+ fmt = "Euclidean Jordan algebra of degree {} over {}"
+ return fmt.format(self.degree(), self.base_ring())
+
+ def rank(self):
+ """
+ Return the rank of this EJA.
+ """
+ if self._rank is None:
+ raise ValueError("no rank specified at genesis")
+ else:
+ return self._rank
+
+
+ class Element(FiniteDimensionalAlgebraElement):
+ """
+ An element of a Euclidean Jordan algebra.
+ """
+
+ def __pow__(self, n):
+ """
+ Return ``self`` raised to the power ``n``.
+
+ 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: x = random_eja().random_element()
+ sage: x.matrix()*x.vector() == (x**2).vector()
+ True
+
+ """
+ A = self.parent()
+ if n == 0:
+ return A.one()
+ elif n == 1:
+ return self
+ else:
+ return A.element_class(A, (self.matrix()**(n-1))*self.vector())
+
+
+ def characteristic_polynomial(self):
+ """
+ Return my characteristic polynomial (if I'm a regular
+ element).
+
+ Eventually this should be implemented in terms of the parent
+ algebra's characteristic polynomial that works for ALL
+ elements.
+ """
+ if self.is_regular():
+ return self.minimal_polynomial()
+ else:
+ raise NotImplementedError('irregular element')
+
+
+ def det(self):
+ """
+ Return my determinant, the product of my eigenvalues.
+
+ EXAMPLES::
+
+ sage: J = eja_ln(2)
+ sage: e0,e1 = J.gens()
+ sage: x = e0 + e1
+ sage: x.det()
+ 0
+ sage: J = eja_ln(3)
+ sage: e0,e1,e2 = J.gens()
+ sage: x = e0 + e1 + e2
+ sage: x.det()
+ -1
+
+ """
+ cs = self.characteristic_polynomial().coefficients(sparse=False)
+ r = len(cs) - 1
+ if r >= 0:
+ return cs[0] * (-1)**r
+ else:
+ raise ValueError('charpoly had no coefficients')
+
+
+ def is_nilpotent(self):
+ """
+ Return whether or not some power of this element is zero.
+
+ The superclass method won't work unless we're in an
+ associative algebra, and we aren't. However, we generate
+ an assocoative subalgebra and we're nilpotent there if and
+ only if we're nilpotent here (probably).
+
+ TESTS:
+
+ The identity element is never nilpotent::
+
+ sage: set_random_seed()
+ sage: random_eja().one().is_nilpotent()
+ False
+
+ The additive identity is always nilpotent::
+
+ sage: set_random_seed()
+ sage: random_eja().zero().is_nilpotent()
+ True
+
+ """
+ # The element we're going to call "is_nilpotent()" on.
+ # Either myself, interpreted as an element of a finite-
+ # dimensional algebra, or an element of an associative
+ # subalgebra.
+ elt = None
+
+ if self.parent().is_associative():
+ elt = FiniteDimensionalAlgebraElement(self.parent(), self)
+ else:
+ V = self.span_of_powers()
+ assoc_subalg = self.subalgebra_generated_by()
+ # Mis-design warning: the basis used for span_of_powers()
+ # and subalgebra_generated_by() must be the same, and in
+ # the same order!
+ elt = assoc_subalg(V.coordinates(self.vector()))
+
+ # Recursive call, but should work since elt lives in an
+ # associative algebra.
+ return elt.is_nilpotent()
+
+
+ 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 degree(self):
+ """
+ Compute the degree of this element the straightforward way
+ according to the definition; by appending powers to a list
+ and figuring out its dimension (that is, whether or not
+ they're linearly dependent).
+
+ EXAMPLES::
+
+ sage: J = eja_ln(4)
+ sage: J.one().degree()
+ 1
+ sage: e0,e1,e2,e3 = J.gens()
+ sage: (e0 - e1).degree()
+ 2
+
+ In the spin factor algebra (of rank two), all elements that
+ aren't multiples of the identity are regular::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,10).abs()
+ sage: J = eja_ln(n)
+ sage: x = J.random_element()
+ sage: x == x.coefficient(0)*J.one() or x.degree() == 2
+ True
+
+ """
+ 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 minimal_polynomial(self):
+ """
+ EXAMPLES::
+
+ sage: set_random_seed()
+ sage: x = random_eja().random_element()
+ sage: x.degree() == x.minimal_polynomial().degree()
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: x = random_eja().random_element()
+ sage: x.degree() == x.minimal_polynomial().degree()
+ True
+
+ The minimal polynomial and the characteristic polynomial coincide
+ and are known (see Alizadeh, Example 11.11) for all elements of
+ the spin factor algebra that aren't scalar multiples of the
+ identity::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(2,10).abs()
+ sage: J = eja_ln(n)
+ sage: y = J.random_element()
+ sage: while y == y.coefficient(0)*J.one():
+ ....: y = J.random_element()
+ sage: y0 = y.vector()[0]
+ sage: y_bar = y.vector()[1:]
+ sage: actual = y.minimal_polynomial()
+ sage: x = SR.symbol('x', domain='real')
+ sage: expected = x^2 - 2*y0*x + (y0^2 - norm(y_bar)^2)
+ sage: bool(actual == expected)
+ True
+
+ """
+ # The element we're going to call "minimal_polynomial()" on.
+ # Either myself, interpreted as an element of a finite-
+ # dimensional algebra, or an element of an associative
+ # subalgebra.
+ elt = None