Twisted $Gamma$-Lie algebras and their vertex operator representations

Let $Gamma$ be a generic subgroup of the multiplicative group $mathbb{C}^*$ of nonzero complex numbers. We define a class of Lie algebras associated to $Gamma$, called twisted $Gamma$-Lie algebras, which is a natural generalization of the twisted affine Lie algebras. Starting from an arbitrary even sublattice $Q$ of $mathbb Z^N$ and an arbitrary finite order isometry of $mathbb Z^N$ preserving $Q$, we construct a family of twisted $Gamma$-vertex operators acting on generalized Fock spaces which afford irreducible representations for certain twisted $Gamma$-Lie algebras. As application, this recovers a number of known vertex operator realizations for infinite dimensional Lie algebras, such as twisted affine Lie algebras, extended affine Lie algebras of type $A$, trigonometric Lie algebras of series $A$ and $B$, unitary Lie algebras, and $BC$-graded Lie algebras.