AXIOMATIC G₁-VERTEX ALGEBRAS

Abstract

Inspired by 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 G1. Specifically, we formulate a notion of axiomatic G1-vertex algebra, prove certain basic properties and give certain examples, where the notion of axiomatic G1-vertex algebra is a nonlocal generalization of the notion of vertex algebra. We also show how to construct axiomatic G1-vertex algebras from a set of compatible G1-vertex operators.

Some Quantum Vertex Algebras of Zamolodchikov–faddeev Type

“…For a(x), b(x) ∈ T with the above information, we define It was proved in [Li6] that any S-local subset of E(W ) generates a weak quantum vertex algebra with W as a canonical module. This particular result generalizes a result of [Li1], which states that for any vector space W , every local subset of E(W ) generates a vertex algebra with W as a module (see [Li2], [Li3], [Li7], and [Li10] for similar results). Regarding quantum vertex algebras, a variant, which was formulated in [Li6], of ( [EK], Proposition 1.11), is that if a weak quantum vertex algebra V is nondegenerate in the sense of [EK], V is a quantum vertex algebra with S(x) uniquely determined.…”

Constructing Quantum Vertex Algebras

“…Denote by ι x 1 ,x 2 the natural embedding of C * (x 1 , x 2 ) into the field C((x 1 ))((x 2 )). Throughout this paper, nonlocal vertex algebras are synonymous to G 1 -vertex algebras studied in [Li3] and they are also essentially field algebras studied in [BK] (cf. [K]).…”

“…[BK], [Li3]). We prove that this rather trivial nonlocal vertex algebra actually is a nondegenerate quantum vertex algebra with a nontrivial quantum Yang-Baxter operator.…”

Associativity of Field Algebras

“…Examples also include the quantum vertex algebras of Etingof and Kazhdan [6], and the smash product of a vertex algebra and its finite automorphism group [7]. Bakalov and Kac defined field algebras using formal calculus in [7], and Li studied similar notions in [8,9], although there are differences in terminology.…”

“…Our definition of a field algebra is slightly different from those given in [7][8][9], but it is easy to establish equivalence using only the identity and associativity axioms. We extend the notion of associativity to fields of several variables, treating the ordinary fields as a special case.…”