X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feuclidean_jordan_algebra.py;h=15ff26ca909cd1fbec4a2f1fc95e69b69f5bd02f;hb=b82bd087507b8f727c10ca0eb55f9a1d15ed3438;hp=24eaf50f381fbab7107bf5f3512354c221148199;hpb=633d193089c3f42aa2e14e0342d5e7030ad74be8;p=sage.d.git diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py index 24eaf50..15ff26c 100644 --- a/mjo/eja/euclidean_jordan_algebra.py +++ b/mjo/eja/euclidean_jordan_algebra.py @@ -21,8 +21,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): assume_associative=False, category=None, rank=None, - natural_basis=None, - inner_product=None): + natural_basis=None): n = len(mult_table) mult_table = [b.base_extend(field) for b in mult_table] for b in mult_table: @@ -46,18 +45,17 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): names=names, category=cat, rank=rank, - natural_basis=natural_basis, - inner_product=inner_product) + natural_basis=natural_basis) - def __init__(self, field, + def __init__(self, + field, mult_table, names='e', assume_associative=False, category=None, rank=None, - natural_basis=None, - inner_product=None): + natural_basis=None): """ EXAMPLES: @@ -73,7 +71,6 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): """ self._rank = rank self._natural_basis = natural_basis - self._inner_product = inner_product fda = super(FiniteDimensionalEuclideanJordanAlgebra, self) fda.__init__(field, mult_table, @@ -93,7 +90,9 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): """ The inner product associated with this Euclidean Jordan algebra. - Will default to the trace inner product if nothing else. + Defaults to the trace inner product, but can be overridden by + subclasses if they are sure that the necessary properties are + satisfied. EXAMPLES: @@ -111,10 +110,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): """ if (not x in self) or (not y in self): raise TypeError("arguments must live in this algebra") - if self._inner_product is None: - return x.trace_inner_product(y) - else: - return self._inner_product(x,y) + return x.trace_inner_product(y) def natural_basis(self): @@ -132,7 +128,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = RealSymmetricSimpleEJA(2) + sage: J = RealSymmetricEJA(2) sage: J.basis() Family (e0, e1, e2) sage: J.natural_basis() @@ -143,7 +139,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): :: - sage: J = JordanSpinSimpleEJA(2) + sage: J = JordanSpinEJA(2) sage: J.basis() Family (e0, e1) sage: J.natural_basis() @@ -180,14 +176,14 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): The identity in `S^n` is converted to the identity in the EJA:: - sage: J = RealSymmetricSimpleEJA(3) + sage: J = RealSymmetricEJA(3) sage: I = identity_matrix(QQ,3) sage: J(I) == J.one() True This skew-symmetric matrix can't be represented in the EJA:: - sage: J = RealSymmetricSimpleEJA(3) + sage: J = RealSymmetricEJA(3) sage: A = matrix(QQ,3, lambda i,j: i-j) sage: J(A) Traceback (most recent call last): @@ -295,7 +291,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): inner product on `R^n` (this example only works because the basis for the Jordan algebra is the standard basis in `R^n`):: - sage: J = JordanSpinSimpleEJA(3) + sage: J = JordanSpinEJA(3) sage: x = vector(QQ,[1,2,3]) sage: y = vector(QQ,[4,5,6]) sage: x.inner_product(y) @@ -308,7 +304,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): so the inner product of the identity matrix with itself should be the `n`:: - sage: J = RealSymmetricSimpleEJA(3) + sage: J = RealSymmetricEJA(3) sage: J.one().inner_product(J.one()) 3 @@ -317,13 +313,13 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): part because the product of Hermitian matrices may not be Hermitian:: - sage: J = ComplexHermitianSimpleEJA(3) + sage: J = ComplexHermitianEJA(3) sage: J.one().inner_product(J.one()) 3 Ditto for the quaternions:: - sage: J = QuaternionHermitianSimpleEJA(3) + sage: J = QuaternionHermitianEJA(3) sage: J.one().inner_product(J.one()) 3 @@ -390,12 +386,12 