From: Michael Orlitzky Date: Thu, 9 Apr 2026 00:29:34 +0000 (-0400) Subject: mjo/hurwitz.py: delete custom Octonions implementation X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=8e029c3b6f335dc1a6f553390f28c359a8216f56;p=sage.d.git mjo/hurwitz.py: delete custom Octonions implementation Everything has been updated to use the Sage OctonionAlgebra. --- diff --git a/mjo/hurwitz.py b/mjo/hurwitz.py index 6d02168..467cf73 100644 --- a/mjo/hurwitz.py +++ b/mjo/hurwitz.py @@ -1,303 +1,7 @@ -from sage.misc.cachefunc import cached_method -from sage.combinat.free_module import CombinatorialFreeModule -from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement from sage.rings.all import AA from mjo.matrix_algebra import MatrixAlgebra, MatrixAlgebraElement -class Octonion(IndexedFreeModuleElement): - def conjugate(self): - r""" - SETUP:: - - sage: from mjo.hurwitz import Octonions - - EXAMPLES:: - - sage: O = Octonions() - sage: x = sum(O.gens()) - sage: x.conjugate() - e0 - e1 - e2 - e3 - e4 - e5 - e6 - e7 - - TESTS:: - - Conjugating twice gets you the original element:: - - sage: O = Octonions() - sage: x = O.random_element() - sage: x.conjugate().conjugate() == x - True - - """ - from sage.rings.all import ZZ - from sage.matrix.matrix_space import MatrixSpace - C = MatrixSpace(ZZ,8).diagonal_matrix((1,-1,-1,-1,-1,-1,-1,-1)) - return self.parent().from_vector(C*self.to_vector()) - - def real(self): - r""" - Return the real part of this octonion. - - The real part of an octonion is its projection onto the span - of the first generator. In other words, the "first dimension" - is real and the others are imaginary. - - SETUP:: - - sage: from mjo.hurwitz import Octonions - - EXAMPLES:: - - sage: O = Octonions() - sage: x = sum(O.gens()) - sage: x.real() - e0 - - TESTS: - - This method is idempotent:: - - sage: O = Octonions() - sage: x = O.random_element() - sage: x.real().real() == x.real() - True - - """ - return (self + self.conjugate())/2 - - def imag(self): - r""" - Return the imaginary part of this octonion. - - The imaginary part of an octonion is its projection onto the - orthogonal complement of the span of the first generator. In - other words, the "first dimension" is real and the others are - imaginary. - - SETUP:: - - sage: from mjo.hurwitz import Octonions - - EXAMPLES:: - - sage: O = Octonions() - sage: x = sum(O.gens()) - sage: x.imag() - e1 + e2 + e3 + e4 + e5 + e6 + e7 - - TESTS: - - This method is idempotent:: - - sage: O = Octonions() - sage: x = O.random_element() - sage: x.imag().imag() == x.imag() - True - - """ - return (self - self.conjugate())/2 - - def _norm_squared(self): - return (self*self.conjugate()).coefficient(0) - - def norm(self): - r""" - Return the norm of this octonion. - - SETUP:: - - sage: from mjo.hurwitz import Octonions - - EXAMPLES:: - - sage: O = Octonions() - sage: O.one().norm() - 1 - - TESTS: - - The norm is nonnegative and belongs to the base field:: - - sage: O = Octonions() - sage: n = O.random_element().norm() - sage: n >= 0 and n in O.base_ring() - True - - The norm is homogeneous:: - - sage: O = Octonions() - sage: x = O.random_element() - sage: alpha = O.base_ring().random_element() - sage: (alpha*x).norm() == alpha.abs()*x.norm() - True - - """ - return self._norm_squared().sqrt() - - # The absolute value notation is typically used for complex numbers... - # and norm() isn't supported in AA, so this lets us use abs() in all - # of the division algebras we need. - abs = norm - - def inverse(self): - r""" - Return the inverse of this element if it exists. - - SETUP:: - - sage: from mjo.hurwitz import Octonions - - EXAMPLES:: - - sage: O = Octonions() - sage: x = sum(O.gens()) - sage: x*x.inverse() == O.one() - True - - :: - - sage: O = Octonions() - sage: O.one().inverse() == O.one() - True - - TESTS:: - - sage: O = Octonions() - sage: