_lift_idempotent for elements of sub-algebras

sagemath · Jul 25, 2014 · 75e4f5a · 75e4f5a
1 parent a203460
commit 75e4f5a
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Showing 3 changed files with 42 additions and 6 deletions.
diff --git a/src/sage/categories/algebras.py b/src/sage/categories/algebras.py
@@ -241,3 +241,29 @@ def _mul_(self, right):
                 """
                 p = self.parent()
                 return p.retract( self.lift() * right.lift())
+
+            def _lift_idempotent(self):
+                r"""
+                Lift an idempotent of parent of ``self`` into an indempotent of
+                parent of ``self.lift()``
+
+                EXAMPLES::
+
+                    sage: A3 = SymmetricGroup(3).algebra(QQ)
+                    sage: Z3 = A3.center()
+                    sage: Z3._refine_category_(SemisimpleAlgebras(QQ))
+                    sage: orth = Z3.orthogonal_idempotents()
+                    sage: x = orth[2]._lift_idempotent()
+                    sage: x not in Z3 and x in A3
+                    True
+
+                TODO: better documentation.
+                """
+                idempOld = None
+                idemp = self.lift()
+                p = idemp.parent()
+                while idemp <> idempOld:
+                    tmp = idemp
+                    idemp = (p.one() - (p.one() - idemp**2)**2)
+                    idempOld = tmp
+                return idemp
diff --git a/src/sage/categories/finite_dimensional_algebras_with_basis.py b/src/sage/categories/finite_dimensional_algebras_with_basis.py
@@ -343,6 +343,10 @@ def center(self):
             center.rename("Center of {}".format(self))
             return center
 
+        # def orthogonal_idempotents(self):
+        #     pass
+
+
     class ElementMethods:
         def to_matrix(self, base_ring=None, action=operator.mul, side='left'):
             """

diff --git a/src/sage/categories/semisimple_algebras.py b/src/sage/categories/semisimple_algebras.py
@@ -84,9 +84,6 @@ def _semi_simple_commutative_decomposition_generators(self, listGen=None, topLev
                     sage: G5 = SymmetricGroup(5)
                     sage: A5 = G5.algebra(QQ)
                     sage: Z5 = A5.center()
-                    sage: Z5.an_element() ** 2
-                    179*B[0] + 44*B[1] + 38*B[2] + 39*B[3] + 36*B[4] + 24*B[5]
-                    + 45*B[6]
                     sage: Z5._refine_category_(SemisimpleAlgebras(QQ))
                     sage: gens = Z5._semi_simple_commutative_decomposition_generators()
                     sage: sorted(gens, key=str)
@@ -143,14 +140,19 @@ def orthogonal_idempotents(self):
                 r"""
                 Return the minimal orthogonal idempotents of ``self``.
 
+                INPUT::
+
+                    - ``self`` a commutative semisimple algebra
+
+                OUTPUT::
+
+                    - list of idempotents of ``self``
+
                 EXAMPLES::
 
                     sage: G5 = SymmetricGroup(5)
                     sage: A5 = G5.algebra(QQ)
                     sage: Z5 = A5.center()
-                    sage: Z5.an_element() ** 2
-                    179*B[0] + 44*B[1] + 38*B[2] + 39*B[3] + 36*B[4] + 24*B[5] +
-                    45*B[6]
                     sage: Z5._refine_category_(SemisimpleAlgebras(QQ))
                     sage: orth = Z5.orthogonal_idempotents()
                     sage: orth = Z5.orthogonal_idempotents()
@@ -164,6 +166,10 @@ def orthogonal_idempotents(self):
                     5/24*B[0] + 1/24*B[1] + 1/24*B[2] - 1/24*B[3] + 1/24*B[4] -
                     1/24*B[5], 5/24*B[0] - 1/24*B[1] + 1/24*B[2] - 1/24*B[3] -
                     1/24*B[4] + 1/24*B[5]]
+                    sage: orth[2] * orth[4]
+                    0
+                    sage: orth[1] ** 2 == orth[1]
+                    True
                 """
                 return [(e.leading_coefficient()/(e*e).leading_coefficient())*e
                         for e in