from sage.rings.all import (ZZ, QQ, AA, QQbar, RR, RLF, CLF,
PolynomialRing,
QuadraticField)
-from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
-from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator
+from mjo.eja.eja_element import FiniteDimensionalEJAElement
+from mjo.eja.eja_operator import FiniteDimensionalEJAOperator
from mjo.eja.eja_utils import _mat2vec
-class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
+class FiniteDimensionalEJA(CombinatorialFreeModule):
r"""
- The lowest-level class for representing a Euclidean Jordan algebra.
+ A finite-dimensional Euclidean Jordan algebra.
+
+ INPUT:
+
+ - basis -- a tuple of basis elements in their matrix form.
+
+ - jordan_product -- function of two elements (in matrix form)
+ that returns their jordan product in this algebra; this will
+ be applied to ``basis`` to compute a multiplication table for
+ the algebra.
+
+ - inner_product -- function of two elements (in matrix form) that
+ returns their inner product. This will be applied to ``basis`` to
+ compute an inner-product table (basically a matrix) for this algebra.
+
"""
+ Element = FiniteDimensionalEJAElement
+
+ def __init__(self,
+ basis,
+ jordan_product,
+ inner_product,
+ field=AA,
+ orthonormalize=True,
+ associative=False,
+ check_field=True,
+ check_axioms=True,
+ prefix='e'):
+
+ 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")
+
+ # If the basis given to us wasn't over the field that it's
+ # supposed to be over, fix that. Or, you know, crash.
+ basis = tuple( b.change_ring(field) for b in basis )
+
+ 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()
+ if associative:
+ # Element subalgebras can take advantage of this.
+ category = category.Associative()
+
+ # Call the superclass constructor so that we can use its from_vector()
+ # method to build our multiplication table.
+ n = len(basis)
+ super().__init__(field,
+ range(n),
+ prefix=prefix,
+ category=category,
+ bracket=False)
+
+ # Now comes all of the hard work. We'll be constructing an
+ # ambient vector space V that our (vectorized) basis lives in,
+ # 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*e1 + 2*e2.
+ vector_basis = basis
+
+ degree = 0
+ if n > 0:
+ # Works on both column and square matrices...
+ degree = len(basis[0].list())
+
+ # Build an ambient space that fits our matrix basis when
+ # written out as "long vectors."
+ V = VectorSpace(field, degree)
+
+ # The matrix that will hole the orthonormal -> unorthonormal
+ # coordinate transformation.
+ self._deortho_matrix = None
+
+ if orthonormalize:
+ # Save a copy of the un-orthonormalized basis for later.
+ # Convert it to ambient V (vector) coordinates while we're
+ # at it, because we'd have to do it later anyway.
+ deortho_vector_basis = tuple( V(b.list()) for b in basis )
+
+ from mjo.eja.eja_utils import gram_schmidt
+ basis = gram_schmidt(basis, inner_product)
+
+ # Save the (possibly orthonormalized) matrix basis for
+ # later...
+ self._matrix_basis = basis
+
+ # 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(b.list()) for b in basis )
+ W = 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.
+ U = V.span_of_basis( deortho_vector_basis, check=check_axioms)
+ self._deortho_matrix = matrix( 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.zero(field, n)
+ self._multiplication_table = [ [0 for j in range(i+1)]
+ for i in range(n) ]
+
+ # Note: the Jordan and inner-products are defined in terms
+ # of the ambient basis. It's important that their arguments
+ # are in ambient coordinates as well.
+ for i in range(n):
+ for j in range(i+1):
+ # ortho basis w.r.t. ambient coords
+ q_i = basis[i]
+ q_j = basis[j]
+
+ elt = jordan_product(q_i, q_j)
+ ip = inner_product(q_i, q_j)
+
+ # The jordan product returns a matrixy answer, so we
+ # have to convert it to the algebra coordinates.
+ elt = W.coordinate_vector(V(elt.list()))
+ self._multiplication_table[i][j] = self.from_vector(elt)
+ self._inner_product_matrix[i,j] = ip
+ self._inner_product_matrix[j,i] = ip
+
+ self._inner_product_matrix._cache = {'hermitian': True}
+ self._inner_product_matrix.set_immutable()
+
+ if check_axioms:
+ if not self._is_jordanian():
+ raise ValueError("Jordan identity does not hold")
+ if not self._inner_product_is_associative():
+ raise ValueError("inner product is not associative")
+
+
def _coerce_map_from_base_ring(self):
"""
Disable the map from the base ring into the algebra.
"""
return None
- def __init__(self,
- field,
- multiplication_table,
- inner_product_table,
- prefix='e',
- category=None,
- matrix_basis=None,
- check_field=True,
- check_axioms=True):
- """
- INPUT:
- * field -- the scalar field for this algebra (must be real)
+ def product_on_basis(self, i, j):
+ # We only stored the lower-triangular portion of the
+ # multiplication table.
+ if j <= i:
+ return self._multiplication_table[i][j]
+ else:
+ return self._multiplication_table[j][i]
+
+ def inner_product(self, x, y):
+ """
+ The inner product associated with this Euclidean Jordan algebra.
- * multiplication_table -- the multiplication table for this
- algebra's implicit basis. Only the lower-triangular portion
- of the table is used, since the multiplication is assumed
- to be commutative.
+ Defaults to the trace inner product, but can be overridden by
+ subclasses if they are sure that the necessary properties are
+ satisfied.
SETUP::
- sage: from mjo.eja.eja_algebra import (
- ....: FiniteDimensionalEuclideanJordanAlgebra,
- ....: JordanSpinEJA,
- ....: random_eja)
+ sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: HadamardEJA,
+ ....: BilinearFormEJA)
EXAMPLES:
- By definition, Jordan multiplication commutes::
+ 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 = J.random_elements(2)
- sage: x*y == y*x
+ sage: x,y,z = J.random_elements(3)
+ sage: (x*y).inner_product(z) == y.inner_product(x*z)
True
- An error is raised if the Jordan product is not commutative::
-
- sage: JP = ((1,2),(0,0))
- sage: IP = ((1,0),(0,1))
- sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,JP,IP)
- Traceback (most recent call last):
- ...
- ValueError: Jordan product is not commutative
-
- An error is raised if the inner-product is not commutative::
+ TESTS:
- sage: JP = ((1,0),(0,1))
- sage: IP = ((1,2),(0,0))
- sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,JP,IP)
- Traceback (most recent call last):
- ...
