def positive_roots(self):
        """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]}

        """

        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
dataset/ETHPy150Open sympy/sympy/sympy/liealgebras/type_f.py/TypeF.positive_roots/original