associative=False,
check_field=True,
check_axioms=True,
- prefix='e'):
+ prefix='e',
+ category=None):
if check_field:
if not field.is_subring(RR):
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()
+ if category is None:
+ 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.
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)
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)
SETUP::
- sage: from mjo.eja.eja_algebra import (CartesianProductEJA,
+ sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: CartesianProductEJA,
....: HadamardEJA,
....: JordanSpinEJA,
....: RealSymmetricEJA)
The ability to retrieve the original factors is implemented by our
CombinatorialFreeModule Cartesian product superclass::
- sage: J1 = HadamardEJA(2, field=QQ)
- sage: J2 = JordanSpinEJA(3, field=QQ)
- sage: J = cartesian_product([J1,J2])
- sage: J.cartesian_factors()
- (Euclidean Jordan algebra of dimension 2 over Rational Field,
- Euclidean Jordan algebra of dimension 3 over Rational Field)
+ sage: J1 = HadamardEJA(2, field=QQ)
+ sage: J2 = JordanSpinEJA(3, field=QQ)
+ sage: J = cartesian_product([J1,J2])
+ sage: J.cartesian_factors()
+ (Euclidean Jordan algebra of dimension 2 over Rational Field,
+ Euclidean Jordan algebra of dimension 3 over Rational Field)
+
+ You can provide more than two factors::
+
+ sage: J1 = HadamardEJA(2)
+ sage: J2 = JordanSpinEJA(3)
+ sage: J3 = RealSymmetricEJA(3)
+ sage: cartesian_product([J1,J2,J3])
+ Euclidean Jordan algebra of dimension 2 over Algebraic Real
+ Field (+) Euclidean Jordan algebra of dimension 3 over Algebraic
+ 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
TESTS:
Traceback (most recent call last):
...
ValueError: all factors must share the same base field
+
+ The "cached" Jordan and inner products are the componentwise
+ ones::
+
+ 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)
+ True
+
+ 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: 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):
- CombinatorialFreeModule_CartesianProduct.__init__(self, modules, **kwargs)
+ CombinatorialFreeModule_CartesianProduct.__init__(self,
+ modules,
+ **kwargs)
field = modules[0].base_ring()
if not all( J.base_ring() == field for J in modules ):
raise ValueError("all factors must share the same base field")
- M = cartesian_product( [J.matrix_space() for J in modules] )
+ basis = tuple( b.to_vector().column() for b in self.basis() )
- m = len(modules)
- W = VectorSpace(field,m)
- self._matrix_basis = []
- for k in range(m):
- for a in modules[k].matrix_basis():
- v = W.zero().list()
- v[k] = a
- self._matrix_basis.append(M(v))
+ # 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()
- self._matrix_basis = tuple(self._matrix_basis)
+ 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)
- n = len(self._matrix_basis)
- # TODO:
- #
- # Initialize the FDEJA class, too. Does this override the
- # initialization that we did for the
- # CombinatorialFreeModule_CartesianProduct class? If not, we
- # will probably have to duplicate some of the work (i.e. one
- # of the constructors). Since the CartesianProduct one is
- # smaller, that makes the most sense to copy/paste if it comes
- # down to that.
+ # 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.
#
-
+ # If you want the basis to be orthonormalized, orthonormalize
+ # the factors.
+ FiniteDimensionalEJA.__init__(self,
+ basis,
+ jordan_product,
+ inner_product,
+ field=field,
+ orthonormalize=False,
+ check_field=False,
+ check_axioms=False,
+ category=self.category())
+
+ ones = tuple(J.one() for J in modules)
+ self.one.set_cache(self._cartesian_product_of_elements(ones))
self.rank.set_cache(sum(J.rank() for J in modules))
+ # Now that everything else is ready, we clobber our computed
+ # matrix basis with the "correct" one consisting of ordered
+ # tuples. Since we didn't orthonormalize our basis, we can
+ # create these from the basis that was handed to us; that is,
+ # we don't need to use the one that the earlier __init__()
+ # method came up with.
