we implement our own "random vector" method,
+ # and then coerce that into the algebra. We use the vector
+ # space degree here instead of the dimension because a
+ # subalgebra could (for example) be spanned by only two
+ # vectors, each with five coordinates. We need to
+ # generate all five coordinates.
+ if thorough:
+ v *= QQbar.random_element().real()
+ else:
+ v *= QQ.random_element()
+
+ return self.from_vector(V.coordinate_vector(v))
+
+ def random_elements(self, count, thorough=False):
"""
Return ``count`` random elements as a tuple.
+ INPUT:
+
+ - ``thorough`` -- (boolean; default False) whether or not we
+ should generate irrational coefficients for the random
+ elements when our base ring is irrational; this slows the
+ algebra operations to a crawl, but any truly random method
+ should include them
+
SETUP::
sage: from mjo.eja.eja_algebra import JordanSpinEJA
True
"""
- return tuple( self.random_element() for idx in xrange(count) )
+ return tuple( self.random_element(thorough)
+ for idx in range(count) )
@classmethod
- def random_instance(cls, field=QQ, **kwargs):
+ def random_instance(cls, field=AA, **kwargs):
"""
Return a random instance of this type of algebra.
- In subclasses for algebras that we know how to construct, this
- is a shortcut for constructing test cases and examples.
+ Beware, this will crash for "most instances" because the
+ constructor below looks wrong.
"""
- if cls is FiniteDimensionalEuclideanJordanAlgebra:
- # Red flag! But in theory we could do this I guess. The
- # only finite-dimensional exceptional EJA is the
- # octononions. So, we could just create an EJA from an
- # associative matrix algebra (generated by a subset of
- # elements) with the symmetric product. Or, we could punt
- # to random_eja() here, override it in our subclasses, and
- # not worry about it.
- raise NotImplementedError
+ if cls is TrivialEJA:
+ # The TrivialEJA class doesn't take an "n" argument because
+ # there's only one.
+ return cls(field)
- n = ZZ.random_element(cls._max_test_case_size()) + 1
+ n = ZZ.random_element(cls._max_test_case_size() + 1)
return cls(n, field, **kwargs)
+ @cached_method
+ def _charpoly_coefficients(self):
+ r"""
+ The `r` polynomial coefficients of the "characteristic polynomial
+ of" function.
+ """
+ n = self.dimension()
+ var_names = [ "X" + str(z) for z in range(1,n+1) ]
+ R = PolynomialRing(self.base_ring(), var_names)
+ vars = R.gens()
+ F = R.fraction_field()
+
+ def L_x_i_j(i,j):
+ # From a result in my book, these are the entries of the
+ # basis representation of L_x.
+ return sum( vars[k]*self.monomial(k).operator().matrix()[i,j]
+ for k in range(n) )
+
+ L_x = matrix(F, n, n, L_x_i_j)
+
+ r = None
+ if self.rank.is_in_cache():
+ r = self.rank()
+ # There's no need to pad the system with redundant
+ # columns if we *know* they'll be redundant.
+ n = r
+
+ # Compute an extra power in case the rank is equal to
+ # the dimension (otherwise, we would stop at x^(r-1)).
+ x_powers = [ (L_x**k)*self.one().to_vector()
+ for k in range(n+1) ]
+ A = matrix.column(F, x_powers[:n])
+ AE = A.extended_echelon_form()
+ E = AE[:,n:]
+ A_rref = AE[:,:n]
+ if r is None:
+ r = A_rref.rank()
+ b = x_powers[r]
+
+ # The theory says that only the first "r" coefficients are
+ # nonzero, and they actually live in the original polynomial
+ # ring and not the fraction field. We negate them because
+ # in the actual characteristic polynomial, they get moved
+ # to the other side where x^r lives.
+ return -A_rref.solve_right(E*b).change_ring(R)[:r]
+ @cached_method
def rank(self):
- """
+ r"""
Return the rank of this EJA.
- ALGORITHM:
-
- The author knows of no algorithm to compute the rank of an EJA
- where only the multiplication table is known. In lieu of one, we
- require the rank to be specified when the algebra is created,
- and simply pass along that number here.
