]> gitweb.michael.orlitzky.com - sage.d.git/blobdiff - mjo/eja/eja_algebra.py
eja: pass check=False for known-good constructions.
[sage.d.git] / mjo / eja / eja_algebra.py
index 7ae61d513d53246f588a9610f20c5acb105d069b..b681296b698287bdb506ee0519fcf098711b380b 100644 (file)
@@ -61,7 +61,10 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
         """
         SETUP::
 
-            sage: from mjo.eja.eja_algebra import (JordanSpinEJA, random_eja)
+            sage: from mjo.eja.eja_algebra import (
+            ....:   FiniteDimensionalEuclideanJordanAlgebra,
+            ....:   JordanSpinEJA,
+            ....:   random_eja)
 
         EXAMPLES:
 
@@ -75,13 +78,20 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
 
         TESTS:
 
-        The ``field`` we're given must be real::
+        The ``field`` we're given must be real with ``check=True``::
 
             sage: JordanSpinEJA(2,QQbar)
             Traceback (most recent call last):
             ...
             ValueError: field is not real
 
+        The multiplication table must be square with ``check=True``::
+
+            sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,((),()))
+            Traceback (most recent call last):
+            ...
+            ValueError: multiplication table is not square
+
         """
         if check:
             if not field.is_subring(RR):
@@ -96,9 +106,15 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
             category = MagmaticAlgebras(field).FiniteDimensional()
             category = category.WithBasis().Unital()
 
+        # The multiplication table had better be square
+        n = len(mult_table)
+        if check:
+            if not all( len(l) == n for l in mult_table ):
+                raise ValueError("multiplication table is not square")
+
         fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
         fda.__init__(field,
-                     range(len(mult_table)),
+                     range(n),
                      prefix=prefix,
                      category=category)
         self.print_options(bracket='')
@@ -114,6 +130,13 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
             for ls in mult_table
         ]
 
+        if check:
+            if not self._is_commutative():
+                raise ValueError("algebra is not commutative")
+            if not self._is_jordanian():
+                raise ValueError("Jordan identity does not hold")
+            if not self._inner_product_is_associative():
+                raise ValueError("inner product is not associative")
 
     def _element_constructor_(self, elt):
         """
@@ -235,6 +258,67 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
     def product_on_basis(self, i, j):
         return self._multiplication_table[i][j]
 
+    def _is_commutative(self):
+        r"""
+        Whether or not this algebra's multiplication table is commutative.
+
+        This method should of course always return ``True``, unless
+        this algebra was constructed with ``check=False`` and passed
+        an invalid multiplication table.
+        """
+        return all( self.product_on_basis(i,j) == self.product_on_basis(i,j)
+                    for i in range(self.dimension())
+                    for j in range(self.dimension()) )
+
+    def _is_jordanian(self):
+        r"""
+        Whether or not this algebra's multiplication table respects the
+        Jordan identity `(x^{2})(xy) = x(x^{2}y)`.
+
+        We only check one arrangement of `x` and `y`, so for a
+        ``True`` result to be truly true, you should also check
+        :meth:`_is_commutative`. This method should of course always
+        return ``True``, unless this algebra was constructed with
+        ``check=False`` and passed an invalid multiplication table.
+        """
+        return all( (self.monomial(i)**2)*(self.monomial(i)*self.monomial(j))
+                    ==
+                    (self.monomial(i))*((self.monomial(i)**2)*self.monomial(j))
+                    for i in range(self.dimension())
+                    for j in range(self.dimension()) )
+
+    def _inner_product_is_associative(self):
+        r"""
+        Return whether or not this algebra's inner product `B` is
+        associative; that is, whether or not `B(xy,z) = B(x,yz)`.
+
+        This method should of course always return ``True``, unless
+        this algebra was constructed with ``check=False`` and passed
+        an invalid multiplication table.
+        """
+
+        # Used to check whether or not something is zero in an inexact
+        # ring. This number is sufficient to allow the construction of
+        # QuaternionHermitianEJA(2, RDF) with check=True.
+        epsilon = 1e-16
+
+        for i in range(self.dimension()):
+            for j in range(self.dimension()):
+                for k in range(self.dimension()):
+                    x = self.monomial(i)
+                    y = self.monomial(j)
+                    z = self.monomial(k)
+                    diff = (x*y).inner_product(z) - x.inner_product(y*z)
+
+                    if self.base_ring().is_exact():
+                        if diff != 0:
+                            return False
+                    else:
+                        if diff.abs() > epsilon:
+                            return False
+
+        return True
+
     @cached_method
     def characteristic_polynomial_of(self):
         """
@@ -965,8 +1049,10 @@ class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra):
         mult_table = [ [ V.gen(i)*(i == j) for j in range(n) ]
                        for i in range(n) ]
 
-        fdeja = super(HadamardEJA, self)
-        fdeja.__init__(field, mult_table, **kwargs)
+        super(HadamardEJA, self).__init__(field,
+                                          mult_table,
+                                          check=False,
+                                          **kwargs)
         self.rank.set_cache(n)
 
     def inner_product(self, x, y):
@@ -1055,9 +1141,10 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra):
 
         Qs = self.multiplication_table_from_matrix_basis(basis)
 