- ValueError: inner-product is not commutative
+ Ensure that this is the usual inner product for the algebras
+ over `R^n`::
- TESTS:
+ sage: set_random_seed()
+ sage: J = HadamardEJA.random_instance()
+ sage: x,y = J.random_elements(2)
+ sage: actual = x.inner_product(y)
+ sage: expected = x.to_vector().inner_product(y.to_vector())
+ sage: actual == expected
+ True
- The ``field`` we're given must be real with ``check_field=True``::
+ Ensure that this is one-half of the trace inner-product in a
+ BilinearFormEJA that isn't just the reals (when ``n`` isn't
+ one). This is in Faraut and Koranyi, and also my "On the
+ symmetry..." paper::
- sage: JordanSpinEJA(2, field=QQbar)
- Traceback (most recent call last):
- ...
- ValueError: scalar field is not real
- sage: JordanSpinEJA(2, field=QQbar, check_field=False)
- Euclidean Jordan algebra of dimension 2 over Algebraic Field
+ sage: set_random_seed()
+ sage: J = BilinearFormEJA.random_instance()
+ sage: n = J.dimension()
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: (n == 1) or (x.inner_product(y) == (x*y).trace()/2)
+ True
+ """
+ B = self._inner_product_matrix
+ return (B*x.to_vector()).inner_product(y.to_vector())
- The multiplication table must be square with ``check_axioms=True``::
- sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,((),()),((1,),))
- Traceback (most recent call last):
- ...
- ValueError: multiplication table is not square
+ def _is_commutative(self):
+ r"""
+ Whether or not this algebra's multiplication table is commutative.
- The multiplication and inner-product tables must be the same
- size (and in particular, the inner-product table must also be
- square) with ``check_axioms=True``::
+ This method should of course always return ``True``, unless
+ this algebra was constructed with ``check_axioms=False`` and
+ passed an invalid multiplication table.
+ """
+ return all( self.product_on_basis(i,j) == self.product_on_basis(i,j)
+ for i in range(self.dimension())
+ for j in range(self.dimension()) )
- sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,((1,),),(()))
- Traceback (most recent call last):
- ...
- ValueError: multiplication and inner-product tables are
- different sizes
- sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,((1,),),((1,2),))
- Traceback (most recent call last):
- ...
- ValueError: multiplication and inner-product tables are
- different sizes
+ def _is_jordanian(self):
+ r"""
+ Whether or not this algebra's multiplication table respects the
+ Jordan identity `(x^{2})(xy) = x(x^{2}y)`.
+ We only check one arrangement of `x` and `y`, so for a
+ ``True`` result to be truly true, you should also check
+ :meth:`_is_commutative`. This method should of course always
+ return ``True``, unless this algebra was constructed with
+ ``check_axioms=False`` and passed an invalid multiplication table.
"""
- 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")
+ return all( (self.monomial(i)**2)*(self.monomial(i)*self.monomial(j))
+ ==
+ (self.monomial(i))*((self.monomial(i)**2)*self.monomial(j))
+ for i in range(self.dimension())
+ for j in range(self.dimension()) )
+ def _inner_product_is_associative(self):
+ r"""
+ Return whether or not this algebra's inner product `B` is
+ associative; that is, whether or not `B(xy,z) = B(x,yz)`.
- # The multiplication and inner-product tables should be square
- # if the user wants us to verify them. And we verify them as
- # soon as possible, because we want to exploit their symmetry.
- n = len(multiplication_table)
- if check_axioms:
- if not all( len(l) == n for l in multiplication_table ):
- raise ValueError("multiplication table is not square")
-
- # If the multiplication table is square, we can check if
- # the inner-product table is square by comparing it to the
- # multiplication table's dimensions.
- msg = "multiplication and inner-product tables are different sizes"
- if not len(inner_product_table) == n:
- raise ValueError(msg)
-
- if not all( len(l) == n for l in inner_product_table ):
- raise ValueError(msg)
-
- # Check commutativity of the Jordan product (symmetry of
- # the multiplication table) and the commutativity of the
- # inner-product (symmetry of the inner-product table)
- # first if we're going to check them at all.. This has to
- # be done before we define product_on_basis(), because
- # that method assumes that self._multiplication_table is
- # symmetric. And it has to be done before we build
- # self._inner_product_matrix, because the process used to
- # construct it assumes symmetry as well.
- if not all( multiplication_table[j][i]
- == multiplication_table[i][j]
- for i in range(n)
- for j in range(i+1) ):
- raise ValueError("Jordan product is not commutative")
+ This method should of course always return ``True``, unless
+ this algebra was constructed with ``check_axioms=False`` and
+ passed an invalid Jordan or inner-product.
+ """
- if not all( inner_product_table[j][i]
- == inner_product_table[i][j]
- for i in range(n)
- for j in range(i+1) ):
- raise ValueError("inner-product is not commutative")
+ # Used to check whether or not something is zero in an inexact
+ # ring. This number is sufficient to allow the construction of
+ # QuaternionHermitianEJA(2, field=RDF) with check_axioms=True.
+ epsilon = 1e-16
- self._matrix_basis = matrix_basis
-
- if category is None:
- category = MagmaticAlgebras(field).FiniteDimensional()
- category = category.WithBasis().Unital()
-
- fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
- fda.__init__(field,
- range(n),
- prefix=prefix,
- category=category)
- self.print_options(bracket='')
-
- # The multiplication table we're given is necessarily in terms
- # of vectors, because we don't have an algebra yet for
- # anything to be an element of. However, it's faster in the
- # long run to have the multiplication table be in terms of
- # algebra elements. We do this after calling the superclass
- # constructor so that from_vector() knows what to do.
- #
- # Note: we take advantage of symmetry here, and only store
- # the lower-triangular portion of the table.
- self._multiplication_table = [ [ self.vector_space().zero()
- for j in range(i+1) ]
- for i in range(n) ]
+ for i in range(self.dimension()):
+ for j in range(self.dimension()):
+ for k in range(self.dimension()):
+ x = self.monomial(i)
+ y = self.monomial(j)
+ z = self.monomial(k)
+ diff = (x*y).inner_product(z) - x.inner_product(y*z)
- for i in range(n):
- for j in range(i+1):
- elt = self.from_vector(multiplication_table[i][j])
- self._multiplication_table[i][j] = elt
-
- self._multiplication_table = tuple(map(tuple, self._multiplication_table))
-
- # Save our inner product as a matrix, since the efficiency of
- # matrix multiplication will usually outweigh the fact that we
- # have to store a redundant upper- or lower-triangular part.
- # Pre-cache the fact that these are Hermitian (real symmetric,
- # in fact) in case some e.g. matrix multiplication routine can
- # take advantage of it.
