assume_associative=False,
category=None,
rank=None):
+ """
+ EXAMPLES:
+
+ By definition, Jordan multiplication commutes::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: x*y == y*x
+ True
+
+ """
self._rank = rank
fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
fda.__init__(field,
instead of column vectors! We, on the other hand, assume column
vectors everywhere.
- EXAMPLES:
+ EXAMPLES::
+
+ sage: set_random_seed()
+ sage: x = random_eja().random_element()
+ sage: x.matrix()*x.vector() == (x^2).vector()
+ True
+
+ A few examples of power-associativity::
+
+ sage: set_random_seed()
+ sage: x = random_eja().random_element()
+ sage: x*(x*x)*(x*x) == x^5
+ True
+ sage: (x*x)*(x*x*x) == x^5
+ True
+
+ We also know that powers operator-commute (Koecher, Chapter
+ III, Corollary 1)::
sage: set_random_seed()
sage: x = random_eja().random_element()
- sage: x.matrix()*x.vector() == (x**2).vector()
+ sage: m = ZZ.random_element(0,10)
+ sage: n = ZZ.random_element(0,10)
+ sage: Lxm = (x^m).matrix()
+ sage: Lxn = (x^n).matrix()
+ sage: Lxm*Lxn == Lxn*Lxm
True
"""
EXAMPLES::
- sage: J = eja_ln(2)
+ sage: J = JordanSpinSimpleEJA(2)
sage: e0,e1 = J.gens()
sage: x = e0 + e1
sage: x.det()
0
- sage: J = eja_ln(3)
+ sage: J = JordanSpinSimpleEJA(3)
sage: e0,e1,e2 = J.gens()
sage: x = e0 + e1 + e2
sage: x.det()
raise ValueError('charpoly had no coefficients')
+ def inverse(self):
+ """
+ Return the Jordan-multiplicative inverse of this element.
+
+ We can't use the superclass method because it relies on the
+ algebra being associative.
+
+ EXAMPLES:
+
+ The inverse in the spin factor algebra is given in Alizadeh's
+ Example 11.11::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,10)
+ sage: J = JordanSpinSimpleEJA(n)
+ sage: x = J.random_element()
+ sage: while x.is_zero():
+ ....: x = J.random_element()
+ sage: x_vec = x.vector()
+ sage: x0 = x_vec[0]
+ sage: x_bar = x_vec[1:]
+ sage: coeff = 1/(x0^2 - x_bar.inner_product(x_bar))
+ sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
+ sage: x_inverse = coeff*inv_vec
+ sage: x.inverse() == J(x_inverse)
+ True
+
+ TESTS:
+
+ The identity element is its own inverse::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: J.one().inverse() == J.one()
+ True
+
+ If an element has an inverse, it acts like one. TODO: this
+ can be a lot less ugly once ``is_invertible`` doesn't crash
+ on irregular elements::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: try:
+ ....: x.inverse()*x == J.one()
+ ....: except:
+ ....: True
+ True
+
+ """
+ if self.parent().is_associative():
+ elt = FiniteDimensionalAlgebraElement(self.parent(), self)
+ return elt.inverse()
+
+ # TODO: we can do better once the call to is_invertible()
+ # doesn't crash on irregular elements.
+ #if not self.is_invertible():
+ # raise ArgumentError('element is not invertible')
+
+ # We do this a little different than the usual recursive
+ # call to a finite-dimensional algebra element, because we
+ # wind up with an inverse that lives in the subalgebra and
+ # we need information about the parent to convert it back.
+ V = self.span_of_powers()
+ assoc_subalg = self.subalgebra_generated_by()
+ # Mis-design warning: the basis used for span_of_powers()
+ # and subalgebra_generated_by() must be the same, and in
+ # the same order!
+ elt = assoc_subalg(V.coordinates(self.vector()))
+
+ # This will be in the subalgebra's coordinates...
+ fda_elt = FiniteDimensionalAlgebraElement(assoc_subalg, elt)
+ subalg_inverse = fda_elt.inverse()
+
+ # So we have to convert back...
+ basis = [ self.parent(v) for v in V.basis() ]
+ pairs = zip(subalg_inverse.vector(), basis)
+ return self.parent().linear_combination(pairs)
+
+
+ def is_invertible(self):
+ """
+ Return whether or not this element is invertible.
+
+ We can't use the superclass method because it relies on
+ the algebra being associative.
+ """
+ return not self.det().is_zero()
+
+
def is_nilpotent(self):
"""
Return whether or not some power of this element is zero.
