Hirota equations associated with simply laced affine Lie algebras: The cuspidal class E₆ 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 |${\mathfrak {e}}_6$|$e6$⁠.