- ip_matrix_constructor = lambda i,j: inner_product_table[i][j] if j <= i else inner_product_table[j][i]
- self._inner_product_matrix = matrix(field, n, ip_matrix_constructor)
- self._inner_product_matrix._cache = {'hermitian': True}
- self._inner_product_matrix.set_immutable()
+ if self.base_ring().is_exact():
+ if diff != 0:
+ return False
+ else:
+ if diff.abs() > epsilon:
+ return False
- if check_axioms:
- if not self._is_jordanian():
- raise ValueError("Jordan identity does not hold")
- if not self._inner_product_is_associative():
- raise ValueError("inner product is not associative")
+ return True
def _element_constructor_(self, elt):
"""
# that the integer 3 belongs to the space of 2-by-2 matrices.
raise ValueError(msg)
+ try:
+ elt = elt.column()
+ except (AttributeError, TypeError):
+ # Try to convert a vector into a column-matrix
+ pass
+
if elt not in self.matrix_space():
raise ValueError(msg)
fmt = "Euclidean Jordan algebra of dimension {} over {}"
return fmt.format(self.dimension(), self.base_ring())
- def product_on_basis(self, i, j):
- # We only stored the lower-triangular portion of the
- # multiplication table.
- if j <= i:
- return self._multiplication_table[i][j]
- else:
- return self._multiplication_table[j][i]
-
- def _is_commutative(self):
- r"""
- Whether or not this algebra's multiplication table is commutative.
-
- This method should of course always return ``True``, unless
- this algebra was constructed with ``check_axioms=False`` and
- passed an invalid multiplication table.
- """
- return all( self.product_on_basis(i,j) == self.product_on_basis(i,j)
- for i in range(self.dimension())
- for j in range(self.dimension()) )
-
- def _is_jordanian(self):
- r"""
- Whether or not this algebra's multiplication table respects the
- Jordan identity `(x^{2})(xy) = x(x^{2}y)`.
-
- We only check one arrangement of `x` and `y`, so for a
- ``True`` result to be truly true, you should also check
- :meth:`_is_commutative`. This method should of course always
- return ``True``, unless this algebra was constructed with
- ``check_axioms=False`` and passed an invalid multiplication table.
- """
- return all( (self.monomial(i)**2)*(self.monomial(i)*self.monomial(j))
- ==
- (self.monomial(i))*((self.monomial(i)**2)*self.monomial(j))
- for i in range(self.dimension())
- for j in range(self.dimension()) )
-
- def _inner_product_is_associative(self):
- r"""
- Return whether or not this algebra's inner product `B` is
- associative; that is, whether or not `B(xy,z) = B(x,yz)`.
-
- This method should of course always return ``True``, unless
- this algebra was constructed with ``check_axioms=False`` and
- passed an invalid multiplication table.
- """
-
- # Used to check whether or not something is zero in an inexact
- # ring. This number is sufficient to allow the construction of
- # QuaternionHermitianEJA(2, field=RDF) with check_axioms=True.
- epsilon = 1e-16
-
- for i in range(self.dimension()):
- for j in range(self.dimension()):
- for k in range(self.dimension()):
- x = self.monomial(i)
- y = self.monomial(j)
- z = self.monomial(k)
- diff = (x*y).inner_product(z) - x.inner_product(y*z)
-
- if self.base_ring().is_exact():
- if diff != 0:
- return False
- else:
- if diff.abs() > epsilon:
- return False
-
- return True
@cached_method
def characteristic_polynomial_of(self):
"""
n = self.dimension()
- M = [ [ self.zero() for j in range(n) ]
- for i in range(n) ]
- for i in range(n):
- for j in range(i+1):
- M[i][j] = self._multiplication_table[i][j]
- M[j][i] = M[i][j]
+ # Prepend the header row.
+ M = [["*"] + list(self.gens())]
- for i in range(n):
- # Prepend the left "header" column entry Can't do this in
- # the loop because it messes up the symmetry.
- M[i] = [self.monomial(i)] + M[i]
+ # And to each subsequent row, prepend an entry that belongs to
+ # the left-side "header column."
+ M += [ [self.monomial(i)] + [ self.product_on_basis(i,j)
+ for j in range(n) ]
+ for i in range(n) ]
- # Prepend the header row.
- M = [["*"] + list(self.gens())] + M
return table(M, header_row=True, header_column=True, frame=True)
[0], [1]
)
"""
- if self._matrix_basis is None:
- M = self.matrix_space()
- return tuple( M(b.to_vector()) for b in self.basis() )
- else:
- return self._matrix_basis
+ return self._matrix_basis
def matrix_space(self):
"""
if self.is_trivial():
return MatrixSpace(self.base_ring(), 0)
- elif self._matrix_basis is None or len(self._matrix_basis) == 0:
- return MatrixSpace(self.base_ring(), self.dimension(), 1)
else:
return self._matrix_basis[0].matrix_space()
if not c.is_idempotent():
raise ValueError("element is not idempotent: %s" % c)
- from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
+ from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra
# Default these to what they should be if they turn out to be
# trivial, because eigenspaces_left() won't return eigenvalues
# corresponding to trivial spaces (e.g. it returns only the
# eigenspace corresponding to lambda=1 if you take the
# decomposition relative to the identity element).
- trivial = FiniteDimensionalEuclideanJordanSubalgebra(self, ())
+ trivial = FiniteDimensionalEJASubalgebra(self, ())
J0 = trivial # eigenvalue zero
J5 = VectorSpace(self.base_ring(), 0) # eigenvalue one-half
J1 = trivial # eigenvalue one
J5 = eigspace
else:
gens = tuple( self.from_vector(b) for b in eigspace.basis() )
- subalg = FiniteDimensionalEuclideanJordanSubalgebra(self,
- gens,
- check_axioms=False)
+ subalg = FiniteDimensionalEJASubalgebra(self,
+ gens,
+ check_axioms=False)
if eigval == 0:
J0 = subalg
elif eigval == 1:
return self.zero().to_vector().parent().ambient_vector_space()
- Element = FiniteDimensionalEuclideanJordanAlgebraElement
+ Element = FiniteDimensionalEJAElement
-class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra):
+class RationalBasisEJA(FiniteDimensionalEJA):
r"""
New class for algebras whose supplied basis elements have all rational entries.
jordan_product,
inner_product,
field=AA,
- orthonormalize=True,
- prefix='e',
- category=None,
check_field=True,
- check_axioms=True):
+ **kwargs):
if check_field:
# Abuse the check_field parameter to check that the entries of
if not all( all(b_i in QQ for b_i in b.list()) for b in basis ):
raise TypeError("basis not rational")
- # Temporary(?) hack to ensure that the matrix and vector bases
- # are over the same ring.