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = JordanSpinSimpleEJA(2) + sage: J = JordanSpinEJA(2) sage: e0,e1 = J.gens() sage: x = e0 + e1 sage: x.det() 0 - sage: J = JordanSpinSimpleEJA(3) + sage: J = JordanSpinEJA(3) sage: e0,e1,e2 = J.gens() sage: x = e0 + e1 + e2 sage: x.det() @@ -424,7 +420,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: set_random_seed() sage: n = ZZ.random_element(1,10) - sage: J = JordanSpinSimpleEJA(n) + sage: J = JordanSpinEJA(n) sage: x = J.random_element() sage: while x.is_zero(): ....: x = J.random_element() @@ -554,7 +550,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): The identity element always has degree one, but any element linearly-independent from it is regular:: - sage: J = JordanSpinSimpleEJA(5) + sage: J = JordanSpinEJA(5) sage: J.one().is_regular() False sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity @@ -579,7 +575,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = JordanSpinSimpleEJA(4) + sage: J = JordanSpinEJA(4) sage: J.one().degree() 1 sage: e0,e1,e2,e3 = J.gens() @@ -591,7 +587,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: set_random_seed() sage: n = ZZ.random_element(1,10) - sage: J = JordanSpinSimpleEJA(n) + sage: J = JordanSpinEJA(n) sage: x = J.random_element() sage: x == x.coefficient(0)*J.one() or x.degree() == 2 True @@ -623,7 +619,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: set_random_seed() sage: n = ZZ.random_element(2,10) - sage: J = JordanSpinSimpleEJA(n) + sage: J = JordanSpinEJA(n) sage: y = J.random_element() sage: while y == y.coefficient(0)*J.one(): ....: y = J.random_element() @@ -669,7 +665,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = ComplexHermitianSimpleEJA(3) + sage: J = ComplexHermitianEJA(3) sage: J.one() e0 + e5 + e8 sage: J.one().natural_representation() @@ -682,7 +678,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): :: - sage: J = QuaternionHermitianSimpleEJA(3) + sage: J = QuaternionHermitianEJA(3) sage: J.one() e0 + e9 + e14 sage: J.one().natural_representation() @@ -783,7 +779,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: set_random_seed() sage: n = ZZ.random_element(1,10) - sage: J = JordanSpinSimpleEJA(n) + sage: J = JordanSpinEJA(n) sage: x = J.random_element() sage: x_vec = x.vector() sage: x0 = x_vec[0] @@ -926,11 +922,11 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): TESTS:: sage: set_random_seed() - sage: J = eja_rn(5) + sage: J = RealCartesianProductEJA(5) sage: c = J.random_element().subalgebra_idempotent() sage: c^2 == c True - sage: J = JordanSpinSimpleEJA(5) + sage: J = JordanSpinEJA(5) sage: c = J.random_element().subalgebra_idempotent() sage: c^2 == c True @@ -986,7 +982,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = JordanSpinSimpleEJA(3) + sage: J = JordanSpinEJA(3) sage: e0,e1,e2 = J.gens() sage: x = e0 + e1 + e2 sage: x.trace() @@ -1010,16 +1006,20 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): return (self*other).trace() -def eja_rn(dimension, field=QQ): +class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra): """ Return the Euclidean Jordan Algebra corresponding to the set `R^n` under the Hadamard product. + 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. + EXAMPLES: This multiplication table can be verified by hand:: - sage: J = eja_rn(3) + sage: J = RealCartesianProductEJA(3) sage: e0,e1,e2 = J.gens() sage: e0*e0 e0 @@ -1035,19 +1035,21 @@ def eja_rn(dimension, field=QQ): e2 """ - # The FiniteDimensionalAlgebra constructor takes a list of - # matrices, the ith representing right multiplication by the ith - # basis element in the vector space. So if e_1 = (1,0,0), then - # right (Hadamard) multiplication of x by e_1 picks out the first - # component of x; and likewise for the ith basis element e_i. - Qs = [ matrix(field, dimension, dimension, lambda k,j: 1*(k == j == i)) - for i in xrange(dimension) ] - - return FiniteDimensionalEuclideanJordanAlgebra(field, - Qs, - rank=dimension, - inner_product=_usual_ip) + @staticmethod + def __classcall_private__(cls, n, field=QQ): + # The FiniteDimensionalAlgebra constructor takes a list of + # matrices, the ith representing right multiplication by the ith + # basis element in the vector space. So if e_1 = (1,0,0), then + # right (Hadamard) multiplication of x by e_1 picks out the first + # component of x; and likewise for the ith basis element e_i. + Qs = [ matrix(field, n, n, lambda k,j: 1*(k == j == i)) + for i in xrange(n) ] + + fdeja = super(RealCartesianProductEJA, cls) + return fdeja.