X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=f687e46215a1f1aefb5a5955acffd4fc4c6be301;hb=a167bfc4ffb18296eb75d505112a6c1dd4c7f9ae;hp=4acf513d3a8aa6b498621af5908b11363f77d420;hpb=a26c6d48d496531069c0f89842e60c4407873314;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 4acf513..f687e46 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -123,11 +123,11 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): vector representations) back and forth faithfully:: sage: set_random_seed() - sage: J = RealCartesianProductEJA(5) + sage: J = RealCartesianProductEJA.random_instance() sage: x = J.random_element() sage: J(x.to_vector().column()) == x True - sage: J = JordanSpinEJA(5) + sage: J = JordanSpinEJA.random_instance() sage: x = J.random_element() sage: J(x.to_vector().column()) == x True @@ -155,6 +155,26 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): return self.from_vector(coords) + @staticmethod + def _max_test_case_size(): + """ + Return an integer "size" that is an upper bound on the size of + this algebra when it is used in a random test + case. Unfortunately, the term "size" is quite vague -- when + dealing with `R^n` under either the Hadamard or Jordan spin + product, the "size" refers to the dimension `n`. When dealing + with a matrix algebra (real symmetric or complex/quaternion + Hermitian), it refers to the size of the matrix, which is + far less than the dimension of the underlying vector space. + + We default to five in this class, which is safe in `R^n`. The + matrix algebra subclasses (or any class where the "size" is + interpreted to be far less than the dimension) should override + with a smaller number. + """ + return 5 + + def _repr_(self): """ Return a string representation of ``self``. @@ -435,7 +455,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): """ X = x.natural_representation() Y = y.natural_representation() - return self.__class__.natural_inner_product(X,Y) + return self.natural_inner_product(X,Y) def is_trivial(self): @@ -638,6 +658,28 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): return s.random_element() + @classmethod + def random_instance(cls, field=QQ, **kwargs): + """ + Return a random instance of this type of algebra. + + In subclasses for algebras that we know how to construct, this + is a shortcut for constructing test cases and examples. + """ + if cls is FiniteDimensionalEuclideanJordanAlgebra: + # Red flag! But in theory we could do this I guess. The + # only finite-dimensional exceptional EJA is the + # octononions. So, we could just create an EJA from an + # associative matrix algebra (generated by a subset of + # elements) with the symmetric product. Or, we could punt + # to random_eja() here, override it in our subclasses, and + # not worry about it. + raise NotImplementedError + + n = ZZ.random_element(1, cls._max_test_case_size()) + return cls(n, field, **kwargs) + + def rank(self): """ Return the rank of this EJA. @@ -761,8 +803,7 @@ class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra): Our inner product satisfies the Jordan axiom:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = RealCartesianProductEJA(n) + sage: J = RealCartesianProductEJA.random_instance() sage: x = J.random_element() sage: y = J.random_element() sage: z = J.random_element() @@ -792,13 +833,12 @@ class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra): over `R^n`:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = RealCartesianProductEJA(n) + sage: J = RealCartesianProductEJA.random_instance() sage: x = J.random_element() sage: y = J.random_element() sage: X = x.natural_representation() sage: Y = y.natural_representation() - sage: x.inner_product(y) == J.