- basis = tuple( b.change_ring(field) for b in basis )
-
- n = len(basis)
- vector_basis = basis
-
- from sage.structure.element import is_Matrix
- basis_is_matrices = False
-
- degree = 0
- if n > 0:
- if is_Matrix(basis[0]):
- basis_is_matrices = True
- from mjo.eja.eja_utils import _vec2mat
- vector_basis = tuple( map(_mat2vec,basis) )
- degree = basis[0].nrows()**2
- else:
- degree = basis[0].degree()
-
- V = VectorSpace(field, degree)
-
- # Save a copy of an algebra with the original, rational basis
- # and over QQ where computations are fast.
- self._rational_algebra = None
-
if field is not QQ:
# There's no point in constructing the extra algebra if this
# one is already rational.
# 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 = RationalBasisEuclideanJordanAlgebra(
+ self._rational_algebra = FiniteDimensionalEJA(
basis,
jordan_product,
inner_product,
field=QQ,
orthonormalize=False,
- prefix=prefix,
- category=category,
check_field=False,
check_axioms=False)
- if orthonormalize:
- # Compute the deorthonormalized tables before we orthonormalize
- # the given basis. The "check" parameter here guarantees that
- # the basis is linearly-independent.
- W = V.span_of_basis( vector_basis, check=check_axioms)
-
- # Note: the Jordan and inner-products are defined in terms
- # of the ambient basis. It's important that their arguments
- # are in ambient coordinates as well.
- for i in range(n):
- for j in range(i+1):
- # given basis w.r.t. ambient coords
- q_i = vector_basis[i]
- q_j = vector_basis[j]
-
- if basis_is_matrices:
- q_i = _vec2mat(q_i)
- q_j = _vec2mat(q_j)
-
- elt = jordan_product(q_i, q_j)
- ip = inner_product(q_i, q_j)
-
- if basis_is_matrices:
- # do another mat2vec because the multiplication
- # table is in terms of vectors
- elt = _mat2vec(elt)
-
- # We overwrite the name "vector_basis" in a second, but never modify it
- # in place, to this effectively makes a copy of it.
- deortho_vector_basis = vector_basis
- self._deortho_matrix = None
-
- if orthonormalize:
- from mjo.eja.eja_utils import gram_schmidt
- if basis_is_matrices:
- vector_ip = lambda x,y: inner_product(_vec2mat(x), _vec2mat(y))
- vector_basis = gram_schmidt(vector_basis, vector_ip)
- else:
- vector_basis = gram_schmidt(vector_basis, inner_product)
-
- # Normalize the "matrix" basis, too!
- basis = vector_basis
-
- if basis_is_matrices:
- basis = tuple( map(_vec2mat,basis) )
-
- W = V.span_of_basis( vector_basis, check=check_axioms)
-
- # 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.
- U = V.span_of_basis( deortho_vector_basis, check=check_axioms)
- self._deortho_matrix = matrix( U.coordinate_vector(q)
- for q in vector_basis )
-
- # If the superclass constructor is going to verify the
- # symmetry of this table, it has better at least be
- # square...
- if check_axioms:
- mult_table = [ [0 for j in range(n)] for i in range(n) ]
- ip_table = [ [0 for j in range(n)] for i in range(n) ]
- else:
- mult_table = [ [0 for j in range(i+1)] for i in range(n) ]
- ip_table = [ [0 for j in range(i+1)] for i in range(n) ]
-
- # Note: the Jordan and inner-products are defined in terms
- # of the ambient basis. It's important that their arguments
- # are in ambient coordinates as well.
- for i in range(n):
- for j in range(i+1):
- # ortho basis w.r.t. ambient coords
- q_i = vector_basis[i]
- q_j = vector_basis[j]
-
- if basis_is_matrices:
- q_i = _vec2mat(q_i)
- q_j = _vec2mat(q_j)
-
- elt = jordan_product(q_i, q_j)
- ip = inner_product(q_i, q_j)
-
- if basis_is_matrices:
- # do another mat2vec because the multiplication
- # table is in terms of vectors
- elt = _mat2vec(elt)
-
- elt = W.coordinate_vector(elt)
- mult_table[i][j] = elt
- ip_table[i][j] = ip
- if check_axioms:
- # The tables are square if we're verifying that they
- # are commutative.
- mult_table[j][i] = elt
- ip_table[j][i] = ip
-
- if basis_is_matrices:
- for m in basis:
- m.set_immutable()
- else:
- basis = tuple( x.column() for x in basis )
-
- super().__init__(field,
- mult_table,
- ip_table,
- prefix,
- category,
- basis, # matrix basis
- check_field,
- check_axioms)
+ super().__init__(basis,
+ jordan_product,
+ inner_product,
+ field=field,
+ check_field=check_field,
+ **kwargs)
@cached_method
def _charpoly_coefficients(self):
# 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.
- superclass = super(RationalBasisEuclideanJordanAlgebra, self)
- return superclass._charpoly_coefficients()
+ 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.
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 ConcreteEuclideanJordanAlgebra(RationalBasisEuclideanJordanAlgebra):
+class ConcreteEJA(RationalBasisEJA):
r"""
A class for the Euclidean Jordan algebras that we know by name.
SETUP::
- sage: from mjo.eja.eja_algebra import ConcreteEuclideanJordanAlgebra
+ sage: from mjo.eja.eja_algebra import ConcreteEJA
TESTS:
product, unless we specify otherwise::
sage: set_random_seed()
- sage: J = ConcreteEuclideanJordanAlgebra.random_instance()
+ sage: J = ConcreteEJA.random_instance()
sage: all( b.norm() == 1 for b in J.gens() )
True
EJA the operator is self-adjoint by the Jordan axiom::
sage: set_random_seed()
- sage: J = ConcreteEuclideanJordanAlgebra.random_instance()
+ sage: J = ConcreteEJA.random_instance()
sage: x = J.random_element()
sage: x.operator().is_self_adjoint()
True
from sage.misc.prandom import choice
eja_class = choice(cls.__subclasses__())
- # These all bubble up to the RationalBasisEuclideanJordanAlgebra
- # superclass constructor, so any (kw)args valid there are also
- # valid here.
