X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=e32bc24cbb2741fca9de2285de703b7adf1497af;hb=3c7644ecfe369b6f83aa707b87d7a1f9aa246e27;hp=107adccdd6c6c44262fda89752f604c9b93c50e8;hpb=472c151eae2e95d5611cf3761dfe98a3fe386400;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 107adcc..e32bc24 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -119,10 +119,12 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): The ``field`` we're given must be real with ``check_field=True``:: - sage: JordanSpinEJA(2,QQbar) + sage: JordanSpinEJA(2, field=QQbar) Traceback (most recent call last): ... ValueError: scalar field is not real + sage: JordanSpinEJA(2, field=QQbar, check_field=False) + Euclidean Jordan algebra of dimension 2 over Algebraic Field The multiplication table must be square with ``check_axioms=True``:: @@ -291,6 +293,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: x = J.random_element() sage: J(x.to_vector().column()) == x True + """ msg = "not an element of this algebra" if elt == 0: @@ -313,8 +316,12 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): # element's ring because the basis space might be an algebraic # closure whereas the base ring of the 3-by-3 identity matrix # could be QQ instead of QQbar. + # + # We pass check=False because the matrix basis is "guaranteed" + # to be linearly independent... right? Ha ha. V = VectorSpace(self.base_ring(), elt.nrows()*elt.ncols()) - W = V.span_of_basis( _mat2vec(s) for s in self.matrix_basis() ) + W = V.span_of_basis( (_mat2vec(s) for s in self.matrix_basis()), + check=False) try: coords = W.coordinate_vector(_mat2vec(elt)) @@ -393,7 +400,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): # Used to check whether or not something is zero in an inexact # ring. This number is sufficient to allow the construction of - # QuaternionHermitianEJA(2, RDF) with check_axioms=True. + # QuaternionHermitianEJA(2, field=RDF) with check_axioms=True. epsilon = 1e-16 for i in range(self.dimension()): @@ -490,7 +497,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: J = HadamardEJA(2) sage: J.coordinate_polynomial_ring() Multivariate Polynomial Ring in X1, X2... - sage: J = RealSymmetricEJA(3,QQ,orthonormalize=False) + sage: J = RealSymmetricEJA(3,field=QQ,orthonormalize=False) sage: J.coordinate_polynomial_ring() Multivariate Polynomial Ring in X1, X2, X3, X4, X5, X6... @@ -1155,16 +1162,26 @@ class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebr """ def __init__(self, - field, basis, jordan_product, inner_product, + field=AA, orthonormalize=True, prefix='e', category=None, check_field=True, check_axioms=True): + 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 ): + raise TypeError("basis not rational") + + # Temporary(?) hack to ensure that the matrix and vector bases + # are over the same ring. + basis = tuple( b.change_ring(field) for b in basis ) + n = len(basis) vector_basis = basis @@ -1189,19 +1206,34 @@ class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebr # deorthonormalized basis. These can be used later to # construct a deorthonormalized copy of this algebra over QQ # in which several operations are much faster. - self._deortho_multiplication_table = None - self._deortho_inner_product_table = None + self._rational_algebra = None if orthonormalize: - # Compute the deorthonormalized tables before we orthonormalize - # the given basis. - W = V.span_of_basis( vector_basis ) + if self.base_ring() is not QQ: + # There's no point in constructing the extra algebra if this + # one is already rational. If the original basis is rational + # but normalization would make it irrational, then this whole + # constructor will just fail anyway as it tries to stick an + # irrational number into a rational algebra. + # + # 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 = RationalBasisEuclideanJordanAlgebra( + basis, + jordan_product, + inner_product, + field=QQ, + orthonormalize=False, + prefix=prefix, + category=category, + check_field=False, + check_axioms=False) - # TODO: use symmetry - self._deortho_multiplication_table = [ [0 for j in range(n)] - for i in range(n) ] - self._deortho_inner_product_table = [ [0 for j in range(n)] - for i in range(n) ] + # Compute the deorthonormalized tables before we orthonormalize + # the given basis. The "check" parameter here guarantees that + # the basis is linearly-independent. + W = V.span_of_basis( vector_basis, check=check_axioms) # Note: the Jordan and inner-products are defined in terms # of the ambient basis. It's important that their arguments @@ -1224,18 +1256,6 @@ class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebr # table is in terms of vectors elt = _mat2vec(elt) - # TODO: use symmetry - elt = W.coordinate_vector(elt) - self._deortho_multiplication_table[i][j] = elt - self._deortho_multiplication_table[j][i] = elt - self._deortho_inner_product_table[i][j] = ip - self._deortho_inner_product_table[j][i] = ip - - if self._deortho_multiplication_table is not None: - self._deortho_multiplication_table = tuple(map(tuple, self._deortho_multiplication_table)) - if self._deortho_inner_product_table is not None: - self._deortho_inner_product_table = tuple(map(tuple, self._deortho_inner_product_table)) - # We overwrite the name "vector_basis" in a second, but never modify it # in place, to this effectively makes a copy of it. deortho_vector_basis = vector_basis @@ -1249,28 +1269,32 @@ class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebr else: vector_basis = gram_schmidt(vector_basis, inner_product) - W = V.span_of_basis( vector_basis ) - # Normalize the "matrix" basis, too! basis = vector_basis if basis_is_matrices: basis = tuple( map(_vec2mat,basis) ) - W = V.span_of_basis( vector_basis ) + W = V.span_of_basis( vector_basis, check=check_axioms) # Now "W" is the vector space of our algebra coordinates. The # variables "X1", "X2",... refer to the entries of vectors in # W. Thus to convert back and forth between the orthonormal # coordinates and the given ones, we need to stick the original # basis in W. - U = V.span_of_basis( deortho_vector_basis ) + U = V.span_of_basis( deortho_vector_basis, check=check_axioms) self._deortho_matrix = matrix( U.coordinate_vector(q) for q in vector_basis ) - # TODO: use symmetry - mult_table = [ [0 for j in range(n)] for i in range(n) ] - ip_table = [ [0 for j in range(n)] for i in range(n) ] + # If the superclass constructor is going to verify the + # symmetry of this table, it has better at least be + # square... + if check_axioms: + mult_table = [ [0 for j in range(n)] for i in range(n) ] + ip_table = [ [0 for j in range(n)] for i in range(n) ] + else: + mult_table = [ [0 for j in range(i+1)] for i in range(n) ] + ip_table = [ [0 for j in range(i+1)] for i in range(n) ] # Note: the Jordan and inner-products are defined in terms # of the ambient basis. It's important that their arguments @@ -1293,12 +1317,14 @@ class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebr # table is in terms of vectors elt = _mat2vec(elt) - # TODO: use symmetry elt = W.coordinate_vector(elt) mult_table[i][j] = elt - mult_table[j][i] = elt ip_table[i][j] = ip - ip_table[j][i] = ip + if check_axioms: + # The tables are square if we're verifying that they + # are commutative. + mult_table[j][i] = elt + ip_table[j][i] = ip if basis_is_matrices: for m in basis: @@ -1339,21 +1365,19 @@ class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebr Algebraic Real Field """ - if self.base_ring() is QQ: + if self.base_ring() is QQ or self._rational_algebra is None: # There's no need to construct *another* algebra over the - # rationals if this one is already over the rationals. + # rationals if this one is already over the + # rationals. Likewise, if we never orthonormalized our + # basis, we might as well just use the given one. superclass = super(RationalBasisEuclideanJordanAlgebra, self) return superclass._charpoly_coefficients() # Do the computation over the rationals. The answer will be # the same, because all we've done is a change of basis. - J = FiniteDimensionalEuclideanJordanAlgebra(QQ, - self._deortho_multiplication_table, - self._deortho_inner_product_table) - - # Change back from QQ to our real base ring + # Then, change back from QQ to our real base ring a = ( a_i.change_ring(self.base_ring()) - for a_i in J._charpoly_coefficients() ) + for a_i in self._rational_algebra._charpoly_coefficients() ) # Now convert the coordinate variables back to the # deorthonormalized ones. @@ -1520,9 +1544,9 @@ class RealSymmetricEJA(ConcreteEuclideanJordanAlgebra, In theory, our "field" can be any subfield of the reals:: - sage: RealSymmetricEJA(2, RDF) + sage: RealSymmetricEJA(2, field=RDF) Euclidean Jordan algebra of dimension 3 over Real Double Field - sage: RealSymmetricEJA(2, RR) + sage: RealSymmetricEJA(2, field=RR) Euclidean Jordan algebra of dimension 3 over Real Field with 53 bits of precision @@ -1563,7 +1587,7 @@ class RealSymmetricEJA(ConcreteEuclideanJordanAlgebra, """ @classmethod - def _denormalized_basis(cls, n, field): + def _denormalized_basis(cls, n): """ Return a basis for the space of real symmetric n-by-n matrices. @@ -1575,7 +1599,7 @@ class RealSymmetricEJA(ConcreteEuclideanJordanAlgebra, sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: B = RealSymmetricEJA._denormalized_basis(n,QQ) + sage: B = RealSymmetricEJA._denormalized_basis(n) sage: all( M.is_symmetric() for M in B) True @@ -1585,13 +1609,13 @@ class RealSymmetricEJA(ConcreteEuclideanJordanAlgebra, S = [] for i in range(n): for j in range(i+1): - Eij = matrix(field, n, lambda k,l: k==i and l==j) + Eij = matrix(ZZ, n, lambda k,l: k==i and l==j) if i == j: Sij = Eij else: Sij = Eij + Eij.transpose() S.append(Sij) - return S + return tuple(S) @staticmethod @@ -1599,22 +1623,20 @@ class RealSymmetricEJA(ConcreteEuclideanJordanAlgebra, return 4 # Dimension 10 @classmethod - def random_instance(cls, field=AA, **kwargs): + 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, field, **kwargs) + return cls(n, **kwargs) - def __init__(self, n, field=AA, **kwargs): - basis = self._denormalized_basis(n, field) - super(RealSymmetricEJA, self).__init__(field, - basis, + def __init__(self, n, **kwargs): + super(RealSymmetricEJA, self).