+ m = len(self.cartesian_factors())
+ MS = self.matrix_space()
+ self._matrix_basis = tuple(
+ MS(tuple( self.cartesian_projection(i)(b).to_matrix()
+ for i in range(m) ))
+ for b in self.basis()
+ )
+
+ 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):
r"""
SETUP::
sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: JordanSpinEJA,
....: HadamardEJA,
- ....: RealSymmetricEJA)
+ ....: RealSymmetricEJA,
+ ....: ComplexHermitianEJA)
- EXAMPLES::
+ EXAMPLES:
+
+ The projection morphisms are Euclidean Jordan algebra
+ operators::
sage: J1 = HadamardEJA(2)
sage: J2 = RealSymmetricEJA(2)
Codomain: Euclidean Jordan algebra of dimension 3 over Algebraic
Real Field
+ The projections work the way you'd expect on the vector
+ representation of an element::
+
+ sage: J1 = JordanSpinEJA(2)
+ sage: J2 = ComplexHermitianEJA(2)
+ sage: J = cartesian_product([J1,J2])
+ sage: pi_left = J.cartesian_projection(0)
+ sage: pi_right = J.cartesian_projection(1)
+ sage: pi_left(J.one()).to_vector()
+ (1, 0)
+ sage: pi_right(J.one()).to_vector()
+ (1, 0, 0, 1)
+ sage: J.one().to_vector()
+ (1, 0, 1, 0, 0, 1)
+
TESTS:
The answer never changes::
"""
Ji = self.cartesian_factors()[i]
- # We reimplement the CombinatorialFreeModule superclass method
- # because if we don't, something gets messed up with the caching
- # and the answer changes the second time you run it. See the TESTS.
- Pi = self._module_morphism(lambda j_t: Ji.monomial(j_t[1])
- if i == j_t[0] else Ji.zero(),
- codomain=Ji)
+ # Requires the fix on Trac 31421/31422 to work!
+ Pi = super().cartesian_projection(i)
return FiniteDimensionalEJAOperator(self,Ji,Pi.matrix())
@cached_method
SETUP::
sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: JordanSpinEJA,
....: HadamardEJA,
....: RealSymmetricEJA)
- EXAMPLES::
+ EXAMPLES:
+
+ The embedding morphisms are Euclidean Jordan algebra
+ operators::
sage: J1 = HadamardEJA(2)
sage: J2 = RealSymmetricEJA(2)
sage: J = cartesian_product([J1,J2])
- sage: J
- foo
- sage: J.cartesian_embedding
- bar
sage: J.cartesian_embedding(0)
Linear operator between finite-dimensional Euclidean Jordan
algebras represented by the matrix:
Algebraic Real Field (+) Euclidean Jordan algebra of
dimension 3 over Algebraic Real Field
+ The embeddings work the way you'd expect on the vector
+ representation of an element::
+
+ sage: J1 = JordanSpinEJA(3)
+ sage: J2 = RealSymmetricEJA(2)
+ sage: J = cartesian_product([J1,J2])
+ sage: iota_left = J.cartesian_embedding(0)
+ sage: iota_right = J.cartesian_embedding(1)
+ 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:
The answer never changes::
sage: E0 == E1
True
+ 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 = cartesian_product([J1,J2])
+ sage: iota_left = J.cartesian_embedding(0)
+ sage: iota_right = J.cartesian_embedding(1)
+ sage: pi_left = J.cartesian_projection(0)
+ sage: pi_right = J.cartesian_projection(1)
+ 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
+
"""
Ji = self.cartesian_factors()[i]
- # We reimplement the CombinatorialFreeModule superclass method
- # because if we don't, something gets messed up with the caching
- # and the answer changes the second time you run it. See the TESTS.
- Ei = Ji._module_morphism(lambda t: self.monomial((i, t)), codomain=self)
+ # Requires the fix on Trac 31421/31422 to work!
+ Ei = super().cartesian_embedding(i)
return FiniteDimensionalEJAOperator(Ji,self,Ei.matrix())
-FiniteDimensionalEJA.CartesianProduct = CartesianProductEJA
+ def cartesian_jordan_product(self, x, y):
+ r"""
+ The componentwise Jordan product.
+
+ 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.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: JordanSpinEJA)
+
+ EXAMPLE::
+
+ 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.
-# 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))
+ 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: 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
+ """
+ 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 )
+
+
+ def _element_constructor_(self, elt):
+ r"""
+ Construct an element of this algebra from an ordered tuple.
+
+ 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,
+ ....: RealSymmetricEJA)
+
+ EXAMPLES::
+
+ sage: J1 = HadamardEJA(3)
+ sage: J2 = RealSymmetricEJA(2)
+ sage: J = cartesian_product([J1,J2])
+ sage: J( (J1.matrix_basis()[1], J2.matrix_basis()[2]) )
+ e(0, 1) + e(1, 2)
+ """
+ m = len(self.cartesian_factors())
+ z = tuple( self.cartesian_factors()[i](elt[i]) for i in range(m) )
+ return self._cartesian_product_of_elements(z)
+
+ Element = FiniteDimensionalEJAElement
+
+
+FiniteDimensionalEJA.CartesianProduct = CartesianProductEJA
random_eja = ConcreteEJA.random_instance
projects / sage.d.git / blobdiff