The identity element always has degree one, but any element
linearly-independent from it is regular::
- sage: J = eja_ln(5)
+ sage: J = JordanSpinSimpleEJA(5)
sage: J.one().is_regular()
False
sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
EXAMPLES::
- sage: J = eja_ln(4)
+ sage: J = JordanSpinSimpleEJA(4)
sage: J.one().degree()
1
sage: e0,e1,e2,e3 = J.gens()
aren't multiples of the identity are regular::
sage: set_random_seed()
- sage: n = ZZ.random_element(1,10).abs()
- sage: J = eja_ln(n)
+ sage: n = ZZ.random_element(1,10)
+ sage: J = JordanSpinSimpleEJA(n)
sage: x = J.random_element()
sage: x == x.coefficient(0)*J.one() or x.degree() == 2
True
We have to override this because the superclass method
returns a matrix that acts on row vectors (that is, on
the right).
+
+ EXAMPLES:
+
+ Test the first polarization identity from my notes, Koecher Chapter
+ III, or from Baes (2.3)::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: Lx = x.matrix()
+ sage: Ly = y.matrix()
+ sage: Lxx = (x*x).matrix()
+ sage: Lxy = (x*y).matrix()
+ sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
+ True
+
+ Test the second polarization identity from my notes or from
+ Baes (2.4)::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: z = J.random_element()
+ sage: Lx = x.matrix()
+ sage: Ly = y.matrix()
+ sage: Lz = z.matrix()
+ sage: Lzy = (z*y).matrix()
+ sage: Lxy = (x*y).matrix()
+ sage: Lxz = (x*z).matrix()
+ sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
+ True
+
+ Test the third polarization identity from my notes or from
+ Baes (2.5)::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: u = J.random_element()
+ sage: y = J.random_element()
+ sage: z = J.random_element()
+ sage: Lu = u.matrix()
+ sage: Ly = y.matrix()
+ sage: Lz = z.matrix()
+ sage: Lzy = (z*y).matrix()
+ sage: Luy = (u*y).matrix()
+ sage: Luz = (u*z).matrix()
+ sage: Luyz = (u*(y*z)).matrix()
+ sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
+ sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
+ sage: bool(lhs == rhs)
+ True
+
"""
fda_elt = FiniteDimensionalAlgebraElement(self.parent(), self)
return fda_elt.matrix().transpose()
identity::
sage: set_random_seed()
- sage: n = ZZ.random_element(2,10).abs()
- sage: J = eja_ln(n)
+ sage: n = ZZ.random_element(2,10)
+ sage: J = JordanSpinSimpleEJA(n)
sage: y = J.random_element()
sage: while y == y.coefficient(0)*J.one():
....: y = J.random_element()
return elt.minimal_polynomial()
- def quadratic_representation(self):
+ def quadratic_representation(self, other=None):
"""
Return the quadratic representation of this element.
The explicit form in the spin factor algebra is given by
Alizadeh's Example 11.12::
- sage: n = ZZ.random_element(1,10).abs()
- sage: J = eja_ln(n)
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,10)
+ sage: J = JordanSpinSimpleEJA(n)
sage: x = J.random_element()
sage: x_vec = x.vector()
sage: x0 = x_vec[0]
sage: Q == x.quadratic_representation()
True
+ Test all of the properties from Theorem 11.2 in Alizadeh::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+
+ Property 1:
+
+ sage: actual = x.quadratic_representation(y)
+ sage: expected = ( (x+y).quadratic_representation()
+ ....: -x.quadratic_representation()
+ ....: -y.quadratic_representation() ) / 2
+ sage: actual == expected
+ True
+
+ Property 2:
+
+ sage: alpha = QQ.random_element()
+ sage: actual = (alpha*x).quadratic_representation()
+ sage: expected = (alpha^2)*x.quadratic_representation()
+ sage: actual == expected
+ True
+
+ Property 5:
+
+ sage: Qy = y.quadratic_representation()
+ sage: actual = J(Qy*x.vector()).quadratic_representation()
+ sage: expected = Qy*x.quadratic_representation()*Qy
+ sage: actual == expected
+ True
+
+ Property 6:
+
+ sage: k = ZZ.random_element(1,10)
+ sage: actual = (x^k).quadratic_representation()
+ sage: expected = (x.quadratic_representation())^k
+ sage: actual == expected
+ True
+
"""
- return 2*(self.matrix()**2) - (self**2).matrix()
+ if other is None:
+ other=self
+ elif not other in self.parent():
+ raise ArgumentError("'other' must live in the same algebra")
+
+ return ( self.matrix()*other.matrix()
+ + other.matrix()*self.matrix()
+ - (self*other).matrix() )
def span_of_powers(self):
sage: c = J.random_element().subalgebra_idempotent()
sage: c^2 == c
True
- sage: J = eja_ln(5)
+ sage: J = JordanSpinSimpleEJA(5)
sage: c = J.random_element().subalgebra_idempotent()
sage: c^2 == c
True
EXAMPLES::
- sage: J = eja_ln(3)
+ sage: J = JordanSpinSimpleEJA(3)
sage: e0,e1,e2 = J.gens()
sage: x = e0 + e1 + e2
sage: x.trace()
return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=dimension)
-def eja_ln(dimension, field=QQ):
- """
- Return the Jordan algebra corresponding to the Lorentz "ice cream"
- cone of the given ``dimension``.