+ This is a cached method because we know the rank a priori for
+ all of the algebras we can construct. Thus we can avoid the
+ expensive ``_charpoly_coefficients()`` call unless we truly
+ need to compute the whole characteristic polynomial.
SETUP::
- sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: JordanSpinEJA,
....: RealSymmetricEJA,
....: ComplexHermitianEJA,
....: QuaternionHermitianEJA,
TESTS:
Ensure that every EJA that we know how to construct has a
- positive integer rank::
+ positive integer rank, unless the algebra is trivial in
+ which case its rank will be zero::
sage: set_random_seed()
- sage: r = random_eja().rank()
- sage: r in ZZ and r > 0
+ sage: J = random_eja()
+ sage: r = J.rank()
+ sage: r in ZZ
True
+ sage: r > 0 or (r == 0 and J.is_trivial())
+ True
+
+ Ensure that computing the rank actually works, since the ranks
+ of all simple algebras are known and will be cached by default::
+
+ sage: J = HadamardEJA(4)
+ sage: J.rank.clear_cache()
+ sage: J.rank()
+ 4
+
+ ::
+ sage: J = JordanSpinEJA(4)
+ sage: J.rank.clear_cache()
+ sage: J.rank()
+ 2
+
+ ::
+
+ sage: J = RealSymmetricEJA(3)
+ sage: J.rank.clear_cache()
+ sage: J.rank()
+ 3
+
+ ::
+
+ sage: J = ComplexHermitianEJA(2)
+ sage: J.rank.clear_cache()
+ sage: J.rank()
+ 2
+
+ ::
+
+ sage: J = QuaternionHermitianEJA(2)
+ sage: J.rank.clear_cache()
+ sage: J.rank()
+ 2
"""
- return self._rank
+ return len(self._charpoly_coefficients())
def vector_space(self):
Element = FiniteDimensionalEuclideanJordanAlgebraElement
-class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra):
+class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra):
"""
Return the Euclidean Jordan Algebra corresponding to the set
`R^n` under the Hadamard product.
SETUP::
- sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
+ sage: from mjo.eja.eja_algebra import HadamardEJA
EXAMPLES:
This multiplication table can be verified by hand::
- sage: J = RealCartesianProductEJA(3)
+ sage: J = HadamardEJA(3)
sage: e0,e1,e2 = J.gens()
sage: e0*e0
e0
We can change the generator prefix::
- sage: RealCartesianProductEJA(3, prefix='r').gens()
+ sage: HadamardEJA(3, prefix='r').gens()
(r0, r1, r2)
"""
- def __init__(self, n, field=QQ, **kwargs):
+ def __init__(self, n, field=AA, **kwargs):
V = VectorSpace(field, n)
- mult_table = [ [ V.gen(i)*(i == j) for j in xrange(n) ]
- for i in xrange(n) ]
+ mult_table = [ [ V.gen(i)*(i == j) for j in range(n) ]
+ for i in range(n) ]
- fdeja = super(RealCartesianProductEJA, self)
- return fdeja.__init__(field, mult_table, rank=n, **kwargs)
+ super(HadamardEJA, self).__init__(field,
+ mult_table,
+ check_axioms=False,
+ **kwargs)
+ self.rank.set_cache(n)
def inner_product(self, x, y):
"""
SETUP::
- sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
+ sage: from mjo.eja.eja_algebra import HadamardEJA
TESTS:
over `R^n`::
sage: set_random_seed()
- sage: J = RealCartesianProductEJA.random_instance()
+ sage: J = HadamardEJA.random_instance()
sage: x,y = J.random_elements(2)
sage: X = x.natural_representation()
sage: Y = y.natural_representation()
return x.to_vector().inner_product(y.to_vector())
-def random_eja():
+def random_eja(field=AA):
"""
Return a "random" finite-dimensional Euclidean Jordan Algebra.
- ALGORITHM:
-
- For now, we choose a random natural number ``n`` (greater than zero)
- and then give you back one of the following:
-
- * The cartesian product of the rational numbers ``n`` times; this is
- ``QQ^n`` with the Hadamard product.
-
- * The Jordan spin algebra on ``QQ^n``.
-
- * The ``n``-by-``n`` rational symmetric matrices with the symmetric
- product.
