Hirota equations associated with simply laced affine Lie algebras: The cuspidal class E 6 of ${\mathfrak {e}}_6^{(1)}$e6(1)

In this paper we derive Hirota equations associated with the simply laced affine Lie algebras ${\mathfrak {g}}^{(1)}$g(1), where ${\mathfrak {g}}$g is one of the simply laced complex Lie algebras ${\mathfrak {a}}_n, {\mathfrak {d}}_n, {\mathfrak {e}}_6, {\mathfrak {e}}_7$an,dn,e6,e7 or ${\mathfrak {e}}_8$e8, defined by finite order automorphisms of ${\mathfrak {g}}$g which we call Lepowsky automorphisms. In particular, we investigate the Hirota equations for Lepowsky automorphisms of ${\mathfrak {e}}_6$e6 defined by the cuspidal class E6 of the Weyl group W(E6) of ${\mathfrak {e}}_6$e6. We also investigate the relationship between the Lepowsky automorphisms of the simply laced complex Lie algebras ${\mathfrak {g}}$g and the conjugate canonical automorphisms defined by Kac. This analysis is applied to identify the canonical automorphisms for the cuspidal class E6 of ${\mathfrak {e}}_6$e6.