-"""
+r"""
Representations and constructions for Euclidean Jordan algebras.
A Euclidean Jordan algebra is a Jordan algebra that has some
* :class:`RealSymmetricEJA`
* :class:`ComplexHermitianEJA`
* :class:`QuaternionHermitianEJA`
+ * :class:`OctonionHermitianEJA`
-Missing from this list is the algebra of three-by-three octononion
-Hermitian matrices, as there is (as of yet) no implementation of the
-octonions in SageMath. In addition to these, we provide two other
-example constructions,
+In addition to these, we provide a few other example constructions,
+ * :class:`JordanSpinEJA`
* :class:`HadamardEJA`
+ * :class:`AlbertEJA`
* :class:`TrivialEJA`
+ * :class:`ComplexSkewSymmetricEJA`
The Jordan spin algebra is a bilinear form algebra where the bilinear
form is the identity. The Hadamard EJA is simply a Cartesian product
-of one-dimensional spin algebras. And last but not least, the trivial
-EJA is exactly what you think. Cartesian products of these are also
-supported using the usual ``cartesian_product()`` function; as a
-result, we support (up to isomorphism) all Euclidean Jordan algebras
-that don't involve octonions.
+of one-dimensional spin algebras. The Albert EJA is simply a special
+case of the :class:`OctonionHermitianEJA` where the matrices are
+three-by-three and the resulting space has dimension 27. And
+last/least, the trivial EJA is exactly what you think it is; it could
+also be obtained by constructing a dimension-zero instance of any of
+the other algebras. Cartesian products of these are also supported
+using the usual ``cartesian_product()`` function; as a result, we
+support (up to isomorphism) all Euclidean Jordan algebras.
+
+At a minimum, the following are required to construct a Euclidean
+Jordan algebra:
+
+ * A basis of matrices, column vectors, or MatrixAlgebra elements
+ * A Jordan product defined on the basis
+ * Its inner product defined on the basis
+
+The real numbers form a Euclidean Jordan algebra when both the Jordan
+and inner products are the usual multiplication. We use this as our
+example, and demonstrate a few ways to construct an EJA.
+
+First, we can use one-by-one SageMath matrices with algebraic real
+entries to represent real numbers. We define the Jordan and inner
+products to be essentially real-number multiplication, with the only
+difference being that the Jordan product again returns a one-by-one
+matrix, whereas the inner product must return a scalar. Our basis for
+the one-by-one matrices is of course the set consisting of a single
+matrix with its sole entry non-zero::
+
+ sage: from mjo.eja.eja_algebra import EJA
+ sage: jp = lambda X,Y: X*Y
+ sage: ip = lambda X,Y: X[0,0]*Y[0,0]
+ sage: b1 = matrix(AA, [[1]])
+ sage: J1 = EJA((b1,), jp, ip)
+ sage: J1
+ Euclidean Jordan algebra of dimension 1 over Algebraic Real Field
+
+In fact, any positive scalar multiple of that inner-product would work::
+
+ sage: ip2 = lambda X,Y: 16*ip(X,Y)
+ sage: J2 = EJA((b1,), jp, ip2)
+ sage: J2
+ Euclidean Jordan algebra of dimension 1 over Algebraic Real Field
+
+But beware that your basis will be orthonormalized _with respect to the
+given inner-product_ unless you pass ``orthonormalize=False`` to the
+constructor. For example::
+
+ sage: J3 = EJA((b1,), jp, ip2, orthonormalize=False)
+ sage: J3
+ Euclidean Jordan algebra of dimension 1 over Algebraic Real Field
+
+To see the difference, you can take the first and only basis element
+of the resulting algebra, and ask for it to be converted back into
+matrix form::
+
+ sage: J1.basis()[0].to_matrix()
+ [1]
+ sage: J2.basis()[0].to_matrix()
+ [1/4]
+ sage: J3.basis()[0].to_matrix()
+ [1]
+
+Since square roots are used in that process, the default scalar field
+that we use is the field of algebraic real numbers, ``AA``. You can
+also Use rational numbers, but only if you either pass
+``orthonormalize=False`` or know that orthonormalizing your basis
+won't stray beyond the rational numbers. The example above would
+have worked only because ``sqrt(16) == 4`` is rational.
+
+Another option for your basis is to use elemebts of a
+:class:`MatrixAlgebra`::
+
+ sage: from mjo.matrix_algebra import MatrixAlgebra
+ sage: A = MatrixAlgebra(1,AA,AA)
+ sage: J4 = EJA(A.gens(), jp, ip)
+ sage: J4
+ Euclidean Jordan algebra of dimension 1 over Algebraic Real Field
+ sage: J4.basis()[0].to_matrix()
+ +---+
+ | 1 |
+ +---+
+
+An easier way to view the entire EJA basis in its original (but
+perhaps orthonormalized) matrix form is to use the ``matrix_basis``
+method::
+
+ sage: J4.matrix_basis()
+ (+---+
+ | 1 |
+ +---+,)
+
+In particular, a :class:`MatrixAlgebra` is needed to work around the
+fact that matrices in SageMath must have entries in the same
+(commutative and associative) ring as its scalars. There are many
+Euclidean Jordan algebras whose elements are matrices that violate
+those assumptions. The complex, quaternion, and octonion Hermitian
+matrices all have entries in a ring (the complex numbers, quaternions,
+or octonions...) that differs from the algebra's scalar ring (the real
+numbers). Quaternions are also non-commutative; the octonions are
+neither commutative nor associative.
SETUP::
Euclidean Jordan algebra of dimension...
"""
-from itertools import repeat
-
from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra
from sage.categories.magmatic_algebras import MagmaticAlgebras
from sage.categories.sets_cat import cartesian_product
from sage.rings.all import (ZZ, QQ, AA, QQbar, RR, RLF, CLF,
PolynomialRing,
QuadraticField)
-from mjo.eja.eja_element import FiniteDimensionalEJAElement
-from mjo.eja.eja_operator import FiniteDimensionalEJAOperator
-from mjo.eja.eja_utils import _all2list, _mat2vec
+from mjo.eja.eja_element import (CartesianProductEJAElement,
+ EJAElement)
+from mjo.eja.eja_operator import EJAOperator
+from mjo.eja.eja_utils import _all2list
+
+def EuclideanJordanAlgebras(field):
+ r"""
+ The category of Euclidean Jordan algebras over ``field``, which
+ must be a subfield of the real numbers. For now this is just a
+ convenient wrapper around all of the other category axioms that
+ apply to all EJAs.
+ """
+ category = MagmaticAlgebras(field).FiniteDimensional()
+ category = category.WithBasis().Unital().Commutative()
+ return category
-class FiniteDimensionalEJA(CombinatorialFreeModule):
+class EJA(CombinatorialFreeModule):
r"""
A finite-dimensional Euclidean Jordan algebra.
product. This will be applied to ``basis`` to compute an
inner-product table (basically a matrix) for this algebra.
+ - ``matrix_space`` -- the space that your matrix basis lives in,
+ or ``None`` (the default). So long as your basis does not have
+ length zero you can omit this. But in trivial algebras, it is
+ required.
+
- ``field`` -- a subfield of the reals (default: ``AA``); the scalar
field for the algebra.
We should compute that an element subalgebra is associative even
if we circumvent the element method::
- sage: set_random_seed()
sage: J = random_eja(field=QQ,orthonormalize=False)
sage: x = J.random_element()
sage: A = x.subalgebra_generated_by(orthonormalize=False)
sage: basis = tuple(b.superalgebra_element() for b in A.basis())
sage: J.subalgebra(basis, orthonormalize=False).is_associative()
True
-
"""
- Element = FiniteDimensionalEJAElement
+ Element = EJAElement
+
+ @staticmethod
+ def _check_input_field(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")
+
+ @staticmethod
+ def _check_input_axioms(basis, jordan_product, inner_product):
+ if not all( jordan_product(bi,bj) == jordan_product(bj,bi)
+ for bi in basis
+ for bj in basis ):
+ raise ValueError("Jordan product is not commutative")
+
+ if not all( inner_product(bi,bj) == inner_product(bj,bi)
+ for bi in basis
+ for bj in basis ):
+ raise ValueError("inner-product is not commutative")
def __init__(self,
basis,
jordan_product,
inner_product,
field=AA,
+ matrix_space=None,
orthonormalize=True,
associative=None,
- cartesian_product=False,
check_field=True,
check_axioms=True,
prefix="b"):
n = len(basis)
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")
+ self._check_input_field(field)
if check_axioms:
# Check commutativity of the Jordan and inner-products.
# This has to be done before we build the multiplication
# and inner-product tables/matrices, because we take
# advantage of symmetry in the process.
- if not all( jordan_product(bi,bj) == jordan_product(bj,bi)
- for bi in basis
- for bj in basis ):
- raise ValueError("Jordan product is not commutative")
-
- if not all( inner_product(bi,bj) == inner_product(bj,bi)
- for bi in basis
- for bj in basis ):
- raise ValueError("inner-product is not commutative")
-
-
- category = MagmaticAlgebras(field).FiniteDimensional()
- category = category.WithBasis().Unital().Commutative()
+ self._check_input_axioms(basis, jordan_product, inner_product)
+ if n <= 1:
+ # All zero- and one-dimensional algebras are just the real
+ # numbers with (some positive multiples of) the usual
+ # multiplication as its Jordan and inner-product.
+ associative = True
if associative is None:
# We should figure it out. As with check_axioms, we have to do
# this without the help of the _jordan_product_is_associative()
for bj in basis
for bk in basis)
+ category = EuclideanJordanAlgebras(field)
+
if associative:
# Element subalgebras can take advantage of this.
category = category.Associative()
- if cartesian_product:
- # Use join() here because otherwise we only get the
- # "Cartesian product of..." and not the things themselves.
- category = category.join([category,
- category.CartesianProducts()])
# Call the superclass constructor so that we can use its from_vector()
# method to build our multiplication table.
# as well as a subspace W of V spanned by those (vectorized)
# basis elements. The W-coordinates are the coefficients that
# we see in things like x = 1*b1 + 2*b2.
- vector_basis = basis
degree = 0
if n > 0:
# written out as "long vectors."
V = VectorSpace(field, degree)
- # The matrix that will hole the orthonormal -> unorthonormal
- # coordinate transformation.
- self._deortho_matrix = None
+ # The matrix that will hold the orthonormal -> unorthonormal
+ # coordinate transformation. Default to an identity matrix of
+ # the appropriate size to avoid special cases for None
+ # everywhere.
+ self._deortho_matrix = matrix.identity(field,n)
if orthonormalize:
# Save a copy of the un-orthonormalized basis for later.
basis = tuple(gram_schmidt(basis, inner_product))
# Save the (possibly orthonormalized) matrix basis for
- # later...
+ # later, as well as the space that its elements live in.
+ # In most cases we can deduce the matrix space, but when
+ # n == 0 (that is, there are no basis elements) we cannot.
self._matrix_basis = basis
+ if matrix_space is None:
+ self._matrix_space = self._matrix_basis[0].parent()
+ else:
+ self._matrix_space = matrix_space
# Now create the vector space for the algebra, which will have
# its own set of non-ambient coordinates (in terms of the
# supplied basis).
vector_basis = tuple( V(_all2list(b)) for b in basis )
- W = V.span_of_basis( vector_basis, check=check_axioms)
+
+ # Save the span of our matrix basis (when written out as long
+ # vectors) because otherwise we'll have to reconstruct it
+ # every time we want to coerce a matrix into the algebra.
+ self._matrix_span = V.span_of_basis( vector_basis, check=check_axioms)
if orthonormalize:
- # Now "W" is the vector space of our algebra coordinates. The
- # variables "X1", "X2",... refer to the entries of vectors in
- # W. Thus to convert back and forth between the orthonormal
- # coordinates and the given ones, we need to stick the original
- # basis in W.
+ # Now "self._matrix_span" is the vector space of our
+ # algebra coordinates. The variables "X0", "X1",... refer
+ # to the entries of vectors in self._matrix_span. Thus to
+ # convert back and forth between the orthonormal
+ # coordinates and the given ones, we need to stick the
+ # original basis in self._matrix_span.
