## Equivalence of Borcherds G-vertex algebras and axiomatic vertex algebras (1999)

Citations: | 6 - 1 self |

### BibTeX

@MISC{Snydal99equivalenceof,

author = {Craig T. Snydal},

title = {Equivalence of Borcherds G-vertex algebras and axiomatic vertex algebras},

year = {1999}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper we build an abstract description of vertex algebras from their basic axioms. Starting with Borcherds ’ notion of a vertex group, we naturally construct a family of multilinear singular maps parameterised by trees. These singular maps are defined in a way which focusses on the relations of singularities to their inputs. In particular we show that this description of a vertex algebra allows us to present generalised notions of rationality, commutativity and associativity as natural consequences of the definition. Finally, we show that for a certain choice of vertex group, axiomatic vertex algebras correspond bijectively to algebras in the relaxed multilinear category of representations of a vertex group.

### Citations

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(Show Context)
Citation Context ... (cdot, y1)Y (·, x1)·. Then the locality condition reduces to the requirement that f be symmetric (this is equivalent to the rationality and commutativity of products of Frenkel, Huang, and Lepowsky, =-=[7]-=-). In fact, the refinement, V � V ����� V � V V �� x1 ����� � �� 0 z2� z1 �� −→ �� � � 0 ������ V � �� y1 �� 0 �� V is interpreted as giving a map from f to Y (Y (·, x1)·, y1)·, and thus formalises th... |

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(Show Context)
Citation Context .... For our purposes we take the definition due to Tom Leinster [13] which seems to arise most naturally when dealing with higher dimensional categories. (For possibly different definitions see [16] or =-=[5]-=-.) In this categorical structure, we define a single morphism from a tree q to a tree p exactly when p is a refinement of q. So in equation (A.1) there exists an arrow from the second tree to the firs... |

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16 |
Vertex algebras for beginners, volume 10 of University Lecture Series
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(Show Context)
Citation Context ...ning it closely and paying special attention to its interaction with the infinitesimal translation operator. Before beginning we recall the definition of a vertex algebra (following the definition in =-=[11]-=-). Definition 2.1. A vertex algebra consists of a complex vector space V (the state space) together with an endomorphism, T : V → V (the infinitesimal translation operator), a distinguished vector den... |

15 |
Vertex algebras
- Borcherds
- 1998
(Show Context)
Citation Context ...ns, and after considering composition of these singular functions we define spaces of singular functions parameterised by binary trees. Our definitions, while motivated by the Borcherds definition in =-=[3]-=-, are different because they emphasise the relation of singularities on inputs. For the classical vertex group, we demonstrate the correspondence between a binary singular functions and vertex operato... |

9 | General operads and multicategories - Leinster - 1998 |

7 |
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- 1999
(Show Context)
Citation Context ...ments of one another. There are a number of (possibly inequivalent) ways of giving the collection of all trees the structure of a category. For our purposes we take the definition due to Tom Leinster =-=[13]-=- which seems to arise most naturally when dealing with higher dimensional categories. (For possibly different definitions see [16] or [5].) In this categorical structure, we define a single morphism f... |

5 | On the D-module and formal-variable approaches to vertex algebras, in: Topics in Geometry - Huang, Lepowsky - 1996 |

5 | Meromorphic tensor categories
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(Show Context)
Citation Context ...category. For our purposes we take the definition due to Tom Leinster [13] which seems to arise most naturally when dealing with higher dimensional categories. (For possibly different definitions see =-=[16]-=- or [5].) In this categorical structure, we define a single morphism from a tree q to a tree p exactly when p is a refinement of q. So in equation (A.1) there exists an arrow from the second tree to t... |

1 | What is a vertex algebra? q-alg/9709033 - Borcherds - 1997 |

1 | Formal Groups and Applications, chapter 36 - Hazewinkel - 1978 |

1 |
Representations of vertex groups
- Snydal
(Show Context)
Citation Context ...ll allow us to naturally add the elementary vertex structure. This treatment will avoid a very general abstract treatment, instead aiming to provide the flavour of theory. The details can be found in =-=[15]-=-. Given any linear map between G-modules, f : A → B, we define a unique G-linear map ̂ f : A → HomR(G, B) by: ̂f(a)(g) = f(g·a). 8This pairing between linear maps and G-linear maps is actually an iso... |