'This method generates all the positive roots of
A_n.  This is half of all of the roots of A_n;
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):
        for j in range((i + 1), (n + 1)):
            k += 1
            posroots[k] = self.basic_root(i, j)
    return posroots
