  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.