__class__.natural_inner_product(X,Y) + sage: x.inner_product(y) == J.natural_inner_product(X,Y) True """ @@ -841,18 +881,12 @@ def random_eja(): Euclidean Jordan algebra of dimension... """ - - # The max_n component lets us choose different upper bounds on the - # value "n" that gets passed to the constructor. This is needed - # because e.g. R^{10} is reasonable to test, while the Hermitian - # 10-by-10 quaternion matrices are not. - (constructor, max_n) = choice([(RealCartesianProductEJA, 6), - (JordanSpinEJA, 6), - (RealSymmetricEJA, 5), - (ComplexHermitianEJA, 4), - (QuaternionHermitianEJA, 3)]) - n = ZZ.random_element(1, max_n) - return constructor(n, field=QQ) + classname = choice([RealCartesianProductEJA, + JordanSpinEJA, + RealSymmetricEJA, + ComplexHermitianEJA, + QuaternionHermitianEJA]) + return classname.random_instance() @@ -1032,7 +1066,8 @@ def _embed_complex_matrix(M): SETUP:: - sage: from mjo.eja.eja_algebra import _embed_complex_matrix + sage: from mjo.eja.eja_algebra import (_embed_complex_matrix, + ....: ComplexHermitianEJA) EXAMPLES:: @@ -1054,7 +1089,8 @@ def _embed_complex_matrix(M): Embedding is a homomorphism (isomorphism, in fact):: sage: set_random_seed() - sage: n = ZZ.random_element(5) + sage: n_max = ComplexHermitianEJA._max_test_case_size() + sage: n = ZZ.random_element(n_max) sage: F = QuadraticField(-1, 'i') sage: X = random_matrix(F, n) sage: Y = random_matrix(F, n) @@ -1146,7 +1182,8 @@ def _embed_quaternion_matrix(M): SETUP:: - sage: from mjo.eja.eja_algebra import _embed_quaternion_matrix + sage: from mjo.eja.eja_algebra import (_embed_quaternion_matrix, + ....: QuaternionHermitianEJA) EXAMPLES:: @@ -1163,7 +1200,8 @@ def _embed_quaternion_matrix(M): Embedding is a homomorphism (isomorphism, in fact):: sage: set_random_seed() - sage: n = ZZ.random_element(5) + sage: n_max = QuaternionHermitianEJA._max_test_case_size() + sage: n = ZZ.random_element(n_max) sage: Q = QuaternionAlgebra(QQ,-1,-1) sage: X = random_matrix(Q, n) sage: Y = random_matrix(Q, n) @@ -1232,7 +1270,10 @@ def _unembed_quaternion_matrix(M): if not n.mod(4).is_zero(): raise ValueError("the matrix 'M' must be a complex embedding") - Q = QuaternionAlgebra(QQ,-1,-1) + # Use the base ring of the matrix to ensure that its entries can be + # multiplied by elements of the quaternion algebra. + field = M.base_ring() + Q = QuaternionAlgebra(field,-1,-1) i,j,k = Q.gens() # Go top-left to bottom-right (reading order), converting every @@ -1246,21 +1287,15 @@ def _unembed_quaternion_matrix(M): 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() + submat[0,0].imag()*i - z += submat[0,1].real()*j + submat[0,1].imag()*k + z = submat[0,0].vector()[0] # real part + z += submat[0,0].vector()[1]*i # imag part + z += submat[0,1].vector()[0]*j # real part + z += submat[0,1].vector()[1]*k # imag part elements.append(z) return matrix(Q, n/4, elements) -# The inner product used for the real symmetric simple EJA. -# We keep it as a separate function because e.g. the complex -# algebra uses the same inner product, except divided by 2. -def _matrix_ip(X,Y): - X_mat = X.natural_representation() - Y_mat = Y.natural_representation() - return (X_mat*Y_mat).trace() - class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): """ @@ -1288,7 +1323,8 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): The dimension of this algebra is `(n^2 + n) / 2`:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) + sage: n_max = RealSymmetricEJA._max_test_case_size() + sage: n = ZZ.random_element(1, n_max) sage: J = RealSymmetricEJA(n) sage: J.dimension() == (n^2 + n)/2 True @@ -1296,8 +1332,7 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): The Jordan multiplication is what we think it is:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = RealSymmetricEJA(n) + sage: J = RealSymmetricEJA.random_instance() sage: x = J.random_element() sage: y = J.random_element() sage: actual = (x*y).natural_representation() @@ -1317,8 +1352,7 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): Our inner product satisfies the Jordan axiom:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = RealSymmetricEJA(n) + sage: J = RealSymmetricEJA.random_instance() sage: x = J.random_element() sage: y = J.random_element() sage: z = J.random_element() @@ -1329,8 +1363,7 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): product unless we specify otherwise:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = RealSymmetricEJA(n) + sage: J = RealSymmetricEJA.random_instance() sage: all( b.norm() == 1 for b in J.gens() ) True @@ -1341,8 +1374,7 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): the operator is self-adjoint by the Jordan axiom:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: x = RealSymmetricEJA(n).random_element() + sage: x = RealSymmetricEJA.random_instance().random_element() sage: x.operator().matrix().is_symmetric() True @@ -1359,10 +1391,9 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): p = z**2 - 2 if p.is_irreducible(): field = NumberField(p, 'sqrt2', embedding=RLF(2).sqrt()) - S = [ s.change_ring(field) for s in S ] + S = [ s.change_ring(field) for s in S ] self._basis_normalizers = tuple( - ~(self.