-        fdeja = super(MatrixEuclideanJordanAlgebra, self)
-        fdeja.__init__(field, Qs, natural_basis=basis, **kwargs)
-        return
+        super(MatrixEuclideanJordanAlgebra, self).__init__(field,
+                                                           Qs,
+                                                           natural_basis=basis,
+                                                           **kwargs)
 
 
     @cached_method
@@ -1076,10 +1163,13 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra):
 
             # Do this over the rationals and convert back at the end.
             # Only works because we know the entries of the basis are
-            # integers.
+            # integers. The argument ``check=False`` is required
+            # because the trace inner-product method for this
+            # class is a stub and can't actually be checked.
             J = MatrixEuclideanJordanAlgebra(QQ,
                                              basis,
-                                             normalize_basis=False)
+                                             normalize_basis=False,
+                                             check=False)
             a = J._charpoly_coefficients()
 
             # Unfortunately, changing the basis does change the
@@ -1322,7 +1412,10 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra):
 
     def __init__(self, n, field=AA, **kwargs):
         basis = self._denormalized_basis(n, field)
-        super(RealSymmetricEJA, self).__init__(field, basis, **kwargs)
+        super(RealSymmetricEJA, self).__init__(field,
+                                               basis,
+                                               check=False,
+                                               **kwargs)
         self.rank.set_cache(n)
 
 
@@ -1618,7 +1711,10 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra):
 
     def __init__(self, n, field=AA, **kwargs):
         basis = self._denormalized_basis(n,field)
-        super(ComplexHermitianEJA,self).__init__(field, basis, **kwargs)
+        super(ComplexHermitianEJA,self).__init__(field,
+                                                 basis,
+                                                 check=False,
+                                                 **kwargs)
         self.rank.set_cache(n)
 
 
@@ -1919,7 +2015,10 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra):
 
     def __init__(self, n, field=AA, **kwargs):
         basis = self._denormalized_basis(n,field)
-        super(QuaternionHermitianEJA,self).__init__(field, basis, **kwargs)
+        super(QuaternionHermitianEJA,self).__init__(field,
+                                                    basis,
+                                                    check=False,
+                                                    **kwargs)
         self.rank.set_cache(n)
 
 
@@ -2002,8 +2101,10 @@ class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra):
         # The rank of this algebra is two, unless we're in a
         # one-dimensional ambient space (because the rank is bounded
         # by the ambient dimension).
-        fdeja = super(BilinearFormEJA, self)
-        fdeja.__init__(field, mult_table, **kwargs)
+        super(BilinearFormEJA, self).__init__(field,
+                                              mult_table,
+                                              check=False,
+                                              **kwargs)
         self.rank.set_cache(min(n,2))
 
     def inner_product(self, x, y):
@@ -2095,7 +2196,7 @@ class JordanSpinEJA(BilinearFormEJA):
     def __init__(self, n, field=AA, **kwargs):
         # This is a special case of the BilinearFormEJA with the identity
         # matrix as its bilinear form.
-        return super(JordanSpinEJA, self).__init__(n, field, **kwargs)
+        super(JordanSpinEJA, self).__init__(n, field, **kwargs)
 
 
 class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra):
@@ -2129,8 +2230,59 @@ class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra):
     """
     def __init__(self, field=AA, **kwargs):
         mult_table = []
-        fdeja = super(TrivialEJA, self)
+        super(TrivialEJA, self).__init__(field,
+                                         mult_table,
+                                         check=False,
+                                         **kwargs)
         # The rank is zero using my definition, namely the dimension of the
         # largest subalgebra generated by any element.
-        fdeja.__init__(field, mult_table, **kwargs)
         self.rank.set_cache(0)
+
+
+class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra):
+    r"""
+    The external (orthogonal) direct sum of two other Euclidean Jordan
+    algebras. Essentially the Cartesian product of its two factors.
+    Every Euclidean Jordan algebra decomposes into an orthogonal
+    direct sum of simple Euclidean Jordan algebras, so no generality
+    is lost by providing only this construction.
+
+    SETUP::
+
+        sage: from mjo.eja.eja_algebra import (HadamardEJA,
+        ....:                                  RealSymmetricEJA,
+        ....:                                  DirectSumEJA)
+
+    EXAMPLES::
+
+        sage: J1 = HadamardEJA(2)
+        sage: J2 = RealSymmetricEJA(3)
+        sage: J = DirectSumEJA(J1,J2)
+        sage: J.dimension()
+        8
+        sage: J.rank()
+        5
+
+    """
+    def __init__(self, J1, J2, field=AA, **kwargs):
+        n1 = J1.dimension()
+        n2 = J2.dimension()
+        n = n1+n2
+        V = VectorSpace(field, n)
+        mult_table = [ [ V.zero() for j in range(n) ]
+                       for i in range(n) ]
+        for i in range(n1):
+            for j in range(n1):
+                p = (J1.monomial(i)*J1.monomial(j)).to_vector()
+                mult_table[i][j] = V(p.list() + [field.zero()]*n2)
+
+        for i in range(n2):
+            for j in range(n2):
+                p = (J2.monomial(i)*J2.monomial(j)).to_vector()
+                mult_table[n1+i][n1+j] = V([field.zero()]*n1 + p.list())
+
+        super(DirectSumEJA, self).__init__(field,
+                                           mult_table,
+                                           check=False,
+                                           **kwargs)
+        self.rank.set_cache(J1.rank() + J2.rank())