U = V.span_of_basis( deortho_vector_basis, check=check_axioms)
- self._deortho_matrix = matrix( U.coordinate_vector(q)
- for q in vector_basis )
+ self._deortho_matrix = matrix.column( U.coordinate_vector(q)
+ for q in vector_basis )
# Now we actually compute the multiplication and inner-product
# tables/matrices using the possibly-orthonormalized basis.
self._inner_product_matrix = matrix.identity(field, n)
- self._multiplication_table = [ [0 for j in range(i+1)]
+ zed = self.zero()
+ self._multiplication_table = [ [zed for j in range(i+1)]
for i in range(n) ]
# Note: the Jordan and inner-products are defined in terms
# The jordan product returns a matrixy answer, so we
# have to convert it to the algebra coordinates.
elt = jordan_product(q_i, q_j)
- elt = W.coordinate_vector(V(_all2list(elt)))
+ elt = self._matrix_span.coordinate_vector(V(_all2list(elt)))
self._multiplication_table[i][j] = self.from_vector(elt)
if not orthonormalize:
TESTS::
- sage: set_random_seed()
sage: J = random_eja()
sage: J(1)
Traceback (most recent call last):
TESTS::
- sage: set_random_seed()
sage: J = random_eja()
sage: n = J.dimension()
sage: bi = J.zero()
Our inner product is "associative," which means the following for
a symmetric bilinear form::
- sage: set_random_seed()
sage: J = random_eja()
sage: x,y,z = J.random_elements(3)
sage: (x*y).inner_product(z) == y.inner_product(x*z)
Ensure that this is the usual inner product for the algebras
over `R^n`::
- sage: set_random_seed()
sage: J = HadamardEJA.random_instance()
sage: x,y = J.random_elements(2)
sage: actual = x.inner_product(y)
one). This is in Faraut and Koranyi, and also my "On the
symmetry..." paper::
- sage: set_random_seed()
sage: J = BilinearFormEJA.random_instance()
sage: n = J.dimension()
sage: x = J.random_element()
The values we've presupplied to the constructors agree with
the computation::
- sage: set_random_seed()
sage: J = random_eja()
sage: J.is_associative() == J._jordan_product_is_associative()
True
def _element_constructor_(self, elt):
"""
- Construct an element of this algebra from its vector or matrix
- representation.
+ Construct an element of this algebra or a subalgebra from its
+ EJA element, vector, or matrix representation.
This gets called only after the parent element _call_ method
fails to find a coercion for the argument.
sage: J( (J1.matrix_basis()[1], J2.matrix_basis()[2]) )
b1 + b5
+ Subalgebra elements are embedded into the superalgebra::
+
+ sage: J = JordanSpinEJA(3)
+ sage: J.one()
+ b0
+ sage: x = sum(J.gens())
+ sage: A = x.subalgebra_generated_by()
+ sage: J(A.one())
+ b0
+
TESTS:
Ensure that we can convert any element back and forth
faithfully between its matrix and algebra representations::
- sage: set_random_seed()
sage: J = random_eja()
sage: x = J.random_element()
sage: J(x.to_matrix()) == x
Traceback (most recent call last):
...
ValueError: not an element of this algebra
+
"""
msg = "not an element of this algebra"
if elt in self.base_ring():
# that the integer 3 belongs to the space of 2-by-2 matrices.
raise ValueError(msg)
- try:
- # Try to convert a vector into a column-matrix...
+ if hasattr(elt, 'superalgebra_element'):
+ # Handle subalgebra elements
+ if elt.parent().superalgebra() == self:
+ return elt.superalgebra_element()
+
+ if hasattr(elt, 'sparse_vector'):
+ # Convert a vector into a column-matrix. We check for
+ # "sparse_vector" and not "column" because matrices also
+ # have a "column" method.
elt = elt.column()
- except (AttributeError, TypeError):
- # and ignore failure, because we weren't really expecting
- # a vector as an argument anyway.
- pass
if elt not in self.matrix_space():
raise ValueError(msg)
# is that we're already converting everything to long vectors,
# and that strategy works for tuples as well.
#
- # We pass check=False because the matrix basis is "guaranteed"
- # to be linearly independent... right? Ha ha.
- elt = _all2list(elt)
- V = VectorSpace(self.base_ring(), len(elt))
- W = V.span_of_basis( (V(_all2list(s)) for s in self.matrix_basis()),
- check=False)
+ elt = self._matrix_span.ambient_vector_space()(_all2list(elt))
try:
- coords = W.coordinate_vector(V(elt))
+ coords = self._matrix_span.coordinate_vector(elt)
except ArithmeticError: # vector is not in free module
raise ValueError(msg)
sage: J = JordanSpinEJA(3)
sage: p = J.characteristic_polynomial_of(); p
- X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
+ X0^2 - X1^2 - X2^2 + (-2*t)*X0 + t^2
sage: xvec = J.one().to_vector()
sage: p(*xvec)
t^2 - 2*t + 1
sage: J = HadamardEJA(2)
sage: J.coordinate_polynomial_ring()
- Multivariate Polynomial Ring in X1, X2...
+ Multivariate Polynomial Ring in X0, X1...
sage: J = RealSymmetricEJA(3,field=QQ,orthonormalize=False)
sage: J.coordinate_polynomial_ring()
- Multivariate Polynomial Ring in X1, X2, X3, X4, X5, X6...
+ Multivariate Polynomial Ring in X0, X1, X2, X3, X4, X5...
"""
- var_names = tuple( "X%d" % z for z in range(1, self.dimension()+1) )
+ var_names = tuple( "X%d" % z for z in range(self.dimension()) )
return PolynomialRing(self.base_ring(), var_names)
def inner_product(self, x, y):
Our inner product is "associative," which means the following for
a symmetric bilinear form::
- sage: set_random_seed()
sage: J = random_eja()
sage: x,y,z = J.random_elements(3)
sage: (x*y).inner_product(z) == y.inner_product(x*z)
Ensure that this is the usual inner product for the algebras
over `R^n`::
- sage: set_random_seed()
sage: J = HadamardEJA.random_instance()
sage: x,y = J.random_elements(2)
sage: actual = x.inner_product(y)
one). This is in Faraut and Koranyi, and also my "On the
symmetry..." paper::
- sage: set_random_seed()
sage: J = BilinearFormEJA.random_instance()
sage: n = J.dimension()
sage: x = J.random_element()
sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
sage: J.matrix_space()
- Full MatrixSpace of 4 by 4 dense matrices over Rational Field
+ Module of 2 by 2 matrices with entries in Algebraic Field over
+ the scalar ring Rational Field
sage: J = QuaternionHermitianEJA(1,field=QQ,orthonormalize=False)
sage: J.matrix_space()
- Full MatrixSpace of 4 by 4 dense matrices over Rational Field
+ Module of 1 by 1 matrices with entries in Quaternion
+ Algebra (-1, -1) with base ring Rational Field over
+ the scalar ring Rational Field
"""
- if self.is_trivial():
- return MatrixSpace(self.base_ring(), 0)
- else:
- return self.matrix_basis()[0].parent()
+ return self._matrix_space
@cached_method
The identity element acts like the identity, regardless of
whether or not we orthonormalize::
- sage: set_random_seed()
sage: J = random_eja()
sage: x = J.random_element()
sage: J.one()*x == x and x*J.one() == x
True
- sage: A = x.subalgebra_generated_by()
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
sage: y = A.random_element()
sage: A.one()*y == y and y*A.one() == y
True
::
- sage: set_random_seed()
sage: J = random_eja(field=QQ, orthonormalize=False)
sage: x = J.random_element()
sage: J.one()*x == x and x*J.one() == x
regardless of the base field and whether or not we
orthonormalize::
- sage: set_random_seed()
sage: J = random_eja()
sage: actual = J.one().operator().matrix()
sage: expected = matrix.identity(J.base_ring(), J.dimension())
sage: actual == expected
True
sage: x = J.random_element()
- sage: A = x.subalgebra_generated_by()
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
sage: actual = A.one().operator().matrix()
sage: expected = matrix.identity(A.base_ring(), A.dimension())
sage: actual == expected
::
- sage: set_random_seed()
sage: J = random_eja(field=QQ, orthonormalize=False)
sage: actual = J.one().operator().matrix()
sage: expected = matrix.identity(J.base_ring(), J.dimension())
Ensure that the cached unit element (often precomputed by
hand) agrees with the computed one::
- sage: set_random_seed()
sage: J = random_eja()
sage: cached = J.one()
sage: J.one.clear_cache()
::
- sage: set_random_seed()
sage: J = random_eja(field=QQ, orthonormalize=False)
sage: cached = J.one()
sage: J.one.clear_cache()
#
# Of course, matrices aren't vectors in sage, so we have to
# appeal to the "long vectors" isometry.
- oper_vecs = [ _mat2vec(g.operator().matrix()) for g in self.gens() ]
+
+ V = VectorSpace(self.base_ring(), self.dimension()**2)
+ oper_vecs = [ V(g.operator().matrix().list()) for g in self.gens() ]
# Now we use basic linear algebra to find the coefficients,
# of the matrices-as-vectors-linear-combination, which should
# We used the isometry on the left-hand side already, but we
# still need to do it for the right-hand side. Recall that we
# wanted something that summed to the identity matrix.
- b = _mat2vec( matrix.identity(self.base_ring(), self.dimension()) )
+ b = V( matrix.identity(self.base_ring(), self.dimension()).list() )
# Now if there's an identity element in the algebra, this
# should work. We solve on the left to avoid having to
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
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():
# 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 = self.subalgebra(())
+ trivial = self.subalgebra((), check_axioms=False)
J0 = trivial # eigenvalue zero
J5 = VectorSpace(self.base_ring(), 0) # eigenvalue one-half
J1 = trivial # eigenvalue one
# 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()
+ if self.base_ring() is AA and not thorough:
+ # Now that AA generates actually random random elements
+ # (post Trac 30875), we only need to de-thorough the
+ # randomness when asked to.
+ V = V.change_ring(QQ)
+ v = V.random_element()
return self.from_vector(V.coordinate_vector(v))
def random_elements(self, count, thorough=False):
for idx in range(count) )
+ def operator_polynomial_matrix(self):
+ r"""
+ Return the matrix of polynomials (over this algebra's
+ :meth:`coordinate_polynomial_ring`) that, when evaluated at
+ the basis coordinates of an element `x`, produces the basis
+ representation of `L_{x}`.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: JordanSpinEJA)
+
+ EXAMPLES::
+
+ sage: J = HadamardEJA(4)
+ sage: L_x = J.operator_polynomial_matrix()
+ sage: L_x
+ [X0 0 0 0]
+ [ 0 X1 0 0]
+ [ 0 0 X2 0]
+ [ 0 0 0 X3]
+ sage: x = J.one()
+ sage: d = zip(J.coordinate_polynomial_ring().gens(), x.to_vector())
+ sage: L_x.subs(dict(d))
+ [1 0 0 0]
+ [0 1 0 0]
+ [0 0 1 0]
+ [0 0 0 1]
+
+ ::
+
+ sage: J = JordanSpinEJA(4)
+ sage: L_x = J.operator_polynomial_matrix()
+ sage: L_x
+ [X0 X1 X2 X3]
+ [X1 X0 0 0]
+ [X2 0 X0 0]
+ [X3 0 0 X0]
+ sage: x = J.one()
+ sage: d = zip(J.coordinate_polynomial_ring().gens(), x.to_vector())
+ sage: L_x.subs(dict(d))
+ [1 0 0 0]
+ [0 1 0 0]
+ [0 0 1 0]
+ [0 0 0 1]
+
+ """
+ R = self.coordinate_polynomial_ring()
+
+ 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( v*self.monomial(k).operator().matrix()[i,j]
+ for (k,v) in enumerate(R.gens()) )
+
+ n = self.dimension()
+ return matrix(R, n, n, L_x_i_j)
+
@cached_method
def _charpoly_coefficients(self):
r"""
The theory shows that these are all homogeneous polynomials of
a known degree::
- sage: set_random_seed()
sage: J = random_eja()
sage: all(p.is_homogeneous() for p in J._charpoly_coefficients())
True
"""
n = self.dimension()
R = self.coordinate_polynomial_ring()
- 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)
+ L_x = self.operator_polynomial_matrix()
r = None
if self.rank.is_in_cache():
positive integer rank, unless the algebra is trivial in
which case its rank will be zero::
- sage: set_random_seed()
sage: J = random_eja()
sage: r = J.rank()
sage: r in ZZ
Ensure that computing the rank actually works, since the ranks
of all simple algebras are known and will be cached by default::
- sage: set_random_seed() # long time
sage: J = random_eja() # long time
sage: cached = J.rank() # long time
sage: J.rank.clear_cache() # long time
r"""
Create a subalgebra of this algebra from the given basis.
"""
- from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra
- return FiniteDimensionalEJASubalgebra(self, basis, **kwargs)
+ from mjo.eja.eja_subalgebra import EJASubalgebra
+ return EJASubalgebra(self, basis, **kwargs)
def vector_space(self):
-class RationalBasisEJA(FiniteDimensionalEJA):
+class RationalBasisEJA(EJA):
r"""
- New class for algebras whose supplied basis elements have all rational entries.
+ Algebras whose supplied basis elements have all rational entries.
SETUP::
if check_field:
# Abuse the check_field parameter to check that the entries of
# out basis (in ambient coordinates) are in the field QQ.
- if not all( all(b_i in QQ for b_i in b.list()) for b in basis ):
+ # Use _all2list to get the vector coordinates of octonion
+ # entries and not the octonions themselves (which are not
+ # rational).
