return (J0, J5, J1)
- def a_jordan_frame(self):
- r"""
- Generate a Jordan frame for this algebra.
-
- This implementation is based on the so-called "central
- orthogonal idempotents" implemented for (semisimple) centers
- of SageMath ``FiniteDimensionalAlgebrasWithBasis``. Since all
- Euclidean Jordan algebas are commutative (and thus equal to
- their own centers) and semisimple, the method should work more
- or less as implemented, if it ever worked in the first place.
- (I don't know the justification for the original implementation.
- yet).
-
- How it works: we loop through the algebras generators, looking
- for their eigenspaces. If there's more than one eigenspace,
- and if they result in more than one subalgebra, then we split
- those subalgebras recursively until we get to subalgebras of
- dimension one (whose idempotent is the unit element). Why does
- some generator have to produce at least two subalgebras? I
- dunno. But it seems to work.
-
- Beware that Koecher defines the "center" of a Jordan algebra to
- be something else, because the usual definition is stupid in a
- (necessarily commutative) Jordan algebra.
+ def random_elements(self, count):
+ """
+ Return ``count`` random elements as a tuple.
SETUP::
- sage: from mjo.eja.eja_algebra import (random_eja,
- ....: JordanSpinEJA,
- ....: TrivialEJA)
-
- EXAMPLES:
-
- A Jordan frame for the trivial algebra has to be empty
- (zero-length) since its rank is zero. More to the point, there
- are no non-zero idempotents in the trivial EJA. This does not
- cause any problems so long as we adopt the convention that the
- empty sum is zero, since then the sole element of the trivial
- EJA has an (empty) spectral decomposition::
-
- sage: J = TrivialEJA()
- sage: J.a_jordan_frame()
- ()
-
- A one-dimensional algebra has rank one (equal to its dimension),
- and only one primitive idempotent, namely the algebra's unit
- element::
-
- sage: J = JordanSpinEJA(1)
- sage: J.a_jordan_frame()
- (e0,)
+ sage: from mjo.eja.eja_algebra import JordanSpinEJA
- TESTS::
+ EXAMPLES::
- sage: J = random_eja()
- sage: c = J.a_jordan_frame()
- sage: all( x^2 == x for x in c )
+ sage: J = JordanSpinEJA(3)
+ sage: x,y,z = J.random_elements(3)
+ sage: all( [ x in J, y in J, z in J ])
True
- sage: r = len(c)
- sage: all( c[i]*c[j] == c[i]*(i==j) for i in range(r)
- ....: for j in range(r) )
+ sage: len( J.random_elements(10) ) == 10
True
"""
- if self.dimension() == 0:
- return ()
- if self.dimension() == 1:
- return (self.one(),)
-
- for g in self.gens():
- eigenpairs = g.operator().matrix().right_eigenspaces()
- if len(eigenpairs) >= 2:
- subalgebras = []
- for eigval, eigspace in eigenpairs:
- # Make sub-EJAs from the matrix eigenspaces...
- sb = tuple( self.from_vector(b) for b in eigspace.basis() )
- try:
- # This will fail if e.g. the eigenspace basis
- # contains two elements and their product
- # isn't a linear combination of the two of
- # them (i.e. the generated EJA isn't actually
- # two dimensional).
- s = FiniteDimensionalEuclideanJordanSubalgebra(self, sb)
- subalgebras.append(s)
- except ArithmeticError as e:
- if str(e) == "vector is not in free module":
- # Ignore only the "not a sub-EJA" error
- pass
-
- if len(subalgebras) >= 2:
- # apply this method recursively.
- return tuple( c.superalgebra_element()
- for subalgebra in subalgebras
- for c in subalgebra.a_jordan_frame() )
-
- # If we got here, the algebra didn't decompose, at least not when we looked at
- # the eigenspaces corresponding only to basis elements of the algebra. The
- # implementation I stole says that this should work because of Schur's Lemma,
- # so I personally blame Schur's Lemma if it does not.
- raise Exception("Schur's Lemma didn't work!")
+ return tuple( self.random_element() for idx in range(count) )
- def random_elements(self, count):
- """
- Return ``count`` random elements as a tuple.
+ def _rank_computation1(self):
+ r"""
+ Compute the rank of this algebra using highly suspicious voodoo.
+
+ ALGORITHM:
+
+ We first compute the basis representation of the operator L_x
+ using polynomial indeterminates are placeholders for the
+ coordinates of "x", which is arbitrary. We then use that
+ matrix to compute the (polynomial) entries of x^0, x^1, ...,
+ x^d,... for increasing values of "d", starting at zero. The
+ idea is that. If we also add "coefficient variables" a_0,
+ a_1,... to the ring, we can form the linear combination
+ a_0*x^0 + ... + a_d*x^d = 0, and ask what dimension the
+ solution space has as an affine variety. When "d" is smaller
+ than the rank, we expect that dimension to be the number of
+ coordinates of "x", since we can set *those* to whatever we
+ want, but linear independence forces the coefficients a_i to
+ be zero. Eventually, when "d" passes the rank, the dimension
+ of the solution space begins to grow, because we can *still*
+ set the coordinates of "x" arbitrarily, but now there are some
+ coefficients that make the sum zero as well. So, when the
+ dimension of the variety jumps, we return the corresponding
+ "d" as the rank of the algebra. This appears to work.
