assume_associative=False,
category=None,
rank=None,
- natural_basis=None):
+ natural_basis=None,
+ inner_product=None):
n = len(mult_table)
mult_table = [b.base_extend(field) for b in mult_table]
for b in mult_table:
names=names,
category=cat,
rank=rank,
- natural_basis=natural_basis)
+ natural_basis=natural_basis,
+ inner_product=inner_product)
- def __init__(self, field,
+ def __init__(self,
+ field,
mult_table,
names='e',
assume_associative=False,
category=None,
rank=None,
- natural_basis=None):
+ natural_basis=None,
+ inner_product=None):
"""
EXAMPLES:
"""
self._rank = rank
self._natural_basis = natural_basis
+ self._inner_product = inner_product
fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
fda.__init__(field,
mult_table,
return fmt.format(self.degree(), self.base_ring())
+ def inner_product(self, x, y):
+ """
+ The inner product associated with this Euclidean Jordan algebra.
+
+ Will default to the trace inner product if nothing else.
+
+ EXAMPLES:
+
+ The inner product must satisfy its axiom for this algebra to truly
+ be a Euclidean Jordan Algebra::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: z = J.random_element()
+ sage: (x*y).inner_product(z) == y.inner_product(x*z)
+ True
+
+ """
+ if (not x in self) or (not y in self):
+ raise TypeError("arguments must live in this algebra")
+ if self._inner_product is None:
+ return x.trace_inner_product(y)
+ else:
+ return self._inner_product(x,y)
+
+
def natural_basis(self):
"""
Return a more-natural representation of this algebra's basis.
EXAMPLES::
- sage: J = RealSymmetricSimpleEJA(2)
+ sage: J = RealSymmetricEJA(2)
sage: J.basis()
Family (e0, e1, e2)
sage: J.natural_basis()
::
- sage: J = JordanSpinSimpleEJA(2)
+ sage: J = JordanSpinEJA(2)
sage: J.basis()
Family (e0, e1)
sage: J.natural_basis()
An element of a Euclidean Jordan algebra.
"""
+ def __init__(self, A, elt=None):
+ """
+ EXAMPLES:
+
+ The identity in `S^n` is converted to the identity in the EJA::
+
+ sage: J = RealSymmetricEJA(3)
+ sage: I = identity_matrix(QQ,3)
+ sage: J(I) == J.one()
+ True
+
+ This skew-symmetric matrix can't be represented in the EJA::
+
+ sage: J = RealSymmetricEJA(3)
+ sage: A = matrix(QQ,3, lambda i,j: i-j)
+ sage: J(A)
+ Traceback (most recent call last):
+ ...
+ ArithmeticError: vector is not in free module
+
+ """
+ # Goal: if we're given a matrix, and if it lives in our
+ # parent algebra's "natural ambient space," convert it
+ # into an algebra element.
+ #
+ # The catch is, we make a recursive call after converting
+ # the given matrix into a vector that lives in the algebra.
+ # This we need to try the parent class initializer first,
+ # to avoid recursing forever if we're given something that
+ # already fits into the algebra, but also happens to live
+ # in the parent's "natural ambient space" (this happens with
+ # vectors in R^n).
+ try:
+ FiniteDimensionalAlgebraElement.__init__(self, A, elt)
+ except ValueError:
+ natural_basis = A.natural_basis()
+ if elt in natural_basis[0].matrix_space():
+ # Thanks for nothing! Matrix spaces aren't vector
+ # spaces in Sage, so we have to figure out its
+ # natural-basis coordinates ourselves.
+ V = VectorSpace(elt.base_ring(), elt.nrows()**2)
+ W = V.span( _mat2vec(s) for s in natural_basis )
+ coords = W.coordinates(_mat2vec(elt))
+ FiniteDimensionalAlgebraElement.__init__(self, A, coords)
+
def __pow__(self, n):
"""
Return ``self`` raised to the power ``n``.
raise NotImplementedError('irregular element')
+ def inner_product(self, other):
+ """
+ Return the parent algebra's inner product of myself and ``other``.
