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_element import (CartesianProductEJAElement,
+ FiniteDimensionalEJAElement)
from mjo.eja.eja_operator import FiniteDimensionalEJAOperator
from mjo.eja.eja_utils import _mat2vec
INPUT:
- - basis -- a tuple of basis elements in their matrix form.
+ - basis -- a tuple of basis elements in "matrix form," which
+ must be the same form as the arguments to ``jordan_product``
+ and ``inner_product``. In reality, "matrix form" can be either
+ vectors, matrices, or a Cartesian product (ordered tuple)
+ of vectors or matrices. All of these would ideally be vector
+ spaces in sage with no special-casing needed; but in reality
+ we turn vectors into column-matrices and Cartesian products
+ `(a,b)` into column matrices `(a,b)^{T}` after converting
+ `a` and `b` themselves.
- jordan_product -- function of two elements (in matrix form)
that returns their jordan product in this algebra; this will
field=AA,
orthonormalize=True,
associative=False,
+ cartesian_product=False,
check_field=True,
check_axioms=True,
- prefix='e',
- category=None):
+ prefix='e'):
if check_field:
if not field.is_subring(RR):
# 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 not cartesian_product:
+ # The field for a cartesian product algebra comes from one
+ # of its factors and is the same for all factors, so
+ # there's no need to "reapply" it on product algebras.
+ basis = tuple( b.change_ring(field) for b in basis )
+
if check_axioms:
# Check commutativity of the Jordan and inner-products.
raise ValueError("inner-product is not commutative")
- if category is None:
- category = MagmaticAlgebras(field).FiniteDimensional()
- category = category.WithBasis().Unital()
- if associative:
- # Element subalgebras can take advantage of this.
- category = category.Associative()
+ category = MagmaticAlgebras(field).FiniteDimensional()
+ category = category.WithBasis().Unital()
+ if associative:
+ # Element subalgebras can take advantage of this.
+ category = category.Associative()
+ if cartesian_product:
+ category = category.CartesianProducts()
# Call the superclass constructor so that we can use its from_vector()
# method to build our multiplication table.
# we see in things like x = 1*e1 + 2*e2.
vector_basis = basis
+ def flatten(b):
+ # flatten a vector, matrix, or cartesian product of those
+ # things into a long list.
+ if cartesian_product:
+ return sum(( b_i.list() for b_i in b ), [])
+ else:
+ return b.list()
+
degree = 0
if n > 0:
- # Works on both column and square matrices...
- degree = len(basis[0].list())
+ degree = len(flatten(basis[0]))
# Build an ambient space that fits our matrix basis when
# written out as "long vectors."
# 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 )
+ deortho_vector_basis = tuple( V(flatten(b)) for b in basis )
from mjo.eja.eja_utils import gram_schmidt
basis = tuple(gram_schmidt(basis, inner_product))
# 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 )
+ vector_basis = tuple( V(flatten(b)) for b in basis )
W = V.span_of_basis( vector_basis, check=check_axioms)
if orthonormalize:
# 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(elt.list()))
+ elt = W.coordinate_vector(V(flatten(elt)))
self._multiplication_table[i][j] = self.from_vector(elt)
if not orthonormalize:
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())
- def _is_commutative(self):
+ def is_associative(self):
r"""
- Whether or not this algebra's multiplication table is commutative.
+ Return whether or not this algebra's Jordan product is associative.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
+
+ EXAMPLES::
+
+ sage: J = ComplexHermitianEJA(3, field=QQ, orthonormalize=False)
+ sage: J.is_associative()
+ False
+ sage: x = sum(J.gens())
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: A.is_associative()
+ 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()) )
+ return "Associative" in self.category().axioms()
def _is_jordanian(self):
r"""
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
+ :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))
+ return all( (self.gens()[i]**2)*(self.gens()[i]*self.gens()[j])
==
- (self.monomial(i))*((self.monomial(i)**2)*self.monomial(j))
+ (self.gens()[i])*((self.gens()[i]**2)*self.gens()[j])
for i in range(self.dimension())
for j in range(self.dimension()) )
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)
+ x = self.gens()[i]
+ y = self.gens()[j]
+ z = self.gens()[k]
diff = (x*y).inner_product(z) - x.inner_product(y*z)
if self.base_ring().is_exact():
# 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) ]
+ M += [ [self.gens()[i]] + [ self.product_on_basis(i,j)
+ for j in range(n) ]
for i in range(n) ]
return table(M, header_row=True, header_column=True, frame=True)
if not c.is_idempotent():
raise ValueError("element is not idempotent: %s" % c)
- 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 = FiniteDimensionalEJASubalgebra(self, ())
+ trivial = self.subalgebra(())
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 = FiniteDimensionalEJASubalgebra(self,
- gens,
- check_axioms=False)
+ subalg = self.subalgebra(gens, check_axioms=False)
if eigval == 0:
J0 = subalg
elif eigval == 1:
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]
+ return sum( vars[k]*self.gens()[k].operator().matrix()[i,j]
for k in range(n) )
L_x = matrix(F, n, n, L_x_i_j)
return len(self._charpoly_coefficients())
+ def subalgebra(self, basis, **kwargs):
+ r"""
+ Create a subalgebra of this algebra from the given basis.
