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``.
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
Qs = [ matrix(field, dimension, dimension, lambda k,j: 1*(k == j == i))
for i in xrange(dimension) ]
- # The usual inner product on R^n.
- ip = lambda x, y: x.vector().inner_product(y.vector())
-
return FiniteDimensionalEuclideanJordanAlgebra(field,
Qs,
rank=dimension,
- inner_product=ip)
+ inner_product=_usual_ip)
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)
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):
"""
return FiniteDimensionalEuclideanJordanAlgebra(field,
Qs,
rank=n,
- natural_basis=T)
+ natural_basis=T,
+ inner_product=_matrix_ip)
def ComplexHermitianSimpleEJA(n, field=QQ):
"""
S = _complex_hermitian_basis(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)
+ natural_basis=T,
+ inner_product=ip)
def QuaternionHermitianSimpleEJA(n):
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
+
"""
Qs = []
id_matrix = identity_matrix(field, n)
Qi[0,0] = Qi[0,0] * ~field(2)
Qs.append(Qi)
- # The usual inner product on R^n.
- ip = lambda x, y: x.vector().inner_product(y.vector())
-
# 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),
- inner_product=ip)
+ inner_product=_usual_ip)
projects / sage.d.git / blobdiff