* :class:`RealSymmetricEJA`
* :class:`ComplexHermitianEJA`
* :class:`QuaternionHermitianEJA`
+ * :class:`OctonionHermitianEJA`
-Missing from this list is the algebra of three-by-three octononion
-Hermitian matrices, as there is (as of yet) no implementation of the
-octonions in SageMath. In addition to these, we provide two other
-example constructions,
+In addition to these, we provide two other example constructions,
+ * :class:`JordanSpinEJA`
* :class:`HadamardEJA`
+ * :class:`AlbertEJA`
* :class:`TrivialEJA`
The Jordan spin algebra is a bilinear form algebra where the bilinear
form is the identity. The Hadamard EJA is simply a Cartesian product
-of one-dimensional spin algebras. And last but not least, the trivial
-EJA is exactly what you think. Cartesian products of these are also
-supported using the usual ``cartesian_product()`` function; as a
-result, we support (up to isomorphism) all Euclidean Jordan algebras
-that don't involve octonions.
+of one-dimensional spin algebras. The Albert EJA is simply a special
+case of the :class:`OctonionHermitianEJA` where the matrices are
+three-by-three and the resulting space has dimension 27. And
+last/least, the trivial EJA is exactly what you think it is; it could
+also be obtained by constructing a dimension-zero instance of any of
+the other algebras. Cartesian products of these are also supported
+using the usual ``cartesian_product()`` function; as a result, we
+support (up to isomorphism) all Euclidean Jordan algebras.
SETUP::
Euclidean Jordan algebra of dimension...
"""
-from itertools import repeat
-
from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra
from sage.categories.magmatic_algebras import MagmaticAlgebras
from sage.categories.sets_cat import cartesian_product
Full MatrixSpace of 4 by 4 dense matrices over Rational Field
sage: J = QuaternionHermitianEJA(1,field=QQ,orthonormalize=False)
sage: J.matrix_space()
- Full MatrixSpace of 4 by 4 dense matrices over Rational Field
+ Module of 1 by 1 matrices with entries in Quaternion
+ Algebra (-1, -1) with base ring Rational Field over
+ the scalar ring Rational Field
"""
if self.is_trivial():
class RationalBasisEJA(FiniteDimensionalEJA):
r"""
- New class for algebras whose supplied basis elements have all rational entries.
+ Algebras whose supplied basis elements have all rational entries.
SETUP::
if check_field:
# Abuse the check_field parameter to check that the entries of
# out basis (in ambient coordinates) are in the field QQ.
- if not all( all(b_i in QQ for b_i in b.list()) for b in basis ):
+ # Use _all2list to get the vector coordinates of octonion
+ # entries and not the octonions themselves (which are not
+ # rational).
+ if not all( all(b_i in QQ for b_i in _all2list(b))
+ for b in basis ):
raise TypeError("basis not rational")
super().__init__(basis,
SETUP::
- sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
- ....: QuaternionHermitianEJA)
+ sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
EXAMPLES::
sage: actual == expected
True
- ::
-
- sage: set_random_seed()
- sage: J = QuaternionHermitianEJA.random_instance()
- sage: x,y = J.random_elements(2)
- sage: Xe = x.to_matrix()
- sage: Ye = y.to_matrix()
- sage: X = J.real_unembed(Xe)
- sage: Y = J.real_unembed(Ye)
- sage: expected = (X*Y).trace().coefficient_tuple()[0]
- sage: actual = J.trace_inner_product(Xe,Ye)
- sage: actual == expected
- True
-
"""
# This does in fact compute the real part of the trace.
# If we compute the trace of e.g. a complex matrix M,
# as a REAL matrix will be 2*a = 2*Re(z_1). And so forth.
return (X*Y).trace()/cls.dimension_over_reals()
-class RealSymmetricEJA(ConcreteEJA, RationalBasisEJA, MatrixEJA):
+class RealSymmetricEJA(RationalBasisEJA, ConcreteEJA, MatrixEJA):
"""
The rank-n simple EJA consisting of real symmetric n-by-n
matrices, the usual symmetric Jordan product, and the trace inner
return matrix(F, n/d, elements)
-class ComplexHermitianEJA(ConcreteEJA, RationalBasisEJA, ComplexMatrixEJA):
+class ComplexHermitianEJA(RationalBasisEJA, ConcreteEJA, ComplexMatrixEJA):
"""
The rank-n simple EJA consisting of complex Hermitian n-by-n
matrices over the real numbers, the usual symmetric Jordan product,
n = ZZ.random_element(cls._max_random_instance_size() + 1)
return cls(n, **kwargs)
-class QuaternionMatrixEJA(RealEmbeddedMatrixEJA):
-
- # A manual dictionary-cache for the quaternion_extension() method,
- # since apparently @classmethods can't also be @cached_methods.