__init__(self._denormalized_basis(n), self.jordan_product, self.trace_inner_product, **kwargs) self.rank.set_cache(n) - self.one.set_cache(self(matrix.identity(field,n))) + self.one.set_cache(self(matrix.identity(ZZ,n))) class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): @@ -1791,9 +1813,9 @@ class ComplexHermitianEJA(ConcreteEuclideanJordanAlgebra, In theory, our "field" can be any subfield of the reals:: - sage: ComplexHermitianEJA(2, RDF) + sage: ComplexHermitianEJA(2, field=RDF) Euclidean Jordan algebra of dimension 4 over Real Double Field - sage: ComplexHermitianEJA(2, RR) + sage: ComplexHermitianEJA(2, field=RR) Euclidean Jordan algebra of dimension 4 over Real Field with 53 bits of precision @@ -1835,7 +1857,7 @@ class ComplexHermitianEJA(ConcreteEuclideanJordanAlgebra, """ @classmethod - def _denormalized_basis(cls, n, field): + def _denormalized_basis(cls, n): """ Returns a basis for the space of complex Hermitian n-by-n matrices. @@ -1854,15 +1876,16 @@ class ComplexHermitianEJA(ConcreteEuclideanJordanAlgebra, sage: set_random_seed() sage: n = ZZ.random_element(1,5) sage: field = QuadraticField(2, 'sqrt2') - sage: B = ComplexHermitianEJA._denormalized_basis(n, field) + sage: B = ComplexHermitianEJA._denormalized_basis(n) sage: all( M.is_symmetric() for M in B) True """ + field = ZZ R = PolynomialRing(field, 'z') z = R.gen() F = field.extension(z**2 + 1, 'I') - I = F.gen() + I = F.gen(1) # This is like the symmetric case, but we need to be careful: # @@ -1888,13 +1911,11 @@ class ComplexHermitianEJA(ConcreteEuclideanJordanAlgebra, return tuple( s.change_ring(field) for s in S ) - def __init__(self, n, field=AA, **kwargs): - basis = self._denormalized_basis(n,field) - super(ComplexHermitianEJA, self).__init__(field, - basis, - self.jordan_product, - self.trace_inner_product, - **kwargs) + def __init__(self, n, **kwargs): + super(ComplexHermitianEJA, self).__init__(self._denormalized_basis(n), + self.jordan_product, + self.trace_inner_product, + **kwargs) self.rank.set_cache(n) # TODO: pre-cache the identity! @@ -1903,12 +1924,12 @@ class ComplexHermitianEJA(ConcreteEuclideanJordanAlgebra, return 3 # Dimension 9 @classmethod - def random_instance(cls, field=AA, **kwargs): + 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, field, **kwargs) + return cls(n, **kwargs) class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): @staticmethod @@ -2088,9 +2109,9 @@ class QuaternionHermitianEJA(ConcreteEuclideanJordanAlgebra, In theory, our "field" can be any subfield of the reals:: - sage: QuaternionHermitianEJA(2, RDF) + sage: QuaternionHermitianEJA(2, field=RDF) Euclidean Jordan algebra of dimension 6 over Real Double Field - sage: QuaternionHermitianEJA(2, RR) + sage: QuaternionHermitianEJA(2, field=RR) Euclidean Jordan algebra of dimension 6 over Real Field with 53 bits of precision @@ -2131,7 +2152,7 @@ class QuaternionHermitianEJA(ConcreteEuclideanJordanAlgebra, """ @classmethod - def _denormalized_basis(cls, n, field): + def _denormalized_basis(cls, n): """ Returns a basis for the space of quaternion Hermitian n-by-n matrices. @@ -2149,11 +2170,12 @@ class QuaternionHermitianEJA(ConcreteEuclideanJordanAlgebra, sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: B = QuaternionHermitianEJA._denormalized_basis(n,QQ) + sage: B = QuaternionHermitianEJA._denormalized_basis(n) sage: all( M.is_symmetric() for M in B ) True """ + field = ZZ Q = QuaternionAlgebra(QQ,-1,-1) I,J,K = Q.gens() @@ -2186,10 +2208,8 @@ class QuaternionHermitianEJA(ConcreteEuclideanJordanAlgebra, return tuple( s.change_ring(field) for s in S ) - def __init__(self, n, field=AA, **kwargs): - basis = self._denormalized_basis(n,field) - super(QuaternionHermitianEJA, self).__init__(field, - basis, + def __init__(self, n, **kwargs): + super(QuaternionHermitianEJA, self).