+
+ This is a simple wrapper around a subalgebra class constructor
+ that can be overridden in subclasses.
+ """
+ from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra
+ return FiniteDimensionalEJASubalgebra(self, basis, **kwargs)
+
+
def vector_space(self):
"""
Return the vector space that underlies this algebra.
if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
column_basis = tuple( b.column() for b in FreeModule(ZZ, n).basis() )
- super().__init__(column_basis, jordan_product, inner_product, **kwargs)
+ super().__init__(column_basis,
+ jordan_product,
+ inner_product,
+ associative=True,
+ **kwargs)
self.rank.set_cache(n)
if n == 0:
Real Field (+) Euclidean Jordan algebra of dimension 6 over
Algebraic Real Field
+ Rank is additive on a Cartesian product::
+
+ sage: J1 = HadamardEJA(1)
+ sage: J2 = RealSymmetricEJA(2)
+ sage: J = cartesian_product([J1,J2])
+ sage: J1.rank.clear_cache()
+ sage: J2.rank.clear_cache()
+ sage: J.rank.clear_cache()
+ sage: J.rank()
+ 3
+ sage: J.rank() == J1.rank() + J2.rank()
+ True
+
+ The same rank computation works over the rationals, with whatever
+ basis you like::
+
+ sage: J1 = HadamardEJA(1, field=QQ, orthonormalize=False)
+ sage: J2 = RealSymmetricEJA(2, field=QQ, orthonormalize=False)
+ sage: J = cartesian_product([J1,J2])
+ sage: J1.rank.clear_cache()
+ sage: J2.rank.clear_cache()
+ sage: J.rank.clear_cache()
+ sage: J.rank()
+ 3
+ sage: J.rank() == J1.rank() + J2.rank()
+ True
+
+ The product algebra will be associative if and only if all of its
+ components are associative::
+
+ sage: J1 = HadamardEJA(2)
+ sage: J1.is_associative()
+ True
+ sage: J2 = HadamardEJA(3)
+ sage: J2.is_associative()
+ True
+ sage: J3 = RealSymmetricEJA(3)
+ sage: J3.is_associative()
+ False
+ sage: CP1 = cartesian_product([J1,J2])
+ sage: CP1.is_associative()
+ True
+ sage: CP2 = cartesian_product([J1,J3])
+ sage: CP2.is_associative()
+ False
+
TESTS:
All factors must share the same base field::
...
ValueError: all factors must share the same base field
- The "cached" Jordan and inner products are the componentwise
- ones::
+ The cached unit element is the same one that would be computed::
- sage: set_random_seed()
- sage: J1 = random_eja()
- sage: J2 = random_eja()
- sage: J = cartesian_product([J1,J2])
- sage: x,y = J.random_elements(2)
- sage: x*y == J.cartesian_jordan_product(x,y)
- True
- sage: x.inner_product(y) == J.cartesian_inner_product(x,y)
+ 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: actual = J.one() # long time
+ sage: J.one.clear_cache() # long time
+ sage: expected = J.one() # long time
+ sage: actual == expected # long time
True
"""
- def __init__(self, modules, **kwargs):
+ def __init__(self, algebras, **kwargs):
CombinatorialFreeModule_CartesianProduct.__init__(self,
- modules,
+ algebras,
**kwargs)
- field = modules[0].base_ring()
- if not all( J.base_ring() == field for J in modules ):
+ field = algebras[0].base_ring()
+ if not all( J.base_ring() == field for J in algebras ):
raise ValueError("all factors must share the same base field")
- basis = tuple( b.to_vector().column() for b in self.basis() )
+ associative = all( m.is_associative() for m in algebras )
+
+ # The definition of matrix_space() and self.basis() relies
+ # only on the stuff in the CFM_CartesianProduct class, which
+ # we've already initialized.
+ Js = self.cartesian_factors()
+ m = len(Js)
+ MS = self.matrix_space()
+ basis = tuple(
+ MS(tuple( self.cartesian_projection(i)(b).to_matrix()
+ for i in range(m) ))
+ for b in self.basis()
+ )
- # Define jordan/inner products that operate on the basis.
- def jordan_product(x_mat,y_mat):
- x = self.from_vector(_mat2vec(x_mat))
- y = self.from_vector(_mat2vec(y_mat))
- return self.cartesian_jordan_product(x,y).to_vector().column()
+ # Define jordan/inner products that operate on that matrix_basis.