- _quaternion_extension = {}
-
- @classmethod
- def quaternion_extension(cls,field):
- r"""
- The quaternion field that we embed/unembed, as an extension
- of the given ``field``.
- """
- if field in cls._quaternion_extension:
- return cls._quaternion_extension[field]
-
- Q = QuaternionAlgebra(field,-1,-1)
-
- cls._quaternion_extension[field] = Q
- return Q
-
- @staticmethod
- def dimension_over_reals():
- return 4
-
- @classmethod
- def real_embed(cls,M):
- """
- Embed the n-by-n quaternion matrix ``M`` into the space of real
- matrices of size 4n-by-4n by first sending each quaternion entry `z
- = a + bi + cj + dk` to the block-complex matrix ``[[a + bi,
- c+di],[-c + di, a-bi]]`, and then embedding those into a real
- matrix.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import QuaternionMatrixEJA
- EXAMPLES::
-
- sage: Q = QuaternionAlgebra(QQ,-1,-1)
- sage: i,j,k = Q.gens()
- sage: x = 1 + 2*i + 3*j + 4*k
- sage: M = matrix(Q, 1, [[x]])
- sage: QuaternionMatrixEJA.real_embed(M)
- [ 1 2 3 4]
- [-2 1 -4 3]
- [-3 4 1 -2]
- [-4 -3 2 1]
-
- Embedding is a homomorphism (isomorphism, in fact)::
-
- sage: set_random_seed()
- sage: n = ZZ.random_element(2)
- sage: Q = QuaternionAlgebra(QQ,-1,-1)
- sage: X = random_matrix(Q, n)
- sage: Y = random_matrix(Q, n)
- sage: Xe = QuaternionMatrixEJA.real_embed(X)
- sage: Ye = QuaternionMatrixEJA.real_embed(Y)
- sage: XYe = QuaternionMatrixEJA.real_embed(X*Y)
- sage: Xe*Ye == XYe
- True
-
- """
- super().real_embed(M)
- quaternions = M.base_ring()
- n = M.nrows()
-
- F = QuadraticField(-1, 'I')
- i = F.gen()
-
- blocks = []
- for z in M.list():
- t = z.coefficient_tuple()
- a = t[0]
- b = t[1]
- c = t[2]
- d = t[3]
- cplxM = matrix(F, 2, [[ a + b*i, c + d*i],
- [-c + d*i, a - b*i]])
- realM = ComplexMatrixEJA.real_embed(cplxM)
- blocks.append(realM)
-
- # We should have real entries by now, so use the realest field
- # we've got for the return value.
- return matrix.block(quaternions.base_ring(), n, blocks)
-
-
-
- @classmethod
- def real_unembed(cls,M):
- """
- The inverse of _embed_quaternion_matrix().
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import QuaternionMatrixEJA
-
- EXAMPLES::
-
- sage: M = matrix(QQ, [[ 1, 2, 3, 4],
- ....: [-2, 1, -4, 3],
- ....: [-3, 4, 1, -2],
- ....: [-4, -3, 2, 1]])
- sage: QuaternionMatrixEJA.real_unembed(M)
- [1 + 2*i + 3*j + 4*k]
-
- TESTS:
-
- Unembedding is the inverse of embedding::
-
- sage: set_random_seed()
- sage: Q = QuaternionAlgebra(QQ, -1, -1)
- sage: M = random_matrix(Q, 3)
- sage: Me = QuaternionMatrixEJA.real_embed(M)
- sage: QuaternionMatrixEJA.real_unembed(Me) == M
- True
-
- """
- super().real_unembed(M)
- n = ZZ(M.nrows())
- d = cls.dimension_over_reals()
-
- # Use the base ring of the matrix to ensure that its entries can be
- # multiplied by elements of the quaternion algebra.