+ if not all( all(b_i in QQ for b_i in _all2list(b))
+ for b in basis ):
raise TypeError("basis not rational")
super().__init__(basis,
# Note: the same Jordan and inner-products work here,
# because they are necessarily defined with respect to
# ambient coordinates and not any particular basis.
- self._rational_algebra = FiniteDimensionalEJA(
+ self._rational_algebra = EJA(
basis,
jordan_product,
inner_product,
field=QQ,
+ matrix_space=self.matrix_space(),
associative=self.is_associative(),
orthonormalize=False,
check_field=False,
check_axioms=False)
+ def rational_algebra(self):
+ # Using None as a flag here (rather than just assigning "self"
+ # to self._rational_algebra by default) feels a little bit
+ # more sane to me in a garbage-collected environment.
+ if self._rational_algebra is None:
+ return self
+ else:
+ return self._rational_algebra
+
@cached_method
def _charpoly_coefficients(self):
r"""
sage: J = JordanSpinEJA(3)
sage: J._charpoly_coefficients()
- (X1^2 - X2^2 - X3^2, -2*X1)
+ (X0^2 - X1^2 - X2^2, -2*X0)
sage: a0 = J._charpoly_coefficients()[0]
sage: J.base_ring()
Algebraic Real Field
Algebraic Real Field
"""
- if self._rational_algebra is None:
- # There's no need to construct *another* algebra over the
- # rationals if this one is already over the
- # rationals. Likewise, if we never orthonormalized our
- # basis, we might as well just use the given one.
+ if self.rational_algebra() is self:
+ # Bypass the hijinks if they won't benefit us.
return super()._charpoly_coefficients()
- # Do the computation over the rationals. The answer will be
- # the same, because all we've done is a change of basis.
- # Then, change back from QQ to our real base ring
+ # Do the computation over the rationals.
a = ( a_i.change_ring(self.base_ring())
- for a_i in self._rational_algebra._charpoly_coefficients() )
+ for a_i in self.rational_algebra()._charpoly_coefficients() )
- if self._deortho_matrix is None:
- # This can happen if our base ring was, say, AA and we
- # chose not to (or didn't need to) orthonormalize. It's
- # still faster to do the computations over QQ even if
- # the numbers in the boxes stay the same.
- return tuple(a)
-
- # Otherwise, convert the coordinate variables back to the
- # deorthonormalized ones.
+ # Convert our coordinate variables into deorthonormalized ones
+ # and substitute them into the deorthonormalized charpoly
+ # coefficients.
R = self.coordinate_polynomial_ring()
from sage.modules.free_module_element import vector
X = vector(R, R.gens())
subs_dict = { X[i]: BX[i] for i in range(len(X)) }
return tuple( a_i.subs(subs_dict) for a_i in a )
-class ConcreteEJA(RationalBasisEJA):
+class ConcreteEJA(EJA):
r"""
A class for the Euclidean Jordan algebras that we know by name.
Our basis is normalized with respect to the algebra's inner
product, unless we specify otherwise::
- sage: set_random_seed()
sage: J = ConcreteEJA.random_instance()
sage: all( b.norm() == 1 for b in J.gens() )
True
natural->EJA basis representation is an isometry and within the
EJA the operator is self-adjoint by the Jordan axiom::
- sage: set_random_seed()
sage: J = ConcreteEJA.random_instance()
sage: x = J.random_element()
sage: x.operator().is_self_adjoint()
"""
@staticmethod
- def _max_random_instance_size():
+ def _max_random_instance_dimension():
+ r"""
+ The maximum dimension of any random instance. Ten dimensions seems
+ to be about the point where everything takes a turn for the
+ worse. And dimension ten (but not nine) allows the 4-by-4 real
+ Hermitian matrices, the 2-by-2 quaternion Hermitian matrices,
+ and the 2-by-2 octonion Hermitian matrices.
+ """
+ return 10
+
+ @staticmethod
+ def _max_random_instance_size(max_dimension):
"""
Return an integer "size" that is an upper bound on the size of
- this algebra when it is used in a random test
- case. Unfortunately, the term "size" is ambiguous -- when
- dealing with `R^n` under either the Hadamard or Jordan spin
- product, the "size" refers to the dimension `n`. When dealing
- with a matrix algebra (real symmetric or complex/quaternion
- Hermitian), it refers to the size of the matrix, which is far
- less than the dimension of the underlying vector space.
+ this algebra when it is used in a random test case. This size
+ (which can be passed to the algebra's constructor) is itself
+ based on the ``max_dimension`` parameter.
This method must be implemented in each subclass.
"""
raise NotImplementedError
@classmethod
- def random_instance(cls, *args, **kwargs):
+ def random_instance(cls, max_dimension=None, *args, **kwargs):
"""
- Return a random instance of this type of algebra.
+ Return a random instance of this type of algebra whose dimension
+ is less than or equal to the lesser of ``max_dimension`` and
+ the value returned by ``_max_random_instance_dimension()``. If
+ the dimension bound is omitted, then only the
+ ``_max_random_instance_dimension()`` is used as a bound.
This method should be implemented in each subclass.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import ConcreteEJA
+
+ TESTS:
+
+ Both the class bound and the ``max_dimension`` argument are upper
+ bounds on the dimension of the algebra returned::
+
+ sage: from sage.misc.prandom import choice
+ sage: eja_class = choice(ConcreteEJA.__subclasses__())
+ sage: class_max_d = eja_class._max_random_instance_dimension()
+ sage: J = eja_class.random_instance(max_dimension=20,
+ ....: field=QQ,
+ ....: orthonormalize=False)
+ sage: J.dimension() <= class_max_d
+ True
+ sage: J = eja_class.random_instance(max_dimension=2,
+ ....: field=QQ,
+ ....: orthonormalize=False)
+ sage: J.dimension() <= 2
+ True
+
"""
from sage.misc.prandom import choice
eja_class = choice(cls.__subclasses__())
# These all bubble up to the RationalBasisEJA superclass
# constructor, so any (kw)args valid there are also valid
# here.
- return eja_class.random_instance(*args, **kwargs)
+ return eja_class.random_instance(max_dimension, *args, **kwargs)
-class MatrixEJA:
+class HermitianMatrixEJA(EJA):
@staticmethod
- def jordan_product(X,Y):
- return (X*Y + Y*X)/2
-
- @staticmethod
- def trace_inner_product(X,Y):
- r"""
- A trace inner-product for matrices that aren't embedded in the
- reals.
+ def _denormalized_basis(A):
"""
- # We take the norm (absolute value) because Octonions() isn't
- # smart enough yet to coerce its one() into the base field.
- return (X*Y).trace().abs()
+ Returns a basis for the given Hermitian matrix space.
-class RealEmbeddedMatrixEJA(MatrixEJA):
- @staticmethod
- def dimension_over_reals():
- r"""
- The dimension of this matrix's base ring over the reals.
+ Why do we embed these? Basically, because all of numerical linear
+ algebra assumes that you're working with vectors consisting of `n`
+ entries from a field and scalars from the same field. There's no way
+ to tell SageMath that (for example) the vectors contain complex
+ numbers, while the scalar field is real.
- The reals are dimension one over themselves, obviously; that's
- just `\mathbb{R}^{1}`. Likewise, the complex numbers `a + bi`
- have dimension two. Finally, the quaternions have dimension
- four over the reals.
+ SETUP::
- This is used to determine the size of the matrix returned from
- :meth:`real_embed`, among other things.
- """
- raise NotImplementedError
+ sage: from mjo.hurwitz import (ComplexMatrixAlgebra,
+ ....: QuaternionMatrixAlgebra,
+ ....: OctonionMatrixAlgebra)
+ sage: from mjo.eja.eja_algebra import HermitianMatrixEJA
- @classmethod
- def real_embed(cls,M):
- """
- Embed the matrix ``M`` into a space of real matrices.
+ TESTS::
- The matrix ``M`` can have entries in any field at the moment:
- the real numbers, complex numbers, or quaternions. And although
- they are not a field, we can probably support octonions at some
- point, too. This function returns a real matrix that "acts like"
- the original with respect to matrix multiplication; i.e.
+ sage: n = ZZ.random_element(1,5)
+ sage: A = MatrixSpace(QQ, n)
+ sage: B = HermitianMatrixEJA._denormalized_basis(A)
+ sage: all( M.is_hermitian() for M in B)
+ True
- real_embed(M*N) = real_embed(M)*real_embed(N)
+ ::
- """
- if M.ncols() != M.nrows():
- raise ValueError("the matrix 'M' must be square")
- return M
+ sage: n = ZZ.random_element(1,5)
+ sage: A = ComplexMatrixAlgebra(n, scalars=QQ)
+ sage: B = HermitianMatrixEJA._denormalized_basis(A)
+ sage: all( M.is_hermitian() for M in B)
+ True
+ ::
+
+ sage: n = ZZ.random_element(1,5)
+ sage: A = QuaternionMatrixAlgebra(n, scalars=QQ)
+ sage: B = HermitianMatrixEJA._denormalized_basis(A)
+ sage: all( M.is_hermitian() for M in B )
+ True
+
+ ::
+
+ sage: n = ZZ.random_element(1,5)
+ sage: A = OctonionMatrixAlgebra(n, scalars=QQ)
+ sage: B = HermitianMatrixEJA._denormalized_basis(A)
+ sage: all( M.is_hermitian() for M in B )
+ True
- @classmethod
- def real_unembed(cls,M):
- """
- The inverse of :meth:`real_embed`.
"""
- if M.ncols() != M.nrows():
- raise ValueError("the matrix 'M' must be square")
- if not ZZ(M.nrows()).mod(cls.dimension_over_reals()).is_zero():
- raise ValueError("the matrix 'M' must be a real embedding")
- return M
+ # These work for real MatrixSpace, whose monomials only have
+ # two coordinates (because the last one would always be "1").
+ es = A.base_ring().gens()
+ gen = lambda A,m: A.monomial(m[:2])
+ if hasattr(A, 'entry_algebra_gens'):
+ # We've got a MatrixAlgebra, and its monomials will have
+ # three coordinates.
+ es = A.entry_algebra_gens()
+ gen = lambda A,m: A.monomial(m)
- @classmethod
- def trace_inner_product(cls,X,Y):
+ basis = []
+ for i in range(A.nrows()):
+ for j in range(i+1):
+ if i == j:
+ E_ii = gen(A, (i,j,es[0]))
+ basis.append(E_ii)
+ else:
+ for e in es:
+ E_ij = gen(A, (i,j,e))
+ E_ij += E_ij.conjugate_transpose()
+ basis.append(E_ij)
+
+ return tuple( basis )
+
+ @staticmethod
+ def jordan_product(X,Y):
+ return (X*Y + Y*X)/2
+
+ @staticmethod
+ def trace_inner_product(X,Y):
r"""
- Compute the trace inner-product of two real-embeddings.
+ A trace inner-product for matrices that aren't embedded in the
+ reals. It takes MATRICES as arguments, not EJA elements.
SETUP::
- sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
- ....: QuaternionHermitianEJA)
+ sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
+ ....: ComplexHermitianEJA,
+ ....: QuaternionHermitianEJA,
+ ....: OctonionHermitianEJA)
EXAMPLES::
- sage: set_random_seed()
- sage: J = ComplexHermitianEJA.random_instance()
- sage: x,y = J.random_elements(2)
- sage: Xe = x.to_matrix()
- sage: Ye = y.to_matrix()
- sage: X = J.real_unembed(Xe)
- sage: Y = J.real_unembed(Ye)
- sage: expected = (X*Y).trace().real()
- sage: actual = J.trace_inner_product(Xe,Ye)
- sage: actual == expected
- True
+ sage: J = RealSymmetricEJA(2,field=QQ,orthonormalize=False)
+ sage: I = J.one().to_matrix()
+ sage: J.trace_inner_product(I, -I)
+ -2
::
- sage: set_random_seed()
- sage: J = QuaternionHermitianEJA.random_instance()
- sage: x,y = J.random_elements(2)
- sage: Xe = x.to_matrix()
- sage: Ye = y.to_matrix()
- sage: X = J.real_unembed(Xe)
- sage: Y = J.real_unembed(Ye)
- sage: expected = (X*Y).trace().coefficient_tuple()[0]
- sage: actual = J.trace_inner_product(Xe,Ye)
- sage: actual == expected
- True
+ sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: I = J.one().to_matrix()
+ sage: J.trace_inner_product(I, -I)
+ -2
+
+ ::
+
+ sage: J = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: I = J.one().to_matrix()
+ sage: J.trace_inner_product(I, -I)
+ -2
+
+ ::
+
+ sage: J = OctonionHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: I = J.one().to_matrix()
+ sage: J.trace_inner_product(I, -I)
+ -2
"""
- # This does in fact compute the real part of the trace.