-
- * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
- in the space of ``2n``-by-``2n`` real symmetric matrices.
-
- * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
- in the space of ``4n``-by-``4n`` real symmetric matrices.
-
- Later this might be extended to return Cartesian products of the
- EJAs above.
-
SETUP::
sage: from mjo.eja.eja_algebra import random_eja
Euclidean Jordan algebra of dimension...
"""
- classname = choice([RealCartesianProductEJA,
+ classname = choice([TrivialEJA,
+ HadamardEJA,
JordanSpinEJA,
RealSymmetricEJA,
ComplexHermitianEJA,
QuaternionHermitianEJA])
- return classname.random_instance()
-
-
+ return classname.random_instance(field=field)
# field can have dimension 4 (quaternions) too.
return 2
- @classmethod
- def _denormalized_basis(cls, n, field):
- raise NotImplementedError
-
- def __init__(self, n, field=QQ, normalize_basis=True, **kwargs):
- S = self._denormalized_basis(n, field)
-
- # Used in this class's fast _charpoly_coeff() override.
+ def __init__(self, field, basis, normalize_basis=True, **kwargs):
+ """
+ Compared to the superclass constructor, we take a basis instead of
+ a multiplication table because the latter can be computed in terms
+ of the former when the product is known (like it is here).
+ """
+ # Used in this class's fast _charpoly_coefficients() override.
self._basis_normalizers = None
- if n > 1 and normalize_basis:
+ # We're going to loop through this a few times, so now's a good
+ # time to ensure that it isn't a generator expression.
+ basis = tuple(basis)
+
+ if len(basis) > 1 and normalize_basis:
# We'll need sqrt(2) to normalize the basis, and this
# winds up in the multiplication table, so the whole
# algebra needs to be over the field extension.
z = R.gen()
p = z**2 - 2
if p.is_irreducible():
- field = NumberField(p, 'sqrt2', embedding=RLF(2).sqrt())
- S = [ s.change_ring(field) for s in S ]
+ field = field.extension(p, 'sqrt2', embedding=RLF(2).sqrt())
+ basis = tuple( s.change_ring(field) for s in basis )
self._basis_normalizers = tuple(
- ~(self.natural_inner_product(s,s).sqrt()) for s in S )
- S = tuple( s*c for (s,c) in zip(S,self._basis_normalizers) )
+ ~(self.natural_inner_product(s,s).sqrt()) for s in basis )
+ basis = tuple(s*c for (s,c) in zip(basis,self._basis_normalizers))
- Qs = self.multiplication_table_from_matrix_basis(S)
+ Qs = self.multiplication_table_from_matrix_basis(basis)
- fdeja = super(MatrixEuclideanJordanAlgebra, self)
- return fdeja.__init__(field,
- Qs,
- rank=n,
- natural_basis=S,
- **kwargs)
+ super(MatrixEuclideanJordanAlgebra, self).__init__(field,
+ Qs,
+ natural_basis=basis,
+ **kwargs)
@cached_method
- def _charpoly_coeff(self, i):
- """
+ def _charpoly_coefficients(self):
+ r"""
Override the parent method with something that tries to compute
over a faster (non-extension) field.
"""
if self._basis_normalizers is None:
# We didn't normalize, so assume that the basis we started
# with had entries in a nice field.
- return super(MatrixEuclideanJordanAlgebra, self)._charpoly_coeff(i)
+ return super(MatrixEuclideanJordanAlgebra, self)._charpoly_coefficients()
else:
- # If we didn't unembed first, this number would be wrong
- # by a power-of-two factor for complex/quaternion matrices.
- n = self.real_unembed(self.natural_basis_space().zero()).nrows()
- field = self.base_ring().base_ring() # yeeeeaaaahhh
- J = self.__class__(n, field, False)
- (_,x,_,_) = J._charpoly_matrix_system()
- p = J._charpoly_coeff(i)
- # p might be missing some vars, have to substitute "optionally"
- pairs = izip(x.base_ring().gens(), self._basis_normalizers)
- substitutions = { v: v*c for (v,c) in pairs }
- return p.subs(substitutions)
+ basis = ( (b/n) for (b,n) in zip(self.natural_basis(),
+ self._basis_normalizers) )
+
+ # Do this over the rationals and convert back at the end.
