3-graded decompositions of exceptional Lie algebras \(\mathfrak g\) and group realizations of \(\mathfrak g_{\text{ev}}\), \(\mathfrak g_0\) and \(\mathfrak g_{\text{ed}}\). III: \(G=E_8\)
In [Jap. J. Math., New Ser. 26, No. 1, 31--50 (2000; Zbl 1040.17005), 24, No. 2, 257--296 (1998; Zbl 1016.17008), and 25, No. 1, 155--179 (1999; Zbl 1016.17009)], the authors completed the determination of group realizations \(\mathfrak {g}_{ev}\) and \(\mathfrak{g}_{0}\) of 2-graded decompositions \(\mathfrak g=\mathfrak g_{-2}\oplus \mathfrak g_{-1}\oplus \mathfrak g_0\oplus \mathfrak g_1\oplus \mathfrak g_2\) of exceptional Lie algebras \(\mathfrak g\) for the universal exceptional Lie groups. In the present article, which is a continuation of [J. Math. Kyoto Univ. 41, No. 3, 449--474 (2001; Zbl 1042.17004), J. Math. Kyoto Univ. 46, No. 2, 383--413 (2006; Zbl 1137.17010), 46, No. 4, 805--832 (2006; Zbl 1138.17004)], the authors determine group realizations of subalgebras \(\mathfrak{g}_{ev}, \mathfrak{g}_{0}\) and \(\mathfrak{g}_{ed}\) of 3-graded decompositions of exceptional Lie algebras \(\mathfrak{g}\) for the universal exceptional Lie groups of type \(E_{8}\). This completes the group realizations of Hara's table with respect to 3-graded decompositions of exceptional Lie algebras by the connected exceptional universal linear Lie groups.
scientific article; zbMATH DE number 5735951