- # If we compute the trace of e.g. a complex matrix M,
- # then we do so by adding up its diagonal entries --
- # call them z_1 through z_n. The real embedding of z_1
- # will be a 2-by-2 REAL matrix [a, b; -b, a] whose trace
- # as a REAL matrix will be 2*a = 2*Re(z_1). And so forth.
- return (X*Y).trace()/cls.dimension_over_reals()
+ tr = (X*Y).trace()
+ if hasattr(tr, 'coefficient'):
+ # Works for octonions, and has to come first because they
+ # also have a "real()" method that doesn't return an
+ # element of the scalar ring.
+ return tr.coefficient(0)
+ elif hasattr(tr, 'coefficient_tuple'):
+ # Works for quaternions.
+ return tr.coefficient_tuple()[0]
+
+ # Works for real and complex numbers.
+ return tr.real()
+
+
+ def __init__(self, matrix_space, **kwargs):
+ # We know this is a valid EJA, but will double-check
+ # if the user passes check_axioms=True.
+ if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
-class RealSymmetricEJA(ConcreteEJA, MatrixEJA):
+ super().__init__(self._denormalized_basis(matrix_space),
+ self.jordan_product,
+ self.trace_inner_product,
+ field=matrix_space.base_ring(),
+ matrix_space=matrix_space,
+ **kwargs)
+
+ self.rank.set_cache(matrix_space.nrows())
+ self.one.set_cache( self(matrix_space.one()) )
+
+class RealSymmetricEJA(HermitianMatrixEJA, RationalBasisEJA, ConcreteEJA):
"""
The rank-n simple EJA consisting of real symmetric n-by-n
matrices, the usual symmetric Jordan product, and the trace inner
The dimension of this algebra is `(n^2 + n) / 2`::
- sage: set_random_seed()
- sage: n_max = RealSymmetricEJA._max_random_instance_size()
- sage: n = ZZ.random_element(1, n_max)
+ sage: d = RealSymmetricEJA._max_random_instance_dimension()
+ sage: n = RealSymmetricEJA._max_random_instance_size(d)
sage: J = RealSymmetricEJA(n)
sage: J.dimension() == (n^2 + n)/2
True
The Jordan multiplication is what we think it is::
- sage: set_random_seed()
sage: J = RealSymmetricEJA.random_instance()
sage: x,y = J.random_elements(2)
sage: actual = (x*y).to_matrix()
Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
"""
- @classmethod
- def _denormalized_basis(cls, n, field):
- """
- Return a basis for the space of real symmetric n-by-n matrices.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import RealSymmetricEJA
-
- TESTS::
-
- sage: set_random_seed()
- sage: n = ZZ.random_element(1,5)
- sage: B = RealSymmetricEJA._denormalized_basis(n,ZZ)
- sage: all( M.is_symmetric() for M in B)
- True
-
- """
- # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
- # coordinates.
- S = []
- 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)
-
-
@staticmethod
- def _max_random_instance_size():
- return 4 # Dimension 10
+ def _max_random_instance_size(max_dimension):
+ # Obtained by solving d = (n^2 + n)/2.
+ # The ZZ-int-ZZ thing is just "floor."
+ return ZZ(int(ZZ(8*max_dimension + 1).sqrt()/2 - 1/2))
@classmethod
- def random_instance(cls, **kwargs):
+ def random_instance(cls, max_dimension=None, *args, **kwargs):
"""
Return a random instance of this type of algebra.
"""
- n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ class_max_d = cls._max_random_instance_dimension()
+ if (max_dimension is None or max_dimension > class_max_d):
+ max_dimension = class_max_d
+ max_size = cls._max_random_instance_size(max_dimension)
+ n = ZZ.random_element(max_size + 1)
return cls(n, **kwargs)
def __init__(self, n, field=AA, **kwargs):
- # We know this is a valid EJA, but will double-check
- # if the user passes check_axioms=True.
- if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
+ A = MatrixSpace(field, n)
+ super().__init__(A, **kwargs)
- associative = False
- if n <= 1:
- associative = True
+ from mjo.eja.eja_cache import real_symmetric_eja_coeffs
+ a = real_symmetric_eja_coeffs(self)
+ if a is not None:
+ self.rational_algebra()._charpoly_coefficients.set_cache(a)
- super().__init__(self._denormalized_basis(n,field),
- self.jordan_product,
- self.trace_inner_product,
- field=field,
- associative=associative,
- **kwargs)
- # TODO: this could be factored out somehow, but is left here
- # because the MatrixEJA is not presently a subclass of the
- # FDEJA class that defines rank() and one().
- self.rank.set_cache(n)
- idV = self.matrix_space().one()
- self.one.set_cache(self(idV))
+class ComplexHermitianEJA(HermitianMatrixEJA, RationalBasisEJA, ConcreteEJA):
+ """
+ The rank-n simple EJA consisting of complex Hermitian n-by-n
+ matrices over the real numbers, the usual symmetric Jordan product,
+ and the real-part-of-trace inner product. It has dimension `n^2` over
+ the reals.
+ SETUP::
-class ComplexMatrixEJA(RealEmbeddedMatrixEJA):
- # A manual dictionary-cache for the complex_extension() method,
- # since apparently @classmethods can't also be @cached_methods.
- _complex_extension = {}
+ sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
- @classmethod
- def complex_extension(cls,field):
- r"""
- The complex field that we embed/unembed, as an extension
- of the given ``field``.
- """
- if field in cls._complex_extension:
- return cls._complex_extension[field]
-
- # Sage doesn't know how to adjoin the complex "i" (the root of
- # x^2 + 1) to a field in a general way. Here, we just enumerate
- # all of the cases that I have cared to support so far.
- 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
- elif not field.is_exact():
- # RDF or RR
- F = field.complex_field()
- else:
- # Works for QQ and... maybe some other fields.
- R = PolynomialRing(field, 'z')
- z = R.gen()
- F = field.extension(z**2 + 1, 'I', embedding=CLF(-1).sqrt())
+ EXAMPLES:
- cls._complex_extension[field] = F
- return F
+ In theory, our "field" can be any subfield of the reals, but we
+ can't use inexact real fields at the moment because SageMath
+ doesn't know how to convert their elements into complex numbers,
+ or even into algebraic reals::
- @staticmethod
- def dimension_over_reals():
- return 2
+ sage: QQbar(RDF(1))
+ Traceback (most recent call last):
+ ...
+ TypeError: Illegal initializer for algebraic number
+ sage: AA(RR(1))
+ Traceback (most recent call last):
+ ...
+ TypeError: Illegal initializer for algebraic number
- @classmethod
- def real_embed(cls,M):
- """
- Embed the n-by-n complex matrix ``M`` into the space of real
- matrices of size 2n-by-2n via the map the sends each entry `z = a +
- bi` to the block matrix ``[[a,b],[-b,a]]``.
+ TESTS:
- SETUP::
+ The dimension of this algebra is `n^2`::
- sage: from mjo.eja.eja_algebra import ComplexMatrixEJA
+ sage: d = ComplexHermitianEJA._max_random_instance_dimension()
+ sage: n = ComplexHermitianEJA._max_random_instance_size(d)
+ sage: J = ComplexHermitianEJA(n)
+ sage: J.dimension() == n^2
+ True
- EXAMPLES::
+ The Jordan multiplication is what we think it is::
- sage: F = QuadraticField(-1, 'I')
- sage: x1 = F(4 - 2*i)
- sage: x2 = F(1 + 2*i)
- sage: x3 = F(-i)
- sage: x4 = F(6)
- sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
- sage: ComplexMatrixEJA.real_embed(M)
- [ 4 -2| 1 2]
- [ 2 4|-2 1]
- [-----+-----]
- [ 0 -1| 6 0]
- [ 1 0| 0 6]
+ sage: J = ComplexHermitianEJA.random_instance()
+ sage: x,y = J.random_elements(2)
+ sage: actual = (x*y).to_matrix()
+ sage: X = x.to_matrix()
+ sage: Y = y.to_matrix()
+ sage: expected = (X*Y + Y*X)/2
+ sage: actual == expected
+ True
+ sage: J(expected) == x*y
+ True
- TESTS:
+ We can change the generator prefix::
- Embedding is a homomorphism (isomorphism, in fact)::
-
- sage: set_random_seed()
- sage: n = ZZ.random_element(3)
- sage: F = QuadraticField(-1, 'I')
- sage: X = random_matrix(F, n)
- sage: Y = random_matrix(F, n)
- sage: Xe = ComplexMatrixEJA.real_embed(X)
- sage: Ye = ComplexMatrixEJA.real_embed(Y)
- sage: XYe = ComplexMatrixEJA.real_embed(X*Y)
- sage: Xe*Ye == XYe
- True
+ sage: ComplexHermitianEJA(2, prefix='z').gens()
+ (z0, z1, z2, z3)
- """
- super().real_embed(M)
- n = M.nrows()
+ We can construct the (trivial) algebra of rank zero::
- # We don't need any adjoined elements...
- field = M.base_ring().base_ring()
+ sage: ComplexHermitianEJA(0)
+ Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
- blocks = []
- for z in M.list():
- a = z.real()
- b = z.imag()
- blocks.append(matrix(field, 2, [ [ a, b],
- [-b, a] ]))
+ """
+ def __init__(self, n, field=AA, **kwargs):
+ from mjo.hurwitz import ComplexMatrixAlgebra
+ A = ComplexMatrixAlgebra(n, scalars=field)
+ super().__init__(A, **kwargs)
- return matrix.block(field, n, blocks)
+ from mjo.eja.eja_cache import complex_hermitian_eja_coeffs
+ a = complex_hermitian_eja_coeffs(self)
+ if a is not None:
+ self.rational_algebra()._charpoly_coefficients.set_cache(a)
+ @staticmethod
+ def _max_random_instance_size(max_dimension):
+ # Obtained by solving d = n^2.
+ # The ZZ-int-ZZ thing is just "floor."
+ return ZZ(int(ZZ(max_dimension).sqrt()))
@classmethod
- def real_unembed(cls,M):
+ def random_instance(cls, max_dimension=None, *args, **kwargs):
"""
- The inverse of _embed_complex_matrix().
+ Return a random instance of this type of algebra.
+ """
+ class_max_d = cls._max_random_instance_dimension()
+ if (max_dimension is None or max_dimension > class_max_d):
+ max_dimension = class_max_d
+ max_size = cls._max_random_instance_size(max_dimension)
+ n = ZZ.random_element(max_size + 1)
+ return cls(n, **kwargs)
- SETUP::
- sage: from mjo.eja.eja_algebra import ComplexMatrixEJA
-
- EXAMPLES::
-
- sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
- ....: [-2, 1, -4, 3],
- ....: [ 9, 10, 11, 12],
- ....: [-10, 9, -12, 11] ])
- sage: ComplexMatrixEJA.real_unembed(A)
- [ 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: M = random_matrix(F, 3)
- sage: Me = ComplexMatrixEJA.real_embed(M)
- sage: ComplexMatrixEJA.real_unembed(Me) == M
- True
-
- """
- super().real_unembed(M)
- n = ZZ(M.nrows())
- d = cls.dimension_over_reals()
- F = cls.complex_extension(M.base_ring())
- 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 range(n/d):
- for j in range(n/d):
- submat = M[d*k:d*k+d,d*j:d*j+d]
- if submat[0,0] != submat[1,1]:
- raise ValueError('bad on-diagonal submatrix')
- if submat[0,1] != -submat[1,0]:
- raise ValueError('bad off-diagonal submatrix')
- z = submat[0,0] + submat[0,1]*i
- elements.append(z)
-
- return matrix(F, n/d, elements)
-
-
-class ComplexHermitianEJA(ConcreteEJA, ComplexMatrixEJA):
- """
- The rank-n simple EJA consisting of complex Hermitian n-by-n
- matrices over the real numbers, the usual symmetric Jordan product,
- and the real-part-of-trace inner product. It has dimension `n^2` over
- the reals.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
-
- EXAMPLES:
-
- In theory, our "field" can be any subfield of the reals::
-
- sage: ComplexHermitianEJA(2, field=RDF, check_axioms=True)
- Euclidean Jordan algebra of dimension 4 over Real Double Field
- sage: ComplexHermitianEJA(2, field=RR, check_axioms=True)
- Euclidean Jordan algebra of dimension 4 over Real Field with
- 53 bits of precision
-
- TESTS:
-
- The dimension of this algebra is `n^2`::
-
- sage: set_random_seed()
- sage: n_max = ComplexHermitianEJA._max_random_instance_size()
- sage: n = ZZ.random_element(1, n_max)
- sage: J = ComplexHermitianEJA(n)
- sage: J.dimension() == n^2
- True
-
- The Jordan multiplication is what we think it is::
-
- sage: set_random_seed()
- sage: J = ComplexHermitianEJA.random_instance()
- sage: x,y = J.random_elements(2)
- sage: actual = (x*y).to_matrix()
- sage: X = x.to_matrix()
- sage: Y = y.to_matrix()
- sage: expected = (X*Y + Y*X)/2
- sage: actual == expected
- True
- sage: J(expected) == x*y
- True
-
- We can change the generator prefix::
-
- sage: ComplexHermitianEJA(2, prefix='z').gens()
- (z0, z1, z2, z3)
-
- 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):
- """
- Returns a basis for the space of complex Hermitian n-by-n matrices.