+ # 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)
-class MatrixEuclideanJordanAlgebra:
+class MatrixEJA:
@staticmethod
def dimension_over_reals():
r"""
return tr.coefficient_tuple()[0]
-class RealMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra):
+class RealMatrixEJA(MatrixEJA):
@staticmethod
def dimension_over_reals():
return 1
-class RealSymmetricEJA(ConcreteEuclideanJordanAlgebra,
- RealMatrixEuclideanJordanAlgebra):
+class RealSymmetricEJA(ConcreteEJA, RealMatrixEJA):
"""
The rank-n simple EJA consisting of real symmetric n-by-n
matrices, the usual symmetric Jordan product, and the trace inner
**kwargs)
# TODO: this could be factored out somehow, but is left here
- # because the MatrixEuclideanJordanAlgebra is not presently
- # a subclass of the FDEJA class that defines rank() and one().
+ # 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))
-class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra):
+class ComplexMatrixEJA(MatrixEJA):
@staticmethod
def dimension_over_reals():
return 2
SETUP::
- sage: from mjo.eja.eja_algebra import \
- ....: ComplexMatrixEuclideanJordanAlgebra
+ sage: from mjo.eja.eja_algebra import ComplexMatrixEJA
EXAMPLES::
sage: x3 = F(-i)
sage: x4 = F(6)
sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
- sage: ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
+ sage: ComplexMatrixEJA.real_embed(M)
[ 4 -2| 1 2]
[ 2 4|-2 1]
[-----+-----]
sage: F = QuadraticField(-1, 'I')
sage: X = random_matrix(F, n)
sage: Y = random_matrix(F, n)
- sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X)
- sage: Ye = ComplexMatrixEuclideanJordanAlgebra.real_embed(Y)
- sage: XYe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X*Y)
+ sage: Xe = ComplexMatrixEJA.real_embed(X)
+ sage: Ye = ComplexMatrixEJA.real_embed(Y)
+ sage: XYe = ComplexMatrixEJA.real_embed(X*Y)
sage: Xe*Ye == XYe
True
"""
- super(ComplexMatrixEuclideanJordanAlgebra,cls).real_embed(M)
+ super(ComplexMatrixEJA,cls).real_embed(M)
n = M.nrows()
# We don't need any adjoined elements...
SETUP::
- sage: from mjo.eja.eja_algebra import \
- ....: ComplexMatrixEuclideanJordanAlgebra
+ sage: from mjo.eja.eja_algebra import ComplexMatrixEJA
EXAMPLES::
....: [-2, 1, -4, 3],
....: [ 9, 10, 11, 12],
....: [-10, 9, -12, 11] ])
- sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A)
+ sage: ComplexMatrixEJA.real_unembed(A)
[ 2*I + 1 4*I + 3]
[ 10*I + 9 12*I + 11]
sage: set_random_seed()
sage: F = QuadraticField(-1, 'I')
sage: M = random_matrix(F, 3)
- sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
- sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
+ sage: Me = ComplexMatrixEJA.real_embed(M)
+ sage: ComplexMatrixEJA.real_unembed(Me) == M
True
"""
- super(ComplexMatrixEuclideanJordanAlgebra,cls).real_unembed(M)
+ super(ComplexMatrixEJA,cls).real_unembed(M)
n = ZZ(M.nrows())
d = cls.dimension_over_reals()
return matrix(F, n/d, elements)
-class ComplexHermitianEJA(ConcreteEuclideanJordanAlgebra,
- ComplexMatrixEuclideanJordanAlgebra):
+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,
self.trace_inner_product,
**kwargs)
# TODO: this could be factored out somehow, but is left here
- # because the MatrixEuclideanJordanAlgebra is not presently
- # a subclass of the FDEJA class that defines rank() and one().
+ # 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))
n = ZZ.random_element(cls._max_random_instance_size() + 1)
return cls(n, **kwargs)
-class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra):
+class QuaternionMatrixEJA(MatrixEJA):
@staticmethod
def dimension_over_reals():
return 4
SETUP::
- sage: from mjo.eja.eja_algebra import \
- ....: QuaternionMatrixEuclideanJordanAlgebra
+ sage: from mjo.eja.eja_algebra import QuaternionMatrixEJA
EXAMPLES::
sage: i,j,k = Q.gens()
sage: x = 1 + 2*i + 3*j + 4*k
sage: M = matrix(Q, 1, [[x]])
- sage: QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
+ sage: QuaternionMatrixEJA.real_embed(M)
[ 1 2 3 4]
[-2 1 -4 3]
[-3 4 1 -2]
sage: Q = QuaternionAlgebra(QQ,-1,-1)
sage: X = random_matrix(Q, n)
sage: Y = random_matrix(Q, n)
- sage: Xe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X)
- sage: Ye = QuaternionMatrixEuclideanJordanAlgebra.real_embed(Y)
- sage: XYe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X*Y)
+ 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(QuaternionMatrixEuclideanJordanAlgebra,cls).real_embed(M)
+ super(QuaternionMatrixEJA,cls).real_embed(M)
quaternions = M.base_ring()
n = M.nrows()
d = t[3]
cplxM = matrix(F, 2, [[ a + b*i, c + d*i],
[-c + d*i, a - b*i]])
- realM = ComplexMatrixEuclideanJordanAlgebra.real_embed(cplxM)
+ realM = ComplexMatrixEJA.real_embed(cplxM)
blocks.append(realM)
# We should have real entries by now, so use the realest field
SETUP::
- sage: from mjo.eja.eja_algebra import \
- ....: QuaternionMatrixEuclideanJordanAlgebra
+ sage: from mjo.eja.eja_algebra import QuaternionMatrixEJA
EXAMPLES::
....: [-2, 1, -4, 3],
....: [-3, 4, 1, -2],
....: [-4, -3, 2, 1]])
- sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(M)
+ sage: QuaternionMatrixEJA.real_unembed(M)
[1 + 2*i + 3*j + 4*k]
TESTS:
sage: set_random_seed()
sage: Q = QuaternionAlgebra(QQ, -1, -1)
sage: M = random_matrix(Q, 3)
- sage: Me = QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
- sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
+ sage: Me = QuaternionMatrixEJA.real_embed(M)
+ sage: QuaternionMatrixEJA.real_unembed(Me) == M
True
"""
- super(QuaternionMatrixEuclideanJordanAlgebra,cls).real_unembed(M)
+ super(QuaternionMatrixEJA,cls).real_unembed(M)
n = ZZ(M.nrows())
d = cls.dimension_over_reals()
elements = []
for l in range(n/d):
for m in range(n/d):
- submat = ComplexMatrixEuclideanJordanAlgebra.real_unembed(
+ 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')
return matrix(Q, n/d, elements)
-class QuaternionHermitianEJA(ConcreteEuclideanJordanAlgebra,
- QuaternionMatrixEuclideanJordanAlgebra):
+class QuaternionHermitianEJA(ConcreteEJA, QuaternionMatrixEJA):
r"""
The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
matrices, the usual symmetric Jordan product, and the
self.trace_inner_product,
**kwargs)
# TODO: this could be factored out somehow, but is left here
- # because the MatrixEuclideanJordanAlgebra is not presently
- # a subclass of the FDEJA class that defines rank() and one().
