-"""
+r"""
Representations and constructions for Euclidean Jordan algebras.
A Euclidean Jordan algebra is a Jordan algebra that has some
* :class:`QuaternionHermitianEJA`
* :class:`OctonionHermitianEJA`
-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
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 FiniteDimensionalEJA
+ 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 = FiniteDimensionalEJA((b1,), jp, ip)
+ 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 = FiniteDimensionalEJA((b1,), jp, ip2)
+ sage: J2 = EJA((b1,), jp, ip2)
sage: J2
Euclidean Jordan algebra of dimension 1 over Algebraic Real Field
given inner-product_ unless you pass ``orthonormalize=False`` to the
constructor. For example::
- sage: J3 = FiniteDimensionalEJA((b1,), jp, ip2, orthonormalize=False)
+ sage: J3 = EJA((b1,), jp, ip2, orthonormalize=False)
sage: J3
Euclidean Jordan algebra of dimension 1 over Algebraic Real Field
sage: from mjo.matrix_algebra import MatrixAlgebra
sage: A = MatrixAlgebra(1,AA,AA)
- sage: J4 = FiniteDimensionalEJA(A.gens(), jp, ip)
+ 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()
PolynomialRing,
QuadraticField)
from mjo.eja.eja_element import (CartesianProductEJAElement,
- FiniteDimensionalEJAElement)
-from mjo.eja.eja_operator import FiniteDimensionalEJAOperator
+ EJAElement)
+from mjo.eja.eja_operator import EJAOperator
from mjo.eja.eja_utils import _all2list
def EuclideanJordanAlgebras(field):
category = category.WithBasis().Unital().Commutative()
return category
-class FiniteDimensionalEJA(CombinatorialFreeModule):
+class EJA(CombinatorialFreeModule):
r"""
A finite-dimensional Euclidean Jordan algebra.
sage: J.subalgebra(basis, orthonormalize=False).is_associative()
True
"""
- Element = FiniteDimensionalEJAElement
+ Element = EJAElement
@staticmethod
def _check_input_field(field):
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: 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
# 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):
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"""
Algebras whose supplied basis elements have all rational entries.
# 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,
# 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() )
- # 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(FiniteDimensionalEJA):
+class ConcreteEJA(EJA):
r"""
A class for the Euclidean Jordan algebras that we know by name.
return eja_class.random_instance(max_dimension, *args, **kwargs)
-class HermitianMatrixEJA(FiniteDimensionalEJA):
+class HermitianMatrixEJA(EJA):
@staticmethod
def _denormalized_basis(A):
"""
...
TypeError: Illegal initializer for algebraic number
- This causes the following error when we try to scale a matrix of
- complex numbers by an inexact real number::
-
- sage: ComplexHermitianEJA(2,field=RR)
- Traceback (most recent call last):
- ...
- TypeError: Unable to coerce entries (=(1.00000000000000,
- -0.000000000000000)) to coefficients in Algebraic Real Field
-
TESTS:
The dimension of this algebra is `n^2`::
r"""
SETUP::
- sage: from mjo.eja.eja_algebra import (FiniteDimensionalEJA,
+ sage: from mjo.eja.eja_algebra import (EJA,
....: OctonionHermitianEJA)
sage: from mjo.hurwitz import Octonions, OctonionMatrixAlgebra
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 = FiniteDimensionalEJA(basis,
+ sage: J = EJA(basis,
....: jp,
....: ip,
....: field=QQ,
@staticmethod
def _max_random_instance_size(max_dimension):
r"""
- The maximum rank of a random QuaternionHermitianEJA.
+ 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.
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
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):
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 (FiniteDimensionalEJA,
+ sage: from mjo.eja.eja_algebra import (EJA,
....: HadamardEJA,
....: JordanSpinEJA,
....: RealSymmetricEJA)
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 = FiniteDimensionalEJA((b1,), jp, ip)
+ 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
return cartesian_product([J1,J2])
-class ComplexSkewHermitianEJA(RationalBasisEJA):
+class ComplexSkewSymmetricEJA(RationalBasisEJA, ConcreteEJA):
r"""
- The EJA described in Faraut and Koranyi's Exercise III.1.b.
+ 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 ComplexSkewHermitianEJA
+ sage: from mjo.eja.eja_algebra import ComplexSkewSymmetricEJA
- TESTS::
+ TESTS:
- sage: n = ZZ.random_element(1,2)
+ 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 = ComplexSkewHermitianEJA._denormalized_basis(A)
- sage: all( M.is_skew_hermitian() for M in B)
+ 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
"""
# The size of the blocks. We're going to treat these thing as
# 2x2 block matrices,
#
- # [ x1 x2 ]
- # [ -x2^* x1-conj ]
+ # [ x1 x2 ]
+ # [ -x2-conj x1-conj ]
#
- # where x1 is skew-Hermitian and x2 is symmetric.
+ # 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-Hermitian.
+ # 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:
- # Top-left block's entry.
- E_ii = gen(A, (i,j,es[1]))
-
- # Bottom-right block's entry.
- E_ii += gen(A, (i+m,j+m,es[1])).conjugate()
- basis.append(E_ii)
- else:
+ 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 -= E_ij.conjugate_transpose()
+ E_ij -= gen(A, (j,i,e))
# Bottom-right block's entry.
F_ij = gen(A, (i+m,j+m,e)).conjugate()
- F_ij -= F_ij.conjugate_transpose()
+ F_ij -= gen(A, (j+m,i+m,e)).conjugate()
basis.append(E_ij + F_ij)
- # Now do the top-right block, which is symmetric, and compute
+ # 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:
- # A symmetric (not Hermitian!) complex matrix can
- # have both real and complex entries on its
- # diagonal.
- for e in es:
- # Top-right block's entry.
- E_ii = gen(A, (i,j,e))
+ # Hermitian matrices have real diagonal entries.
+ # Top-right block's entry.
+ E_ii = gen(A, (i,j,es[0]))
- # Bottom-left block's entry.
- E_ii -= gen(A, (i-m,j-m,e)).conjugate()
- basis.append(E_ii)
+ # 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
E_ij = gen(A, (i,j,e))
# Shift it back to non-offset coords, transpose,
- # and put it back:
+ # 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))
+ 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)).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
+ #if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
from mjo.hurwitz import ComplexMatrixAlgebra
A = ComplexMatrixAlgebra(2*n, scalars=field)
-
- I_n = matrix.identity(ZZ, n)
- J = matrix.block(ZZ, 2, 2, (0, I_n, -I_n, 0), subdivide=False)
- J = A.from_list(J.rows())
+ 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*Y.conjugate_transpose()).trace().real()
+ return (X.conjugate_transpose()*Y).trace().real()
super().__init__(self._denormalized_basis(A),
jordan_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