from .cartan_type import Standard_Cartan from sympy.core.backend import eye, Rational class TypeE(Standard_Cartan): def __new__(cls, n): if n < 6 or n > 8: raise ValueError("Invalid value of n") return Standard_Cartan.__new__(cls, "E", n) def dimension(self): """Dimension of the vector space V underlying the Lie algebra Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType("E6") >>> c.dimension() 8 """ return 8 def basic_root(self, i, j): """ This is a method just to generate roots with a -1 in the ith position and a 1 in the jth position. """ root = [0]*8 root[i] = -1 root[j] = 1 return root def simple_root(self, i): """ Every lie algebra has a unique root system. Given a root system Q, there is a subset of the roots such that an element of Q is called a simple root if it cannot be written as the sum of two elements in Q. If we let D denote the set of simple roots, then it is clear that every element of Q can be written as a linear combination of elements of D with all coefficients non-negative. This method returns the ith simple root for E_n. Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType("E6") >>> c.simple_root(2) [1, 1, 0, 0, 0, 0, 0, 0] """ n = self.n if i == 1: root = [-0.5]*8 root[0] = 0.5 root[7] = 0.5 return root elif i == 2: root = [0]*8 root[1] = 1 root[0] = 1 return root else: if i in (7, 8) and n == 6: raise ValueError("E6 only has six simple roots!") if i == 8 and n == 7: raise ValueError("E7 has only 7 simple roots!") return self.basic_root(i - 3, i - 2) def positive_roots(self): """ This method generates all the positive roots of A_n. This is half of all of the roots of E_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 if n == 6: posroots = {} k = 0 for i in range(n-1): for j in range(i+1, n-1): k += 1 root = self.basic_root(i, j) posroots[k] = root k += 1 root = self.basic_root(i, j) root[i] = 1 posroots[k] = root root = [Rational(1, 2), Rational(1, 2), Rational(1, 2), Rational(1, 2), Rational(1, 2), Rational(-1, 2), Rational(-1, 2), Rational(1, 2)] for a in range(0, 2): for b in range(0, 2): for c in range(0, 2): for d in range(0, 2): for e in range(0, 2): if (a + b + c + d + e)%2 == 0: k += 1 if a == 1: root[0] = Rational(-1, 2) if b == 1: root[1] = Rational(-1, 2) if c == 1: root[2] = Rational(-1, 2) if d == 1: root[3] = Rational(-1, 2) if e == 1: root[4] = Rational(-1, 2) posroots[k] = root return posroots if n == 7: posroots = {} k = 0 for i in range(n-1): for j in range(i+1, n-1): k += 1 root = self.basic_root(i, j) posroots[k] = root k += 1 root = self.basic_root(i, j) root[i] = 1 posroots[k] = root k += 1 posroots[k] = [0, 0, 0, 0, 0, 1, 1, 0] root = [Rational(1, 2), Rational(1, 2), Rational(1, 2), Rational(1, 2), Rational(1, 2), Rational(-1, 2), Rational(-1, 2), Rational(1, 2)] for a in range(0, 2): for b in range(0, 2): for c in range(0, 2): for d in range(0, 2): for e in range(0, 2): for f in range(0, 2): if (a + b + c + d + e + f)%2 == 0: k += 1 if a == 1: root[0] = Rational(-1, 2) if b == 1: root[1] = Rational(-1, 2) if c == 1: root[2] = Rational(-1, 2) if d == 1: root[3] = Rational(-1, 2) if e == 1: root[4] = Rational(-1, 2) if f == 1: root[5] = Rational(1, 2) posroots[k] = root return posroots if n == 8: posroots = {} k = 0 for i in range(n): for j in range(i+1, n): k += 1 root = self.basic_root(i, j) posroots[k] = root k += 1 root = self.basic_root(i, j) root[i] = 1 posroots[k] = root root = [Rational(1, 2), Rational(1, 2), Rational(1, 2), Rational(1, 2), Rational(1, 2), Rational(-1, 2), Rational(-1, 2), Rational(1, 2)] for a in range(0, 2): for b in range(0, 2): for c in range(0, 2): for d in range(0, 2): for e in range(0, 2): for f in range(0, 2): for g in range(0, 2): if (a + b + c + d + e + f + g)%2 == 0: k += 1 if a == 1: root[0] = Rational(-1, 2) if b == 1: root[1] = Rational(-1, 2) if c == 1: root[2] = Rational(-1, 2) if d == 1: root[3] = Rational(-1, 2) if e == 1: root[4] = Rational(-1, 2) if f == 1: root[5] = Rational(1, 2) if g == 1: root[6] = Rational(1, 2) posroots[k] = root return posroots def roots(self): """ Returns the total number of roots of E_n """ n = self.n if n == 6: return 72 if n == 7: return 126 if n == 8: return 240 def cartan_matrix(self): """ Returns the Cartan matrix for G_2 The Cartan matrix matrix for a Lie algebra is generated by assigning an ordering to the simple roots, (alpha[1], ...., alpha[l]). Then the ijth entry of the Cartan matrix is (). Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType('A4') >>> c.cartan_matrix() Matrix([ [ 2, -1, 0, 0], [-1, 2, -1, 0], [ 0, -1, 2, -1], [ 0, 0, -1, 2]]) """ n = self.n m = 2*eye(n) i = 3 while i < n-1: m[i, i+1] = -1 m[i, i-1] = -1 i += 1 m[0, 2] = m[2, 0] = -1 m[1, 3] = m[3, 1] = -1 m[2, 3] = -1 m[n-1, n-2] = -1 return m def basis(self): """ Returns the number of independent generators of E_n """ n = self.n if n == 6: return 78 if n == 7: return 133 if n == 8: return 248 def dynkin_diagram(self): n = self.n diag = " "*8 + str(2) + "\n" diag += " "*8 + "0\n" diag += " "*8 + "|\n" diag += " "*8 + "|\n" diag += "---".join("0" for i in range(1, n)) + "\n" diag += "1 " + " ".join(str(i) for i in range(3, n+1)) return diag