'Generate all the positive roots of A_n
This is half of all of the roots of F_4; by multiplying all the
positive roots by -1 we get the negative roots.
Examples
>>> from sympy.liealgebras.cartan_type import CartanType
>>> c = CartanType("A3")
>>> c.positive_roots()
{1: [1, -1, 0, 0], 2: [1, 0, -1, 0], 3: [1, 0, 0, -1], 4: [0, 1, -1, 0],
5: [0, 1, 0, -1], 6: [0, 0, 1, -1]}'

def positive_roots(self):

    n = self.n
    posroots = {}
    k = 0
    for i in range(0, (n - 1)):
        for j in range((i + 1), n):
            k += 1
            posroots[k] = self.basic_root(i, j)
            k += 1
            root = self.basic_root(i, j)
            root[j] = 1
            posroots[k] = root
    for i in range(0, n):
        k += 1
        root = ([0] * n)
        root[i] = 1
        posroots[k] = root
    k += 1
    root = ([Rational(1, 2)] * n)
    posroots[k] = root
    for i in range(1, 4):
        k += 1
        root = ([Rational(1, 2)] * n)
        root[i] = Rational((-1), 2)
        posroots[k] = root
    posroots[(k + 1)] = [Rational(1, 2), Rational(1, 2), Rational((-1), 2), Rational((-1), 2)]
    posroots[(k + 2)] = [Rational(1, 2), Rational((-1), 2), Rational(1, 2), Rational((-1), 2)]
    posroots[(k + 3)] = [Rational(1, 2), Rational((-1), 2), Rational((-1), 2), Rational(1, 2)]
    posroots[(k + 4)] = [Rational(1, 2), Rational((-1), 2), Rational((-1), 2), Rational((-1), 2)]
    return posroots
