From 008446f3a13b4fc117e1adfbc66f86784a6495c9 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Sat, 9 Nov 2019 19:42:25 -0500 Subject: [PATCH] eja: refactor the element subalgebra stuff into generic subalgebra. A messy job, but I got the tests passing. They certainly need some cleanup, and we should test non-element subalgebras too. --- mjo/eja/eja_element_subalgebra.py | 152 +------------- mjo/eja/eja_subalgebra.py | 324 ++++++++++++++++++++++++++++++ 2 files changed, 330 insertions(+), 146 deletions(-) create mode 100644 mjo/eja/eja_subalgebra.py diff --git a/mjo/eja/eja_element_subalgebra.py b/mjo/eja/eja_element_subalgebra.py index 2a82940..7cf3f37 100644 --- a/mjo/eja/eja_element_subalgebra.py +++ b/mjo/eja/eja_element_subalgebra.py @@ -1,132 +1,15 @@ from sage.matrix.constructor import matrix -from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra -from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement +from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra -class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement): - """ - SETUP:: - sage: from mjo.eja.eja_algebra import random_eja - - TESTS:: - - The natural representation of an element in the subalgebra is - the same as its natural representation in the superalgebra:: - - sage: set_random_seed() - sage: A = random_eja().random_element().subalgebra_generated_by() - sage: y = A.random_element() - sage: actual = y.natural_representation() - sage: expected = y.superalgebra_element().natural_representation() - sage: actual == expected - True - - The left-multiplication-by operator for elements in the subalgebra - works like it does in the superalgebra, even if we orthonormalize - our basis:: - - sage: set_random_seed() - sage: x = random_eja(AA).random_element() - sage: A = x.subalgebra_generated_by(orthonormalize_basis=True) - sage: y = A.random_element() - sage: y.operator()(A.one()) == y - True - - """ - - def superalgebra_element(self): - """ - Return the object in our algebra's superalgebra that corresponds - to myself. - - SETUP:: - - sage: from mjo.eja.eja_algebra import (RealSymmetricEJA, - ....: random_eja) - - EXAMPLES:: - - sage: J = RealSymmetricEJA(3) - sage: x = sum(J.gens()) - sage: x - e0 + e1 + e2 + e3 + e4 + e5 - sage: A = x.subalgebra_generated_by() - sage: A(x) - f1 - sage: A(x).superalgebra_element() - e0 + e1 + e2 + e3 + e4 + e5 - - TESTS: - - We can convert back and forth faithfully:: - - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: A = x.subalgebra_generated_by() - sage: A(x).superalgebra_element() == x - True - sage: y = A.random_element() - sage: A(y.superalgebra_element()) == y - True - - """ - return self.parent().superalgebra().linear_combination( - zip(self.parent()._superalgebra_basis, self.to_vector()) ) - - - - -class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanAlgebra): - """ - The subalgebra of an EJA generated by a single element. - - SETUP:: - - sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, - ....: JordanSpinEJA) - - TESTS: - - Ensure that our generator names don't conflict with the superalgebra:: - - sage: J = JordanSpinEJA(3) - sage: J.one().subalgebra_generated_by().gens() - (f0,) - sage: J = JordanSpinEJA(3, prefix='f') - sage: J.one().subalgebra_generated_by().gens() - (g0,) - sage: J = JordanSpinEJA(3, prefix='b') - sage: J.one().subalgebra_generated_by().gens() - (c0,) - - Ensure that we can find subalgebras of subalgebras:: - - sage: A = ComplexHermitianEJA(3).one().subalgebra_generated_by() - sage: B = A.one().subalgebra_generated_by() - sage: B.dimension() - 1 - - """ +class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanSubalgebra): def __init__(self, elt, orthonormalize_basis): self._superalgebra = elt.parent() category = self._superalgebra.category().Associative() V = self._superalgebra.vector_space() field = self._superalgebra.base_ring() - # A half-assed attempt to ensure that we don't collide with - # the superalgebra's prefix (ignoring the fact that there - # could be super-superelgrbas in scope). If possible, we - # try to "increment" the parent algebra's prefix, although - # this idea goes out the window fast because some prefixen - # are off-limits. - prefixen = [ 'f', 'g', 'h', 'a', 'b', 'c', 'd' ] - try: - prefix = prefixen[prefixen.index(self._superalgebra.prefix()) + 1] - except ValueError: - prefix = prefixen[0] - # This list is guaranteed to contain all independent powers, # because it's the maximal set of powers that could possibly # be independent (by a dimension argument). @@ -165,17 +48,6 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide for b in basis_vectors ] W = V.span_of_basis( V.from_vector(v) for v in basis_vectors ) - n = len(superalgebra_basis) - mult_table = [[W.zero() for i in range(n)] for j in range(n)] - for i in range(n): - for j in range(n): - product = superalgebra_basis[i]*superalgebra_basis[j] - # product.to_vector() might live in a vector subspace - # if our parent algebra is already a subalgebra. We - # use V.from_vector() to make it "the right size" in - # that case. - product_vector = V.from_vector(product.to_vector()) - mult_table[i][j] = W.coordinate_vector(product_vector) # The rank is the highest possible degree of a minimal # polynomial, and is bounded above by the dimension. We know @@ -185,21 +57,11 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide # its rank too. rank = W.dimension() - natural_basis = tuple( b.natural_representation() - for b in superalgebra_basis ) - - - self._vector_space = W - self._superalgebra_basis = superalgebra_basis - - fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self) - return fdeja.