eja: add is_regular() method on elements.

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

diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py
index 2ec45cf537661f359ef6cf4e34c578c238f80b31..60a7ba1ede07bac242eb9aef6bcf9b722340c595 100644 (file)
--- a/mjo/eja/euclidean_jordan_algebra.py
+++ b/mjo/eja/euclidean_jordan_algebra.py
@@ -101,6 +101,21 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
 
             Jordan algebras are always power-associative; see for
             example Faraut and Koranyi, Proposition II.1.2 (ii).
+
+            .. WARNING:
+
+                We have to override this because our superclass uses row vectors
+                instead of column vectors! We, on the other hand, assume column
+                vectors everywhere.
+
+            EXAMPLES:
+
+                sage: set_random_seed()
+                sage: J = eja_ln(5)
+                sage: x = J.random_element()
+                sage: x.matrix()*x.vector() == (x**2).vector()
+                True
+
             """
             A = self.parent()
             if n == 0:
@@ -108,8 +123,32 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
             elif n == 1:
                 return self
             else:
-                return A.element_class(A, self.vector()*(self.matrix()**(n-1)))
+                return A.element_class(A, (self.matrix()**(n-1))*self.vector())
+
+
+        def is_regular(self):
+            """
+            Return whether or not this is a regular element.
+
+            EXAMPLES:
+
+            The identity element always has degree one, but any element
+            linearly-independent from it is regular::
 
+                sage: J = eja_ln(5)
+                sage: J.one().is_regular()
+                False
+                sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
+                sage: for x in J.gens():
+                ....:     (J.one() + x).is_regular()
+                False
+                True
+                True
+                True
+                True
+
+            """
+            return self.degree() == self.parent().rank()
 
         def span_of_powers(self):
             """
@@ -153,6 +192,19 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
             return self.span_of_powers().dimension()
 
 
+        def matrix(self):
+            """
+            Return the matrix that represents left- (or right-)
+            multiplication by this element in the parent algebra.
+
+            We have to override this because the superclass method
+            returns a matrix that acts on row vectors (that is, on
+            the right).
+            """
+            fda_elt = FiniteDimensionalAlgebraElement(self.parent(), self)
+            return fda_elt.matrix().transpose()
+
+
         def subalgebra_generated_by(self):
             """
             Return the associative subalgebra of the parent EJA generated
@@ -171,6 +223,15 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
                 sage: x.subalgebra_generated_by().is_associative()
                 True
 
+            Squaring in the subalgebra should be the same thing as
+            squaring in the superalgebra::
+
+                sage: J = eja_ln(5)
+                sage: x = J.random_element()
+                sage: u = x.subalgebra_generated_by().random_element()
+                sage: u.matrix()*u.vector() == (u**2).vector()
+                True
+
             """
             # First get the subspace spanned by the powers of myself...
             V = self.span_of_powers()
@@ -187,6 +248,9 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
                 # b1 is what we get if we apply that matrix to b1. The
                 # second row of the right multiplication matrix by b1
                 # is what we get when we apply that matrix to b2...
+                #
+                # IMPORTANT: this assumes that all vectors are COLUMN
+                # vectors, unlike our superclass (which uses row vectors).
                 for b_left in V.basis():
                     eja_b_left = self.parent()(b_left)
                     # Multiply in the original EJA, but then get the
@@ -324,6 +388,19 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
             """
             Find an idempotent in the associative subalgebra I generate
             using Proposition 2.3.5 in Baes.
+
+            TESTS::
+
+                sage: set_random_seed()
+                sage: J = eja_rn(5)
+                sage: c = J.random_element().subalgebra_idempotent()
+                sage: c^2 == c
+                True
+                sage: J = eja_ln(5)
+                sage: c = J.random_element().subalgebra_idempotent()
+                sage: c^2 == c
+                True
+
             """
             if self.is_nilpotent():
                 raise ValueError("this only works with non-nilpotent elements!")
@@ -353,6 +430,9 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
             # subspace... or do we? Can't we just solve, knowing that
             # A(c) = u^(s+1) should have a solution in the big space,
             # too?
+            #
+            # Beware, solve_right() means that we're using COLUMN vectors.
+            # Our FiniteDimensionalAlgebraElement superclass uses rows.
             u_next = u**(s+1)
             A = u_next.matrix()
             c_coordinates = A.solve_right(u_next.vector())
@@ -362,10 +442,6 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
             #
             # We need the basis for J, but as elements of the parent algebra.
             #
-            #
-            # TODO: this is buggy, but it's probably because the
-            # multiplication table for the subalgebra is wrong! The
-            # matrices should be symmetric I bet.
             basis = [self.parent(v) for v in V.basis()]
             return self.parent().linear_combination(zip(c_coordinates, basis))