names='e',
assume_associative=False,
category=None,
- rank=None):
+ rank=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:
assume_associative=assume_associative,
names=names,
category=cat,
- rank=rank)
+ rank=rank,
+ natural_basis=natural_basis,
+ inner_product=inner_product)
def __init__(self, field,
names='e',
assume_associative=False,
category=None,
- rank=None):
+ rank=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,
fmt = "Euclidean Jordan algebra of degree {} over {}"
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.
+ """
+ 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.
+
+ 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" basis
+ for our underlying vector space. (Typically, the natural basis
+ is used to construct the multiplication table in the first place.)
+
+ Note that this will always return a matrix. The standard basis
+ in `R^n` will be returned as `n`-by-`1` column matrices.
+
+ EXAMPLES::
+
+ sage: J = RealSymmetricSimpleEJA(2)
+ sage: J.basis()
+ Family (e0, e1, e2)
+ sage: J.natural_basis()
+ (
+ [1 0] [0 1] [0 0]
+ [0 0], [1 0], [0 1]
+ )
+
+ ::
+
+ sage: J = JordanSpinSimpleEJA(2)
+ sage: J.basis()
+ Family (e0, e1)
+ sage: J.natural_basis()
+ (
+ [1] [0]
+ [0], [1]
+ )
+
+ """
+ if self._natural_basis is None:
+ return tuple( b.vector().column() for b in self.basis() )
+ else:
+ return self._natural_basis
+
+
def rank(self):
"""
Return the rank of this EJA.
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 = RealSymmetricSimpleEJA(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 = RealSymmetricSimpleEJA(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 = JordanSpinSimpleEJA(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 = RealSymmetricSimpleEJA(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 = ComplexHermitianSimpleEJA(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()
# 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
return self.span_of_powers().dimension()
- def matrix(self):
+ 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 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 = ComplexHermitianSimpleEJA(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]
+
+ """
+ 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-)
multiplication by this element in the parent algebra.
fda_elt = FiniteDimensionalAlgebraElement(self.parent(), self)
return fda_elt.matrix().transpose()
- #
- # The plan is to eventually phase out "matrix()", which sounds
- # too much like "matrix_representation()", in favor of the more-
- # accurate "operator_matrix()". But we need to override matrix()
- # to keep parent class methods happy in the meantime.
- #
- operator_matrix = matrix
-
-
- 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):
"""
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()
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)
# Beware, orthogonal but not normalized!
Sij = Eij + Eij.transpose()
S.append(Sij)
- return S
+ return tuple(S)
def _complex_hermitian_basis(n, field=QQ):
S.append(Sij_real)
Sij_imag = _embed_complex_matrix(I*Eij - I*Eij.transpose())
S.append(Sij_imag)
- return S
+ 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
multiplication on the right is matrix multiplication. Given a basis
for the underlying matrix space, this function returns a
multiplication table (obtained by looping through the basis
- elements) for an algebra of those matrices.
+ elements) for an algebra of those matrices. A reordered copy
+ of the basis is also returned to work around the fact that
+ the ``span()`` in this function will change the order of the basis
+ from what we think it is, to... something else.
"""
# In S^2, for example, we nominally have four coordinates even
# though the space is of dimension three only. The vector space V
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 = [ 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)
- return Qs
+ return (Qs, S)
def _embed_complex_matrix(M):
"""
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():
"""
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 real submatrix')
if submat[0,1] != -submat[1,0]:
- raise ArgumentError('bad imag submatrix')
+ raise ValueError('bad imag submatrix')
z = submat[0,0] + submat[1,0]*i
elements.append(z)
return matrix(F, n/2, 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()
+
def RealSymmetricSimpleEJA(n, field=QQ):
"""
"""
S = _real_symmetric_basis(n, field=field)
- Qs = _multiplication_table_from_matrix_basis(S)
+ (Qs, T) = _multiplication_table_from_matrix_basis(S)
- return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=n)
+ return FiniteDimensionalEuclideanJordanAlgebra(field,
+ Qs,
+ rank=n,
+ natural_basis=T,
+ inner_product=_matrix_ip)
def ComplexHermitianSimpleEJA(n, field=QQ):
"""
S = _complex_hermitian_basis(n)
- Qs = _multiplication_table_from_matrix_basis(S)
- return FiniteDimensionalEuclideanJordanAlgebra(field, Qs, rank=n)
+ (Qs, T) = _multiplication_table_from_matrix_basis(S)
+
+ # 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.
+ ip = lambda X,Y: _matrix_ip(X,Y)/2
+
+ return FiniteDimensionalEuclideanJordanAlgebra(field,
+ Qs,
+ rank=n,
+ natural_basis=T,
+ inner_product=ip)
def QuaternionHermitianSimpleEJA(n):
In one dimension, this is the reals under multiplication::
- sage: J1 = JordanSpinSimpleEJA(1)
- sage: J2 = eja_rn(1)
- sage: J1 == J2
- True
+ sage: J1 = JordanSpinSimpleEJA(1)
+ sage: J2 = eja_rn(1)
+ sage: J1 == J2
+ True
"""
Qs = []
# 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))
+ return FiniteDimensionalEuclideanJordanAlgebra(field,
+ Qs,
+ rank=min(n,2),
+ inner_product=_usual_ip)
projects / sage.d.git / blobdiff