SETUP::
- sage: from mjo.eja.eja_algebra import JordanSpinEJA
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: JordanSpinEJA,
+ ....: RealSymmetricEJA,
+ ....: ComplexHermitianEJA,
+ ....: QuaternionHermitianEJA)
EXAMPLES::
- sage: J = JordanSpinEJA(3)
- sage: x,y,z = J.random_elements(3)
- sage: all( [ x in J, y in J, z in J ])
+ sage: J = HadamardEJA(5)
+ sage: J._rank_computation() == J.rank()
True
- sage: len( J.random_elements(10) ) == 10
+ sage: J = JordanSpinEJA(5)
+ sage: J._rank_computation() == J.rank()
+ True
+ sage: J = RealSymmetricEJA(4)
+ sage: J._rank_computation() == J.rank()
+ True
+ sage: J = ComplexHermitianEJA(3)
+ sage: J._rank_computation() == J.rank()
+ True
+ sage: J = QuaternionHermitianEJA(2)
+ sage: J._rank_computation() == J.rank()
True
"""
- return tuple( self.random_element() for idx in range(count) )
+ n = self.dimension()
+ var_names = [ "X" + str(z) for z in range(1,n+1) ]
+ d = 0
+ ideal_dim = len(var_names)
+ def L_x_i_j(i,j):
+ # From a result in my book, these are the entries of the
+ # basis representation of L_x.
+ return sum( vars[d+k]*self.monomial(k).operator().matrix()[i,j]
+ for k in range(n) )
+
+ while ideal_dim == len(var_names):
+ coeff_names = [ "a" + str(z) for z in range(d) ]
+ R = PolynomialRing(self.base_ring(), coeff_names + var_names)
+ vars = R.gens()
+ L_x = matrix(R, n, n, L_x_i_j)
+ x_powers = [ vars[k]*(L_x**k)*self.one().to_vector()
+ for k in range(d) ]
+ eqs = [ sum(x_powers[k][j] for k in range(d)) for j in range(n) ]
+ ideal_dim = R.ideal(eqs).dimension()
+ d += 1
+
+ # Subtract one because we increment one too many times, and
+ # subtract another one because "d" is one greater than the
+ # answer anyway; when d=3, we go up to x^2.
+ return d-2
+
+ def _rank_computation2(self):
+ r"""
+ Instead of using the dimension of an ideal, find the rank of a
+ matrix containing polynomials.
+ """
+ n = self.dimension()
+ var_names = [ "X" + str(z) for z in range(1,n+1) ]
+ R = PolynomialRing(self.base_ring(), var_names)
+ vars = R.gens()
+ def L_x_i_j(i,j):
+ # From a result in my book, these are the entries of the
+ # basis representation of L_x.
+ return sum( vars[k]*self.monomial(k).operator().matrix()[i,j]
+ for k in range(n) )
+
+ L_x = matrix(R, n, n, L_x_i_j)
+ x_powers = [ (vars[k]*(L_x**k)*self.one().to_vector()).row()
+ for k in range(n) ]
+
+ from sage.matrix.constructor import block_matrix
+ M = block_matrix(n,1,x_powers)
+ return M.rank()
+
+ def _rank_computation3(self):
+ r"""
+ Similar to :meth:`_rank_computation2`, but it stops echelonizing
+ as soon as it hits the first zero row.
+ """
+ n = self.dimension()
+ if n == 0:
+ return 0
+ elif n == 1:
+ return 1
+
+ var_names = [ "X" + str(z) for z in range(1,n+1) ]
+ R = PolynomialRing(self.base_ring(), var_names)
+ vars = R.gens()
+
+ def L_x_i_j(i,j):
+ # From a result in my book, these are the entries of the
+ # basis representation of L_x.
+ return sum( vars[k]*self.monomial(k).operator().matrix()[i,j]
+ for k in range(n) )
+
+ L_x = matrix(R, n, n, L_x_i_j)
+ x_powers = [ vars[k]*(L_x**k)*self.one().to_vector()
+ for k in range(n) ]
+
+ # Can assume n >= 2
+ rows = [x_powers[0]]
+ M = matrix(rows)
+ old_rank = 1
+
+ for d in range(1,n):
+ rows = M.rows() + [x_powers[d]]
+ M = matrix(rows)
+ M.echelonize()
+ new_rank = M.rank()
+ if new_rank == old_rank:
+ return new_rank
+ else:
+ old_rank = new_rank
def rank(self):
"""
projects / sage.d.git / blobdiff