+ # Only works because we know the entries of the basis are
+ # integers. The argument ``check_axioms=False`` is required
+ # because the trace inner-product method for this
+ # class is a stub and can't actually be checked.
+ J = MatrixEuclideanJordanAlgebra(QQ,
+ basis,
+ normalize_basis=False,
+ check_field=False,
+ check_axioms=False)
+ a = J._charpoly_coefficients()
+
+ # Unfortunately, changing the basis does change the
+ # coefficients of the characteristic polynomial, but since
+ # these are really the coefficients of the "characteristic
+ # polynomial of" function, everything is still nice and
+ # unevaluated. It's therefore "obvious" how scaling the
+ # basis affects the coordinate variables X1, X2, et
+ # cetera. Scaling the first basis vector up by "n" adds a
+ # factor of 1/n into every "X1" term, for example. So here
+ # we simply undo the basis_normalizer scaling that we
+ # performed earlier.
+ #
+ # The a[0] access here is safe because trivial algebras
+ # won't have any basis normalizers and therefore won't
+ # make it to this "else" branch.
+ XS = a[0].parent().gens()
+ subs_dict = { XS[i]: self._basis_normalizers[i]*XS[i]
+ for i in range(len(XS)) }
+ return tuple( a_i.subs(subs_dict) for a_i in a )
@staticmethod
# is supposed to hold the entire long vector, and the subspace W
# of V will be spanned by the vectors that arise from symmetric
# matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
+ if len(basis) == 0:
+ return []
+
field = basis[0].base_ring()
dimension = basis[0].nrows()
V = VectorSpace(field, dimension**2)
W = V.span_of_basis( _mat2vec(s) for s in basis )
n = len(basis)
- mult_table = [[W.zero() for j in xrange(n)] for i in xrange(n)]
- for i in xrange(n):
- for j in xrange(n):
+ mult_table = [[W.zero() for j in range(n)] for i in range(n)]
+ for i in range(n):
+ for j in range(n):
mat_entry = (basis[i]*basis[j] + basis[j]*basis[i])/2
mult_table[i][j] = W.coordinate_vector(_mat2vec(mat_entry))
Xu = cls.real_unembed(X)
Yu = cls.real_unembed(Y)
tr = (Xu*Yu).trace()
- if tr in RLF:
- # It's real already.
- return tr
- # Otherwise, try the thing that works for complex numbers; and
- # if that doesn't work, the thing that works for quaternions.
try:
- return tr.vector()[0] # real part, imag part is index 1
+ # Works in QQ, AA, RDF, et cetera.
+ return tr.real()
except AttributeError:
- # A quaternions doesn't have a vector() method, but does
+ # A quaternion doesn't have a real() method, but does
# have coefficient_tuple() method that returns the
# coefficients of 1, i, j, and k -- in that order.
return tr.coefficient_tuple()[0]
sage: e2*e2
e2
+ In theory, our "field" can be any subfield of the reals::
+
+ sage: RealSymmetricEJA(2, RDF)
+ Euclidean Jordan algebra of dimension 3 over Real Double Field
+ sage: RealSymmetricEJA(2, RR)
+ Euclidean Jordan algebra of dimension 3 over Real Field with
+ 53 bits of precision
+
TESTS:
The dimension of this algebra is `(n^2 + n) / 2`::
sage: x.operator().matrix().is_symmetric()
True
+ We can construct the (trivial) algebra of rank zero::
+
+ sage: RealSymmetricEJA(0)
+ Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
+
"""
@classmethod
def _denormalized_basis(cls, n, field):
