class MatrixEJA:
+ @staticmethod
+ def jordan_product(X,Y):
+ return (X*Y + Y*X)/2
+
+ @staticmethod
+ def trace_inner_product(X,Y):
+ r"""
+ A trace inner-product for matrices that aren't embedded in the
+ reals.
+ """
+ # We take the norm (absolute value) because Octonions() isn't
+ # smart enough yet to coerce its one() into the base field.
+ return (X*Y).trace().real().abs()
+
+class RealEmbeddedMatrixEJA(MatrixEJA):
@staticmethod
def dimension_over_reals():
r"""
raise ValueError("the matrix 'M' must be a real embedding")
return M
- @staticmethod
- def jordan_product(X,Y):
- return (X*Y + Y*X)/2
@classmethod
def trace_inner_product(cls,X,Y):
SETUP::
- sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
- ....: ComplexHermitianEJA,
+ sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
....: QuaternionHermitianEJA)
EXAMPLES::
- This gives the same answer as it would if we computed the trace
- from the unembedded (original) matrices::
-
- sage: set_random_seed()
- sage: J = RealSymmetricEJA.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()
- sage: actual = J.trace_inner_product(Xe,Ye)
- sage: actual == expected
- True
-
- ::
-
sage: set_random_seed()
sage: J = ComplexHermitianEJA.random_instance()
sage: x,y = J.random_elements(2)
# as a REAL matrix will be 2*a = 2*Re(z_1). And so forth.
return (X*Y).trace()/cls.dimension_over_reals()
-
-class RealMatrixEJA(MatrixEJA):
- @staticmethod
- def dimension_over_reals():
- return 1
-
-
-class RealSymmetricEJA(ConcreteEJA, RealMatrixEJA):
+class RealSymmetricEJA(ConcreteEJA, MatrixEJA):
"""
The rank-n simple EJA consisting of real symmetric n-by-n
matrices, the usual symmetric Jordan product, and the trace inner
# 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))
-class ComplexMatrixEJA(MatrixEJA):
+class ComplexMatrixEJA(RealEmbeddedMatrixEJA):
# A manual dictionary-cache for the complex_extension() method,
# since apparently @classmethods can't also be @cached_methods.
_complex_extension = {}
n = ZZ.random_element(cls._max_random_instance_size() + 1)
return cls(n, **kwargs)
-class QuaternionMatrixEJA(MatrixEJA):
+class QuaternionMatrixEJA(RealEmbeddedMatrixEJA):
# A manual dictionary-cache for the quaternion_extension() method,
# since apparently @classmethods can't also be @cached_methods.
n = ZZ.random_element(cls._max_random_instance_size() + 1)
return cls(n, **kwargs)
+class OctonionHermitianEJA(FiniteDimensionalEJA, MatrixEJA):
+ r"""
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (FiniteDimensionalEJA,
+ ....: OctonionHermitianEJA)
+
+ EXAMPLES:
+
+ The 3-by-3 algebra satisfies the axioms of an EJA::
+
+ sage: OctonionHermitianEJA(3, # long time
+ ....: field=QQ, # long time
+ ....: orthonormalize=False, # long time
+ ....: check_axioms=True) # long time
+ Euclidean Jordan algebra of dimension 27 over Rational Field
+
+ After a change-of-basis, the 2-by-2 algebra has the same
+ multiplication table as the ten-dimensional Jordan spin algebra::
+
+ sage: b = OctonionHermitianEJA._denormalized_basis(2,QQ)
+ 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,
+ ....: jp,
+ ....: ip,
+ ....: field=QQ,
+ ....: orthonormalize=False)
+ sage: J.multiplication_table()
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | * || b0 | b1 | b2 | b3 | b4 | b5 | b6 | b7 | b8 | b9 |
+ +====++====+====+====+====+====+====+====+====+====+====+
+ | b0 || b0 | b1 | b2 | b3 | b4 | b5 | b6 | b7 | b8 | b9 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b1 || b1 | b0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b2 || b2 | 0 | b0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b3 || b3 | 0 | 0 | b0 | 0 | 0 | 0 | 0 | 0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b4 || b4 | 0 | 0 | 0 | b0 | 0 | 0 | 0 | 0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b5 || b5 | 0 | 0 | 0 | 0 | b0 | 0 | 0 | 0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b6 || b6 | 0 | 0 | 0 | 0 | 0 | b0 | 0 | 0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b7 || b7 | 0 | 0 | 0 | 0 | 0 | 0 | b0 | 0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b8 || b8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | b0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b9 || b9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | b0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+
+ TESTS:
+
+ We can actually construct the 27-dimensional Albert algebra,
+ and we get the right unit element if we recompute it::
+
+ sage: J = OctonionHermitianEJA(3, # long time
+ ....: field=QQ, # long time
+ ....: orthonormalize=False) # long time
+ sage: J.one.clear_cache() # long time
+ sage: J.one() # long time
+ b0 + b9 + b26
+ sage: J.one().to_matrix() # long time
+ +----+----+----+
+ | e0 | 0 | 0 |
+ +----+----+----+
+ | 0 | e0 | 0 |
+ +----+----+----+
+ | 0 | 0 | e0 |
+ +----+----+----+
+
+ The 2-by-2 algebra is isomorphic to the ten-dimensional Jordan
+ spin algebra, but just to be sure, we recompute its rank::
+
+ sage: J = OctonionHermitianEJA(2, # long time
+ ....: field=QQ, # long time
+ ....: orthonormalize=False) # long time
+ sage: J.rank.clear_cache() # long time
+ sage: J.rank() # long time
+ 2
+ """
+ def __init__(self, n, field=AA, **kwargs):
+ if n > 3:
+ # Otherwise we don't get an EJA.
+ raise ValueError("n cannot exceed 3")
+
+ # 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
+
+ super().__init__(self._denormalized_basis(n,field),
+ self.jordan_product,
+ self.trace_inner_product,
+ field=field,
+ **kwargs)
+
+ # TODO: this could be factored out somehow, but is left here
+ # because the MatrixEJA is not presently a subclass of the
+ # FDEJA class that defines rank() and one().
+ self.rank.set_cache(n)
+ idV = self.matrix_space().one()
+ self.one.set_cache(self(idV))
+
+
+ @classmethod
+ def _denormalized_basis(cls, n, field):
+ """
+ Returns a basis for the space of octonion Hermitian n-by-n
+ matrices.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import OctonionHermitianEJA
+
+ EXAMPLES::
+
+ sage: B = OctonionHermitianEJA._denormalized_basis(3,QQ)
+ sage: all( M.is_hermitian() for M in B )
+ True
+ sage: len(B)
+ 27
+
+ """
+ from mjo.octonions import OctonionMatrixAlgebra
+ MS = OctonionMatrixAlgebra(n, scalars=field)
+ es = MS.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]) )
+ basis.append(E_ii)
+ else:
+ for e in es:
+ E_ij = MS.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) )
+ else:
+ E_ij -= MS.monomial( (j,i,-ec) )
+ basis.append(E_ij)
+
+ return tuple( basis )
+
class HadamardEJA(ConcreteEJA):
"""
projects / sage.d.git / blobdiff