+ # 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))
return cls(n, **kwargs)
-class HadamardEJA(ConcreteEuclideanJordanAlgebra):
+class HadamardEJA(ConcreteEJA):
"""
Return the Euclidean Jordan Algebra corresponding to the set
`R^n` under the Hadamard product.
"""
def __init__(self, n, **kwargs):
- def jordan_product(x,y):
- P = x.parent()
- return P(tuple( xi*yi for (xi,yi) in zip(x,y) ))
- def inner_product(x,y):
- return x.inner_product(y)
+ 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) )
+
+ def inner_product(x,y):
+ return (x.T*y)[0,0]
# New defaults for keyword arguments. Don't orthonormalize
# because our basis is already orthonormal with respect to our
if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False
if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
-
- standard_basis = FreeModule(ZZ, n).basis()
- super(HadamardEJA, self).__init__(standard_basis,
- jordan_product,
- inner_product,
- **kwargs)
+ column_basis = tuple( b.column() for b in FreeModule(ZZ, n).basis() )
+ super().__init__(column_basis, jordan_product, inner_product, **kwargs)
self.rank.set_cache(n)
if n == 0:
return cls(n, **kwargs)
-class BilinearFormEJA(ConcreteEuclideanJordanAlgebra):
+class BilinearFormEJA(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 =
....: for j in range(n-1) ]
sage: actual == expected
True
+
"""
def __init__(self, B, **kwargs):
- if not B.is_positive_definite():
- raise ValueError("bilinear form is not positive-definite")
+ # The matrix "B" is supplied by the user in most cases,
+ # so it makes sense to check whether or not its positive-
+ # definite unless we are specifically asked not to...
+ if ("check_axioms" not in kwargs) or kwargs["check_axioms"]:
+ if not B.is_positive_definite():
+ raise ValueError("bilinear form is not positive-definite")
+
+ # However, all of the other data for this EJA is computed
+ # by us in manner that guarantees the axioms are
+ # satisfied. So, again, unless we are specifically asked to
+ # verify things, we'll skip the rest of the checks.
+ if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
def inner_product(x,y):
- return (B*x).inner_product(y)
+ return (y.T*B*x)[0,0]
def jordan_product(x,y):
P = x.parent()
- x0 = x[0]
- xbar = x[1:]
- y0 = y[0]
- ybar = y[1:]
- z0 = inner_product(x,y)
+ 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,) + tuple(zbar))
-
- # 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
+ return P([z0] + zbar.list())
n = B.nrows()
- standard_basis = FreeModule(ZZ, n).basis()
- super(BilinearFormEJA, self).__init__(standard_basis,
+ column_basis = tuple( b.column() for b in FreeModule(ZZ, n).basis() )
+ super(BilinearFormEJA, self).__init__(column_basis,
jordan_product,
inner_product,
**kwargs)
return cls(n, **kwargs)
-class TrivialEJA(ConcreteEuclideanJordanAlgebra):
+class TrivialEJA(ConcreteEJA):
"""
The trivial Euclidean Jordan algebra consisting of only a zero element.
# inappropriate for us.
return cls(**kwargs)
-class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra):
- r"""
- The external (orthogonal) direct sum of two other Euclidean Jordan
- algebras. Essentially the Cartesian product of its two factors.
- Every Euclidean Jordan algebra decomposes into an orthogonal
- direct sum of simple Euclidean Jordan algebras, so no generality
- is lost by providing only this construction.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import (random_eja,
- ....: HadamardEJA,
- ....: RealSymmetricEJA,
- ....: DirectSumEJA)
-
- EXAMPLES::
-
- sage: J1 = HadamardEJA(2)
- sage: J2 = RealSymmetricEJA(3)
- sage: J = DirectSumEJA(J1,J2)
- sage: J.dimension()
- 8
- sage: J.rank()
- 5
-
- TESTS:
-
- The external direct sum construction is only valid when the two factors
- have the same base ring; an error is raised otherwise::
-
- sage: set_random_seed()
- sage: J1 = random_eja(field=AA)
- sage: J2 = random_eja(field=QQ,orthonormalize=False)
- sage: J = DirectSumEJA(J1,J2)
- Traceback (most recent call last):
- ...
- ValueError: algebras must share the same base field
-
- """
- def __init__(self, J1, J2, **kwargs):
- if J1.base_ring() != J2.base_ring():
- raise ValueError("algebras must share the same base field")
- field = J1.base_ring()
-
- self._factors = (J1, J2)
- n1 = J1.dimension()
- n2 = J2.dimension()
- n = n1+n2
- V = VectorSpace(field, n)
- mult_table = [ [ V.zero() for j in range(i+1) ]
- for i in range(n) ]
- for i in range(n1):
- for j in range(i+1):
- p = (J1.monomial(i)*J1.monomial(j)).to_vector()
- mult_table[i][j] = V(p.list() + [field.zero()]*n2)
-
- for i in range(n2):
- for j in range(i+1):
- p = (J2.monomial(i)*J2.monomial(j)).to_vector()
- mult_table[n1+i][n1+j] = V([field.zero()]*n1 + p.list())
-
- # TODO: build the IP table here from the two constituent IP
- # matrices (it'll be block diagonal, I think).
- ip_table = [ [ field.zero() for j in range(i+1) ]
- for i in range(n) ]
- super(DirectSumEJA, self).__init__(field,
- mult_table,
- ip_table,
- check_axioms=False,
- **kwargs)
- self.rank.set_cache(J1.rank() + J2.rank())
-
-
- def factors(self):
- r"""
- Return the pair of this algebra's factors.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import (HadamardEJA,
- ....: JordanSpinEJA,
- ....: DirectSumEJA)
-
- EXAMPLES::
-
- sage: J1 = HadamardEJA(2, field=QQ)
- sage: J2 = JordanSpinEJA(3, field=QQ)
- sage: J = DirectSumEJA(J1,J2)
- sage: J.factors()
- (Euclidean Jordan algebra of dimension 2 over Rational Field,
- Euclidean Jordan algebra of dimension 3 over Rational Field)
-
- """
- return self._factors
-
- def projections(self):
- r"""
- Return a pair of projections onto this algebra's factors.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
- ....: ComplexHermitianEJA,
- ....: DirectSumEJA)
-
- EXAMPLES::
-
- sage: J1 = JordanSpinEJA(2)
- sage: J2 = ComplexHermitianEJA(2)
- sage: J = DirectSumEJA(J1,J2)
- sage: (pi_left, pi_right) = J.projections()
- sage: J.one().to_vector()
- (1, 0, 1, 0, 0, 1)
- sage: pi_left(J.one()).to_vector()
- (1, 0)
- sage: pi_right(J.one()).to_vector()
- (1, 0, 0, 1)
-
- """
- (J1,J2) = self.factors()
- m = J1.dimension()
- n = J2.dimension()
- V_basis = self.vector_space().basis()
- # Need to specify the dimensions explicitly so that we don't
- # wind up with a zero-by-zero matrix when we want e.g. a
- # zero-by-two matrix (important for composing things).