+
+ EXAMPLES:
+
+ The inner product in the Jordan spin algebra is the usual
+ inner product on `R^n` (this example only works because the
+ basis for the Jordan algebra is the standard basis in `R^n`)::
+
+ sage: J = JordanSpinEJA(3)
+ sage: x = vector(QQ,[1,2,3])
+ sage: y = vector(QQ,[4,5,6])
+ sage: x.inner_product(y)
+ 32
+ sage: J(x).inner_product(J(y))
+ 32
+
+ The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
+ multiplication is the usual matrix multiplication in `S^n`,
+ so the inner product of the identity matrix with itself
+ should be the `n`::
+
+ sage: J = RealSymmetricEJA(3)
+ sage: J.one().inner_product(J.one())
+ 3
+
+ Likewise, the inner product on `C^n` is `<X,Y> =
+ Re(trace(X*Y))`, where we must necessarily take the real
+ part because the product of Hermitian matrices may not be
+ Hermitian::
+
+ sage: J = ComplexHermitianEJA(3)
+ sage: J.one().inner_product(J.one())
+ 3
+
+ Ditto for the quaternions::
+
+ sage: J = QuaternionHermitianEJA(3)
+ sage: J.one().inner_product(J.one())
+ 3
+
+ TESTS:
+
+ Ensure that we can always compute an inner product, and that
+ it gives us back a real number::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: x.inner_product(y) in RR
+ True
+
+ """
+ P = self.parent()
+ if not other in P:
+ raise TypeError("'other' must live in the same algebra")
+
+ return P.inner_product(self, other)
+
+
def operator_commutes_with(self, other):
"""
Return whether or not this element operator-commutes
"""
if not other in self.parent():
- raise ArgumentError("'other' must live in the same algebra")
+ raise TypeError("'other' must live in the same algebra")
A = self.operator_matrix()
B = other.operator_matrix()
EXAMPLES::
- sage: J = JordanSpinSimpleEJA(2)
+ sage: J = JordanSpinEJA(2)
sage: e0,e1 = J.gens()
sage: x = e0 + e1
sage: x.det()
0
- sage: J = JordanSpinSimpleEJA(3)
+ sage: J = JordanSpinEJA(3)
sage: e0,e1,e2 = J.gens()
sage: x = e0 + e1 + e2
sage: x.det()
sage: set_random_seed()
sage: n = ZZ.random_element(1,10)
- sage: J = JordanSpinSimpleEJA(n)
+ sage: J = JordanSpinEJA(n)
sage: x = J.random_element()
sage: while x.is_zero():
....: x = J.random_element()
# 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')
+ # raise ValueError('element is not invertible')
# We do this a little different than the usual recursive
# call to a finite-dimensional algebra element, because we
The identity element always has degree one, but any element
linearly-independent from it is regular::
- sage: J = JordanSpinSimpleEJA(5)
+ sage: J = JordanSpinEJA(5)
sage: J.one().is_regular()
False
sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
EXAMPLES::
- sage: J = JordanSpinSimpleEJA(4)
+ sage: J = JordanSpinEJA(4)
sage: J.one().degree()
1
sage: e0,e1,e2,e3 = J.gens()
sage: set_random_seed()
sage: n = ZZ.random_element(1,10)
- sage: J = JordanSpinSimpleEJA(n)
+ sage: J = JordanSpinEJA(n)
sage: x = J.random_element()
sage: x == x.coefficient(0)*J.one() or x.degree() == 2
True
return self.span_of_powers().dimension()
+ def minimal_polynomial(self):
+ """
+ EXAMPLES::
+
+ sage: set_random_seed()
+ sage: x = random_eja().random_element()
+ sage: x.degree() == x.minimal_polynomial().degree()
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: x = random_eja().random_element()
+ sage: x.degree() == x.minimal_polynomial().degree()
+ True
+
+ The minimal polynomial and the characteristic polynomial coincide
+ and are known (see Alizadeh, Example 11.11) for all elements of
+ the spin factor algebra that aren't scalar multiples of the
+ identity::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(2,10)
+ sage: J = JordanSpinEJA(n)
+ sage: y = J.random_element()
+ sage: while y == y.coefficient(0)*J.one():
+ ....: y = J.random_element()
+ sage: y0 = y.vector()[0]
+ sage: y_bar = y.vector()[1:]
+ sage: actual = y.minimal_polynomial()
+ sage: x = SR.symbol('x', domain='real')
+ sage: expected = x^2 - 2*y0*x + (y0^2 - norm(y_bar)^2)
+ sage: bool(actual == expected)
+ True
+
+ """
+ # The element we're going to call "minimal_polynomial()" on.
+ # Either myself, interpreted as an element of a finite-
+ # dimensional algebra, or an element of an associative
+ # subalgebra.