# The basis of symmetric matrices, as matrices, in their R^(n-by-n)
# coordinates.
S = []
- for i in xrange(n):
- for j in xrange(i+1):
+ for i in range(n):
+ for j in range(i+1):
Eij = matrix(field, n, lambda k,l: k==i and l==j)
if i == j:
Sij = Eij
else:
Sij = Eij + Eij.transpose()
S.append(Sij)
- return tuple(S)
+ return S
@staticmethod
return 4 # Dimension 10
+ def __init__(self, n, field=AA, **kwargs):
+ basis = self._denormalized_basis(n, field)
+ super(RealSymmetricEJA, self).__init__(field,
+ basis,
+ check_axioms=False,
+ **kwargs)
+ self.rank.set_cache(n)
+
class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra):
@staticmethod
EXAMPLES::
- sage: F = QuadraticField(-1, 'i')
+ sage: F = QuadraticField(-1, 'I')
sage: x1 = F(4 - 2*i)
sage: x2 = F(1 + 2*i)
sage: x3 = F(-i)
sage: set_random_seed()
sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size()
sage: n = ZZ.random_element(n_max)
- sage: F = QuadraticField(-1, 'i')
+ sage: F = QuadraticField(-1, 'I')
sage: X = random_matrix(F, n)
sage: Y = random_matrix(F, n)
sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X)
n = M.nrows()
if M.ncols() != n:
raise ValueError("the matrix 'M' must be square")
- field = M.base_ring()
+
+ # We don't need any adjoined elements...
+ field = M.base_ring().base_ring()
+
blocks = []
for z in M.list():
- a = z.vector()[0] # real part, I guess
- b = z.vector()[1] # imag part, I guess
+ a = z.list()[0] # real part, I guess
+ b = z.list()[1] # imag part, I guess
blocks.append(matrix(field, 2, [[a,b],[-b,a]]))
- # We can drop the imaginaries here.
- return matrix.block(field.base_ring(), n, blocks)
+ return matrix.block(field, n, blocks)
@staticmethod
....: [ 9, 10, 11, 12],
....: [-10, 9, -12, 11] ])
sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A)
- [ 2*i + 1 4*i + 3]
- [ 10*i + 9 12*i + 11]
+ [ 2*I + 1 4*I + 3]
+ [ 10*I + 9 12*I + 11]
TESTS:
Unembedding is the inverse of embedding::
sage: set_random_seed()
- sage: F = QuadraticField(-1, 'i')
+ sage: F = QuadraticField(-1, 'I')
sage: M = random_matrix(F, 3)
sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
if not n.mod(2).is_zero():
raise ValueError("the matrix 'M' must be a complex embedding")
- field = M.base_ring() # This should already have sqrt2
+ # If "M" was normalized, its base ring might have roots
+ # adjoined and they can stick around after unembedding.
+ field = M.base_ring()
R = PolynomialRing(field, 'z')
z = R.gen()
- F = NumberField(z**2 + 1,'i', embedding=CLF(-1).sqrt())
+ if field is AA:
+ # Sage doesn't know how to embed AA into QQbar, i.e. how
+ # to adjoin sqrt(-1) to AA.
+ F = QQbar
+ else:
+ F = field.extension(z**2 + 1, 'I', embedding=CLF(-1).sqrt())
i = F.gen()
# Go top-left to bottom-right (reading order), converting every
# 2-by-2 block we see to a single complex element.
elements = []
- for k in xrange(n/2):
- for j in xrange(n/2):
+ for k in range(n/2):
+ for j in range(n/2):
submat = M[2*k:2*k+2,2*j:2*j+2]
if submat[0,0] != submat[1,1]:
raise ValueError('bad on-diagonal submatrix')
sage: Ye = y.natural_representation()
sage: X = ComplexHermitianEJA.real_unembed(Xe)
sage: Y = ComplexHermitianEJA.real_unembed(Ye)
- sage: expected = (X*Y).trace().vector()[0]
+ sage: expected = (X*Y).trace().real()
sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye)
sage: actual == expected
True
sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
+ EXAMPLES:
+
+ In theory, our "field" can be any subfield of the reals::
+
+ sage: ComplexHermitianEJA(2, RDF)
+ Euclidean Jordan algebra of dimension 4 over Real Double Field
+ sage: ComplexHermitianEJA(2, RR)
+ Euclidean Jordan algebra of dimension 4 over Real Field with
+ 53 bits of precision
+
TESTS:
The dimension of this algebra is `n^2`::
sage: x.operator().matrix().is_symmetric()
True
+ We can construct the (trivial) algebra of rank zero::
+
+ sage: ComplexHermitianEJA(0)
+ Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
+
"""
+
@classmethod
def _denormalized_basis(cls, n, field):
"""
"""
R = PolynomialRing(field, 'z')
z = R.gen()
- F = NumberField(z**2 + 1, 'I', embedding=CLF(-1).sqrt())
+ F = field.extension(z**2 + 1, 'I')
I = F.gen()