__init__(field, - mult_table, - rank, - prefix=prefix, - category=category, - natural_basis=natural_basis) + return fdeja.__init__(self._superalgebra, + superalgebra_basis, + rank=rank, + category=category) def _a_regular_element(self): @@ -386,5 +248,3 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide """ return self._vector_space - - Element = FiniteDimensionalEuclideanJordanElementSubalgebraElement diff --git a/mjo/eja/eja_subalgebra.py b/mjo/eja/eja_subalgebra.py new file mode 100644 index 0000000..b07f7e2 --- /dev/null +++ b/mjo/eja/eja_subalgebra.py @@ -0,0 +1,324 @@ +from sage.matrix.constructor import matrix + +from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra +from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement + +class FiniteDimensionalEuclideanJordanSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement): + """ + SETUP:: + + sage: from mjo.eja.eja_algebra import random_eja + + TESTS:: + + The natural representation of an element in the subalgebra is + the same as its natural representation in the superalgebra:: + + sage: set_random_seed() + sage: A = random_eja().random_element().subalgebra_generated_by() + sage: y = A.random_element() + sage: actual = y.natural_representation() + sage: expected = y.superalgebra_element().natural_representation() + sage: actual == expected + True + + The left-multiplication-by operator for elements in the subalgebra + works like it does in the superalgebra, even if we orthonormalize + our basis:: + + sage: set_random_seed() + sage: x = random_eja(AA).random_element() + sage: A = x.subalgebra_generated_by(orthonormalize_basis=True) + sage: y = A.random_element() + sage: y.operator()(A.one()) == y + True + + """ + + def superalgebra_element(self): + """ + Return the object in our algebra's superalgebra that corresponds + to myself. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (RealSymmetricEJA, + ....: random_eja) + + EXAMPLES:: + + sage: J = RealSymmetricEJA(3) + sage: x = sum(J.gens()) + sage: x + e0 + e1 + e2 + e3 + e4 + e5 + sage: A = x.subalgebra_generated_by() + sage: A(x) + f1 + sage: A(x).superalgebra_element() + e0 + e1 + e2 + e3 + e4 + e5 + + TESTS: + + We can convert back and forth faithfully:: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: A = x.subalgebra_generated_by() + sage: A(x).superalgebra_element() == x + True + sage: y = A.random_element() + sage: A(y.superalgebra_element()) == y + True + + """ + return self.parent().superalgebra().linear_combination( + zip(self.parent()._superalgebra_basis, self.to_vector()) ) + + + + +class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJordanAlgebra): + """ + A subalgebra of an EJA with a given basis. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, + ....: JordanSpinEJA) + + TESTS: + + Ensure that our generator names don't conflict with the superalgebra:: + + sage: J = JordanSpinEJA(3) + sage: J.one().subalgebra_generated_by().gens() + (f0,) + sage: J = JordanSpinEJA(3, prefix='f') + sage: J.one().subalgebra_generated_by().gens() + (g0,) + sage: J = JordanSpinEJA(3, prefix='b') + sage: J.one().subalgebra_generated_by().gens() + (c0,) + + Ensure that we can find subalgebras of subalgebras:: + + sage: A = ComplexHermitianEJA(3).one().subalgebra_generated_by() + sage: B = A.one().subalgebra_generated_by() + sage: B.dimension() + 1 + + """ + def __init__(self, superalgebra, basis, rank=None, category=None): + self._superalgebra = superalgebra + V = self._superalgebra.vector_space() + field = self._superalgebra.base_ring() + if category is None: + category = self._superalgebra.category() + + # A half-assed attempt to ensure that we don't collide with + # the superalgebra's prefix (ignoring the fact that there + # could be super-superelgrbas in scope). If possible, we + # try to "increment" the parent algebra's prefix, although + # this idea goes out the window fast because some prefixen + # are off-limits. + prefixen = [ 'f', 'g', 'h', 'a', 'b', 'c', 'd' ] + try: + prefix = prefixen[prefixen.index(self._superalgebra.prefix()) + 1] + except ValueError: + prefix = prefixen[0] + + basis_vectors = [ b.to_vector() for b in basis ] + superalgebra_basis = [ self._superalgebra.from_vector(b) + for b in basis_vectors ] + + W = V.span_of_basis( V.from_vector(v) for v in basis_vectors ) + n = len(superalgebra_basis) + mult_table = [[W.zero() for i in range(n)] for j in range(n)] + for i in range(n): + for j in range(n): + product = superalgebra_basis[i]*superalgebra_basis[j] + # product.to_vector() might live in a vector subspace + # if our parent algebra is already a subalgebra. We + # use V.from_vector() to make it "the right size" in + # that case. + product_vector = V.from_vector(product.to_vector()) + mult_table[i][j] = W.coordinate_vector(product_vector) + + natural_basis = tuple( b.natural_representation() + for b in superalgebra_basis ) + + + self._vector_space = W + self._superalgebra_basis = superalgebra_basis + + + fdeja = super(FiniteDimensionalEuclideanJordanSubalgebra, self) + return fdeja.