+ elt = None
+
+ if self.parent().is_associative():
+ elt = FiniteDimensionalAlgebraElement(self.parent(), self)
+ else:
+ 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()))
+
+ # Recursive call, but should work since elt lives in an
+ # associative algebra.
+ return elt.minimal_polynomial()
+
+
+ def natural_representation(self):
+ """
+ Return a more-natural representation of this element.
+
+ Every finite-dimensional Euclidean Jordan Algebra is a
+ direct sum of five simple algebras, four of which comprise
+ Hermitian matrices. This method returns the original
+ "natural" representation of this element as a Hermitian
+ matrix, if it has one. If not, you get the usual representation.
+
+ EXAMPLES::
+
+ sage: J = ComplexHermitianEJA(3)
+ sage: J.one()
+ e0 + e5 + e8
+ sage: J.one().natural_representation()
+ [1 0 0 0 0 0]
+ [0 1 0 0 0 0]
+ [0 0 1 0 0 0]
+ [0 0 0 1 0 0]
+ [0 0 0 0 1 0]
+ [0 0 0 0 0 1]
+
+ ::
+
+ sage: J = QuaternionHermitianEJA(3)
+ sage: J.one()
+ e0 + e9 + e14
+ sage: J.one().natural_representation()
+ [1 0 0 0 0 0 0 0 0 0 0 0]
+ [0 1 0 0 0 0 0 0 0 0 0 0]
+ [0 0 1 0 0 0 0 0 0 0 0 0]
+ [0 0 0 1 0 0 0 0 0 0 0 0]
+ [0 0 0 0 1 0 0 0 0 0 0 0]
+ [0 0 0 0 0 1 0 0 0 0 0 0]
+ [0 0 0 0 0 0 1 0 0 0 0 0]
+ [0 0 0 0 0 0 0 1 0 0 0 0]
+ [0 0 0 0 0 0 0 0 1 0 0 0]
+ [0 0 0 0 0 0 0 0 0 1 0 0]
+ [0 0 0 0 0 0 0 0 0 0 1 0]
+ [0 0 0 0 0 0 0 0 0 0 0 1]
+
+ """
+ B = self.parent().natural_basis()
+ W = B[0].matrix_space()
+ return W.linear_combination(zip(self.vector(), B))
+
+
def operator_matrix(self):
"""
Return the matrix that represents left- (or right-)
return fda_elt.matrix().transpose()
-
- def minimal_polynomial(self):
- """
- EXAMPLES::
-
- sage: set_random_seed()
- sage: x = random_eja().random_element()
- sage: x.degree() == x.minimal_polynomial().degree()
- True
-
- ::
-
- sage: set_random_seed()
- sage: x = random_eja().random_element()
- sage: x.degree() == x.minimal_polynomial().degree()
- True
-
- The minimal polynomial and the characteristic polynomial coincide
- and are known (see Alizadeh, Example 11.11) for all elements of
- the spin factor algebra that aren't scalar multiples of the
- identity::
-
- sage: set_random_seed()
- 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()
- sage: y0 = y.vector()[0]
- sage: y_bar = y.vector()[1:]
- sage: actual = y.minimal_polynomial()
- sage: x = SR.symbol('x', domain='real')
- sage: expected = x^2 - 2*y0*x + (y0^2 - norm(y_bar)^2)
- sage: bool(actual == expected)
- True
-
- """
- # The element we're going to call "minimal_polynomial()" on.
- # Either myself, interpreted as an element of a finite-
- # dimensional algebra, or an element of an associative
- # subalgebra.
- elt = None
-
- if self.parent().is_associative():
- elt = FiniteDimensionalAlgebraElement(self.parent(), self)
- else:
- 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()))
-
- # Recursive call, but should work since elt lives in an
- # associative algebra.
- return elt.minimal_polynomial()
-
-
def quadratic_representation(self, other=None):
"""
Return the quadratic representation of this element.
sage: set_random_seed()
sage: n = ZZ.random_element(1,10)
- sage: J = JordanSpinSimpleEJA(n)
+ sage: J = JordanSpinEJA(n)
sage: x = J.random_element()
sage: x_vec = x.vector()
sage: x0 = x_vec[0]
if other is None:
other=self
elif not other in self.parent():
- raise ArgumentError("'other' must live in the same algebra")
+ raise TypeError("'other' must live in the same algebra")
L = self.operator_matrix()
M = other.operator_matrix()
sage: c = J.random_element().subalgebra_idempotent()
sage: c^2 == c
True
- sage: J = JordanSpinSimpleEJA(5)
+ sage: J = JordanSpinEJA(5)
sage: c = J.random_element().subalgebra_idempotent()
sage: c^2 == c
True
EXAMPLES::
- sage: J = JordanSpinSimpleEJA(3)
+ sage: J = JordanSpinEJA(3)
sage: e0,e1,e2 = J.gens()
sage: x = e0 + e1 + e2
sage: x.trace()
Return the trace inner product of myself and ``other``.