# This is like the symmetric case, but we need to be careful:
# * The diagonal will (as a result) be real.
#
S = []
- for i in xrange(n):
- for j in xrange(i+1):
+ for i in range(n):
+ for j in range(i+1):
Eij = matrix(F, n, lambda k,l: k==i and l==j)
if i == j:
Sij = cls.real_embed(Eij)
# Since we embedded these, we can drop back to the "field" that we
# started with instead of the complex extension "F".
- return tuple( s.change_ring(field) for s in S )
+ return ( s.change_ring(field) for s in S )
+ def __init__(self, n, field=AA, **kwargs):
+ basis = self._denormalized_basis(n,field)
+ super(ComplexHermitianEJA,self).__init__(field,
+ basis,
+ check_axioms=False,
+ **kwargs)
+ self.rank.set_cache(n)
+
class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra):
@staticmethod
if M.ncols() != n:
raise ValueError("the matrix 'M' must be square")
- F = QuadraticField(-1, 'i')
+ F = QuadraticField(-1, 'I')
i = F.gen()
blocks = []
if M.ncols() != n:
raise ValueError("the matrix 'M' must be square")
if not n.mod(4).is_zero():
- raise ValueError("the matrix 'M' must be a complex embedding")
+ raise ValueError("the matrix 'M' must be a quaternion embedding")
# Use the base ring of the matrix to ensure that its entries can be
# multiplied by elements of the quaternion algebra.
# 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
# quaternion block.
elements = []
- for l in xrange(n/4):
- for m in xrange(n/4):
+ for l in range(n/4):
+ for m in range(n/4):
submat = ComplexMatrixEuclideanJordanAlgebra.real_unembed(
M[4*l:4*l+4,4*m:4*m+4] )
if submat[0,0] != submat[1,1].conjugate():
raise ValueError('bad on-diagonal submatrix')
if submat[0,1] != -submat[1,0].conjugate():
raise ValueError('bad off-diagonal submatrix')
- z = submat[0,0].vector()[0] # real part
- z += submat[0,0].vector()[1]*i # imag part
- z += submat[0,1].vector()[0]*j # real part
- z += submat[0,1].vector()[1]*k # imag part
+ z = submat[0,0].real()
+ z += submat[0,0].imag()*i
+ z += submat[0,1].real()*j
+ z += submat[0,1].imag()*k
elements.append(z)
return matrix(Q, n/4, elements)
sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
+ EXAMPLES:
+
+ In theory, our "field" can be any subfield of the reals::
+
+ sage: QuaternionHermitianEJA(2, RDF)
+ Euclidean Jordan algebra of dimension 6 over Real Double Field
+ sage: QuaternionHermitianEJA(2, RR)
+ Euclidean Jordan algebra of dimension 6 over Real Field with
+ 53 bits of precision
+
TESTS:
The dimension of this algebra is `2*n^2 - n`::
sage: x.operator().matrix().is_symmetric()
True
+ We can construct the (trivial) algebra of rank zero::
+
+ sage: QuaternionHermitianEJA(0)
+ Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
+
"""
@classmethod
def _denormalized_basis(cls, n, field):
# * The diagonal will (as a result) be real.
#
S = []
- for i in xrange(n):
- for j in xrange(i+1):
+ for i in range(n):
+ for j in range(i+1):
Eij = matrix(Q, n, lambda k,l: k==i and l==j)
if i == j:
Sij = cls.real_embed(Eij)
S.append(Sij_J)
Sij_K = cls.real_embed(K*Eij - K*Eij.transpose())
S.append(Sij_K)
- return tuple(S)
+ # Since we embedded these, we can drop back to the "field" that we
+ # started with instead of the quaternion algebra "Q".
