eja: add is_trivial() method and special cases for trivial algebras.

[sage.d.git] / mjo / eja / eja_element.py

diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py
index e38012eb3bf9cda640c8cc4c4ef8d822e194ce43..c47156d31e2c6989e1fe863091794520f395523c 100644 (file)
--- a/mjo/eja/eja_element.py
+++ b/mjo/eja/eja_element.py
@@ -482,15 +482,23 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: J.one().is_invertible()
             True
 
-        The zero element is never invertible::
+        The zero element is never invertible in a non-trivial algebra::
 
             sage: set_random_seed()
             sage: J = random_eja()
-            sage: J.zero().is_invertible()
+            sage: (not J.is_trivial()) and J.zero().is_invertible()
             False
 
         """
-        zero = self.parent().zero()
+        if self.is_zero():
+            if self.parent().is_trivial():
+                return True
+            else:
+                return False
+
+        # In fact, we only need to know if the constant term is non-zero,
+        # so we can pass in the field's zero element instead.
+        zero = self.base_ring().zero()
         p = self.minimal_polynomial()
         return not (p(zero) == zero)
 
@@ -643,6 +651,11 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             True
 
         """
+        if self.is_zero() and not self.parent().is_trivial():
+            # The minimal polynomial of zero in a nontrivial algebra
+            # is "t"; in a trivial algebra it's "1" by convention
+            # (it's an empty product).
+            return 1
         return self.subalgebra_generated_by().dimension()
 
 
@@ -721,6 +734,18 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             0
 
         """
+        if self.is_zero():
+            # We would generate a zero-dimensional subalgebra
+            # where the minimal polynomial would be constant.
+            # That might be correct, but only if *this* algebra
+            # is trivial too.
+            if not self.parent().is_trivial():
+                # Pretty sure we know what the minimal polynomial of
+                # the zero operator is going to be. This ensures
+                # consistency of e.g. the polynomial variable returned
+                # in the "normal" case without us having to think about it.
+                return self.operator().minimal_polynomial()
+
         A = self.subalgebra_generated_by()
         return A(self).operator().minimal_polynomial()
 
@@ -801,10 +826,12 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
         """
         P = self.parent()
+        left_mult_by_self = lambda y: self*y
+        L = P.module_morphism(function=left_mult_by_self, codomain=P)
         return FiniteDimensionalEuclideanJordanAlgebraOperator(
                  P,
                  P,
-                 self.to_matrix() )
+                 L.matrix() )
 
 
     def quadratic_representation(self, other=None):
@@ -960,6 +987,15 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: A(x^2) == A(x)*A(x)
             True
 
+        The subalgebra generated by the zero element is trivial::
+
+            sage: set_random_seed()
+            sage: A = random_eja().zero().subalgebra_generated_by()
+            sage: A
+            Euclidean Jordan algebra of dimension 0 over Rational Field
+            sage: A.one()
+            0
+
         """
         return FiniteDimensionalEuclideanJordanElementSubalgebra(self)