+++ /dev/null
-from sage.misc.cachefunc import cached_method
-from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra
-from sage.combinat.free_module import CombinatorialFreeModule
-from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement
-from sage.categories.magmatic_algebras import MagmaticAlgebras
-from sage.rings.all import AA, ZZ
-from sage.matrix.matrix_space import MatrixSpace
-from sage.misc.table import table
-
-from mjo.matrix_algebra import HurwitzMatrixAlgebra
-
-class Octonion(IndexedFreeModuleElement):
- def conjugate(self):
- r"""
- SETUP::
-
- sage: from mjo.octonions 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: set_random_seed()
- sage: O = Octonions()
- sage: x = O.random_element()
- sage: x.conjugate().conjugate() == x
- True
-
- """
- 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.octonions import Octonions
-
- EXAMPLES::
-
- sage: O = Octonions()
- sage: x = sum(O.gens())
- sage: x.real()
- e0
-
- TESTS:
-
- This method is idempotent::
-
- sage: set_random_seed()
- 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.octonions 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: set_random_seed()
- 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.octonions import Octonions
-
- EXAMPLES::
-
- sage: O = Octonions()
- sage: O.one().norm()
- 1
-
- TESTS:
-
- The norm is nonnegative and belongs to the base field::
-
- sage: set_random_seed()
- sage: O = Octonions()
- sage: n = O.random_element().norm()
- sage: n >= 0 and n in O.base_ring()
- True
-
- The norm is homogeneous::
-
- sage: set_random_seed()
- 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.octonions 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: set_random_seed()
- 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()
-
-
- def cayley_dickson(self, Q=None):
- r"""
- Return the Cayley-Dickson representation of this element in terms
- of the quaternion algebra ``Q``.
-
- The Cayley-Dickson representation is an identification of
- octionions `x` and `y` with pairs of quaternions `(a,b)` and
- `(c,d)` respectively such that:
-
- * `x + y = (a+b, c+d)`
- * `xy` = (ac - \bar{d}*b, da + b\bar{c})`
- * `\bar{x} = (a,-b)`
-
- where `\bar{x}` denotes the conjugate of `x`.
-
- SETUP::
-
- sage: from mjo.octonions import Octonions
-
- EXAMPLES::
-
- sage: O = Octonions()
- sage: x = sum(O.gens())
- sage: x.cayley_dickson()
- (1 + i + j + k, 1 + i + j + k)
-
- """
- if Q is None:
- Q = QuaternionAlgebra(self.base_ring(), -1, -1)
-
- i,j,k = Q.gens()
- a = (self.coefficient(0)*Q.one() +
- self.coefficient(1)*i +
- self.coefficient(2)*j +
- self.coefficient(3)*k )
- b = (self.coefficient(4)*Q.one() +
- self.coefficient(5)*i +
- self.coefficient(6)*j +
- self.coefficient(7)*k )
-
- from sage.categories.sets_cat import cartesian_product
- P = cartesian_product([Q,Q])
- return P((a,b))
-
-
-class Octonions(CombinatorialFreeModule):
- r"""
- SETUP::
-
- sage: from mjo.octonions 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
- 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.octonions import Octonions
-
- TESTS:
-
- This gives the correct unit element::
-
- sage: set_random_seed()
- 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.octonions 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) ]
-
- return table(M, header_row=True, header_column=True, frame=True)
-
-
-
-class OctonionMatrixAlgebra(HurwitzMatrixAlgebra):
- r"""
- The algebra of ``n``-by-``n`` matrices with octonion entries over
- (a subfield of) the real numbers.
-
- The usual matrix spaces in SageMath don't support octonion entries
- because they assume that the entries of the matrix come from a
- commutative and associative ring, and the octonions are neither.
