Hirota equations associated with simply laced affine Lie algebras: The cuspidal class E 6 of ${\mathfrak {e}}_6^{(1)}$ e 6 ( 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 ${\mathfrak {a}}_n, {\mathfrak {d}}_n, {\mathfrak {e}}_6, {\mathfrak {e}}_7$ a n , d n , e 6 , e 7 or ${\mathfrak {e}}_8$ e 8 , 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 ${\mathfrak {e}}_6$ e 6 defined by the cuspidal class E 6 of the Weyl group W(E 6) of ${\mathfrak {e}}_6$ e 6 . 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 E 6 of ${\mathfrak {e}}_6$ e 6 .