  We construct two associative algebras from a vertex operator algebra $V$ and
a general automorphism $g$ of $V$. The first, called $g$-twisted zero-mode
algebra, is a subquotient of what we call $g$-twisted universal enveloping
algebra of $V$. These algebras are generalizations of the corresponding
algebras introduced and studied by Frenkel-Zhu and Nagatomo-Tsuchiya in the
(untwisted) case that $g$ is the identity. The other is a generalization of the
$g$-twisted version of Zhu's algebra for suitable $g$-twisted modules
constructed by Dong-Li-Mason when the order of $g$ is finite. We are mainly
interested in $g$-twisted $V$-modules introduced by the first author in the
case that $g$ is of infinite order and does not act on $V$ semisimply. In this
case, twisted vertex operators in general involve the logarithm of the
variable. We construct functors between categories of suitable modules for
these associative algebras and categories of suitable (logarithmic) $g$-twisted
$V$-modules. Using these functors, we prove that the $g$-twisted zero-mode
algebra and the $g$-twisted generalization of Zhu's algebra are in fact
isomorphic.