__class__.natural_inner_product(s,s).sqrt()) - for s in S ) + ~(self.natural_inner_product(s,s).sqrt()) for s in S ) S = tuple( s*c for (s,c) in zip(S,self._basis_normalizers) ) Qs = _multiplication_table_from_matrix_basis(S) @@ -1374,6 +1405,9 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): natural_basis=S, **kwargs) + @staticmethod + def _max_test_case_size(): + return 5 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): @@ -1392,7 +1426,8 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): The dimension of this algebra is `n^2`:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) + sage: n_max = ComplexHermitianEJA._max_test_case_size() + sage: n = ZZ.random_element(1, n_max) sage: J = ComplexHermitianEJA(n) sage: J.dimension() == n^2 True @@ -1400,8 +1435,7 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): The Jordan multiplication is what we think it is:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = ComplexHermitianEJA(n) + sage: J = ComplexHermitianEJA.random_instance() sage: x = J.random_element() sage: y = J.random_element() sage: actual = (x*y).natural_representation() @@ -1421,8 +1455,7 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): Our inner product satisfies the Jordan axiom:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = ComplexHermitianEJA(n) + sage: J = ComplexHermitianEJA.random_instance() sage: x = J.random_element() sage: y = J.random_element() sage: z = J.random_element() @@ -1433,8 +1466,7 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): product unless we specify otherwise:: sage: set_random_seed() - sage: n = ZZ.random_element(1,4) - sage: J = ComplexHermitianEJA(n) + sage: J = ComplexHermitianEJA.random_instance() sage: all( b.norm() == 1 for b in J.gens() ) True @@ -1445,8 +1477,7 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): the operator is self-adjoint by the Jordan axiom:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: x = ComplexHermitianEJA(n).random_element() + sage: x = ComplexHermitianEJA.random_instance().random_element() sage: x.operator().matrix().is_symmetric() True @@ -1463,10 +1494,9 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): p = z**2 - 2 if p.is_irreducible(): field = NumberField(p, 'sqrt2', embedding=RLF(2).sqrt()) - S = [ s.change_ring(field) for s in S ] + S = [ s.change_ring(field) for s in S ] self._basis_normalizers = tuple( - ~(self.__class__.natural_inner_product(s,s).sqrt()) - for s in S ) + ~(self.natural_inner_product(s,s).sqrt()) for s in S ) S = tuple( s*c for (s,c) in zip(S,self._basis_normalizers) ) Qs = _multiplication_table_from_matrix_basis(S) @@ -1479,6 +1509,10 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): **kwargs) + @staticmethod + def _max_test_case_size(): + return 4 + @staticmethod def natural_inner_product(X,Y): Xu = _unembed_complex_matrix(X) @@ -1486,6 +1520,8 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): # The trace need not be real; consider Xu = (i*I) and Yu = I. return ((Xu*Yu).trace()).vector()[0] # real part, I guess + + class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): """ The rank-n simple EJA consisting of self-adjoint n-by-n quaternion @@ -1499,10 +1535,11 @@ class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): TESTS: - The dimension of this algebra is `n^2`:: + The dimension of this algebra is `2*n^2 - n`:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) + sage: n_max = QuaternionHermitianEJA._max_test_case_size() + sage: n = ZZ.random_element(1, n_max) sage: J = QuaternionHermitianEJA(n) sage: J.dimension() == 2*(n^2) - n True @@ -1510,8 +1547,7 @@ class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): The Jordan multiplication is what we think it is:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = QuaternionHermitianEJA(n) + sage: J = QuaternionHermitianEJA.random_instance() sage: x = J.random_element() sage: y = J.random_element() sage: actual = (x*y).natural_representation() @@ -1531,17 +1567,50 @@ class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): Our inner product satisfies the Jordan axiom:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = QuaternionHermitianEJA(n) + sage: J = QuaternionHermitianEJA.random_instance() sage: x = J.random_element() sage: y = J.random_element() sage: z = J.random_element() sage: (x*y).inner_product(z) == y.inner_product(x*z) True + Our natural basis is normalized with respect to the natural inner + product unless we specify otherwise:: + + sage: set_random_seed() + sage: J = QuaternionHermitianEJA.random_instance() + sage: all( b.norm() == 1 for b in J.gens() ) + True + + Since our natural basis is normalized with respect to the natural + inner product, and since we know that this algebra is an EJA, any + left-multiplication operator's matrix will be symmetric because + natural->EJA basis representation is an isometry and within the EJA + the operator is self-adjoint by the Jordan axiom:: + + sage: set_random_seed() + sage: x = QuaternionHermitianEJA.random_instance().random_element() + sage: x.operator().matrix().is_symmetric() + True + """ def __init__(self, n, field=QQ, normalize_basis=True, **kwargs): - S = _quaternion_hermitian_basis(n, field, normalize_basis) + S = _quaternion_hermitian_basis(n, field) + + if n > 1 and normalize_basis: + # We'll need sqrt(2) to normalize the basis, and this + # winds up in the multiplication table, so the whole + # algebra needs to be over the field extension. + R = PolynomialRing(field, 'z') + z = R.gen() + p = z**2 - 2 + if p.is_irreducible(): + field = NumberField(p, 'sqrt2', embedding=RLF(2).sqrt()) + S = [ s.change_ring(field) for s in S ] + self._basis_normalizers = tuple( + ~(self.natural_inner_product(s,s).sqrt()) for s in S ) + S = tuple( s*c for (s,c) in zip(S,self._basis_normalizers) ) + Qs = _multiplication_table_from_matrix_basis(S) fdeja = super(QuaternionHermitianEJA, self) @@ -1551,17 +1620,20 @@ class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): natural_basis=S, **kwargs) - def inner_product(self, x, y): - # Since a+bi+cj+dk on the diagonal is represented as - # - # a + bi +cj + dk = [ a b c d] - # [ -b a -d c] - # [ -c d a -b] - # [ -d -c b a], - # - # we'll quadruple-count the "a" entries if we take the trace of - # the embedding. - return _matrix_ip(x,y)/4 + @staticmethod + def _max_test_case_size(): + return 3 + + @staticmethod + def natural_inner_product(X,Y): + Xu = _unembed_quaternion_matrix(X) + Yu = _unembed_quaternion_matrix(Y) + # The trace need not be real; consider Xu = (i*I) and Yu = I. + # The result will be a quaternion algebra element, which doesn't + # have a "vector" method, but does have coefficient_tuple() method + # that returns the coefficients of 1, i, j, and k -- in that order. + return ((Xu*Yu).trace()).coefficient_tuple()[0] + class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra): @@ -1604,8 +1676,7 @@ class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra): Our inner product satisfies the Jordan axiom:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = JordanSpinEJA(n) + sage: J = JordanSpinEJA.random_instance() sage: x = J.random_element() sage: y = J.random_element() sage: z = J.random_element() @@ -1650,13 +1721,12 @@ class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra): over `R^n`:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = JordanSpinEJA(n) + sage: J = JordanSpinEJA.random_instance() sage: x = J.random_element() sage: y = J.random_element() sage: X = x.natural_representation() sage: Y = y.natural_representation() - sage: x.inner_product(y) == J.__class__.natural_inner_product(X,Y) + sage: x.inner_product(y) == J.natural_inner_product(X,Y) True """