-
- Why do we embed these? Basically, because all of numerical linear
- algebra assumes that you're working with vectors consisting of `n`
- entries from a field and scalars from the same field. There's no way
- to tell SageMath that (for example) the vectors contain complex
- numbers, while the scalar field is real.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
-
- TESTS::
-
- sage: set_random_seed()
- sage: n = ZZ.random_element(1,5)
- sage: B = ComplexHermitianEJA._denormalized_basis(n,ZZ)
- sage: all( M.is_symmetric() for M in B)
- True
-
- """
- R = PolynomialRing(ZZ, 'z')
- z = R.gen()
- F = ZZ.extension(z**2 + 1, 'I')
- I = F.gen(1)
-
- # This is like the symmetric case, but we need to be careful:
- #
- # * We want conjugate-symmetry, not just symmetry.
- # * The diagonal will (as a result) be real.
- #
- S = []
- Eij = matrix.zero(F,n)
- for i in range(n):
- for j in range(i+1):
- # "build" E_ij
- Eij[i,j] = 1
- if i == j:
- Sij = cls.real_embed(Eij)
- S.append(Sij)
- else:
- # The second one has a minus because it's conjugated.
- Eij[j,i] = 1 # Eij = Eij + Eij.transpose()
- Sij_real = cls.real_embed(Eij)
- S.append(Sij_real)
- # Eij = I*Eij - I*Eij.transpose()
- Eij[i,j] = I
- Eij[j,i] = -I
- Sij_imag = cls.real_embed(Eij)
- S.append(Sij_imag)
- Eij[j,i] = 0
- # "erase" E_ij
- Eij[i,j] = 0
-
- # Since we embedded the entries, we can drop back to the
- # desired real "field" instead of the extension "F".
- return tuple( s.change_ring(field) for s in S )
-
-
- def __init__(self, n, field=AA, **kwargs):
- # We know this is a valid EJA, but will double-check
- # if the user passes check_axioms=True.
- if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
-
- associative = False
- if n <= 1:
- associative = True
-
- super().__init__(self._denormalized_basis(n,field),
- self.jordan_product,
- self.trace_inner_product,
- field=field,
- associative=associative,
- **kwargs)
- # TODO: this could be factored out somehow, but is left here
- # because the MatrixEJA is not presently a subclass of the
- # FDEJA class that defines rank() and one().
- self.rank.set_cache(n)
- idV = matrix.identity(ZZ, self.dimension_over_reals()*n)
- self.one.set_cache(self(idV))
-
- @staticmethod
- def _max_random_instance_size():
- return 3 # Dimension 9
-
- @classmethod
- def random_instance(cls, **kwargs):
- """
- Return a random instance of this type of algebra.
- """
- n = ZZ.random_element(cls._max_random_instance_size() + 1)
- return cls(n, **kwargs)
-
-class QuaternionMatrixEJA(RealEmbeddedMatrixEJA):
-
- # A manual dictionary-cache for the quaternion_extension() method,
- # since apparently @classmethods can't also be @cached_methods.
- _quaternion_extension = {}
-
- @classmethod
- def quaternion_extension(cls,field):
- r"""
- The quaternion field that we embed/unembed, as an extension
- of the given ``field``.
- """
- if field in cls._quaternion_extension:
- return cls._quaternion_extension[field]
-
- Q = QuaternionAlgebra(field,-1,-1)
-
- cls._quaternion_extension[field] = Q
- return Q
-
- @staticmethod
- def dimension_over_reals():
- return 4
-
- @classmethod
- def real_embed(cls,M):
- """
- Embed the n-by-n quaternion matrix ``M`` into the space of real
- matrices of size 4n-by-4n by first sending each quaternion entry `z
- = a + bi + cj + dk` to the block-complex matrix ``[[a + bi,
- c+di],[-c + di, a-bi]]`, and then embedding those into a real
- matrix.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import QuaternionMatrixEJA
-
- EXAMPLES::
-
- sage: Q = QuaternionAlgebra(QQ,-1,-1)
- sage: i,j,k = Q.gens()
- sage: x = 1 + 2*i + 3*j + 4*k
- sage: M = matrix(Q, 1, [[x]])
- sage: QuaternionMatrixEJA.real_embed(M)
- [ 1 2 3 4]
- [-2 1 -4 3]
- [-3 4 1 -2]
- [-4 -3 2 1]
-
- Embedding is a homomorphism (isomorphism, in fact)::
-
- sage: set_random_seed()
- sage: n = ZZ.random_element(2)
- sage: Q = QuaternionAlgebra(QQ,-1,-1)
- sage: X = random_matrix(Q, n)
- sage: Y = random_matrix(Q, n)
- sage: Xe = QuaternionMatrixEJA.real_embed(X)
- sage: Ye = QuaternionMatrixEJA.real_embed(Y)
- sage: XYe = QuaternionMatrixEJA.real_embed(X*Y)
- sage: Xe*Ye == XYe
- True
-
- """
- super().real_embed(M)
- quaternions = M.base_ring()
- n = M.nrows()
-
- F = QuadraticField(-1, 'I')
- i = F.gen()
-
- blocks = []
- for z in M.list():
- t = z.coefficient_tuple()
- a = t[0]
- b = t[1]
- c = t[2]
- d = t[3]
- cplxM = matrix(F, 2, [[ a + b*i, c + d*i],
- [-c + d*i, a - b*i]])
- realM = ComplexMatrixEJA.real_embed(cplxM)
- blocks.append(realM)
-
- # We should have real entries by now, so use the realest field
- # we've got for the return value.
- return matrix.block(quaternions.base_ring(), n, blocks)
-
-
-
- @classmethod
- def real_unembed(cls,M):
- """
- The inverse of _embed_quaternion_matrix().
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import QuaternionMatrixEJA
-
- EXAMPLES::
-
- sage: M = matrix(QQ, [[ 1, 2, 3, 4],
- ....: [-2, 1, -4, 3],
- ....: [-3, 4, 1, -2],
- ....: [-4, -3, 2, 1]])
- sage: QuaternionMatrixEJA.real_unembed(M)
- [1 + 2*i + 3*j + 4*k]
-
- TESTS:
-
- Unembedding is the inverse of embedding::
-
- sage: set_random_seed()
- sage: Q = QuaternionAlgebra(QQ, -1, -1)
- sage: M = random_matrix(Q, 3)
- sage: Me = QuaternionMatrixEJA.real_embed(M)
- sage: QuaternionMatrixEJA.real_unembed(Me) == M
- True
-
- """
- super().real_unembed(M)
- n = ZZ(M.nrows())
- d = cls.dimension_over_reals()
-
- # Use the base ring of the matrix to ensure that its entries can be
- # multiplied by elements of the quaternion algebra.
- Q = cls.quaternion_extension(M.base_ring())
- i,j,k = Q.gens()
-
- # Go top-left to bottom-right (reading order), converting every
- # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
- # quaternion block.
- elements = []
- for l in range(n/d):
- for m in range(n/d):
- submat = ComplexMatrixEJA.real_unembed(
- M[d*l:d*l+d,d*m:d*m+d] )
- 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].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/d, elements)
-
-
-class QuaternionHermitianEJA(ConcreteEJA, QuaternionMatrixEJA):
+class QuaternionHermitianEJA(HermitianMatrixEJA, RationalBasisEJA, ConcreteEJA):
r"""
The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
matrices, the usual symmetric Jordan product, and the
The dimension of this algebra is `2*n^2 - n`::
- sage: set_random_seed()
- sage: n_max = QuaternionHermitianEJA._max_random_instance_size()
- sage: n = ZZ.random_element(1, n_max)
+ sage: d = QuaternionHermitianEJA._max_random_instance_dimension()
+ sage: n = QuaternionHermitianEJA._max_random_instance_size(d)
sage: J = QuaternionHermitianEJA(n)
sage: J.dimension() == 2*(n^2) - n
True
The Jordan multiplication is what we think it is::
- sage: set_random_seed()
sage: J = QuaternionHermitianEJA.random_instance()
sage: x,y = J.random_elements(2)
sage: actual = (x*y).to_matrix()
Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
"""
- @classmethod
- def _denormalized_basis(cls, n, field):
- """
- Returns a basis for the space of quaternion Hermitian n-by-n matrices.
-
- Why do we embed these? Basically, because all of numerical
- linear algebra assumes that you're working with vectors consisting
- of `n` entries from a field and scalars from the same field. There's
- no way to tell SageMath that (for example) the vectors contain
- complex numbers, while the scalar field is real.
-
- SETUP::
+ def __init__(self, n, field=AA, **kwargs):
+ from mjo.hurwitz import QuaternionMatrixAlgebra
+ A = QuaternionMatrixAlgebra(n, scalars=field)
+ super().__init__(A, **kwargs)
- sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
+ from mjo.eja.eja_cache import quaternion_hermitian_eja_coeffs
+ a = quaternion_hermitian_eja_coeffs(self)
+ if a is not None:
+ self.rational_algebra()._charpoly_coefficients.set_cache(a)
- TESTS::
- sage: set_random_seed()
- sage: n = ZZ.random_element(1,5)
- sage: B = QuaternionHermitianEJA._denormalized_basis(n,ZZ)
- sage: all( M.is_symmetric() for M in B )
- True
+ @staticmethod
+ def _max_random_instance_size(max_dimension):
+ r"""
+ The maximum rank of a random QuaternionHermitianEJA.
"""
- Q = QuaternionAlgebra(QQ,-1,-1)
- I,J,K = Q.gens()
+ # Obtained by solving d = 2n^2 - n.
+ # The ZZ-int-ZZ thing is just "floor."
+ return ZZ(int(ZZ(8*max_dimension + 1).sqrt()/4 + 1/4))
- # This is like the symmetric case, but we need to be careful:
- #
- # * We want conjugate-symmetry, not just symmetry.
- # * The diagonal will (as a result) be real.
- #
- S = []
- Eij = matrix.zero(Q,n)
- for i in range(n):
- for j in range(i+1):
- # "build" E_ij
- Eij[i,j] = 1
- if i == j:
- Sij = cls.real_embed(Eij)
- S.append(Sij)
- else:
- # The second, third, and fourth ones have a minus
- # because they're conjugated.
- # Eij = Eij + Eij.transpose()
- Eij[j,i] = 1
- Sij_real = cls.real_embed(Eij)
- S.append(Sij_real)
- # Eij = I*(Eij - Eij.transpose())
- Eij[i,j] = I
- Eij[j,i] = -I
- Sij_I = cls.real_embed(Eij)
- S.append(Sij_I)
- # Eij = J*(Eij - Eij.transpose())
- Eij[i,j] = J
- Eij[j,i] = -J
- Sij_J = cls.real_embed(Eij)
- S.append(Sij_J)
- # Eij = K*(Eij - Eij.transpose())
- Eij[i,j] = K
- Eij[j,i] = -K
- Sij_K = cls.real_embed(Eij)
- S.append(Sij_K)
- Eij[j,i] = 0
- # "erase" E_ij
- Eij[i,j] = 0
-
- # Since we embedded the entries, we can drop back to the
- # desired real "field" instead of the quaternion algebra "Q".
- return tuple( s.change_ring(field) for s in S )
+ @classmethod
+ def random_instance(cls, max_dimension=None, *args, **kwargs):
+ """
+ Return a random instance of this type of algebra.
+ """
+ class_max_d = cls._max_random_instance_dimension()
+ if (max_dimension is None or max_dimension > class_max_d):
+ max_dimension = class_max_d
+ max_size = cls._max_random_instance_size(max_dimension)
+ n = ZZ.random_element(max_size + 1)
+ return cls(n, **kwargs)
+class OctonionHermitianEJA(HermitianMatrixEJA, RationalBasisEJA, ConcreteEJA):
+ r"""
+ SETUP::
- def __init__(self, n, field=AA, **kwargs):
- # We know this is a valid EJA, but will double-check
- # if the user passes check_axioms=True.