- Q = cls.quaternion_extension(M.base_ring())
- i,j,k = Q.gens()
-
- # Go top-left to bottom-right (reading order), converting every
- # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
- # quaternion block.
- elements = []
- for l in range(n/d):
- for m in range(n/d):
- submat = ComplexMatrixEJA.real_unembed(
- M[d*l:d*l+d,d*m:d*m+d] )
- if submat[0,0] != submat[1,1].conjugate():
- raise ValueError('bad on-diagonal submatrix')
- if submat[0,1] != -submat[1,0].conjugate():
- raise ValueError('bad off-diagonal submatrix')
- z = submat[0,0].real()
- z += submat[0,0].imag()*i
- z += submat[0,1].real()*j
- z += submat[0,1].imag()*k
- elements.append(z)
-
- return matrix(Q, n/d, elements)
-
-
-class QuaternionHermitianEJA(ConcreteEJA,
- RationalBasisEJA,
- QuaternionMatrixEJA):
+class QuaternionHermitianEJA(RationalBasisEJA, ConcreteEJA, MatrixEJA):
r"""
The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
matrices, the usual symmetric Jordan product, and the
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
- sage: B = QuaternionHermitianEJA._denormalized_basis(n,ZZ)
- sage: all( M.is_symmetric() for M in B )
+ sage: B = QuaternionHermitianEJA._denormalized_basis(n,QQ)
+ sage: all( M.is_hermitian() for M in B )
True
"""
- Q = QuaternionAlgebra(QQ,-1,-1)
- I,J,K = Q.gens()
+ from mjo.hurwitz import QuaternionMatrixAlgebra
+ A = QuaternionMatrixAlgebra(n, scalars=field)
+ es = A.entry_algebra_gens()
- # This is like the symmetric case, but we need to be careful:
- #
- # * We want conjugate-symmetry, not just symmetry.
- # * The diagonal will (as a result) be real.
- #
- S = []
- Eij = matrix.zero(Q,n)
+ basis = []
for i in range(n):
for j in range(i+1):
- # "build" E_ij
- Eij[i,j] = 1
if i == j:
- Sij = cls.real_embed(Eij)
- S.append(Sij)
+ E_ii = A.monomial( (i,j,es[0]) )
+ basis.append(E_ii)
else:
- # The second, third, and fourth ones have a minus
- # because they're conjugated.
- # Eij = Eij + Eij.transpose()
- Eij[j,i] = 1
- Sij_real = cls.real_embed(Eij)
- S.append(Sij_real)
- # Eij = I*(Eij - Eij.transpose())
- Eij[i,j] = I
- Eij[j,i] = -I
- Sij_I = cls.real_embed(Eij)
- S.append(Sij_I)
- # Eij = J*(Eij - Eij.transpose())
- Eij[i,j] = J
- Eij[j,i] = -J
- Sij_J = cls.real_embed(Eij)
- S.append(Sij_J)
- # Eij = K*(Eij - Eij.transpose())
- Eij[i,j] = K
- Eij[j,i] = -K
- Sij_K = cls.real_embed(Eij)
- S.append(Sij_K)
- Eij[j,i] = 0
- # "erase" E_ij
- Eij[i,j] = 0
+ for e in es:
+ E_ij = A.monomial( (i,j,e) )
+ ec = e.conjugate()
+ # If the conjugate has a negative sign in front
+ # of it, (j,i,ec) won't be a monomial!
+ if (j,i,ec) in A.indices():
+ E_ij += A.monomial( (j,i,ec) )
+ else:
+ E_ij -= A.monomial( (j,i,-ec) )
+ basis.append(E_ij)
- # Since we embedded the entries, we can drop back to the
- # desired real "field" instead of the quaternion algebra "Q".