-
- SETUP::
-
- sage: from mjo.octonions import OctonionMatrixAlgebra
-
- EXAMPLES::
-
- sage: OctonionMatrixAlgebra(3)
- Module of 3 by 3 matrices with entries in Octonion algebra with base
- ring Algebraic Real Field over the scalar ring Algebraic Real Field
- sage: OctonionMatrixAlgebra(3,QQ)
- Module of 3 by 3 matrices with entries in Octonion algebra with base
- ring Rational Field over the scalar ring Rational Field
-
- ::
-
- sage: A = OctonionMatrixAlgebra(2)
- sage: e0,e1,e2,e3,e4,e5,e6,e7 = A.entry_algebra().gens()
- sage: A([ [e0+e4, e1+e5],
- ....: [e2-e6, e3-e7] ])
- +---------+---------+
- | e0 + e4 | e1 + e5 |
- +---------+---------+
- | e2 - e6 | e3 - e7 |
- +---------+---------+
-
- ::
-
- sage: A1 = OctonionMatrixAlgebra(1,QQ)
- sage: A2 = OctonionMatrixAlgebra(1,QQ)
- sage: cartesian_product([A1,A2])
- Module of 1 by 1 matrices with entries in Octonion algebra with
- base ring Rational Field over the scalar ring Rational Field (+)
- Module of 1 by 1 matrices with entries in Octonion algebra with
- base ring Rational Field over the scalar ring Rational Field
-
- TESTS::
-
- sage: set_random_seed()
- sage: A = OctonionMatrixAlgebra(ZZ.random_element(10))
- sage: x = A.random_element()
- sage: x*A.one() == x and A.one()*x == x
- True
-
- """
- def __init__(self, n, scalars=AA, prefix="E", **kwargs):
- super().__init__(Octonions(field=scalars),
- scalars,
- n,
- prefix=prefix,
- **kwargs)
-
-class QuaternionMatrixAlgebra(HurwitzMatrixAlgebra):
- r"""
- The algebra of ``n``-by-``n`` matrices with quaternion entries over
- (a subfield of) the real numbers.
-
- The usual matrix spaces in SageMath don't support quaternion entries
- because they assume that the entries of the matrix come from a
- commutative ring, and the quaternions are not commutative.
-
- SETUP::
-
- sage: from mjo.octonions import QuaternionMatrixAlgebra
-
- EXAMPLES::
-
- sage: QuaternionMatrixAlgebra(3)
- Module of 3 by 3 matrices with entries in Quaternion
- Algebra (-1, -1) with base ring Algebraic Real Field
- over the scalar ring Algebraic Real Field
- sage: QuaternionMatrixAlgebra(3,QQ)
- Module of 3 by 3 matrices with entries in Quaternion
- Algebra (-1, -1) with base ring Rational Field over
- the scalar ring Rational Field
-
- ::
-
- sage: A = QuaternionMatrixAlgebra(2)
- sage: i,j,k = A.entry_algebra().gens()
- sage: A([ [1+i, j-2],
- ....: [k, k+j] ])
- +-------+--------+
- | 1 + i | -2 + j |
- +-------+--------+
- | k | j + k |
- +-------+--------+
-
- ::
-
- sage: A1 = QuaternionMatrixAlgebra(1,QQ)
- sage: A2 = QuaternionMatrixAlgebra(2,QQ)
- sage: cartesian_product([A1,A2])
- Module of 1 by 1 matrices with entries in Quaternion Algebra
- (-1, -1) with base ring Rational Field over the scalar ring
- Rational Field (+) Module of 2 by 2 matrices with entries in
- Quaternion Algebra (-1, -1) with base ring Rational Field over
- the scalar ring Rational Field
-
- TESTS::
-
- sage: set_random_seed()
- sage: A = QuaternionMatrixAlgebra(ZZ.random_element(10))
- sage: x = A.random_element()
- sage: x*A.one() == x and A.one()*x == x
- True
-
- """
- def __init__(self, n, scalars=AA, **kwargs):
- # The -1,-1 gives us the "usual" definition of quaternion
- Q = QuaternionAlgebra(scalars,-1,-1)
- super().__init__(Q, scalars, n, **kwargs)
projects / sage.d.git / blobdiff