__classcall_private__(cls, field, Qs, rank=n) + def inner_product(self, x, y): + return _usual_ip(x,y) def random_eja(): @@ -1083,11 +1085,11 @@ def random_eja(): """ n = ZZ.random_element(1,5) - constructor = choice([eja_rn, - JordanSpinSimpleEJA, - RealSymmetricSimpleEJA, - ComplexHermitianSimpleEJA, - QuaternionHermitianSimpleEJA]) + constructor = choice([RealCartesianProductEJA, + JordanSpinEJA, + RealSymmetricEJA, + ComplexHermitianEJA, + QuaternionHermitianEJA]) return constructor(n, field=QQ) @@ -1464,7 +1466,7 @@ def _matrix_ip(X,Y): return (X_mat*Y_mat).trace() -def RealSymmetricSimpleEJA(n, field=QQ): +class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): """ The rank-n simple EJA consisting of real symmetric n-by-n matrices, the usual symmetric Jordan product, and the trace inner @@ -1472,7 +1474,7 @@ def RealSymmetricSimpleEJA(n, field=QQ): EXAMPLES:: - sage: J = RealSymmetricSimpleEJA(2) + sage: J = RealSymmetricEJA(2) sage: e0, e1, e2 = J.gens() sage: e0*e0 e0 @@ -1487,7 +1489,7 @@ def RealSymmetricSimpleEJA(n, field=QQ): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: J = RealSymmetricSimpleEJA(n) + sage: J = RealSymmetricEJA(n) sage: J.degree() == (n^2 + n)/2 True @@ -1495,7 +1497,7 @@ def RealSymmetricSimpleEJA(n, field=QQ): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: J = RealSymmetricSimpleEJA(n) + sage: J = RealSymmetricEJA(n) sage: x = J.random_element() sage: y = J.random_element() sage: actual = (x*y).natural_representation() @@ -1508,17 +1510,23 @@ def RealSymmetricSimpleEJA(n, field=QQ): True """ - S = _real_symmetric_basis(n, field=field) - (Qs, T) = _multiplication_table_from_matrix_basis(S) + @staticmethod + def __classcall_private__(cls, n, field=QQ): + S = _real_symmetric_basis(n, field=field) + (Qs, T) = _multiplication_table_from_matrix_basis(S) - return FiniteDimensionalEuclideanJordanAlgebra(field, - Qs, - rank=n, - natural_basis=T, - inner_product=_matrix_ip) + fdeja = super(RealSymmetricEJA, cls) + return fdeja.__classcall_private__(cls, + field, + Qs, + rank=n, + natural_basis=T) + + def inner_product(self, x, y): + return _matrix_ip(x,y) -def ComplexHermitianSimpleEJA(n, field=QQ): +class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): """ The rank-n simple EJA consisting of complex Hermitian n-by-n matrices over the real numbers, the usual symmetric Jordan product, @@ -1531,7 +1539,7 @@ def ComplexHermitianSimpleEJA(n, field=QQ): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: J = ComplexHermitianSimpleEJA(n) + sage: J = ComplexHermitianEJA(n) sage: J.degree() == n^2 True @@ -1539,7 +1547,7 @@ def ComplexHermitianSimpleEJA(n, field=QQ): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: J = ComplexHermitianSimpleEJA(n) + sage: J = ComplexHermitianEJA(n) sage: x = J.random_element() sage: y = J.random_element() sage: actual = (x*y).natural_representation() @@ -1552,26 +1560,30 @@ def ComplexHermitianSimpleEJA(n, field=QQ): True """ - S = _complex_hermitian_basis(n) - (Qs, T) = _multiplication_table_from_matrix_basis(S) + @staticmethod + def __classcall_private__(cls, n, field=QQ): + S = _complex_hermitian_basis(n) + (Qs, T) = _multiplication_table_from_matrix_basis(S) - # Since a+bi on the diagonal is represented as - # - # a + bi = [ a b ] - # [ -b a ], - # - # we'll double-count the "a" entries if we take the trace of - # the embedding. - ip = lambda X,Y: _matrix_ip(X,Y)/2 + fdeja = super(ComplexHermitianEJA, cls) + return fdeja.