x = O.random_element() - sage: x.is_zero() or ( x*x.inverse() == O.one() ) - True - - """ - if self.is_zero(): - raise ValueError("zero is not invertible") - return self.conjugate()/self._norm_squared() - - - -class Octonions(CombinatorialFreeModule): - r""" - SETUP:: - - sage: from mjo.hurwitz import Octonions - - EXAMPLES:: - - sage: Octonions() - Octonion algebra with base ring Algebraic Real Field - sage: Octonions(field=QQ) - Octonion algebra with base ring Rational Field - - """ - def __init__(self, - field=AA, - prefix="e"): - - # Not associative, not commutative - from sage.categories.magmatic_algebras import MagmaticAlgebras - category = MagmaticAlgebras(field).FiniteDimensional() - category = category.WithBasis().Unital() - - super().__init__(field, - range(8), - element_class=Octonion, - category=category, - prefix=prefix, - bracket=False) - - # The product of each basis element is plus/minus another - # basis element that can simply be looked up on - # https://en.wikipedia.org/wiki/Octonion - e0, e1, e2, e3, e4, e5, e6, e7 = self.gens() - self._multiplication_table = ( - (e0, e1, e2, e3, e4, e5, e6, e7), - (e1,-e0, e3,-e2, e5,-e4,-e7, e6), - (e2,-e3,-e0, e1, e6, e7,-e4,-e5), - (e3, e2,-e1,-e0, e7,-e6, e5,-e4), - (e4,-e5,-e6,-e7,-e0, e1, e2, e3), - (e5, e4,-e7, e6,-e1,-e0,-e3, e2), - (e6, e7, e4,-e5,-e2, e3,-e0,-e1), - (e7,-e6, e5, e4,-e3,-e2, e1,-e0), - ) - - def product_on_basis(self, i, j): - return self._multiplication_table[i][j] - - def one_basis(self): - r""" - Return the monomial index (basis element) corresponding to the - octonion unit element. - - SETUP:: - - sage: from mjo.hurwitz import Octonions - - TESTS: - - This gives the correct unit element:: - - sage: O = Octonions() - sage: x = O.random_element() - sage: x*O.one() == x and O.one()*x == x - True - - """ - return 0 - - def _repr_(self): - return ("Octonion algebra with base ring %s" % self.base_ring()) - - def multiplication_table(self): - """ - Return a visual representation of this algebra's multiplication - table (on basis elements). - - SETUP:: - - sage: from mjo.hurwitz import Octonions - - EXAMPLES: - - The multiplication table is what Wikipedia says it is:: - - sage: Octonions().multiplication_table() - ┌────╥────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┐ - │ * ║ e0 │ e1 │ e2 │ e3 │ e4 │ e5 │ e6 │ e7 │ - ╞════╫════╪═════╪═════╪═════╪═════╪═════╪═════╪═════╡ - │ e0 ║ e0 │ e1 │ e2 │ e3 │ e4 │ e5 │ e6 │ e7 │ - ├────╫────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┤ - │ e1 ║ e1 │ -e0 │ e3 │ -e2 │ e5 │ -e4 │ -e7 │ e6 │ - ├────╫────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┤ - │ e2 ║ e2 │ -e3 │ -e0 │ e1 │ e6 │ e7 │ -e4 │ -e5 │ - ├────╫────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┤ - │ e3 ║ e3 │ e2 │ -e1 │ -e0 │ e7 │ -e6 │ e5 │ -e4 │ - ├────╫────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┤ - │ e4 ║ e4 │ -e5 │ -e6 │ -e7 │ -e0 │ e1 │ e2 │ e3 │ - ├────╫────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┤ - │ e5 ║ e5 │ e4 │ -e7 │ e6 │ -e1 │ -e0 │ -e3 │ e2 │ - ├────╫────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┤ - │ e6 ║ e6 │ e7 │ e4 │ -e5 │ -e2 │ e3 │ -e0 │ -e1 │ - ├────╫────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┤ - │ e7 ║ e7 │ -e6 │ e5 │ e4 │ -e3 │ -e2 │ e1 │ -e0 │ - └────╨────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┘ - - """ - n = self.dimension() - # Prepend the header row. - M = [["*"] + list(self.gens())] - - # And to each subsequent row, prepend an entry that belongs to - # the left-side "header column." - M += [ [self.monomial(i)] + [ self.monomial(i)*self.monomial(j) - for j in range(n) ] - for i in range(n) ] - - from sage.misc.table import table - return table(M, header_row=True, header_column=True, frame=True) - - - - - class HurwitzMatrixAlgebraElement(MatrixAlgebraElement): def conjugate(self): r"""