Axiomatic $G_{1}$-vertex algebras
  Inspired by the Borcherds' work on ``$G$-vertex algebras,'' we formulate and study an axiomatic counterpart of Borcherds' notion of $G$-vertex algebra for the simplest nontrivial elementary vertex group, which we denote by $G_{1}$. Specifically, we formulate a notion of axiomatic $G_{1}$-vertex algebra, prove certain basic properties and give certain examples, where the notion of axiomatic $G_{1}$-vertex algebra is a nonlocal generalization of the notion of vertex algebra. We also show how to construct axiomatic $G_{1}$-vertex algebras from a set of compatible $G_{1}$-vertex operators.   The results of this paper were reported in June 2001, at the International Conference on Lie Algebras in the Morningside center, Beijing, China, and were reported on November 30, 2001, in the Quantum Mathematics Seminar, at Rutgers-New Brunswick. We noticed that a paper of Bakalov and Kac appeared today (math.QA/0204282) on noncommutative generalizations of vertex algebras, which has certain overlaps with the current paper. On the other hand, most of their results are orthogonal to the results of this paper. 
Haisheng Li