- if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
+ sage: from mjo.eja.eja_algebra import (EJA,
+ ....: OctonionHermitianEJA)
+ sage: from mjo.hurwitz import Octonions, OctonionMatrixAlgebra
- associative = False
- if n <= 1:
- associative = True
+ EXAMPLES:
- super().__init__(self._denormalized_basis(n,field),
- self.jordan_product,
- self.trace_inner_product,
- field=field,
- associative=associative,
- **kwargs)
+ The 3-by-3 algebra satisfies the axioms of an EJA::
+
+ sage: OctonionHermitianEJA(3, # long time
+ ....: field=QQ, # long time
+ ....: orthonormalize=False, # long time
+ ....: check_axioms=True) # long time
+ Euclidean Jordan algebra of dimension 27 over Rational Field
+
+ After a change-of-basis, the 2-by-2 algebra has the same
+ multiplication table as the ten-dimensional Jordan spin algebra::
+
+ sage: A = OctonionMatrixAlgebra(2,Octonions(QQ),QQ)
+ sage: b = OctonionHermitianEJA._denormalized_basis(A)
+ sage: basis = (b[0] + b[9],) + b[1:9] + (b[0] - b[9],)
+ sage: jp = OctonionHermitianEJA.jordan_product
+ sage: ip = OctonionHermitianEJA.trace_inner_product
+ sage: J = EJA(basis,
+ ....: jp,
+ ....: ip,
+ ....: field=QQ,
+ ....: orthonormalize=False)
+ sage: J.multiplication_table()
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | * || b0 | b1 | b2 | b3 | b4 | b5 | b6 | b7 | b8 | b9 |
+ +====++====+====+====+====+====+====+====+====+====+====+
+ | b0 || b0 | b1 | b2 | b3 | b4 | b5 | b6 | b7 | b8 | b9 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b1 || b1 | b0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b2 || b2 | 0 | b0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b3 || b3 | 0 | 0 | b0 | 0 | 0 | 0 | 0 | 0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b4 || b4 | 0 | 0 | 0 | b0 | 0 | 0 | 0 | 0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b5 || b5 | 0 | 0 | 0 | 0 | b0 | 0 | 0 | 0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b6 || b6 | 0 | 0 | 0 | 0 | 0 | b0 | 0 | 0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b7 || b7 | 0 | 0 | 0 | 0 | 0 | 0 | b0 | 0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b8 || b8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | b0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b9 || b9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | b0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
- # TODO: this could be factored out somehow, but is left here
- # because the MatrixEJA is not presently a subclass of the
- # FDEJA class that defines rank() and one().
- self.rank.set_cache(n)
- idV = matrix.identity(ZZ, self.dimension_over_reals()*n)
- self.one.set_cache(self(idV))
+ TESTS:
+ We can actually construct the 27-dimensional Albert algebra,
+ and we get the right unit element if we recompute it::
+
+ sage: J = OctonionHermitianEJA(3, # long time
+ ....: field=QQ, # long time
+ ....: orthonormalize=False) # long time
+ sage: J.one.clear_cache() # long time
+ sage: J.one() # long time
+ b0 + b9 + b26
+ sage: J.one().to_matrix() # long time
+ +----+----+----+
+ | e0 | 0 | 0 |
+ +----+----+----+
+ | 0 | e0 | 0 |
+ +----+----+----+
+ | 0 | 0 | e0 |
+ +----+----+----+
+
+ The 2-by-2 algebra is isomorphic to the ten-dimensional Jordan
+ spin algebra, but just to be sure, we recompute its rank::
+
+ sage: J = OctonionHermitianEJA(2, # long time
+ ....: field=QQ, # long time
+ ....: orthonormalize=False) # long time
+ sage: J.rank.clear_cache() # long time
+ sage: J.rank() # long time
+ 2
+ """
@staticmethod
- def _max_random_instance_size():
+ def _max_random_instance_size(max_dimension):
r"""
- The maximum rank of a random QuaternionHermitianEJA.
- """
- return 2 # Dimension 6
+ The maximum rank of a random OctonionHermitianEJA.
+ """
+ # There's certainly a formula for this, but with only four
+ # cases to worry about, I'm not that motivated to derive it.
+ if max_dimension >= 27:
+ return 3
+ elif max_dimension >= 10:
+ return 2
+ elif max_dimension >= 1:
+ return 1
+ else:
+ return 0
@classmethod
- def random_instance(cls, **kwargs):
+ def random_instance(cls, max_dimension=None, *args, **kwargs):
"""
Return a random instance of this type of algebra.
"""
- n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ class_max_d = cls._max_random_instance_dimension()
+ if (max_dimension is None or max_dimension > class_max_d):
+ max_dimension = class_max_d
+ max_size = cls._max_random_instance_size(max_dimension)
+ n = ZZ.random_element(max_size + 1)
return cls(n, **kwargs)
-class OctonionHermitianEJA(FiniteDimensionalEJA, MatrixEJA):
-
def __init__(self, n, field=AA, **kwargs):
if n > 3:
# Otherwise we don't get an EJA.
# if the user passes check_axioms=True.
if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
- super().__init__(self._denormalized_basis(n,field),
- self.jordan_product,
- self.trace_inner_product,
- **kwargs)
+ from mjo.hurwitz import OctonionMatrixAlgebra
+ A = OctonionMatrixAlgebra(n, scalars=field)
+ super().__init__(A, **kwargs)
- # TODO: this could be factored out somehow, but is left here
- # because the MatrixEJA is not presently a subclass of the
- # FDEJA class that defines rank() and one().
- self.rank.set_cache(n)
- idV = self.matrix_space().one()
- self.one.set_cache(self(idV))
+ from mjo.eja.eja_cache import octonion_hermitian_eja_coeffs
+ a = octonion_hermitian_eja_coeffs(self)
+ if a is not None:
+ self.rational_algebra()._charpoly_coefficients.set_cache(a)
- @classmethod
- def _denormalized_basis(cls, n, field):
- """
- Returns a basis for the space of octonion Hermitian n-by-n
- matrices.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import OctonionHermitianEJA
+class AlbertEJA(OctonionHermitianEJA):
+ r"""
+ The Albert algebra is the algebra of three-by-three Hermitian
+ matrices whose entries are octonions.
- EXAMPLES::
+ SETUP::
- sage: B = OctonionHermitianEJA._denormalized_basis(3)
- sage: all( M.is_hermitian() for M in B )
- True
- sage: len(B)
- 27
+ sage: from mjo.eja.eja_algebra import AlbertEJA
- """
- from mjo.octonions import OctonionMatrixAlgebra
- MS = OctonionMatrixAlgebra(n, scalars=field)
- es = MS.entry_algebra().gens()
+ EXAMPLES::
- basis = []
- for i in range(n):
- for j in range(i+1):
- if i == j:
- E_ii = MS.monomial( (i,j,es[0]) )
- basis.append(E_ii)
- else:
- for e in es:
- E_ij = MS.monomial( (i,j,e) )
- E_ij += MS.monomial( (j,i,e.conjugate()) )
- basis.append(E_ij)
+ sage: AlbertEJA(field=QQ, orthonormalize=False)
+ Euclidean Jordan algebra of dimension 27 over Rational Field
+ sage: AlbertEJA() # long time
+ Euclidean Jordan algebra of dimension 27 over Algebraic Real Field
- return tuple( basis )
+ """
+ def __init__(self, *args, **kwargs):
+ super().__init__(3, *args, **kwargs)
-class HadamardEJA(ConcreteEJA):
+class HadamardEJA(RationalBasisEJA, ConcreteEJA):
"""
- Return the Euclidean Jordan Algebra corresponding to the set
- `R^n` under the Hadamard product.
+ Return the Euclidean Jordan algebra on `R^n` with the Hadamard
+ (pointwise real-number multiplication) Jordan product and the
+ usual inner-product.
- Note: this is nothing more than the Cartesian product of ``n``
- copies of the spin algebra. Once Cartesian product algebras
- are implemented, this can go.
+ This is nothing more than the Cartesian product of ``n`` copies of
+ the one-dimensional Jordan spin algebra, and is the most common
+ example of a non-simple Euclidean Jordan algebra.
SETUP::
sage: HadamardEJA(3, prefix='r').gens()
(r0, r1, r2)
-
"""
def __init__(self, n, field=AA, **kwargs):
+ MS = MatrixSpace(field, n, 1)
+
if n == 0:
jordan_product = lambda x,y: x
inner_product = lambda x,y: x
else:
def jordan_product(x,y):
- P = x.parent()
- return P( xi*yi for (xi,yi) in zip(x,y) )
+ return MS( xi*yi for (xi,yi) in zip(x,y) )
def inner_product(x,y):
return (x.T*y)[0,0]
if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False
if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
- column_basis = tuple( b.column()
- for b in FreeModule(field, n).basis() )
+ column_basis = tuple( MS(b) for b in FreeModule(field, n).basis() )
super().__init__(column_basis,
jordan_product,
inner_product,
field=field,
+ matrix_space=MS,
associative=True,
**kwargs)
self.rank.set_cache(n)
- if n == 0:
- self.one.set_cache( self.zero() )
- else:
- self.one.set_cache( sum(self.gens()) )
+ self.one.set_cache( self.sum(self.gens()) )
@staticmethod
- def _max_random_instance_size():
+ def _max_random_instance_dimension():
r"""
- The maximum dimension of a random HadamardEJA.
+ There's no reason to go higher than five here. That's
+ enough to get the point across.
"""
return 5
+ @staticmethod
+ def _max_random_instance_size(max_dimension):
+ r"""
+ The maximum size (=dimension) of a random HadamardEJA.
+ """
+ return max_dimension
+
@classmethod
- def random_instance(cls, **kwargs):
+ def random_instance(cls, max_dimension=None, *args, **kwargs):
"""
Return a random instance of this type of algebra.
"""
- n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ class_max_d = cls._max_random_instance_dimension()
+ if (max_dimension is None or max_dimension > class_max_d):
+ max_dimension = class_max_d
+ max_size = cls._max_random_instance_size(max_dimension)
+ n = ZZ.random_element(max_size + 1)
return cls(n, **kwargs)
-class BilinearFormEJA(ConcreteEJA):
+class BilinearFormEJA(RationalBasisEJA, ConcreteEJA):
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 =
matrix. We opt not to orthonormalize the basis, because if we
did, we would have to normalize the `s_{i}` in a similar manner::
- sage: set_random_seed()
sage: n = ZZ.random_element(5)
sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
sage: B11 = matrix.identity(QQ,1)
# verify things, we'll skip the rest of the checks.
if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
+ n = B.nrows()
+ MS = MatrixSpace(field, n, 1)
+
def inner_product(x,y):
return (y.T*B*x)[0,0]
def jordan_product(x,y):
- P = x.parent()
x0 = x[0,0]
xbar = x[1:,0]
y0 = y[0,0]
ybar = y[1:,0]
z0 = inner_product(y,x)
zbar = y0*xbar + x0*ybar
- return P([z0] + zbar.list())
+ return MS([z0] + zbar.list())
- n = B.nrows()
- column_basis = tuple( b.column()
- for b in FreeModule(field, n).basis() )
+ column_basis = tuple( MS(b) for b in FreeModule(field, n).basis() )
# TODO: I haven't actually checked this, but it seems legit.
associative = False
jordan_product,
inner_product,
field=field,
+ matrix_space=MS,
associative=associative,
**kwargs)
# one-dimensional ambient space (because the rank is bounded
# by the ambient dimension).
self.rank.set_cache(min(n,2))
-
if n == 0:
self.one.set_cache( self.zero() )
else:
self.one.set_cache( self.monomial(0) )
@staticmethod
- def _max_random_instance_size():
+ def _max_random_instance_dimension():
r"""
- The maximum dimension of a random BilinearFormEJA.
+ There's no reason to go higher than five here. That's
+ enough to get the point across.
"""
return 5
+ @staticmethod
+ def _max_random_instance_size(max_dimension):
+ r"""
+ The maximum size (=dimension) of a random BilinearFormEJA.
+ """
+ return max_dimension
+
@classmethod
- def random_instance(cls, **kwargs):
+ def random_instance(cls, max_dimension=None, *args, **kwargs):
"""
Return a random instance of this algebra.
"""
- n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ class_max_d = cls._max_random_instance_dimension()
+ if (max_dimension is None or max_dimension > class_max_d):
+ max_dimension = class_max_d
+ max_size = cls._max_random_instance_size(max_dimension)
+ n = ZZ.random_element(max_size + 1)
+
if n.is_zero():
B = matrix.identity(ZZ, n)
return cls(B, **kwargs)
alpha = ZZ.zero()
while alpha.is_zero():
alpha = ZZ.random_element().abs()
+
B22 = M.transpose()*M + alpha*I
from sage.matrix.special import block_matrix
Ensure that we have the usual inner product on `R^n`::
- sage: set_random_seed()
sage: J = JordanSpinEJA.random_instance()
sage: x,y = J.random_elements(2)
sage: actual = x.inner_product(y)
# can pass in a field!
super().__init__(B, *args, **kwargs)
- @staticmethod
- def _max_random_instance_size():
- r"""
- The maximum dimension of a random JordanSpinEJA.
- """
- return 5
-
@classmethod
- def random_instance(cls, **kwargs):
+ def random_instance(cls, max_dimension=None, *args, **kwargs):
"""
Return a random instance of this type of algebra.