"""
if not other in self.parent():
- raise ArgumentError("'other' must live in the same algebra")
+ raise TypeError("'other' must live in the same algebra")
return (self*other).trace()
Qs = [ matrix(field, dimension, dimension, lambda k,j: 1*(k == j == i))
for i in xrange(dimension) ]
- return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=dimension)
+ return FiniteDimensionalEuclideanJordanAlgebra(field,
+ Qs,
+ rank=dimension,
+ inner_product=_usual_ip)
* The ``n``-by-``n`` rational symmetric matrices with the symmetric
product.
+ * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
+ in the space of ``2n``-by-``2n`` real symmetric matrices.
+
+ * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
+ in the space of ``4n``-by-``4n`` real symmetric matrices.
+
Later this might be extended to return Cartesian products of the
EJAs above.
"""
n = ZZ.random_element(1,5)
constructor = choice([eja_rn,
- JordanSpinSimpleEJA,
- RealSymmetricSimpleEJA,
- ComplexHermitianSimpleEJA])
+ JordanSpinEJA,
+ RealSymmetricEJA,
+ ComplexHermitianEJA,
+ QuaternionHermitianEJA])
return constructor(n, field=QQ)
return tuple(S)
+def _quaternion_hermitian_basis(n, field=QQ):
+ """
+ Returns a basis for the space of quaternion 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 _quaternion_hermitian_basis(n) )
+ True
+
+ """
+ Q = QuaternionAlgebra(QQ,-1,-1)
+ I,J,K = Q.gens()
+
+ # 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(Q, n, lambda k,l: k==i and l==j)
+ if i == j:
+ Sij = _embed_quaternion_matrix(Eij)
+ S.append(Sij)
+ else:
+ # Beware, orthogonal but not normalized! The second,
+ # third, and fourth ones have a minus because they're
+ # conjugated.
+ Sij_real = _embed_quaternion_matrix(Eij + Eij.transpose())
+ S.append(Sij_real)
+ Sij_I = _embed_quaternion_matrix(I*Eij - I*Eij.transpose())
+ S.append(Sij_I)
+ Sij_J = _embed_quaternion_matrix(J*Eij - J*Eij.transpose())
+ S.append(Sij_J)
+ Sij_K = _embed_quaternion_matrix(K*Eij - K*Eij.transpose())
+ S.append(Sij_K)
+ return tuple(S)
+
+
+def _mat2vec(m):
+ return vector(m.base_ring(), m.list())
+
+def _vec2mat(v):
+ return matrix(v.base_ring(), sqrt(v.degree()), v.list())
+
def _multiplication_table_from_matrix_basis(basis):
"""
At least three of the five simple Euclidean Jordan algebras have the
field = basis[0].base_ring()
dimension = basis[0].nrows()
- def mat2vec(m):
- return vector(field, m.list())
-
- def vec2mat(v):
- return matrix(field, dimension, v.list())
-
V = VectorSpace(field, dimension**2)
- W = V.span( mat2vec(s) for s in basis )
+ W = V.span( _mat2vec(s) for s in basis )
# Taking the span above reorders our basis (thanks, jerk!) so we
# need to put our "matrix basis" in the same order as the
# (reordered) vector basis.
- S = tuple( vec2mat(b) for b in W.basis() )
+ S = tuple( _vec2mat(b) for b in W.basis() )
Qs = []
for s in S:
# why we're computing rows here and not columns.
Q_rows = []
for t in S:
- this_row = mat2vec((s*t + t*s)/2)
+ this_row = _mat2vec((s*t + t*s)/2)
Q_rows.append(W.coordinates(this_row))
Q = matrix(field, W.dimension(), Q_rows)
Qs.append(Q)
sage: x2 = F(1 + 2*i)
sage: x3 = F(-i)
sage: x4 = F(6)
- sage: M = matrix(F,2,[x1,x2,x3,x4])
+ sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
sage: _embed_complex_matrix(M)
- [ 4 2| 1 -2]
- [-2 4| 2 1]
+ [ 4 -2| 1 2]
+ [ 2 4|-2 1]
[-----+-----]
- [ 0 1| 6 0]
- [-1 0| 0 6]
+ [ 0 -1| 6 0]
+ [ 1 0| 0 6]
+
+ TESTS:
+
+ Embedding is a homomorphism (isomorphism, in fact)::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(5)
+ sage: F = QuadraticField(-1, 'i')
+ sage: X = random_matrix(F, n)
+ sage: Y = random_matrix(F, n)
+ sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
+ sage: expected = _embed_complex_matrix(X*Y)
+ sage: actual == expected
+ True
"""
n = M.nrows()
if M.ncols() != n:
- raise ArgumentError("the matrix 'M' must be square")
+ raise ValueError("the matrix 'M' must be square")
field = M.base_ring()
blocks = []
for z in M.list():
a = z.real()
b = z.imag()
- blocks.append(matrix(field, 2, [[a,-b],[b,a]]))
+ blocks.append(matrix(field, 2, [[a,b],[-b,a]]))
# We can drop the imaginaries here.
return block_matrix(field.base_ring(), n, blocks)
....: [ 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]
+ [ 2*i + 1 4*i + 3]
+ [ 10*i + 9 12*i + 11]
+
+ TESTS:
+
+ Unembedding is the inverse of embedding::
+
+ sage: set_random_seed()
+ sage: F = QuadraticField(-1, 'i')
+ sage: M = random_matrix(F, 3)
+ sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
+ True
+
"""
n = ZZ(M.nrows())
if M.ncols() != n:
- raise ArgumentError("the matrix 'M' must be square")
+ raise ValueError("the matrix 'M' must be square")
if not n.mod(2).is_zero():
- raise ArgumentError("the matrix 'M' must be a complex embedding")
+ raise ValueError("the matrix 'M' must be a complex embedding")
F = QuadraticField(-1, 'i')
i = F.gen()
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')
+ raise ValueError('bad on-diagonal submatrix')
if submat[0,1] != -submat[1,0]:
- raise ArgumentError('bad imag submatrix')
- z = submat[0,0] + submat[1,0]*i
+ raise ValueError('bad off-diagonal submatrix')
+ z = submat[0,0] + submat[0,1]*i
elements.append(z)
return matrix(F, n/2, elements)
-def RealSymmetricSimpleEJA(n, field=QQ):
+def _embed_quaternion_matrix(M):
+ """
+ Embed the n-by-n quaternion matrix ``M`` into the space of real
+ matrices of size 4n-by-4n by first sending each quaternion entry
+ `z = a + bi + cj + dk` to the block-complex matrix
+ ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
+ a real matrix.
+
+ EXAMPLES::
+
+ sage: Q = QuaternionAlgebra(QQ,-1,-1)
+ sage: i,j,k = Q.gens()
+ sage: x = 1 + 2*i + 3*j + 4*k
+ sage: M = matrix(Q, 1, [[x]])
+ sage: _embed_quaternion_matrix(M)
+ [ 1 2 3 4]
+ [-2 1 -4 3]
+ [-3 4 1 -2]
+ [-4 -3 2 1]
+
+ Embedding is a homomorphism (isomorphism, in fact)::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(5)
+ sage: Q = QuaternionAlgebra(QQ,-1,-1)
+ sage: X = random_matrix(Q, n)
+ sage: Y = random_matrix(Q, n)
+ sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
+ sage: expected = _embed_quaternion_matrix(X*Y)
+ sage: actual == expected
+ True
+
+ """
+ quaternions = M.base_ring()
+ n = M.nrows()
+ if M.ncols() != n:
+ raise ValueError("the matrix 'M' must be square")
+
+ F = QuadraticField(-1, 'i')
+ i = F.gen()
+
+ blocks = []
+ for z in M.list():
+ t = z.coefficient_tuple()
+ a = t[0]
+ b = t[1]
+ c = t[2]
+ d = t[3]
+ cplx_matrix = matrix(F, 2, [[ a + b*i, c + d*i],
+ [-c + d*i, a - b*i]])
+ blocks.append(_embed_complex_matrix(cplx_matrix))
+
+ # We should have real entries by now, so use the realest field
+ # we've got for the return value.
