arXiv subjects
Publications and source records attributed to Ichiro Yokota.
We know that any element $X$ of the exceptional Jordan algebra $\gJ$ is transformed to a diagonal form by the compact exceptional Lie group $F_4$. However, its proof is used the method which is reduced a contradiction. In this paper, we give a direct and constructive proof.
Let J be the exceptional Jordan algebra over R and J^C its complexification. Then the simply connected compact exceptional Lie group F_4 acts on J and F_4 has three orbit types which are F_4/F_4, F_4/Spin(9), F_4/Spin(8). Similarly the simply connected compact exceptional Lie group E_6 acts on J^C and E_6 has five orbit types which are E_6/E_6, E_6/F_4, E_6/Spin(10), E_6/Spin(9), E_6/Spin(8) . In this paper, we determine the orbit types of the simply connected compact exceptional Lie group E_7 in the complex Freudenthal vector space P^C. As results, E_7 has seven orbit types which are E_7/E_7, E_7/E_6, E_7/F_4, E_7/Spin(11), E_7/Spin(10), E_7/Spin(9), E_7/Spin(8).
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}.
We describe simply connected compact exceptional simple Lie groups in very elementary way. We first construct all simply connected compact exceptional Lie groups G concretely. Next, we find all involutive automorphisms of G, and determine the group structures of the fixed points subgroup. They correspond to the classification of all irreducible compact symmetric spaces of exceptional type, and that they also correspond to classification of all non-compact exceptionalsimple Lie groups. Finally, we determined the group structures of the maximal subgroups of maximal rank. At any rate, we would like this book to be used in mathematics and physics.