- P1 = matrix(self.base_ring(), m, m+n, V_basis[:m])
- P2 = matrix(self.base_ring(), n, m+n, V_basis[m:])
- pi_left = FiniteDimensionalEuclideanJordanAlgebraOperator(self,J1,P1)
- pi_right = FiniteDimensionalEuclideanJordanAlgebraOperator(self,J2,P2)
- return (pi_left, pi_right)
-
- def inclusions(self):
- r"""
- Return the pair of inclusion maps from our factors into us.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import (random_eja,
- ....: JordanSpinEJA,
- ....: RealSymmetricEJA,
- ....: DirectSumEJA)
-
- EXAMPLES::
-
- sage: J1 = JordanSpinEJA(3)
- sage: J2 = RealSymmetricEJA(2)
- sage: J = DirectSumEJA(J1,J2)
- sage: (iota_left, iota_right) = J.inclusions()
- sage: iota_left(J1.zero()) == J.zero()
- True
- sage: iota_right(J2.zero()) == J.zero()
- True
- sage: J1.one().to_vector()
- (1, 0, 0)
- sage: iota_left(J1.one()).to_vector()
- (1, 0, 0, 0, 0, 0)
- sage: J2.one().to_vector()
- (1, 0, 1)
- sage: iota_right(J2.one()).to_vector()
- (0, 0, 0, 1, 0, 1)
- sage: J.one().to_vector()
- (1, 0, 0, 1, 0, 1)
-
- TESTS:
-
- Composing a projection with the corresponding inclusion should
- 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 = DirectSumEJA(J1,J2)
- sage: (iota_left, iota_right) = J.inclusions()
- sage: (pi_left, pi_right) = J.projections()
- sage: pi_left*iota_left == J1.one().operator()
- True
- sage: pi_right*iota_right == J2.one().operator()
- True
- sage: (pi_left*iota_right).is_zero()
- True
- sage: (pi_right*iota_left).is_zero()
- True
-
- """
- (J1,J2) = self.factors()
- m = J1.dimension()
- n = J2.dimension()
- V_basis = self.vector_space().basis()
- # Need to specify the dimensions explicitly so that we don't
- # wind up with a zero-by-zero matrix when we want e.g. a
- # two-by-zero matrix (important for composing things).
- I1 = matrix.column(self.base_ring(), m, m+n, V_basis[:m])
- I2 = matrix.column(self.base_ring(), n, m+n, V_basis[m:])
- iota_left = FiniteDimensionalEuclideanJordanAlgebraOperator(J1,self,I1)
- iota_right = FiniteDimensionalEuclideanJordanAlgebraOperator(J2,self,I2)
- return (iota_left, iota_right)
-
- def inner_product(self, x, y):
- r"""
- The standard Cartesian inner-product.
-
- We project ``x`` and ``y`` onto our factors, and add up the
- inner-products from the subalgebras.
-
- SETUP::
-
-
- sage: from mjo.eja.eja_algebra import (HadamardEJA,
- ....: QuaternionHermitianEJA,
- ....: DirectSumEJA)
-
- EXAMPLE::
-
- sage: J1 = HadamardEJA(3,field=QQ)
- sage: J2 = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False)
- sage: J = DirectSumEJA(J1,J2)
- sage: x1 = J1.one()
- sage: x2 = x1
- sage: y1 = J2.one()
- sage: y2 = y1
- sage: x1.inner_product(x2)
- 3
- sage: y1.inner_product(y2)
- 2
- sage: J.one().inner_product(J.one())
- 5
-
- """
- (pi_left, pi_right) = self.projections()
- x1 = pi_left(x)
- x2 = pi_right(x)
- y1 = pi_left(y)
- y2 = pi_right(y)
-
- return (x1.inner_product(y1) + x2.inner_product(y2))
-
-
-
-random_eja = ConcreteEuclideanJordanAlgebra.random_instance
+# class DirectSumEJA(ConcreteEJA):
+# r"""
+# The external (orthogonal) direct sum of two other Euclidean Jordan
+# algebras. Essentially the Cartesian product of its two factors.
+# Every Euclidean Jordan algebra decomposes into an orthogonal
+# direct sum of simple Euclidean Jordan algebras, so no generality
+# is lost by providing only this construction.
+
+# SETUP::
+
+# sage: from mjo.eja.eja_algebra import (random_eja,
+# ....: HadamardEJA,
+# ....: RealSymmetricEJA,
+# ....: DirectSumEJA)
+
+# EXAMPLES::
+
+# sage: J1 = HadamardEJA(2)
+# sage: J2 = RealSymmetricEJA(3)
+# sage: J = DirectSumEJA(J1,J2)
+# sage: J.dimension()
+# 8
+# sage: J.rank()
+# 5
+
+# TESTS:
+
+# The external direct sum construction is only valid when the two factors
+# have the same base ring; an error is raised otherwise::
+
+# sage: set_random_seed()
+# sage: J1 = random_eja(field=AA)
+# sage: J2 = random_eja(field=QQ,orthonormalize=False)
+# sage: J = DirectSumEJA(J1,J2)
+# Traceback (most recent call last):
+# ...
+# ValueError: algebras must share the same base field
+
+# """
+# def __init__(self, J1, J2, **kwargs):
+# if J1.base_ring() != J2.base_ring():
+# raise ValueError("algebras must share the same base field")
+# field = J1.base_ring()
+
+# self._factors = (J1, J2)
+# n1 = J1.dimension()
+# n2 = J2.dimension()
+# n = n1+n2
+# V = VectorSpace(field, n)
+# mult_table = [ [ V.zero() for j in range(i+1) ]
+# for i in range(n) ]
+# for i in range(n1):
+# for j in range(i+1):
+# p = (J1.monomial(i)*J1.monomial(j)).to_vector()
+# mult_table[i][j] = V(p.list() + [field.zero()]*n2)
+
+# for i in range(n2):
+# for j in range(i+1):
+# p = (J2.monomial(i)*J2.monomial(j)).to_vector()
+# mult_table[n1+i][n1+j] = V([field.zero()]*n1 + p.list())
+
+# # TODO: build the IP table here from the two constituent IP
+# # matrices (it'll be block diagonal, I think).