+ return block_matrix(quaternions.base_ring(), n, blocks)
+
+
+def _unembed_quaternion_matrix(M):
+ """
+ The inverse of _embed_quaternion_matrix().
+
+ EXAMPLES::
+
+ sage: M = matrix(QQ, [[ 1, 2, 3, 4],
+ ....: [-2, 1, -4, 3],
+ ....: [-3, 4, 1, -2],
+ ....: [-4, -3, 2, 1]])
+ sage: _unembed_quaternion_matrix(M)
+ [1 + 2*i + 3*j + 4*k]
+
+ TESTS:
+
+ Unembedding is the inverse of embedding::
+
+ sage: set_random_seed()
+ sage: Q = QuaternionAlgebra(QQ, -1, -1)
+ sage: M = random_matrix(Q, 3)
+ sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
+ True
+
+ """
+ n = ZZ(M.nrows())
+ if M.ncols() != n:
+ raise ValueError("the matrix 'M' must be square")
+ if not n.mod(4).is_zero():
+ raise ValueError("the matrix 'M' must be a complex embedding")
+
+ Q = QuaternionAlgebra(QQ,-1,-1)
+ i,j,k = Q.gens()
+
+ # Go top-left to bottom-right (reading order), converting every
+ # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
+ # quaternion block.
+ elements = []
+ for l in xrange(n/4):
+ for m in xrange(n/4):
+ submat = _unembed_complex_matrix(M[4*l:4*l+4,4*m:4*m+4])
+ if submat[0,0] != submat[1,1].conjugate():
+ raise ValueError('bad on-diagonal submatrix')
+ if submat[0,1] != -submat[1,0].conjugate():
+ raise ValueError('bad off-diagonal submatrix')
+ z = submat[0,0].real() + submat[0,0].imag()*i
+ z += submat[0,1].real()*j + submat[0,1].imag()*k
+ elements.append(z)
+
+ return matrix(Q, n/4, elements)
+
+
+# The usual inner product on R^n.
+def _usual_ip(x,y):
+ return x.vector().inner_product(y.vector())
+
+# The inner product used for the real symmetric simple EJA.
+# We keep it as a separate function because e.g. the complex
+# algebra uses the same inner product, except divided by 2.
+def _matrix_ip(X,Y):
+ X_mat = X.natural_representation()
+ Y_mat = Y.natural_representation()
+ return (X_mat*Y_mat).trace()
+
+
+class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra):
"""
The rank-n simple EJA consisting of real symmetric n-by-n
matrices, the usual symmetric Jordan product, and the trace inner
EXAMPLES::
- sage: J = RealSymmetricSimpleEJA(2)
+ sage: J = RealSymmetricEJA(2)
sage: e0, e1, e2 = J.gens()
sage: e0*e0
e0
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
- sage: J = RealSymmetricSimpleEJA(n)
+ sage: J = RealSymmetricEJA(n)
sage: J.degree() == (n^2 + n)/2
True
+ The Jordan multiplication is what we think it is::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: J = RealSymmetricEJA(n)
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: actual = (x*y).natural_representation()
+ sage: X = x.natural_representation()
+ sage: Y = y.natural_representation()
+ sage: expected = (X*Y + Y*X)/2
+ sage: actual == expected
+ True
+ sage: J(expected) == x*y
+ True
+
"""
- S = _real_symmetric_basis(n, field=field)
- (Qs, T) = _multiplication_table_from_matrix_basis(S)
+ @staticmethod
+ def __classcall_private__(cls, n, field=QQ):
+ S = _real_symmetric_basis(n, field=field)
+ (Qs, T) = _multiplication_table_from_matrix_basis(S)
- return FiniteDimensionalEuclideanJordanAlgebra(field,
- Qs,
- rank=n,
- natural_basis=T)
+ fdeja = super(RealSymmetricEJA, cls)
+ return fdeja.__classcall_private__(cls,
+ field,
+ Qs,
+ rank=n,
+ natural_basis=T)
+
+ def inner_product(self, x, y):
+ return _matrix_ip(x,y)
-def ComplexHermitianSimpleEJA(n, field=QQ):
+class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
"""
The rank-n simple EJA consisting of complex Hermitian n-by-n
matrices over the real numbers, the usual symmetric Jordan product,
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
- sage: J = ComplexHermitianSimpleEJA(n)
+ sage: J = ComplexHermitianEJA(n)
sage: J.degree() == n^2
True
- """
- S = _complex_hermitian_basis(n)
- (Qs, T) = _multiplication_table_from_matrix_basis(S)