+ return ( s.change_ring(field) for s in S )
+
+
+ def __init__(self, n, field=AA, **kwargs):
+ basis = self._denormalized_basis(n,field)
+ super(QuaternionHermitianEJA,self).__init__(field,
+ basis,
+ check_axioms=False,
+ **kwargs)
+ self.rank.set_cache(n)
+
+
+class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra):
+ r"""
+ The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
+ with the half-trace inner product and jordan product ``x*y =
+ (x0*y0 + <B*x_bar,y_bar>, x0*y_bar + y0*x_bar)`` where ``B`` is a
+ symmetric positive-definite "bilinear form" matrix. It has
+ dimension `n` over the reals, and reduces to the ``JordanSpinEJA``
+ when ``B`` is the identity matrix of order ``n-1``.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (BilinearFormEJA,
+ ....: JordanSpinEJA)
+
+ EXAMPLES:
+
+ When no bilinear form is specified, the identity matrix is used,
+ and the resulting algebra is the Jordan spin algebra::
+
+ sage: J0 = BilinearFormEJA(3)
+ sage: J1 = JordanSpinEJA(3)
+ sage: J0.multiplication_table() == J0.multiplication_table()
+ True
+
+ TESTS:
+
+ We can create a zero-dimensional algebra::
+
+ sage: J = BilinearFormEJA(0)
+ sage: J.basis()
+ Finite family {}
+
+ We can check the multiplication condition given in the Jordan, von
+ Neumann, and Wigner paper (and also discussed on my "On the
+ symmetry..." paper). Note that this relies heavily on the standard
+ choice of basis, as does anything utilizing the bilinear form matrix::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(5)
+ sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
+ sage: B = M.transpose()*M
+ sage: J = BilinearFormEJA(n, B=B)
+ sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis()
+ sage: V = J.vector_space()
+ sage: sis = [ J.from_vector(V([0] + (M.inverse()*ei).list()))
+ ....: for ei in eis ]
+ sage: actual = [ sis[i]*sis[j]
+ ....: for i in range(n-1)
+ ....: for j in range(n-1) ]
+ sage: expected = [ J.one() if i == j else J.zero()
+ ....: for i in range(n-1)
+ ....: for j in range(n-1) ]
+ sage: actual == expected
+ True
+ """
+ def __init__(self, n, field=AA, B=None, **kwargs):
+ if B is None:
+ self._B = matrix.identity(field, max(0,n-1))
+ else:
+ self._B = B
+
+ V = VectorSpace(field, n)
+ mult_table = [[V.zero() for j in range(n)] for i in range(n)]
+ for i in range(n):
+ for j in range(n):
+ x = V.gen(i)
+ y = V.gen(j)
+ x0 = x[0]
+ xbar = x[1:]
+ y0 = y[0]
+ ybar = y[1:]
+ z0 = x0*y0 + (self._B*xbar).inner_product(ybar)
+ zbar = y0*xbar + x0*ybar
+ z = V([z0] + zbar.list())
+ mult_table[i][j] = z
+
+ # The rank of this algebra is two, unless we're in a
+ # one-dimensional ambient space (because the rank is bounded
+ # by the ambient dimension).
+ super(BilinearFormEJA, self).__init__(field,
+ mult_table,
+ check_axioms=False,
+ **kwargs)
+ self.rank.set_cache(min(n,2))
+
+ def inner_product(self, x, y):
+ r"""
+ Half of the trace inner product.
+
+ This is defined so that the special case of the Jordan spin
+ algebra gets the usual inner product.
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import BilinearFormEJA
+
+ TESTS:
+
+ Ensure that this is one-half of the trace inner-product when
+ the algebra isn't just the reals (when ``n`` isn't one). This
+ is in Faraut and Koranyi, and also my "On the symmetry..."
+ paper::
-class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra):
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(2,5)
+ sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
+ sage: B = M.transpose()*M
+ sage: J = BilinearFormEJA(n, B=B)
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: x.inner_product(y) == (x*y).trace()/2
+ True
+
+ """
+ xvec = x.to_vector()
+ xbar = xvec[1:]
+ yvec = y.to_vector()
+ ybar = yvec[1:]
+ return x[0]*y[0] + (self._B*xbar).inner_product(ybar)
+
+
+class JordanSpinEJA(BilinearFormEJA):
"""
The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
with the usual inner product and jordan product ``x*y =
- (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
+ (<x,y>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
the reals.