- return tuple( s.change_ring(field) for s in S )
+ return tuple( basis )
+ @staticmethod
+ def trace_inner_product(X,Y):
+ r"""
+ Overload the superclass method because the quaternions are weird
+ and we need to use ``coefficient_tuple()`` to get the realpart.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
+
+ TESTS::
+
+ sage: J = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: I = J.one().to_matrix()
+ sage: J.trace_inner_product(I, -I)
+ -2
+
+ """
+ return (X*Y).trace().coefficient_tuple()[0]
+
def __init__(self, n, field=AA, **kwargs):
# We know this is a valid EJA, but will double-check
# if the user passes check_axioms=True.
# 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)
+ idV = self.matrix_space().one()
self.one.set_cache(self(idV))
n = ZZ.random_element(cls._max_random_instance_size() + 1)
return cls(n, **kwargs)
-class OctonionHermitianEJA(FiniteDimensionalEJA, MatrixEJA):
+class OctonionHermitianEJA(RationalBasisEJA, ConcreteEJA, MatrixEJA):
r"""
SETUP::
sage: J.rank.clear_cache() # long time
sage: J.rank() # long time
2
+
"""
+ @staticmethod
+ def _max_random_instance_size():
+ r"""
+ The maximum rank of a random QuaternionHermitianEJA.
+ """
+ return 1 # Dimension 1
+
+ @classmethod
+ def random_instance(cls, **kwargs):
+ """
+ Return a random instance of this type of algebra.
+ """
+ n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ return cls(n, **kwargs)
+
def __init__(self, n, field=AA, **kwargs):
if n > 3:
# Otherwise we don't get an EJA.
27
"""
- from mjo.octonions import OctonionMatrixAlgebra
- MS = OctonionMatrixAlgebra(n, scalars=field)
- es = MS.entry_algebra().gens()
+ from mjo.hurwitz import OctonionMatrixAlgebra
+ A = OctonionMatrixAlgebra(n, scalars=field)
+ es = A.entry_algebra_gens()
basis = []
for i in range(n):
for j in range(i+1):
if i == j:
- E_ii = MS.monomial( (i,j,es[0]) )
+ E_ii = A.monomial( (i,j,es[0]) )
basis.append(E_ii)
else:
for e in es:
- E_ij = MS.monomial( (i,j,e) )
+ E_ij = A.monomial( (i,j,e) )
ec = e.conjugate()
# If the conjugate has a negative sign in front
# of it, (j,i,ec) won't be a monomial!
- if (j,i,ec) in MS.indices():
- E_ij += MS.monomial( (j,i,ec) )
+ if (j,i,ec) in A.indices():
+ E_ij += A.monomial( (j,i,ec) )
else:
- E_ij -= MS.monomial( (j,i,-ec) )
+ E_ij -= A.monomial( (j,i,-ec) )
basis.append(E_ij)
return tuple( basis )
-2
"""
- return (X*Y).trace().real().coefficient(0)
+ return (X*Y).trace().coefficient(0)
+
+
+class AlbertEJA(OctonionHermitianEJA):
+ r"""
+ The Albert algebra is the algebra of three-by-three Hermitian
+ matrices whose entries are octonions.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import AlbertEJA
+
+ EXAMPLES::
+
+ sage: AlbertEJA(field=QQ, orthonormalize=False)
+ Euclidean Jordan algebra of dimension 27 over Rational Field
+ sage: AlbertEJA() # long time
+ Euclidean Jordan algebra of dimension 27 over Algebraic Real Field
-class HadamardEJA(ConcreteEJA, RationalBasisEJA):
"""
- Return the Euclidean Jordan Algebra corresponding to the set
- `R^n` under the Hadamard product.
+ def __init__(self, *args, **kwargs):
+ super().__init__(3, *args, **kwargs)
- Note: this is nothing more than the Cartesian product of ``n``
- copies of the spin algebra. Once Cartesian product algebras
- are implemented, this can go.
+
+class HadamardEJA(RationalBasisEJA, ConcreteEJA):
+ """
+ Return the Euclidean Jordan algebra on `R^n` with the Hadamard
+ (pointwise real-number multiplication) Jordan product and the
+ usual inner-product.
+
+ This is nothing more than the Cartesian product of ``n`` copies of
+ the one-dimensional Jordan spin algebra, and is the most common
+ example of a non-simple Euclidean Jordan algebra.