-
- EXAMPLES:
-
- This multiplication table can be verified by hand::
-
- sage: J = eja_ln(4)
- sage: e0,e1,e2,e3 = J.gens()
- sage: e0*e0
- e0
- sage: e0*e1
- e1
- sage: e0*e2
- e2
- sage: e0*e3
- e3
- sage: e1*e2
- 0
- sage: e1*e3
- 0
- sage: e2*e3
- 0
-
- In one dimension, this is the reals under multiplication::
-
- sage: J1 = eja_ln(1)
- sage: J2 = eja_rn(1)
- sage: J1 == J2
- True
-
- """
- Qs = []
- id_matrix = identity_matrix(field,dimension)
- for i in xrange(dimension):
- ei = id_matrix.column(i)
- Qi = zero_matrix(field,dimension)
- Qi.set_row(0, ei)
- Qi.set_column(0, ei)
- Qi += diagonal_matrix(dimension, [ei[0]]*dimension)
- # 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).
- rank = min(dimension,2)
- return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=rank)
-
-
-def eja_sn(dimension, field=QQ):
- """
- Return the simple Jordan algebra of ``dimension``-by-``dimension``
- symmetric matrices over ``field``.
-
- EXAMPLES::
-
- sage: J = eja_sn(2)
- sage: e0, e1, e2 = J.gens()
- sage: e0*e0
- e0
- sage: e1*e1
- e0 + e2
- sage: e2*e2
- e2
-
- """
- S = _real_symmetric_basis(dimension, field=field)
- Qs = _multiplication_table_from_matrix_basis(S)
-
- return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=dimension)
-
def random_eja():
"""
Euclidean Jordan algebra of degree...
"""
- n = ZZ.random_element(1,10).abs()
- constructor = choice([eja_rn, eja_ln, eja_sn])
- return constructor(dimension=n, field=QQ)
+ n = ZZ.random_element(1,5)
+ constructor = choice([eja_rn,
+ JordanSpinSimpleEJA,
+ RealSymmetricSimpleEJA,
+ ComplexHermitianSimpleEJA])
+ return constructor(n, field=QQ)
return S
+def _complex_hermitian_basis(n, field=QQ):
+ """
+ Returns a basis for the space of complex Hermitian n-by-n matrices.
+
+ TESTS::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
+ True
+
+ """
+ F = QuadraticField(-1, 'I')
+ I = F.gen()
+
+ # This is like the symmetric case, but we need to be careful:
+ #
+ # * We want conjugate-symmetry, not just symmetry.
+ # * The diagonal will (as a result) be real.
+ #
+ S = []
+ for i in xrange(n):
+ for j in xrange(i+1):
+ Eij = matrix(field, n, lambda k,l: k==i and l==j)
+ if i == j:
+ Sij = _embed_complex_matrix(Eij)
+ S.append(Sij)
+ else:
+ # Beware, orthogonal but not normalized! The second one
+ # has a minus because it's conjugated.
+ Sij_real = _embed_complex_matrix(Eij + Eij.transpose())
+ S.append(Sij_real)
+ Sij_imag = _embed_complex_matrix(I*Eij - I*Eij.transpose())
+ S.append(Sij_imag)
+ return S
+
+
def _multiplication_table_from_matrix_basis(basis):
"""
At least three of the five simple Euclidean Jordan algebras have the
S = [ vec2mat(b) for b in W.basis() ]
Qs = []
- for s in basis:
+ for s in S:
# Brute force the multiplication-by-s matrix by looping
# through all elements of the basis and doing the computation
# to find out what the corresponding row should be. BEWARE:
# constructor uses ROW vectors and not COLUMN vectors. That's
# why we're computing rows here and not columns.
Q_rows = []
- for t in basis:
+ for t in S:
this_row = mat2vec((s*t + t*s)/2)
Q_rows.append(W.coordinates(this_row))
- Q = matrix(field,Q_rows)
+ Q = matrix(field, W.dimension(), Q_rows)
Qs.append(Q)
return Qs
a = z.real()
b = z.imag()
blocks.append(matrix(field, 2, [[a,-b],[b,a]]))
- return block_matrix(field, n, blocks)
+ # We can drop the imaginaries here.