+# ip_table = [ [ field.zero() for j in range(i+1) ]
+# for i in range(n) ]
+# super(DirectSumEJA, self).__init__(field,
+# mult_table,
+# ip_table,
+# check_axioms=False,
+# **kwargs)
+# self.rank.set_cache(J1.rank() + J2.rank())
+
+
+# def factors(self):
+# r"""
+# Return the pair of this algebra's factors.
+
+# SETUP::
+
+# sage: from mjo.eja.eja_algebra import (HadamardEJA,
+# ....: JordanSpinEJA,
+# ....: DirectSumEJA)
+
+# EXAMPLES::
+
+# sage: J1 = HadamardEJA(2, field=QQ)
+# sage: J2 = JordanSpinEJA(3, field=QQ)
+# sage: J = DirectSumEJA(J1,J2)
+# sage: J.factors()
+# (Euclidean Jordan algebra of dimension 2 over Rational Field,
+# Euclidean Jordan algebra of dimension 3 over Rational Field)
+
+# """
+# return self._factors
+
+# def projections(self):
+# r"""
+# Return a pair of projections onto this algebra's factors.
+
+# SETUP::
+
+# sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+# ....: ComplexHermitianEJA,
+# ....: DirectSumEJA)
+
+# EXAMPLES::
+
+# sage: J1 = JordanSpinEJA(2)
+# sage: J2 = ComplexHermitianEJA(2)
+# sage: J = DirectSumEJA(J1,J2)
+# sage: (pi_left, pi_right) = J.projections()
+# sage: J.one().to_vector()
+# (1, 0, 1, 0, 0, 1)
+# sage: pi_left(J.one()).to_vector()
+# (1, 0)
+# sage: pi_right(J.one()).to_vector()
+# (1, 0, 0, 1)
+
+# """
+# (J1,J2) = self.factors()
+# m = J1.dimension()
+# n = J2.dimension()
+# V_basis = self.vector_space().basis()
+# # Need to specify the dimensions explicitly so that we don't
+# # wind up with a zero-by-zero matrix when we want e.g. a
+# # zero-by-two matrix (important for composing things).
+# P1 = matrix(self.base_ring(), m, m+n, V_basis[:m])
+# P2 = matrix(self.base_ring(), n, m+n, V_basis[m:])
+# pi_left = FiniteDimensionalEJAOperator(self,J1,P1)
+# pi_right = FiniteDimensionalEJAOperator(self,J2,P2)
+# return (pi_left, pi_right)
+
+# def inclusions(self):
+# r"""
+# Return the pair of inclusion maps from our factors into us.
+
+# SETUP::
+
+# sage: from mjo.eja.eja_algebra import (random_eja,
+# ....: JordanSpinEJA,
+# ....: RealSymmetricEJA,
+# ....: DirectSumEJA)
+
+# EXAMPLES::
+
+# sage: J1 = JordanSpinEJA(3)
+# sage: J2 = RealSymmetricEJA(2)
+# sage: J = DirectSumEJA(J1,J2)
+# sage: (iota_left, iota_right) = J.inclusions()
+# sage: iota_left(J1.zero()) == J.zero()
+# True
+# sage: iota_right(J2.zero()) == J.zero()
+# True
+# sage: J1.one().to_vector()
+# (1, 0, 0)
+# sage: iota_left(J1.one()).to_vector()
+# (1, 0, 0, 0, 0, 0)
+# sage: J2.one().to_vector()
+# (1, 0, 1)
+# sage: iota_right(J2.one()).to_vector()
+# (0, 0, 0, 1, 0, 1)
+# sage: J.one().to_vector()
+# (1, 0, 0, 1, 0, 1)
+
+# TESTS:
+
+# Composing a projection with the corresponding inclusion should
+# 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 = DirectSumEJA(J1,J2)
+# sage: (iota_left, iota_right) = J.inclusions()
+# sage: (pi_left, pi_right) = J.projections()
+# sage: pi_left*iota_left == J1.one().operator()
+# True
+# sage: pi_right*iota_right == J2.one().operator()
+# True
+# sage: (pi_left*iota_right).is_zero()
+# True
+# sage: (pi_right*iota_left).is_zero()
+# True
+
+# """
+# (J1,J2) = self.factors()
+# m = J1.dimension()
+# n = J2.dimension()
+# V_basis = self.vector_space().basis()
+# # Need to specify the dimensions explicitly so that we don't
+# # wind up with a zero-by-zero matrix when we want e.g. a
+# # two-by-zero matrix (important for composing things).
+# I1 = matrix.column(self.base_ring(), m, m+n, V_basis[:m])
+# I2 = matrix.column(self.base_ring(), n, m+n, V_basis[m:])
+# iota_left = FiniteDimensionalEJAOperator(J1,self,I1)
+# iota_right = FiniteDimensionalEJAOperator(J2,self,I2)
+# return (iota_left, iota_right)
+
+# def inner_product(self, x, y):
+# r"""
+# The standard Cartesian inner-product.
+
+# We project ``x`` and ``y`` onto our factors, and add up the
+# inner-products from the subalgebras.
+
+# SETUP::
+
+
+# sage: from mjo.eja.eja_algebra import (HadamardEJA,
+# ....: QuaternionHermitianEJA,
+# ....: DirectSumEJA)
+
+# EXAMPLE::
+
+# sage: J1 = HadamardEJA(3,field=QQ)
+# sage: J2 = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False)
+# sage: J = DirectSumEJA(J1,J2)
+# sage: x1 = J1.one()
+# sage: x2 = x1
+# sage: y1 = J2.one()
+# sage: y2 = y1
+# sage: x1.inner_product(x2)
+# 3
+# sage: y1.inner_product(y2)
+# 2
+# sage: J.one().inner_product(J.one())
+# 5
+
+# """
+# (pi_left, pi_right) = self.projections()
+# x1 = pi_left(x)
+# x2 = pi_right(x)
+# y1 = pi_left(y)
+# y2 = pi_right(y)
+
+# return (x1.inner_product(y1) + x2.inner_product(y2))
+
+
+
+random_eja = ConcreteEJA.random_instance
projects / sage.d.git / blobdiff