__init__(field, + mult_table, + rank, + prefix=prefix, + category=category, + natural_basis=natural_basis) + + + + def _element_constructor_(self, elt): + """ + Construct an element of this subalgebra from the given one. + The only valid arguments are elements of the parent algebra + that happen to live in this subalgebra. + + SETUP:: + + sage: from mjo.eja.eja_algebra import RealSymmetricEJA + sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra + + EXAMPLES:: + + sage: J = RealSymmetricEJA(3) + sage: x = sum( i*J.gens()[i] for i in range(6) ) + sage: basis = tuple( x^k for k in range(J.rank()) ) + sage: K = FiniteDimensionalEuclideanJordanSubalgebra(J,basis) + sage: [ K(x^k) for k in range(J.rank()) ] + [f0, f1, f2] + + :: + + """ + if elt not in self.superalgebra(): + raise ValueError("not an element of this subalgebra") + + coords = self.vector_space().coordinate_vector(elt.to_vector()) + return self.from_vector(coords) + + + def one(self): + """ + Return the multiplicative identity element of this algebra. + + The superclass method computes the identity element, which is + beyond overkill in this case: the superalgebra identity + restricted to this algebra is its identity. Note that we can't + count on the first basis element being the identity -- it migth + have been scaled if we orthonormalized the basis. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA, + ....: random_eja) + + EXAMPLES:: + + sage: J = RealCartesianProductEJA(5) + sage: J.one() + e0 + e1 + e2 + e3 + e4 + sage: x = sum(J.gens()) + sage: A = x.subalgebra_generated_by() + sage: A.one() + f0 + sage: A.one().superalgebra_element() + e0 + e1 + e2 + e3 + e4 + + TESTS: + + The identity element acts like the identity over the rationals:: + + sage: set_random_seed() + sage: x = random_eja().random_element() + sage: A = x.subalgebra_generated_by() + sage: x = A.random_element() + sage: A.one()*x == x and x*A.one() == x + True + + The identity element acts like the identity over the algebraic + reals with an orthonormal basis:: + + sage: set_random_seed() + sage: x = random_eja(AA).random_element() + sage: A = x.subalgebra_generated_by(orthonormalize_basis=True) + sage: x = A.random_element() + sage: A.one()*x == x and x*A.one() == x + True + + The matrix of the unit element's operator is the identity over + the rationals:: + + sage: set_random_seed() + sage: x = random_eja().random_element() + sage: A = x.subalgebra_generated_by() + sage: actual = A.one().operator().matrix() + sage: expected = matrix.identity(A.base_ring(), A.dimension()) + sage: actual == expected + True + + The matrix of the unit element's operator is the identity over + the algebraic reals with an orthonormal basis:: + + sage: set_random_seed() + sage: x = random_eja(AA).random_element() + sage: A = x.subalgebra_generated_by(orthonormalize_basis=True) + sage: actual = A.one().operator().matrix() + sage: expected = matrix.identity(A.base_ring(), A.dimension()) + sage: actual == expected + True + + """ + if self.dimension() == 0: + return self.zero() + else: + sa_one = self.superalgebra().one().to_vector() + sa_coords = self.vector_space().coordinate_vector(sa_one) + return self.from_vector(sa_coords) + + + def natural_basis_space(self): + """ + Return the natural basis space of this algebra, which is identical + to that of its superalgebra. + + This is correct "by definition," and avoids a mismatch when the + subalgebra is trivial (with no natural basis to infer anything + from) and the parent is not. + """ + return self.superalgebra().natural_basis_space() + + + def superalgebra(self): + """ + Return the superalgebra that this algebra was generated from. + """ + return self._superalgebra + + + def vector_space(self): + """ + SETUP:: + + sage: from mjo.eja.eja_algebra import RealSymmetricEJA + sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra + + EXAMPLES:: + + sage: J = RealSymmetricEJA(3) + sage: x = J.monomial(0) + 2*J.monomial(2) + 5*J.monomial(5) + sage: basis = (x^0, x^1, x^2) + sage: K = FiniteDimensionalEuclideanJordanSubalgebra(J,basis) + sage: K.vector_space() + Vector space of degree 6 and dimension 3 over... + User basis matrix: + [ 1 0 1 0 0 1] + [ 1 0 2 0 0 5] + [ 1 0 4 0 0 25] + sage: (x^0).to_vector() + (1, 0, 1, 0, 0, 1) + sage: (x^1).to_vector() + (1, 0, 2, 0, 0, 5) + sage: (x^2).to_vector() + (1, 0, 4, 0, 0, 25) + + """ + return self._vector_space + + + Element = FiniteDimensionalEuclideanJordanSubalgebraElement -- 2.44.2