def METHOD_NAME(self, index):
    r"""
    The antipode on the basis element indexed by ``index``.
    INPUT:
    - ``index`` -- an element of the index set
    For a graded connected Hopf algebra, we can define
    an antipode recursively by
    .. MATH::
        S(x) := -\sum_{x^L \neq x} S(x^L) \times x^R
    when `|x| > 0`, and by `S(x) = x` when `|x| = 0`.
    TESTS::
        sage: # needs sage.modules
        sage: H = GradedHopfAlgebrasWithBasis(QQ).Connected().example()
        sage: H.monomial(0).antipode()  # indirect doctest
        P0
        sage: H.monomial(1).antipode()  # indirect doctest
        -P1
        sage: H.monomial(2).antipode()  # indirect doctest
        P2
        sage: H.monomial(3).antipode()  # indirect doctest
        -P3
    """
    if self.monomial(index) == self.one():
        return self.one()
    S = self.METHOD_NAME
    x__S_Id = tensor([self, self]).module_morphism(
        lambda ab: S(ab[0]) * self.monomial(ab[1]),
        codomain=self)
    return -x__S_Id(
        self.monomial(index).coproduct()
        - tensor([self.monomial(index), self.one()])
    )