SETUP::
sage: JordanSpinEJA(2, prefix='B').gens()
(B0, B1)
- """
- def __init__(self, n, field=QQ, **kwargs):
- V = VectorSpace(field, n)
- mult_table = [[V.zero() for j in xrange(n)] for i in xrange(n)]
- for i in xrange(n):
- for j in xrange(n):
- x = V.gen(i)
- y = V.gen(j)
- x0 = x[0]
- xbar = x[1:]
- y0 = y[0]
- ybar = y[1:]
- # z = x*y
- z0 = x.inner_product(y)
- zbar = y0*xbar + x0*ybar
- z = V([z0] + zbar.list())
- mult_table[i][j] = z
-
- # The rank of the spin algebra is two, unless we're in a
- # one-dimensional ambient space (because the rank is bounded by
- # the ambient dimension).
- fdeja = super(JordanSpinEJA, self)
- return fdeja.__init__(field, mult_table, rank=min(n,2), **kwargs)
-
- def inner_product(self, x, y):
- """
- Faster to reimplement than to use natural representations.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import JordanSpinEJA
-
- TESTS:
+ TESTS:
- Ensure that this is the usual inner product for the algebras
- over `R^n`::
+ Ensure that we have the usual inner product on `R^n`::
sage: set_random_seed()
sage: J = JordanSpinEJA.random_instance()
sage: x.inner_product(y) == J.natural_inner_product(X,Y)
True
- """
- return x.to_vector().inner_product(y.to_vector())
+ """
+ def __init__(self, n, field=AA, **kwargs):
+ # This is a special case of the BilinearFormEJA with the identity
+ # matrix as its bilinear form.
+ super(JordanSpinEJA, self).__init__(n, field, **kwargs)
+
+
+class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra):
+ """
+ The trivial Euclidean Jordan algebra consisting of only a zero element.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import TrivialEJA
+
+ EXAMPLES::
+
+ sage: J = TrivialEJA()
+ sage: J.dimension()
+ 0
+ sage: J.zero()
+ 0
+ sage: J.one()
+ 0
+ sage: 7*J.one()*12*J.one()
+ 0
+ sage: J.one().inner_product(J.one())
+ 0
+ sage: J.one().norm()
+ 0
+ sage: J.one().subalgebra_generated_by()
+ Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
+ sage: J.rank()
+ 0
+
+ """
+ def __init__(self, field=AA, **kwargs):
+ mult_table = []
+ super(TrivialEJA, self).__init__(field,
+ mult_table,
+ check_axioms=False,
+ **kwargs)
+ # The rank is zero using my definition, namely the dimension of the
+ # largest subalgebra generated by any element.
+ self.rank.set_cache(0)
+
+
+class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra):
+ r"""
+ The external (orthogonal) direct sum of two other Euclidean Jordan
+ algebras. Essentially the Cartesian product of its two factors.
+ Every Euclidean Jordan algebra decomposes into an orthogonal
+ direct sum of simple Euclidean Jordan algebras, so no generality
+ is lost by providing only this construction.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: RealSymmetricEJA,
+ ....: DirectSumEJA)
+
+ EXAMPLES::
+
+ sage: J1 = HadamardEJA(2)
+ sage: J2 = RealSymmetricEJA(3)
+ sage: J = DirectSumEJA(J1,J2)
+ sage: J.dimension()
+ 8
+ sage: J.rank()
+ 5
+
+ """
+ def __init__(self, J1, J2, field=AA, **kwargs):
+ n1 = J1.dimension()
+ n2 = J2.dimension()
+ n = n1+n2
+ V = VectorSpace(field, n)
+ mult_table = [ [ V.zero() for j in range(n) ]
+ for i in range(n) ]
+ for i in range(n1):
+ for j in range(n1):
+ p = (J1.monomial(i)*J1.monomial(j)).to_vector()
+ mult_table[i][j] = V(p.list() + [field.zero()]*n2)
+
+ for i in range(n2):
+ for j in range(n2):
+ p = (J2.monomial(i)*J2.monomial(j)).to_vector()
+ mult_table[n1+i][n1+j] = V([field.zero()]*n1 + p.list())
+
+ super(DirectSumEJA, self).__init__(field,
+ mult_table,
+ check_axioms=False,
+ **kwargs)
+ self.rank.set_cache(J1.rank() + J2.rank())
projects / sage.d.git / blobdiff