+ return block_matrix(field.base_ring(), n, blocks)
-def RealSymmetricSimpleEJA(n):
+
+def _unembed_complex_matrix(M):
+ """
+ The inverse of _embed_complex_matrix().
+
+ EXAMPLES::
+
+ sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
+ ....: [-2, 1, -4, 3],
+ ....: [ 9, 10, 11, 12],
+ ....: [-10, 9, -12, 11] ])
+ sage: _unembed_complex_matrix(A)
+ [ -2*i + 1 -4*i + 3]
+ [ -10*i + 9 -12*i + 11]
+ """
+ n = ZZ(M.nrows())
+ if M.ncols() != n:
+ raise ArgumentError("the matrix 'M' must be square")
+ if not n.mod(2).is_zero():
+ raise ArgumentError("the matrix 'M' must be a complex embedding")
+
+ F = QuadraticField(-1, 'i')
+ i = F.gen()
+
+ # Go top-left to bottom-right (reading order), converting every
+ # 2-by-2 block we see to a single complex element.
+ elements = []
+ for k in xrange(n/2):
+ for j in xrange(n/2):
+ submat = M[2*k:2*k+2,2*j:2*j+2]
+ if submat[0,0] != submat[1,1]:
+ raise ArgumentError('bad real submatrix')
+ if submat[0,1] != -submat[1,0]:
+ raise ArgumentError('bad imag submatrix')
+ z = submat[0,0] + submat[1,0]*i
+ elements.append(z)
+
+ return matrix(F, n/2, elements)
+
+
+def RealSymmetricSimpleEJA(n, field=QQ):
"""
The rank-n simple EJA consisting of real symmetric n-by-n
matrices, the usual symmetric Jordan product, and the trace inner
product. It has dimension `(n^2 + n)/2` over the reals.
+
+ EXAMPLES::
+
+ sage: J = RealSymmetricSimpleEJA(2)
+ sage: e0, e1, e2 = J.gens()
+ sage: e0*e0
+ e0
+ sage: e1*e1
+ e0 + e2
+ sage: e2*e2
+ e2
+
+ TESTS:
+
+ The degree of this algebra is `(n^2 + n) / 2`::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: J = RealSymmetricSimpleEJA(n)
+ sage: J.degree() == (n^2 + n)/2
+ True
+
"""
- pass
+ S = _real_symmetric_basis(n, field=field)
+ Qs = _multiplication_table_from_matrix_basis(S)
+
+ return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=n)
-def ComplexHermitianSimpleEJA(n):
+
+def ComplexHermitianSimpleEJA(n, field=QQ):
"""
The rank-n simple EJA consisting of complex Hermitian n-by-n
matrices over the real numbers, the usual symmetric Jordan product,
- and the real-part-of-trace inner product. It has dimension `n^2 over
+ and the real-part-of-trace inner product. It has dimension `n^2` over
the reals.
+
+ TESTS:
+
+ The degree of this algebra is `n^2`::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: J = ComplexHermitianSimpleEJA(n)
+ sage: J.degree() == n^2
+ True
+
"""
- pass
+ S = _complex_hermitian_basis(n)
+ Qs = _multiplication_table_from_matrix_basis(S)
+ return FiniteDimensionalEuclideanJordanAlgebra(field, Qs, rank=n)
+
def QuaternionHermitianSimpleEJA(n):
"""
n = 3
pass
-def JordanSpinSimpleEJA(n):
+def JordanSpinSimpleEJA(n, field=QQ):
"""
The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
with the usual inner product and jordan product ``x*y =
(<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
the reals.
+
+ EXAMPLES:
+
+ This multiplication table can be verified by hand::
+
+ sage: J = JordanSpinSimpleEJA(4)
+ sage: e0,e1,e2,e3 = J.gens()
+ sage: e0*e0
+ e0
+ sage: e0*e1
+ e1
+ sage: e0*e2
+ e2
+ sage: e0*e3
+ e3
+ sage: e1*e2
+ 0
+ sage: e1*e3
+ 0
+ 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
+
"""
- pass
+ 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))