Spinor-generators of compact exceptional Lie groups

Abstract

We know that any element A of the group SO(3) can be represented as A = A1 A2 A1', where A1, A1' are elements of SO1(2)={A is an element of SO(3) | Ae1=e1}, and SO2(2)={A is an element of SO(3) | Ae2=e2} . This fact is known as Euler's angle. When this situation, a matrix A is called the generator. In the present paper, we shall show firstly that the similar results hold for the groups SU(3), and Sp(3). Secondly, we shall show that any element g of the simply connected compact Lie group F4 (respectively. E6) can be represented g= g1 g2 g1', where g1, g1' are elements of Spin1(9), g2 is an element of Spin2(9) (respectively g1, g1' are elements of Spin1(10), g2 is an element of Spin2(10)), where Spink(9) = {g is an element of F4 | g Ek = Ek} (respectively Spink(10) = {g is an element of E6 | g Ek = Ek}. Lastly, we shall show that any element g of the simply connected compact Lie group E7 can be represented as g = g1 g2 g1 ' g2 ' g1", where g1, g1', g1 " are elements of Spin1(12), g2, g2' are elements of Spin2(12), where Spink(12) = {g is an element of E7 | g K(k)= K(k) g, g M(k) = M(k) g}.