Spinor-generators of compact exceptional Lie groups

TL;DR

This paper extends the concept of Euler's angle decomposition from SO(3) to complex Lie groups like SU(3), Sp(3), and exceptional groups F4, E6, E7, demonstrating that their elements can be expressed as products of elements from specific subgroups.

Contribution

It generalizes the idea of spinor-generators and Euler angle decompositions to higher-dimensional and exceptional Lie groups, providing explicit factorization results.

Findings

Any element of SU(3) and Sp(3) can be decomposed similarly to SO(3).

Elements of F4, E6, and E7 can be expressed as products of elements from certain Spin subgroups.

The results extend classical Euler angle decompositions to complex and 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}.