A few years ago, Borcherds introduced a notion of $G$-vertex
algebra, generalizing the notion of vertex algebra. In the
notion of $G$-vertex algebra, $G$ is what Borcherds called an
elementary vertex group and a $G$-vertex algebra was defined
to be an associative algebra in a relaxed multilinear category
associated to $G$. The formulation of the notion of $G$-vertex
algebra looks (at least superficially) quite different from the
axiomatic formulation of the notion of ordinary vertex algebra.
In this talk, I will talk on axiomatic $G$-vertex algebras of
certain types and present some basic results.
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