__init__(self._denormalized_basis(n), self.jordan_product, self.trace_inner_product, **kwargs) @@ -2204,12 +2224,12 @@ class QuaternionHermitianEJA(ConcreteEuclideanJordanAlgebra, return 2 # Dimension 6 @classmethod - def random_instance(cls, field=AA, **kwargs): + 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, field, **kwargs) + return cls(n, **kwargs) class HadamardEJA(ConcreteEuclideanJordanAlgebra): @@ -2252,19 +2272,25 @@ class HadamardEJA(ConcreteEuclideanJordanAlgebra): (r0, r1, r2) """ - def __init__(self, n, field=AA, **kwargs): - V = VectorSpace(field, n) - basis = V.basis() - + def __init__(self, n, **kwargs): def jordan_product(x,y): - return V([ xi*yi for (xi,yi) in zip(x,y) ]) + P = x.parent() + return P(tuple( xi*yi for (xi,yi) in zip(x,y) )) def inner_product(x,y): return x.inner_product(y) - super(HadamardEJA, self).__init__(field, - basis, + # Don't orthonormalize because our basis is already + # orthonormal with respect to our inner-product. + if not 'orthonormalize' in kwargs: + kwargs['orthonormalize'] = False + + # But also don't pass check_field=False here, because the user + # can pass in a field! + standard_basis = FreeModule(ZZ, n).basis() + super(HadamardEJA, self).__init__(standard_basis, jordan_product, inner_product, + check_axioms=False, **kwargs) self.rank.set_cache(n) @@ -2281,12 +2307,12 @@ class HadamardEJA(ConcreteEuclideanJordanAlgebra): return 5 @classmethod - def random_instance(cls, field=AA, **kwargs): + 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, field, **kwargs) + return cls(n, **kwargs) class BilinearFormEJA(ConcreteEuclideanJordanAlgebra): @@ -2370,27 +2396,26 @@ class BilinearFormEJA(ConcreteEuclideanJordanAlgebra): sage: actual == expected True """ - def __init__(self, B, field=AA, **kwargs): + def __init__(self, B, **kwargs): if not B.is_positive_definite(): raise ValueError("bilinear form is not positive-definite") - n = B.nrows() - V = VectorSpace(field, n) - def inner_product(x,y): return (B*x).inner_product(y) def jordan_product(x,y): + P = x.parent() x0 = x[0] xbar = x[1:] y0 = y[0] ybar = y[1:] z0 = inner_product(x,y) zbar = y0*xbar + x0*ybar - return V([z0] + zbar.list()) + return P((z0,) + tuple(zbar)) - super(BilinearFormEJA, self).__init__(field, - V.basis(), + n = B.nrows() + standard_basis = FreeModule(ZZ, n).basis() + super(BilinearFormEJA, self).__init__(standard_basis, jordan_product, inner_product, **kwargs) @@ -2413,28 +2438,28 @@ class BilinearFormEJA(ConcreteEuclideanJordanAlgebra): return 5 @classmethod - def random_instance(cls, field=AA, **kwargs): + def random_instance(cls, **kwargs): """ Return a random instance of this algebra. """ n = ZZ.random_element(cls._max_random_instance_size() + 1) if n.is_zero(): - B = matrix.identity(field, n) - return cls(B, field, **kwargs) + B = matrix.identity(ZZ, n) + return cls(B, **kwargs) - B11 = matrix.identity(field,1) - M = matrix.random(field, n-1) - I = matrix.identity(field, n-1) - alpha = field.zero() + B11 = matrix.identity(ZZ, 1) + M = matrix.random(ZZ, n-1) + I = matrix.identity(ZZ, n-1) + alpha = ZZ.zero() while alpha.is_zero(): - alpha = field.random_element().abs() + alpha = ZZ.random_element().abs() B22 = M.transpose()*M + alpha*I from sage.matrix.special import block_matrix B = block_matrix(2,2, [ [B11, ZZ(0) ], [ZZ(0), B22 ] ]) - return cls(B, field, **kwargs) + return cls(B, **kwargs) class JordanSpinEJA(BilinearFormEJA): @@ -2487,11 +2512,21 @@ class JordanSpinEJA(BilinearFormEJA): True """ - def __init__(self, n, field=AA, **kwargs): - # This is a special case of the BilinearFormEJA with the identity - # matrix as its bilinear form. - B = matrix.identity(field, n) - super(JordanSpinEJA, self).