3-graded decompositions of exceptional Lie algebras \(\mathfrak g\) and group realizations of \(\mathfrak g_{ev},\mathfrak g_0\) and \(\mathfrak g_{ed}\). I: \(G=G_2,F_4,E_6\)
Let \(\mathfrak g\) be a simple Lie algebra such that \(\mathfrak g = \sum_{k=-3}^3 \mathfrak g_k\); \([\mathfrak g_i; \mathfrak g_j] \subset \mathfrak g_{i+j}\). This is known as a 3-graded decomposition of \(\mathfrak g\). Let \(\mathfrak g_{ev} = \mathfrak g_{-2} \oplus \mathfrak g_0 \oplus \mathfrak g_2\) and \(\mathfrak g_{ed} = \mathfrak g_{-3} \oplus \mathfrak g_0 \oplus \mathfrak g_3\). The author reports that M. Hara has classified the 3-graded decompositions of simple Lie algebras \(\mathfrak g\) and determined the types of subalgebras \(\mathfrak g_{ev}; \mathfrak g_0\); and \(\mathfrak g_{ed}\). He gives a table listing Hara's results for \(\mathfrak g\) of types \(G_2; F_4\); and \(E_6\). He then determines the subgroups \(G_{ev}; G_0; G_{ed}\) of the exceptional Lie groups \(G\) of types \(G_2; F_4\); and \(E_6\) corresponding to the subalgebras \(\mathfrak g_{ev}; \mathfrak g_0; \mathfrak g_{ed}\) of \(\mathfrak G = \text{Lie} ~G\).
scientific article; zbMATH DE number 1803228