- return FiniteDimensionalEuclideanJordanAlgebra(field,
- Qs,
- rank=n,
- natural_basis=T)
+ The Jordan multiplication is what we think it is::
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: J = ComplexHermitianEJA(n)
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: actual = (x*y).natural_representation()
+ sage: X = x.natural_representation()
+ sage: Y = y.natural_representation()
+ sage: expected = (X*Y + Y*X)/2
+ sage: actual == expected
+ True
+ sage: J(expected) == x*y
+ True
-def QuaternionHermitianSimpleEJA(n):
+ """
+ @staticmethod
+ def __classcall_private__(cls, n, field=QQ):
+ S = _complex_hermitian_basis(n)
+ (Qs, T) = _multiplication_table_from_matrix_basis(S)
+
+ fdeja = super(ComplexHermitianEJA, cls)
+ return fdeja.__classcall_private__(cls,
+ field,
+ Qs,
+ rank=n,
+ natural_basis=T)
+
+ def inner_product(self, x, y):
+ # Since a+bi on the diagonal is represented as
+ #
+ # a + bi = [ a b ]
+ # [ -b a ],
+ #
+ # we'll double-count the "a" entries if we take the trace of
+ # the embedding.
+ return _matrix_ip(x,y)/2
+
+
+class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
"""
The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
matrices, the usual symmetric Jordan product, and the
real-part-of-trace inner product. It has dimension `2n^2 - n` over
the reals.
- """
- pass
-def OctonionHermitianSimpleEJA(n):
- """
- This shit be crazy. It has dimension 27 over the reals.
- """
- n = 3
- pass
+ TESTS:
+
+ The degree of this algebra is `n^2`::
-def JordanSpinSimpleEJA(n, field=QQ):
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: J = QuaternionHermitianEJA(n)
+ sage: J.degree() == 2*(n^2) - n
+ True
+
+ The Jordan multiplication is what we think it is::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: J = QuaternionHermitianEJA(n)
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: actual = (x*y).natural_representation()
+ sage: X = x.natural_representation()
+ sage: Y = y.natural_representation()
+ sage: expected = (X*Y + Y*X)/2
+ sage: actual == expected
+ True
+ sage: J(expected) == x*y
+ True
+
+ """
+ @staticmethod
+ def __classcall_private__(cls, n, field=QQ):
+ S = _quaternion_hermitian_basis(n)
+ (Qs, T) = _multiplication_table_from_matrix_basis(S)
+
+ fdeja = super(QuaternionHermitianEJA, cls)
+ return fdeja.__classcall_private__(cls,
+ field,
+ Qs,
+ rank=n,
+ natural_basis=T)
+
+ def inner_product(self, x, y):
+ # Since a+bi+cj+dk on the diagonal is represented as
+ #
+ # a + bi +cj + dk = [ a b c d]
+ # [ -b a -d c]
+ # [ -c d a -b]
+ # [ -d -c b a],
+ #
+ # we'll quadruple-count the "a" entries if we take the trace of
+ # the embedding.
+ return _matrix_ip(x,y)/4
+
+
+class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra):
"""
The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
with the usual inner product and jordan product ``x*y =
This multiplication table can be verified by hand::
- sage: J = JordanSpinSimpleEJA(4)
+ sage: J = JordanSpinEJA(4)
sage: e0,e1,e2,e3 = J.gens()
sage: e0*e0
e0
sage: e2*e3
0
- In one dimension, this is the reals under multiplication::
+ """
+ @staticmethod
+ def __classcall_private__(cls, n, field=QQ):
+ 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)
+
+ fdeja = super(JordanSpinEJA, cls)
+ return fdeja.__classcall_private__(cls, field, Qs)
- sage: J1 = JordanSpinSimpleEJA(1)
- sage: J2 = eja_rn(1)
- sage: J1 == J2
- True
+ def rank(self):
+ """
+ Return the rank of this Jordan Spin Algebra.
- """
- 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))
+ The rank of the spin algebra is two, unless we're in a
+ one-dimensional ambient space (because the rank is bounded by
+ the ambient dimension).
+ """
+ return min(self.dimension(),2)
+
+ def inner_product(self, x, y):
+ return _usual_ip(x,y)
projects / sage.d.git / blobdiff