__classcall_private__(cls, + field, + Qs, + rank=n, + natural_basis=T) - return FiniteDimensionalEuclideanJordanAlgebra(field, - Qs, - rank=n, - natural_basis=T, - inner_product=ip) + def inner_product(self, x, y): + # Since a+bi on the diagonal is represented as + # + # a + bi = [ a b ] + # [ -b a ], + # + # we'll double-count the "a" entries if we take the trace of + # the embedding. + return _matrix_ip(x,y)/2 -def QuaternionHermitianSimpleEJA(n, field=QQ): +class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): """ The rank-n simple EJA consisting of self-adjoint n-by-n quaternion matrices, the usual symmetric Jordan product, and the @@ -1584,7 +1596,7 @@ def QuaternionHermitianSimpleEJA(n, field=QQ): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: J = QuaternionHermitianSimpleEJA(n) + sage: J = QuaternionHermitianEJA(n) sage: J.degree() == 2*(n^2) - n True @@ -1592,7 +1604,7 @@ def QuaternionHermitianSimpleEJA(n, field=QQ): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: J = QuaternionHermitianSimpleEJA(n) + sage: J = QuaternionHermitianEJA(n) sage: x = J.random_element() sage: y = J.random_element() sage: actual = (x*y).natural_representation() @@ -1605,35 +1617,32 @@ def QuaternionHermitianSimpleEJA(n, field=QQ): True """ - S = _quaternion_hermitian_basis(n) - (Qs, T) = _multiplication_table_from_matrix_basis(S) - - # 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. - ip = lambda X,Y: _matrix_ip(X,Y)/4 - - return FiniteDimensionalEuclideanJordanAlgebra(field, - Qs, - rank=n, - natural_basis=T, - inner_product=ip) + @staticmethod + def __classcall_private__(cls, n, field=QQ): + S = _quaternion_hermitian_basis(n) + (Qs, T) = _multiplication_table_from_matrix_basis(S) + fdeja = super(QuaternionHermitianEJA, cls) + return fdeja.__classcall_private__(cls, + field, + Qs, + rank=n, + natural_basis=T) -def OctonionHermitianSimpleEJA(n): - """ - This shit be crazy. It has dimension 27 over the reals. - """ - n = 3 - pass - -def JordanSpinSimpleEJA(n, field=QQ): + 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 + + +class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra): """ The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)`` with the usual inner product and jordan product ``x*y = @@ -1644,7 +1653,7 @@ def JordanSpinSimpleEJA(n, field=QQ): This multiplication table can be verified by hand:: - sage: J = JordanSpinSimpleEJA(4) + sage: J = JordanSpinEJA(4) sage: e0,e1,e2,e3 = J.gens() sage: e0*e0 e0 @@ -1661,31 +1670,27 @@ def JordanSpinSimpleEJA(n, field=QQ): sage: e2*e3 0 - In one dimension, this is the reals under multiplication:: - - sage: J1 = JordanSpinSimpleEJA(1) - sage: J2 = eja_rn(1) - sage: J1 == J2 - True - """ - Qs = [] - id_matrix = identity_matrix(field, n) - for i in xrange(n): - ei = id_matrix.column(i) - Qi = zero_matrix(field, n) - Qi.set_row(0, ei) - Qi.set_column(0, ei) - Qi += diagonal_matrix(n, [ei[0]]*n) - # The addition of the diagonal matrix adds an extra ei[0] in the - # upper-left corner of the matrix. - Qi[0,0] = Qi[0,0] * ~field(2) - Qs.append(Qi) - - # The rank of the spin factor algebra is two, UNLESS we're in a - # one-dimensional ambient space (the rank is bounded by the - # ambient dimension). - return FiniteDimensionalEuclideanJordanAlgebra(field, - Qs, - rank=min(n,2), - inner_product=_usual_ip) + @staticmethod + def __classcall_private__(cls, n, field=QQ): + Qs = [] + id_matrix = identity_matrix(field, n) + for i in xrange(n): + ei = id_matrix.column(i) + Qi = zero_matrix(field, n) + Qi.set_row(0, ei) + Qi.set_column(0, ei) + Qi += diagonal_matrix(n, [ei[0]]*n) + # The addition of the diagonal matrix adds an extra ei[0] in the + # upper-left corner of the matrix. + Qi[0,0] = Qi[0,0] * ~field(2) + Qs.append(Qi) + + # The rank of the spin algebra is two, unless we're in a + # one-dimensional ambient space (because the rank is bounded by + # the ambient dimension). + fdeja = super(JordanSpinEJA, cls) + return fdeja.__classcall_private__(cls, field, Qs, rank=min(n,2)) + + def inner_product(self, x, y): + return _usual_ip(x,y)