+ def jordan_product(x,y):
+ return MS(tuple(
+ (Js[i](x[i])*Js[i](y[i])).to_matrix() for i in range(m)
+ ))
- def inner_product(x_mat, y_mat):
- x = self.from_vector(_mat2vec(x_mat))
- y = self.from_vector(_mat2vec(y_mat))
- return self.cartesian_inner_product(x,y)
+ def inner_product(x, y):
+ return sum(
+ Js[i](x[i]).inner_product(Js[i](y[i])) for i in range(m)
+ )
- # Use whatever category the superclass came up with. Usually
- # some join of the EJA and Cartesian product
- # categories. There's no need to check the field since it
- # already came from an EJA.
+ # 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.
inner_product,
field=field,
orthonormalize=False,
+ associative=associative,
+ cartesian_product=True,
check_field=False,
- check_axioms=False,
- category=self.category())
+ check_axioms=False)
+
+ ones = tuple(J.one() for J in algebras)
+ self.one.set_cache(self._cartesian_product_of_elements(ones))
+ self.rank.set_cache(sum(J.rank() for J in algebras))
+
+ 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::
- self.rank.set_cache(sum(J.rank() for J in modules))
+ 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):
return FiniteDimensionalEJAOperator(Ji,self,Ei.matrix())
- def cartesian_jordan_product(self, x, y):
+ def _element_constructor_(self, elt):
r"""
- The componentwise Jordan product.
+ Construct an element of this algebra from an ordered tuple.
- We project ``x`` and ``y`` onto our factors, and add up the
- Jordan products from the subalgebras. This may still be useful
- after (if) the default Jordan product in the Cartesian product
- algebra is overridden.
+ We just apply the element constructor from each of our factors
+ to the corresponding component of the tuple, and package up
+ the result.
SETUP::
sage: from mjo.eja.eja_algebra import (HadamardEJA,
- ....: JordanSpinEJA)
+ ....: RealSymmetricEJA)
- EXAMPLE::
+ EXAMPLES::
sage: J1 = HadamardEJA(3)
- sage: J2 = JordanSpinEJA(3)
- sage: J = cartesian_product([J1,J2])
- sage: x1 = J1.from_vector(vector(QQ,(1,2,1)))
- sage: y1 = J1.from_vector(vector(QQ,(1,0,2)))
- sage: x2 = J2.from_vector(vector(QQ,(1,2,3)))
- sage: y2 = J2.from_vector(vector(QQ,(1,1,1)))
- sage: z1 = J.from_vector(vector(QQ,(1,2,1,1,2,3)))
- sage: z2 = J.from_vector(vector(QQ,(1,0,2,1,1,1)))
- sage: (x1*y1).to_vector()
- (1, 0, 2)
- sage: (x2*y2).to_vector()
- (6, 3, 4)
- sage: J.cartesian_jordan_product(z1,z2).to_vector()
- (1, 0, 2, 6, 3, 4)
-
- """
- m = len(self.cartesian_factors())
- projections = ( self.cartesian_projection(i) for i in range(m) )
- products = ( P(x)*P(y) for P in projections )
- return self._cartesian_product_of_elements(tuple(products))
-
- def cartesian_inner_product(self, x, y):
- r"""
- The standard componentwise Cartesian inner-product.
-
- We project ``x`` and ``y`` onto our factors, and add up the
- inner-products from the subalgebras. This may still be useful
- after (if) the default inner product in the Cartesian product
- algebra is overridden.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import (HadamardEJA,
- ....: QuaternionHermitianEJA)
-
- EXAMPLE::
-
- sage: J1 = HadamardEJA(3,field=QQ)
- sage: J2 = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: J2 = RealSymmetricEJA(2)
sage: J = cartesian_product([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: z1 = J._cartesian_product_of_elements((x1,y1))
- sage: z2 = J._cartesian_product_of_elements((x2,y2))
- sage: J.cartesian_inner_product(z1,z2)
- 5
-
+ sage: J( (J1.matrix_basis()[1], J2.matrix_basis()[2]) )
+ e(0, 1) + e(1, 2)
"""
m = len(self.cartesian_factors())
- projections = ( self.cartesian_projection(i) for i in range(m) )
- return sum( P(x).inner_product(P(y)) for P in projections )
-
+ try:
+ z = tuple( self.cartesian_factors()[i](elt[i]) for i in range(m) )
+ return self._cartesian_product_of_elements(z)
+ except:
+ raise ValueError("not an element of this algebra")
- Element = FiniteDimensionalEJAElement
+ Element = CartesianProductEJAElement
FiniteDimensionalEJA.CartesianProduct = CartesianProductEJA
+
random_eja = ConcreteEJA.random_instance
+#def random_eja(*args, **kwargs):
+# from sage.categories.cartesian_product import cartesian_product
+# J1 = HadamardEJA(1, **kwargs)
+# J2 = RealSymmetricEJA(2, **kwargs)
+# J = cartesian_product([J1,J2])
+# return J
projects / sage.d.git / blobdiff