## Combinatorics of free vertex algebras

Venue: | J. Algebra |

Citations: | 9 - 5 self |

### BibTeX

@ARTICLE{Roitman_combinatoricsof,

author = {Michael Roitman},

title = {Combinatorics of free vertex algebras},

journal = {J. Algebra},

year = {},

pages = {0103173}

}

### OpenURL

### Abstract

This paper illustrates the combinatorial approach to vertex algebra — study of vertex algebras presented by generators and relations. A necessary ingredient of this method is the notion of free vertex algebra. Borcherds [2] was the first to note that free vertex algebras do not exist in general. The

### Citations

726 |
Infinite dimensional Lie algebras
- Kac
- 1990
(Show Context)
Citation Context ... A is a vacuum element. We prove next that the minimal monomials generate A over the extended Heisenberg algebra ̂ H. It will follow then by the representation theory of Heisenberg algebras (see e.g. =-=[10]-=-) that as a module over ̂ H the vertex algebra A = ⊕ w=µ(w) U( ̂ H)w is decomposed into a direct sum of irreducible highest weight modules generated by minimal monomials. Recall that by (5) a vertex a... |

285 |
On axiomatic approaches to vertex operator algebras and modules
- Frenkel, Huang, et al.
- 1993
(Show Context)
Citation Context ... ]] are local. This is not a minimal set of axioms, in fact conditions V2 and V3 can be weakened. Note that V3 implies that Y (Da) = d dz Y (a). For other axiomatic definitions of vertex algebras see =-=[2, 6, 8, 9]-=-. A homomorphism of two vertex algebras A and B is a map φ : A → B such that φ(1) = 1 and φ(a n b) = φ(a) n φ(b) for all n ∈ Z, in particular φ(Da) = d dz φ(a). A module over a vertex algebra A is a l... |

267 |
Vertex operator algebras and the Monster
- Frenkel, Lepowsky, et al.
- 1998
(Show Context)
Citation Context ...y related to the vertex algebras corresponding to integer lattices. The latter algebras play a very important role in different areas of mathematics and physics. They were extensively studied in e.g. =-=[5, 6, 9, 11, 14]-=-. Here we explore the relation between free vertex algebras and lattice vertex algebras in much detail. These results comply with the use of the word “free” in physical literature refering to some ele... |

202 |
The diamond lemma for ring theory
- Bergman
- 1978
(Show Context)
Citation Context ...o the argument in [15] are needed to show that ̂ R is confluent, i.e. the final result of applications of the rules does not depend on the order in which the rules are applied. Then the Diamond lemma =-=[1, 15]-=- would imply that ρT is a basis of F. However, in §12 we prove that ρT is linearly independent in F by a different method. 12. PROOF OF THEOREM 1, PART II AND PROOF OF THEOREM 2 Let F = FN(B) be a fre... |

174 |
Generalized vertex algebras and relative vertex operators
- Dong, Lepowsky
- 1993
(Show Context)
Citation Context ...y related to the vertex algebras corresponding to integer lattices. The latter algebras play a very important role in different areas of mathematics and physics. They were extensively studied in e.g. =-=[5, 6, 9, 11, 14]-=-. Here we explore the relation between free vertex algebras and lattice vertex algebras in much detail. These results comply with the use of the word “free” in physical literature refering to some ele... |

120 |
Vertex algebras, Kac-Moody algebras and the
- Borcherds
- 1986
(Show Context)
Citation Context ...es the combinatorial approach to vertex algebra — study of vertex algebras presented by generators and relations. A necessary ingredient of this method is the notion of free vertex algebra. Borcherds =-=[2]-=- was the first to note that free vertex algebras do not exist in general. The reason for this is that vertex algebras do not form a variety of algebras, as defined in e.g. [4], because the locality ax... |

111 | Local systems of vertex operators, vertex superalgebras and modules
- Li
- 1996
(Show Context)
Citation Context ... d dz α(z) = α(z) −2 1. We remark that the following identities hold: (Da) n b = −n a n−1 b, a n (Db) = n a n−1 b + D ( a n b ) , in particular, D is a derivation of the products n . The Dong’s lemma =-=[6, 11, 13]-=- states that if a, b, c ∈ F(V ) are three pairwise local fields, then for every n ∈ Z the fields a n b and c are local as well. In §7 we will prove a quantitative version of this lemma. A subspace A ⊂... |

97 |
Vertex algebras associated with even lattices
- Dong
- 1993
(Show Context)
Citation Context ...y related to the vertex algebras corresponding to integer lattices. The latter algebras play a very important role in different areas of mathematics and physics. They were extensively studied in e.g. =-=[5, 6, 9, 11, 14]-=-. Here we explore the relation between free vertex algebras and lattice vertex algebras in much detail. These results comply with the use of the word “free” in physical literature refering to some ele... |

27 | On free conformal and vertex algebras
- Roitman
- 1999
(Show Context)
Citation Context ...ubcategory of vertex algebras, obtained by restricting the order of locality of generators, has a universal object, which we call the free vertex algebra corresponding to the given locality bound. In =-=[15]-=- some free vertex algebras were constructed and in certain special cases their linear bases were found. In this paper we generalize the construction of [15] and find linear bases of an arbitrary free ... |

22 |
A New Family of Algebras Underlying the Rogers-Ramanujan
- Lepowsky, Wilson
- 1981
(Show Context)
Citation Context ...1 and 2. Remark. One can prove Lemma 3 in a different way, using the construction of VΛ by vertex operators, and the so-called method of Z-algebras, that originates to the work of Lepowsky and Wilson =-=[12]-=-, see also [5, 9, 14]. This remark is due to C. Dong. 11. PROOF OF THEOREM 1, PART I Let as before B be a set with a symmetric locality bound N : B × B → Z and let F = FN(B) be the corresponding free ... |

20 | Conformal modules
- Cheng, Kac
- 1997
(Show Context)
Citation Context ...d ω 0 ω = Dω, ω 1 ω = 2ω, ω 2 ω = 0, ω 3 ω = c. Another example of a conformal algebra will be given in §9.8 MICHAEL ROITMAN Conformal operators Consider first a k[D]-module V . A conformal operator =-=[3]-=- on V is a series α = ∑ n∈Z+ α(n)z−n−1 ∈ z −1 (gl V )[[z −1 ]] such that α(n)v = 0 α. For v ∈ V call N(α, v) = min {n ∈ Z+ | α(m)v = 0 ∀ m � n} the order of locality of α and v. Denote by cgl V the sp... |

16 |
Vertex algebras for beginners, volume 10 of University Lecture Series
- Kac
- 1998
(Show Context)
Citation Context |

10 |
Universal algebra, volume 6 of Mathematics and its Applications
- Cohn
- 1981
(Show Context)
Citation Context ...ertex algebra. Borcherds [2] was the first to note that free vertex algebras do not exist in general. The reason for this is that vertex algebras do not form a variety of algebras, as defined in e.g. =-=[4]-=-, because the locality axiom (see §2 below) is not an identity. However, a certain subcategory of vertex algebras, obtained by restricting the order of locality of generators, has a universal object, ... |

9 |
A characterization of vertex algebras associated to even lattices
- Li, Xu
- 1995
(Show Context)
Citation Context |

6 |
Universal enveloping conformal algebras
- Roitman
(Show Context)
Citation Context ...2 then states that ