+ """
+ @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.
+ 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 )
+
+ @staticmethod
+ def trace_inner_product(X,Y):
+ r"""
+ The octonions don't know that the reals are embedded in them,
+ so we have to take the e0 component ourselves.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import OctonionHermitianEJA
+
+ TESTS::
+
+ sage: J = OctonionHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: I = J.one().to_matrix()
+ sage: J.trace_inner_product(I, -I)
+ -2
+
+ """
+ return (X*Y).trace().real().coefficient(0)
+
+
+class AlbertEJA(OctonionHermitianEJA):
+ r"""
+ The Albert algebra is the algebra of three-by-three Hermitian
+ matrices whose entries are octonions.
+
+ SETUP::
+
+ 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
+
+ """
+ def __init__(self, *args, **kwargs):
+ super().__init__(3, *args, **kwargs)
+
+
+class HadamardEJA(ConcreteEJA, RationalBasisEJA):
projects / sage.d.git / blobdiff