__init__(B, field, **kwargs) + def __init__(self, n, **kwargs): + # This is a special case of the BilinearFormEJA with the + # identity matrix as its bilinear form. + B = matrix.identity(ZZ, n) + + # Don't orthonormalize because our basis is already + # orthonormal with respect to our inner-product. + if not 'orthonormalize' in kwargs: + kwargs['orthonormalize'] = False + + # But also don't pass check_field=False here, because the user + # can pass in a field! + super(JordanSpinEJA, self).__init__(B, + check_axioms=False, + **kwargs) @staticmethod def _max_random_instance_size(): @@ -2501,14 +2536,14 @@ class JordanSpinEJA(BilinearFormEJA): return 5 @classmethod - def random_instance(cls, field=AA, **kwargs): + def random_instance(cls, **kwargs): """ Return a random instance of this type of algebra. Needed here to override the implementation for ``BilinearFormEJA``. """ n = ZZ.random_element(cls._max_random_instance_size() + 1) - return cls(n, field, **kwargs) + return cls(n, **kwargs) class TrivialEJA(ConcreteEuclideanJordanAlgebra): @@ -2540,12 +2575,11 @@ class TrivialEJA(ConcreteEuclideanJordanAlgebra): 0 """ - def __init__(self, field=AA, **kwargs): + def __init__(self, **kwargs): jordan_product = lambda x,y: x - inner_product = lambda x,y: field(0) + inner_product = lambda x,y: 0 basis = () - super(TrivialEJA, self).__init__(field, - basis, + super(TrivialEJA, self).__init__(basis, jordan_product, inner_product, **kwargs) @@ -2555,10 +2589,10 @@ class TrivialEJA(ConcreteEuclideanJordanAlgebra): self.one.set_cache( self.zero() ) @classmethod - def random_instance(cls, field=AA, **kwargs): + def random_instance(cls, **kwargs): # We don't take a "size" argument so the superclass method is # inappropriate for us. - return cls(field, **kwargs) + return cls(**kwargs) class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra): r""" @@ -2591,8 +2625,8 @@ class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra): have the same base ring; an error is raised otherwise:: sage: set_random_seed() - sage: J1 = random_eja(AA) - sage: J2 = random_eja(QQ,orthonormalize=False) + sage: J1 = random_eja(field=AA) + sage: J2 = random_eja(field=QQ,orthonormalize=False) sage: J = DirectSumEJA(J1,J2) Traceback (most recent call last): ... @@ -2609,21 +2643,22 @@ class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra): n2 = J2.dimension() n = n1+n2 V = VectorSpace(field, n) - mult_table = [ [ V.zero() for j in range(n) ] + mult_table = [ [ V.zero() for j in range(i+1) ] for i in range(n) ] for i in range(n1): - for j in range(n1): + for j in range(i+1): p = (J1.monomial(i)*J1.monomial(j)).to_vector() mult_table[i][j] = V(p.list() + [field.zero()]*n2) for i in range(n2): - for j in range(n2): + for j in range(i+1): p = (J2.monomial(i)*J2.monomial(j)).to_vector() mult_table[n1+i][n1+j] = V([field.zero()]*n1 + p.list()) # TODO: build the IP table here from the two constituent IP # matrices (it'll be block diagonal, I think). - ip_table = None + ip_table = [ [ field.zero() for j in range(i+1) ] + for i in range(n) ] super(DirectSumEJA, self).__init__(field, mult_table, ip_table, @@ -2644,8 +2679,8 @@ class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra): EXAMPLES:: - sage: J1 = HadamardEJA(2,QQ) - sage: J2 = JordanSpinEJA(3,QQ) + sage: J1 = HadamardEJA(2, field=QQ) + sage: J2 = JordanSpinEJA(3, field=QQ) sage: J = DirectSumEJA(J1,J2) sage: J.factors() (Euclidean Jordan algebra of dimension 2 over Rational Field, @@ -2774,8 +2809,8 @@ class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra): EXAMPLE:: - sage: J1 = HadamardEJA(3,QQ) - sage: J2 = QuaternionHermitianEJA(2,QQ,orthonormalize=False) + 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