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: set_random_seed()
sage: n = ZZ.random_element(1,10).abs()
sage: J = JordanSpinSimpleEJA(n)
sage: x = J.random_element()
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).abs()
+ 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):
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).abs()
+ 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
"""
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).abs()
+ sage: J = ComplexHermitianSimpleEJA(n)
+ sage: J.degree() == n^2
+ True
+
"""
- F = QuadraticField(-1, 'i')
- i = F.gen()
- S = _real_symmetric_basis(n, field=F)
- T = []
- for s in S:
- T.append(s)
- T.append(i*s)
- embed_T = [ _embed_complex_matrix(t) for t in T ]
- Qs = _multiplication_table_from_matrix_basis(embed_T)
+ S = _complex_hermitian_basis(n)
+ Qs = _multiplication_table_from_matrix_basis(S)
return FiniteDimensionalEuclideanJordanAlgebra(field, Qs, rank=n)
+
def QuaternionHermitianSimpleEJA(n):
"""
The rank-n simple EJA consisting of self-adjoint n-by-n quaternion