Needed here to override the implementation for ``BilinearFormEJA``.
"""
- n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ class_max_d = cls._max_random_instance_dimension()
+ if (max_dimension is None or max_dimension > class_max_d):
+ max_dimension = class_max_d
+ max_size = cls._max_random_instance_size(max_dimension)
+ n = ZZ.random_element(max_size + 1)
return cls(n, **kwargs)
-class TrivialEJA(ConcreteEJA):
+class TrivialEJA(RationalBasisEJA, ConcreteEJA):
"""
The trivial Euclidean Jordan algebra consisting of only a zero element.
0
"""
- def __init__(self, **kwargs):
+ def __init__(self, field=AA, **kwargs):
jordan_product = lambda x,y: x
- inner_product = lambda x,y: 0
+ inner_product = lambda x,y: field.zero()
basis = ()
+ MS = MatrixSpace(field,0)
# New defaults for keyword arguments
if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False
jordan_product,
inner_product,
associative=True,
+ field=field,
+ matrix_space=MS,
**kwargs)
# The rank is zero using my definition, namely the dimension of the
self.one.set_cache( self.zero() )
@classmethod
- def random_instance(cls, **kwargs):
+ def random_instance(cls, max_dimension=None, *args, **kwargs):
# We don't take a "size" argument so the superclass method is
- # inappropriate for us.
+ # inappropriate for us. The ``max_dimension`` argument is
+ # included so that if this method is called generically with a
+ # ``max_dimension=<whatever>`` argument, we don't try to pass
+ # it on to the algebra constructor.
return cls(**kwargs)
-class CartesianProductEJA(FiniteDimensionalEJA):
+class CartesianProductEJA(EJA):
r"""
The external (orthogonal) direct sum of two or more Euclidean
Jordan algebras. Every Euclidean Jordan algebra decomposes into an
sage: from mjo.eja.eja_algebra import (random_eja,
....: CartesianProductEJA,
+ ....: ComplexHermitianEJA,
....: HadamardEJA,
....: JordanSpinEJA,
....: RealSymmetricEJA)
The Jordan product is inherited from our factors and implemented by
our CombinatorialFreeModule Cartesian product superclass::
- sage: set_random_seed()
sage: J1 = HadamardEJA(2)
sage: J2 = RealSymmetricEJA(2)
sage: J = cartesian_product([J1,J2])
| b2 || 0 | 0 | b2 |
+----++----+----+----+
+ The "matrix space" of a Cartesian product always consists of
+ ordered pairs (or triples, or...) whose components are the
+ matrix spaces of its factors::
+
+ sage: J1 = HadamardEJA(2)
+ sage: J2 = ComplexHermitianEJA(2)
+ sage: J = cartesian_product([J1,J2])
+ sage: J.matrix_space()
+ The Cartesian product of (Full MatrixSpace of 2 by 1 dense
+ matrices over Algebraic Real Field, Module of 2 by 2 matrices
+ with entries in Algebraic Field over the scalar ring Algebraic
+ Real Field)
+ sage: J.one().to_matrix()[0]
+ [1]
+ [1]
+ sage: J.one().to_matrix()[1]
+ +---+---+
+ | 1 | 0 |
+ +---+---+
+ | 0 | 1 |
+ +---+---+
+
TESTS:
All factors must share the same base field::
The cached unit element is the same one that would be computed::
- sage: set_random_seed() # long time
sage: J1 = random_eja() # long time
sage: J2 = random_eja() # long time
sage: J = cartesian_product([J1,J2]) # long time
sage: expected = J.one() # long time
sage: actual == expected # long time
True
-
"""
- Element = FiniteDimensionalEJAElement
-
-
+ Element = CartesianProductEJAElement
def __init__(self, factors, **kwargs):
m = len(factors)
if m == 0:
if not all( J.base_ring() == field for J in factors ):
raise ValueError("all factors must share the same base field")
+ # Figure out the category to use.
associative = all( f.is_associative() for f in factors )
-
- MS = self.matrix_space()
- basis = []
- zero = MS.zero()
+ category = EuclideanJordanAlgebras(field)
+ if associative: category = category.Associative()
+ category = category.join([category, category.CartesianProducts()])
+
+ # Compute my matrix space. We don't simply use the
+ # ``cartesian_product()`` functor here because it acts
+ # differently on SageMath MatrixSpaces and our custom
+ # MatrixAlgebras, which are CombinatorialFreeModules. We
+ # always want the result to be represented (and indexed) as an
+ # ordered tuple. This category isn't perfect, but is good
+ # enough for what we need to do.
+ MS_cat = MagmaticAlgebras(field).FiniteDimensional().WithBasis()
+ MS_cat = MS_cat.Unital().CartesianProducts()
+ MS_factors = tuple( J.matrix_space() for J in factors )
+ from sage.sets.cartesian_product import CartesianProduct
+ self._matrix_space = CartesianProduct(MS_factors, MS_cat)
+
+ self._matrix_basis = []
+ zero = self._matrix_space.zero()
for i in range(m):
for b in factors[i].matrix_basis():
z = list(zero)
z[i] = b
- basis.append(z)
+ self._matrix_basis.append(z)
- basis = tuple( MS(b) for b in basis )
+ self._matrix_basis = tuple( self._matrix_space(b)
+ for b in self._matrix_basis )
+ n = len(self._matrix_basis)
- # Define jordan/inner products that operate on that matrix_basis.
- def jordan_product(x,y):
- return MS(tuple(
- (factors[i](x[i])*factors[i](y[i])).to_matrix()
- for i in range(m)
- ))
-
- def inner_product(x, y):
- return sum(
- factors[i](x[i]).inner_product(factors[i](y[i]))
- for i in range(m)
- )
+ # We already have what we need for the super-superclass constructor.
+ CombinatorialFreeModule.__init__(self,
+ field,
+ range(n),
+ prefix="b",
+ category=category,
+ bracket=False)
- # There's no need to check the field since it already came
- # from an EJA. Likewise the axioms are guaranteed to be
- # satisfied, unless the guy writing this class sucks.
- #
- # If you want the basis to be orthonormalized, orthonormalize
- # the factors.
- FiniteDimensionalEJA.__init__(self,
- basis,
- jordan_product,
- inner_product,
- field=field,
- orthonormalize=False,
- associative=associative,
- cartesian_product=True,
- check_field=False,
- check_axioms=False)
+ # Now create the vector space for the algebra, which will have
+ # its own set of non-ambient coordinates (in terms of the
+ # supplied basis).
+ degree = sum( f._matrix_span.ambient_vector_space().degree()
+ for f in factors )
+ V = VectorSpace(field, degree)
+ vector_basis = tuple( V(_all2list(b)) for b in self._matrix_basis )
+
+ # Save the span of our matrix basis (when written out as long
+ # vectors) because otherwise we'll have to reconstruct it
+ # every time we want to coerce a matrix into the algebra.
+ self._matrix_span = V.span_of_basis( vector_basis, check=False)
+
+ # Since we don't (re)orthonormalize the basis, the FDEJA
+ # constructor is going to set self._deortho_matrix to the
+ # identity matrix. Here we set it to the correct value using
+ # the deortho matrices from our factors.
+ self._deortho_matrix = matrix.block_diagonal(
+ [J._deortho_matrix for J in factors]
+ )
+
+ self._inner_product_matrix = matrix.block_diagonal(
+ [J._inner_product_matrix for J in factors]
+ )
+ self._inner_product_matrix._cache = {'hermitian': True}
+ self._inner_product_matrix.set_immutable()
+
+ # Building the multiplication table is a bit more tricky
+ # because we have to embed the entries of the factors'
+ # multiplication tables into the product EJA.
+ zed = self.zero()
+ self._multiplication_table = [ [zed for j in range(i+1)]
+ for i in range(n) ]
+
+ # Keep track of an offset that tallies the dimensions of all
+ # previous factors. If the second factor is dim=2 and if the
+ # first one is dim=3, then we want to skip the first 3x3 block
+ # when copying the multiplication table for the second factor.
+ offset = 0
+ for f in range(m):
+ phi_f = self.cartesian_embedding(f)
+ factor_dim = factors[f].dimension()
+ for i in range(factor_dim):
+ for j in range(i+1):
+ f_ij = factors[f]._multiplication_table[i][j]
+ e = phi_f(f_ij)
+ self._multiplication_table[offset+i][offset+j] = e
+ offset += factor_dim
+ self.rank.set_cache(sum(J.rank() for J in factors))
ones = tuple(J.one().to_matrix() for J in factors)
self.one.set_cache(self(ones))
- self.rank.set_cache(sum(J.rank() for J in factors))
+
+ def _sets_keys(self):
+ r"""
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+ ....: RealSymmetricEJA)
+
+ TESTS:
+
+ The superclass uses ``_sets_keys()`` to implement its
+ ``cartesian_factors()`` method::
+
+ sage: J1 = RealSymmetricEJA(2,
+ ....: field=QQ,
+ ....: orthonormalize=False,
+ ....: prefix="a")
+ sage: J2 = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: J = cartesian_product([J1,J2])
+ sage: x = sum(i*J.gens()[i] for i in range(len(J.gens())))
+ sage: x.cartesian_factors()
+ (a1 + 2*a2, 3*b0 + 4*b1 + 5*b2 + 6*b3)
+
+ """
+ # Copy/pasted from CombinatorialFreeModule_CartesianProduct,
+ # but returning a tuple instead of a list.
+ return tuple(range(len(self.cartesian_factors())))
def cartesian_factors(self):
# Copy/pasted from CombinatorialFreeModule_CartesianProduct.
return cartesian_product.symbol.join("%s" % factor
for factor in self._sets)
- def matrix_space(self):
- r"""
- Return the space that our matrix basis lives in as a Cartesian
- product.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import (HadamardEJA,
- ....: RealSymmetricEJA)
-
- EXAMPLES::
-
- sage: J1 = HadamardEJA(1)
- sage: J2 = RealSymmetricEJA(2)
- sage: J = cartesian_product([J1,J2])
- sage: J.matrix_space()
- The Cartesian product of (Full MatrixSpace of 1 by 1 dense
- matrices over Algebraic Real Field, Full MatrixSpace of 2
- by 2 dense matrices over Algebraic Real Field)
-
- """
- from sage.categories.cartesian_product import cartesian_product
- return cartesian_product( [J.matrix_space()
- for J in self.cartesian_factors()] )
@cached_method
def cartesian_projection(self, i):
The answer never changes::
- sage: set_random_seed()
sage: J1 = random_eja()
sage: J2 = random_eja()
sage: J = cartesian_product([J1,J2])
Pi = self._module_morphism(lambda j: Ji.monomial(j - offset),
codomain=Ji)
- return FiniteDimensionalEJAOperator(self,Ji,Pi.matrix())
+ return EJAOperator(self,Ji,Pi.matrix())
@cached_method
def cartesian_embedding(self, i):
The answer never changes::
- sage: set_random_seed()
sage: J1 = random_eja()
sage: J2 = random_eja()
sage: J = cartesian_product([J1,J2])
produce the identity map, and mismatching them should produce
the zero map::
- sage: set_random_seed()
sage: J1 = random_eja()
sage: J2 = random_eja()
sage: J = cartesian_product([J1,J2])
Ji = self.cartesian_factor(i)
Ei = Ji._module_morphism(lambda j: self.monomial(j + offset),
codomain=self)
- return FiniteDimensionalEJAOperator(Ji,self,Ei.matrix())
+ return EJAOperator(Ji,self,Ei.matrix())
+
+
+ def subalgebra(self, basis, **kwargs):
+ r"""
+ Create a subalgebra of this algebra from the given basis.
+
+ Only overridden to allow us to use a special Cartesian product
+ subalgebra class.
