Realizations of inner automorphisms of order 4 and fixed points subgroups by them on the connected compact exceptional lie group $E_8$, Part I

Abstract

The compact simply connected Riemannian 4-symmetric spaces were classified by J. A. Jime ´nez as the type of Lie algebra. Needless to say, these spaces as homogeneous manifolds are of the form G=H, where G is a connected compact simple Lie group with an automorphism g g of oder 4 on G and H is a fixed points subgroup G g of G. In the present article, as Part I, for the connected compact exceptional Lie group E 8 , we give the explicit form of automorphism s s 0 4 of order 4 on E 8 induced by the C-linear transformation s 0 4 of 248-dimensional vector space e C 8 and determine the structure of the group ðE 8 Þ s 0 4 . This amounts to the global realization of one of seven cases with an automorphism of order 4 corresponding to the Lie algebra h ¼ soð6Þ l soð10Þ.

Realizations of inner automorphisms of order four and fixed points subgroups by them on the connected compact exceptional Lie group $E_8$, Part II

“…In this section (also in the next Section 6), again we use the 248-dimensional vector space e 8 C used in Case 1 ( [5]) and the connected compact exceptional Lie group of type E 8 constructed by T. Imai and I. Yokota ([1]).…”

“…In [5], the author showed the group realizations for Case 1 in Table . In the present article, we state the realizations of the group H for Case 2, 3 and 4. The remaining cases will be shown in a forthcoming article [6] by the author.…”

“…This article is a continuation of [5], hence we start from Section 4. We refer the reader to [5] for preliminary results and also to [4], [5], [7], [9] or [10] for notations.…”

“…This article is a continuation of [5], hence we start from Section 4. We refer the reader to [5] for preliminary results and also to [4], [5], [7], [9] or [10] for notations. Note that we change the numbering of Case 5 and Case 6 in [5] to the numbering of Case 3 and Case 4 in the present article, respectively.…”

“…We refer the reader to [5] for preliminary results and also to [4], [5], [7], [9] or [10] for notations. Note that we change the numbering of Case 5 and Case 6 in [5] to the numbering of Case 3 and Case 4 in the present article, respectively. In this case, we will study the connected compact exceptional Lie group of type E 8 constructed by S. Gomyo ([2]).…”