SETUP::
sage: HadamardEJA(3, prefix='r').gens()
(r0, r1, r2)
-
"""
def __init__(self, n, field=AA, **kwargs):
if n == 0:
return cls(n, **kwargs)
-class BilinearFormEJA(ConcreteEJA, RationalBasisEJA):
+class BilinearFormEJA(RationalBasisEJA, ConcreteEJA):
r"""
The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
with the half-trace inner product and jordan product ``x*y =
return cls(n, **kwargs)
-class TrivialEJA(ConcreteEJA, RationalBasisEJA):
+class TrivialEJA(RationalBasisEJA, ConcreteEJA):
"""
The trivial Euclidean Jordan algebra consisting of only a zero element.
Return the space that our matrix basis lives in as a Cartesian
product.
+ We don't simply use the ``cartesian_product()`` functor here
+ because it acts differently on SageMath MatrixSpaces and our
+ custom MatrixAlgebras, which are CombinatorialFreeModules. We
+ always want the result to be represented (and indexed) as
+ an ordered tuple.
+
SETUP::
- sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+ ....: HadamardEJA,
+ ....: OctonionHermitianEJA,
....: RealSymmetricEJA)
EXAMPLES::
matrices over Algebraic Real Field, Full MatrixSpace of 2
by 2 dense matrices over Algebraic Real Field)
+ ::
+
+ sage: J1 = ComplexHermitianEJA(1)
+ sage: J2 = ComplexHermitianEJA(1)
+ sage: J = cartesian_product([J1,J2])
+ sage: J.one().to_matrix()[0]
+ [1 0]
+ [0 1]
+ sage: J.one().to_matrix()[1]
+ [1 0]
+ [0 1]
+
+ ::
+
+ sage: J1 = OctonionHermitianEJA(1)
+ sage: J2 = OctonionHermitianEJA(1)
+ sage: J = cartesian_product([J1,J2])
+ sage: J.one().to_matrix()[0]
+ +----+
+ | e0 |
+ +----+
+ sage: J.one().to_matrix()[1]
+ +----+
+ | e0 |
+ +----+
+
"""
- from sage.categories.cartesian_product import cartesian_product
- return cartesian_product( [J.matrix_space()
- for J in self.cartesian_factors()] )
+ scalars = self.cartesian_factor(0).base_ring()
+
+ # This category isn't perfect, but is good enough for what we
+ # need to do.
+ cat = MagmaticAlgebras(scalars).FiniteDimensional().WithBasis()
+ cat = cat.Unital().CartesianProducts()
+ factors = tuple( J.matrix_space() for J in self.cartesian_factors() )
+
+ from sage.sets.cartesian_product import CartesianProduct
+ return CartesianProduct(factors, cat)
+
@cached_method
def cartesian_projection(self, i):
SETUP::
- sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: JordanSpinEJA,
+ ....: OctonionHermitianEJA,
....: RealSymmetricEJA)
EXAMPLES:
sage: J.rank()
5
+ TESTS:
+
+ The ``cartesian_product()`` function only uses the first factor to
+ decide where the result will live; thus we have to be careful to
+ check that all factors do indeed have a `_rational_algebra` member
+ before we try to access it::
+
+ sage: J1 = OctonionHermitianEJA(1) # no rational basis
+ sage: J2 = HadamardEJA(2)
+ sage: cartesian_product([J1,J2])
+ Euclidean Jordan algebra of dimension 1 over Algebraic Real Field
+ (+) Euclidean Jordan algebra of dimension 2 over Algebraic Real Field
+ sage: cartesian_product([J2,J1])
+ Euclidean Jordan algebra of dimension 2 over Algebraic Real Field
+ (+) Euclidean Jordan algebra of dimension 1 over Algebraic Real Field
+
"""
def __init__(self, algebras, **kwargs):
CartesianProductEJA.__init__(self, algebras, **kwargs)
self._rational_algebra = None
if self.vector_space().base_field() is not QQ:
- self._rational_algebra = cartesian_product([
- r._rational_algebra for r in algebras
- ])
+ if all( hasattr(r, "_rational_algebra") for r in algebras ):
+ self._rational_algebra = cartesian_product([
+ r._rational_algebra for r in algebras
+ ])
RationalBasisEJA.CartesianProduct = RationalBasisCartesianProductEJA
projects / sage.d.git / blobdiff