eja: define subalgebra_generated_by() to contain the identity.

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

diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py
index eee8f69bd76ddfba49e3cb4531f55d0e970ebd1d..ee33e4ba89fcc3c47be1f9467042dae5e678c859 100644 (file)
--- a/mjo/eja/eja_element.py
+++ b/mjo/eja/eja_element.py
@@ -407,7 +407,8 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
         SETUP::
 
-            sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+            sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+            ....:                                  JordanSpinEJA,
             ....:                                  random_eja)
 
         EXAMPLES:
@@ -473,6 +474,22 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             ....:    x.operator().inverse()(J.one()) == x.inverse() )
             True
 
+        Proposition II.2.4 in Faraut and Korányi gives a formula for
+        the inverse based on the characteristic polynomial and the
+        Cayley-Hamilton theorem for Euclidean Jordan algebras::
+
+            sage: set_random_seed()
+            sage: J = ComplexHermitianEJA(3)
+            sage: x = J.random_element()
+            sage: while not x.is_invertible():
+            ....:     x = J.random_element()
+            sage: r = J.rank()
+            sage: a = x.characteristic_polynomial().coefficients(sparse=False)
+            sage: expected  = (-1)^(r+1)/x.det()
+            sage: expected *= sum( a[i+1]*x^i for i in range(r) )
+            sage: x.inverse() == expected
+            True
+
         """
         if not self.is_invertible():
             raise ValueError("element is not invertible")
@@ -1029,6 +1046,13 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         Return the associative subalgebra of the parent EJA generated
         by this element.
 
+        Since our parent algebra is unital, we want "subalgebra" to mean
+        "unital subalgebra" as well; thus the subalgebra that an element
+        generates will itself be a Euclidean Jordan algebra after
+        restricting the algebra operations appropriately. This is the
+        subalgebra that Faraut and Korányi work with in section II.2, for
+        example.
+
         SETUP::
 
             sage: from mjo.eja.eja_algebra import random_eja
@@ -1053,14 +1077,13 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: A(x^2) == A(x)*A(x)
             True
 
-        The subalgebra generated by the zero element is trivial::
+        By definition, the subalgebra generated by the zero element is the
+        one-dimensional algebra generated by the identity element::
 
             sage: set_random_seed()
             sage: A = random_eja().zero().subalgebra_generated_by()
-            sage: A
-            Euclidean Jordan algebra of dimension 0 over...
-            sage: A.one()
-            0
+            sage: A.dimension()
+            1
 
         """
         return FiniteDimensionalEuclideanJordanElementSubalgebra(self, orthonormalize_basis)