docstr: This method generates all the positive roots of
  A_n.  This is half of all of the roots of D_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]}
docstr_tok: ["This","method","generates","all","the","positive","roots","of","A_n",".","This","is","half","of","all","of","the","roots","of","D_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","]","}"]
code: def positive_roots(self):
      """
      This method generates all the positive roots of
      A_n.  This is half of all of the roots of D_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]}
      """
  
      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
      return posroots
  
file_path: sympy/sympy/liealgebras/type_d.py
imports: from .cartan_type import Standard_Cartan
  from sympy.core.backend import eye
label_desc: Use image node_id 5 for calling the TypeD obj's underlying member method code with example usage: obj.positive_roots() and returns: posroots