-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"""
projects / sage.d.git / commitdiff