+
+ SETUP::
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: QuaternionHermitianEJA)
+
+ EXAMPLES:
+
+ Subalgebras of Cartesian product EJAs have a different class
+ than those of non-Cartesian-product EJAs::
+
+ sage: J1 = HadamardEJA(2,field=QQ,orthonormalize=False)
+ sage: J2 = QuaternionHermitianEJA(0,field=QQ,orthonormalize=False)
+ sage: J = cartesian_product([J1,J2])
+ sage: K1 = J1.subalgebra((J1.one(),), orthonormalize=False)
+ sage: K = J.subalgebra((J.one(),), orthonormalize=False)
+ sage: K1.__class__ is K.__class__
+ False
+ """
+ from mjo.eja.eja_subalgebra import CartesianProductEJASubalgebra
+ return CartesianProductEJASubalgebra(self, basis, **kwargs)
-FiniteDimensionalEJA.CartesianProduct = CartesianProductEJA
+EJA.CartesianProduct = CartesianProductEJA
class RationalBasisCartesianProductEJA(CartesianProductEJA,
RationalBasisEJA):
SETUP::
- sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ sage: from mjo.eja.eja_algebra import (EJA,
+ ....: HadamardEJA,
+ ....: JordanSpinEJA,
....: RealSymmetricEJA)
EXAMPLES:
sage: J.rank()
5
+ TESTS:
+
+ The ``cartesian_product()`` function only uses the first factor to
+ decide where the result will live; thus we have to be careful to
+ check that all factors do indeed have a ``rational_algebra()`` method
+ before we construct an algebra that claims to have a rational basis::
+
+ sage: J1 = HadamardEJA(2)
+ sage: jp = lambda X,Y: X*Y
+ sage: ip = lambda X,Y: X[0,0]*Y[0,0]
+ sage: b1 = matrix(QQ, [[1]])
+ sage: J2 = EJA((b1,), jp, ip)
+ sage: cartesian_product([J2,J1]) # factor one not RationalBasisEJA
+ Euclidean Jordan algebra of dimension 1 over Algebraic Real
+ Field (+) Euclidean Jordan algebra of dimension 2 over Algebraic
+ Real Field
+ sage: cartesian_product([J1,J2]) # factor one is RationalBasisEJA
+ Traceback (most recent call last):
+ ...
+ ValueError: factor not a RationalBasisEJA
+
"""
def __init__(self, algebras, **kwargs):
+ if not all( hasattr(r, "rational_algebra") for r in algebras ):
+ raise ValueError("factor not a RationalBasisEJA")
+
CartesianProductEJA.__init__(self, algebras, **kwargs)
- self._rational_algebra = None
- if self.vector_space().base_field() is not QQ:
- self._rational_algebra = cartesian_product([
- r._rational_algebra for r in algebras
- ])
+ @cached_method
+ def rational_algebra(self):
+ if self.base_ring() is QQ:
+ return self
+
+ return cartesian_product([
+ r.rational_algebra() for r in self.cartesian_factors()
+ ])
RationalBasisEJA.CartesianProduct = RationalBasisCartesianProductEJA
-def random_eja(*args, **kwargs):
- J1 = ConcreteEJA.random_instance(*args, **kwargs)
+def random_eja(max_dimension=None, *args, **kwargs):
+ r"""
- # This might make Cartesian products appear roughly as often as
- # any other ConcreteEJA.
- if ZZ.random_element(len(ConcreteEJA.__subclasses__()) + 1) == 0:
- # Use random_eja() again so we can get more than two factors.
- J2 = random_eja(*args, **kwargs)
- J = cartesian_product([J1,J2])
- return J
- else:
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja
+
+ TESTS::
+
+ sage: n = ZZ.random_element(1,5)
+ sage: J = random_eja(max_dimension=n, field=QQ, orthonormalize=False)
+ sage: J.dimension() <= n
+ True
+
+ """
+ # Use the ConcreteEJA default as the total upper bound (regardless
+ # of any whether or not any individual factors set a lower limit).
+ if max_dimension is None:
+ max_dimension = ConcreteEJA._max_random_instance_dimension()
+ J1 = ConcreteEJA.random_instance(max_dimension, *args, **kwargs)
+
+
+ # Roll the dice to see if we attempt a Cartesian product.
+ dice_roll = ZZ.random_element(len(ConcreteEJA.__subclasses__()) + 1)
+ new_max_dimension = max_dimension - J1.dimension()
+ if new_max_dimension == 0 or dice_roll != 0:
+ # If it's already as big as we're willing to tolerate, just
+ # return it and don't worry about Cartesian products.
return J1
+ else:
+ # Use random_eja() again so we can get more than two factors
+ # if the sub-call also Decides on a cartesian product.
+ J2 = random_eja(new_max_dimension, *args, **kwargs)
+ return cartesian_product([J1,J2])
+
+
+class ComplexSkewSymmetricEJA(RationalBasisEJA, ConcreteEJA):
+ r"""
+ The skew-symmetric EJA of order `2n` described in Faraut and
+ Koranyi's Exercise III.1.b. It has dimension `2n^2 - n`.
+
+ It is (not obviously) isomorphic to the QuaternionHermitianEJA of
+ order `n`, as can be inferred by comparing rank/dimension or
+ explicitly from their "characteristic polynomial of" functions,
+ which just so happen to align nicely.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (ComplexSkewSymmetricEJA,
+ ....: QuaternionHermitianEJA)
+ sage: from mjo.eja.eja_operator import EJAOperator
+
+ EXAMPLES:
+
+ This EJA is isomorphic to the quaternions::
+
+ sage: J = ComplexSkewSymmetricEJA(2, field=QQ, orthonormalize=False)
+ sage: K = QuaternionHermitianEJA(2, field=QQ, orthonormalize=False)
+ sage: jordan_isom_matrix = matrix.diagonal(QQ,[-1,1,1,1,1,-1])
+ sage: phi = EJAOperator(J,K,jordan_isom_matrix)
+ sage: all( phi(x*y) == phi(x)*phi(y)
+ ....: for x in J.gens()
+ ....: for y in J.gens() )
+ True
+ sage: x,y = J.random_elements(2)
+ sage: phi(x*y) == phi(x)*phi(y)
+ True
+
+ TESTS:
+
+ Random elements should satisfy the same conditions that the basis
+ elements do::
+
+ sage: K = ComplexSkewSymmetricEJA.random_instance(field=QQ,
+ ....: orthonormalize=False)
+ sage: x,y = K.random_elements(2)
+ sage: z = x*y
+ sage: x = x.to_matrix()
+ sage: y = y.to_matrix()
+ sage: z = z.to_matrix()
+ sage: all( e.is_skew_symmetric() for e in (x,y,z) )
+ True
+ sage: J = -K.one().to_matrix()
+ sage: all( e*J == J*e.conjugate() for e in (x,y,z) )
+ True
+
+ The power law in Faraut & Koranyi's II.7.a is satisfied.
+ We're in a subalgebra of theirs, but powers are still
+ defined the same::
+
+ sage: K = ComplexSkewSymmetricEJA.random_instance(field=QQ,
+ ....: orthonormalize=False)
+ sage: x = K.random_element()
+ sage: k = ZZ.random_element(5)
+ sage: actual = x^k
+ sage: J = -K.one().to_matrix()
+ sage: expected = K(-J*(J*x.to_matrix())^k)
+ sage: actual == expected
+ True
+
+ """
+ @staticmethod
+ def _max_random_instance_size(max_dimension):
+ # Obtained by solving d = 2n^2 - n, which comes from noticing
+ # that, in 2x2 block form, any element of this algebra has a
+ # free skew-symmetric top-left block, a Hermitian top-right
+ # block, and two bottom blocks that are determined by the top.
+ # The ZZ-int-ZZ thing is just "floor."
+ return ZZ(int(ZZ(8*max_dimension + 1).sqrt()/4 + 1/4))
+
+ @classmethod
+ def random_instance(cls, max_dimension=None, *args, **kwargs):
+ """
+ Return a random instance of this type of algebra.
+ """
+ class_max_d = cls._max_random_instance_dimension()
+ if (max_dimension is None or max_dimension > class_max_d):
+ max_dimension = class_max_d
+ max_size = cls._max_random_instance_size(max_dimension)
+ n = ZZ.random_element(max_size + 1)
+ return cls(n, **kwargs)
+
+ @staticmethod
+ def _denormalized_basis(A):
+ """
+ SETUP::
+
+ sage: from mjo.hurwitz import ComplexMatrixAlgebra
+ sage: from mjo.eja.eja_algebra import ComplexSkewSymmetricEJA
+
+ TESTS:
+
+ The basis elements are all skew-Hermitian::
+
+ sage: d_max = ComplexSkewSymmetricEJA._max_random_instance_dimension()
+ sage: n_max = ComplexSkewSymmetricEJA._max_random_instance_size(d_max)
+ sage: n = ZZ.random_element(n_max + 1)
+ sage: A = ComplexMatrixAlgebra(2*n, scalars=QQ)
+ sage: B = ComplexSkewSymmetricEJA._denormalized_basis(A)
+ sage: all( M.is_skew_symmetric() for M in B)
+ True
+
+ The basis elements ``b`` all satisfy ``b*J == J*b.conjugate()``,
+ as in the definition of the algebra::
+
+ sage: d_max = ComplexSkewSymmetricEJA._max_random_instance_dimension()
+ sage: n_max = ComplexSkewSymmetricEJA._max_random_instance_size(d_max)
+ sage: n = ZZ.random_element(n_max + 1)
+ sage: A = ComplexMatrixAlgebra(2*n, scalars=QQ)
+ sage: I_n = matrix.identity(ZZ, n)
+ sage: J = matrix.block(ZZ, 2, 2, (0, I_n, -I_n, 0), subdivide=False)
+ sage: J = A.from_list(J.rows())
+ sage: B = ComplexSkewSymmetricEJA._denormalized_basis(A)
+ sage: all( b*J == J*b.conjugate() for b in B )
+ True
+
+ """
+ es = A.entry_algebra_gens()
+ gen = lambda A,m: A.monomial(m)
+
+ basis = []
+
+ # The size of the blocks. We're going to treat these thing as
+ # 2x2 block matrices,
+ #
+ # [ x1 x2 ]
+ # [ -x2-conj x1-conj ]
+ #
+ # where x1 is skew-symmetric and x2 is Hermitian.
+ #
+ m = A.nrows()/2
+
+ # We only loop through the top half of the matrix, because the
+ # bottom can be constructed from the top.
+ for i in range(m):
+ # First do the top-left block, which is skew-symmetric.
+ # We can compute the bottom-right block in the process.
+ for j in range(i+1):
+ if i != j:
+ # Skew-symmetry implies zeros for (i == j).
+ for e in es:
+ # Top-left block's entry.
+ E_ij = gen(A, (i,j,e))
+ E_ij -= gen(A, (j,i,e))
+
+ # Bottom-right block's entry.
+ F_ij = gen(A, (i+m,j+m,e)).conjugate()
+ F_ij -= gen(A, (j+m,i+m,e)).conjugate()
+
+ basis.append(E_ij + F_ij)
+
+ # Now do the top-right block, which is Hermitian, and compute
+ # the bottom-left block along the way.
+ for j in range(m,i+m+1):
+ if (i+m) == j:
+ # Hermitian matrices have real diagonal entries.
+ # Top-right block's entry.
+ E_ii = gen(A, (i,j,es[0]))
+
+ # Bottom-left block's entry. Don't conjugate
+ # 'cause it's real.
+ E_ii -= gen(A, (i+m,j-m,es[0]))
+ basis.append(E_ii)
+ else:
+ for e in es:
+ # Top-right block's entry. BEWARE! We're not
+ # reflecting across the main diagonal as in
+ # (i,j)~(j,i). We're only reflecting across
+ # the diagonal for the top-right block.
+ E_ij = gen(A, (i,j,e))
+
+ # Shift it back to non-offset coords, transpose,
+ # conjugate, and put it back:
+ #
+ # (i,j) -> (i,j-m) -> (j-m, i) -> (j-m, i+m)
+ E_ij += gen(A, (j-m,i+m,e)).conjugate()
+
+ # Bottom-left's block's below-diagonal entry.
+ # Just shift the top-right coords down m and
+ # left m.
+ F_ij = -gen(A, (i+m,j-m,e)).conjugate()
+ F_ij += -gen(A, (j,i,e)) # double-conjugate cancels
+
+ basis.append(E_ij + F_ij)
+
+ return tuple( basis )
+
+ @staticmethod
+ @cached_method
+ def _J_matrix(matrix_space):
+ n = matrix_space.nrows() // 2
+ F = matrix_space.base_ring()
+ I_n = matrix.identity(F, n)
+ J = matrix.block(F, 2, 2, (0, I_n, -I_n, 0), subdivide=False)
+ return matrix_space.from_list(J.rows())
+
+ def J_matrix(self):
+ return ComplexSkewSymmetricEJA._J_matrix(self.matrix_space())
+
+ def __init__(self, n, field=AA, **kwargs):
+ # New code; always check the axioms.
+ #if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
+
+ from mjo.hurwitz import ComplexMatrixAlgebra
+ A = ComplexMatrixAlgebra(2*n, scalars=field)
+ J = ComplexSkewSymmetricEJA._J_matrix(A)
+
+ def jordan_product(X,Y):
+ return (X*J*Y + Y*J*X)/2
+
+ def inner_product(X,Y):
+ return (X.conjugate_transpose()*Y).trace().real()
+
+ super().__init__(self._denormalized_basis(A),
+ jordan_product,
+ inner_product,
+ field=field,
+ matrix_space=A,
+ **kwargs)
+
+ # This algebra is conjectured (by me) to be isomorphic to
+ # the quaternion Hermitian EJA of size n, and the rank
+ # would follow from that.
+ #self.rank.set_cache(n)
+ self.one.set_cache( self(-J) )
projects / sage.d.git / blobdiff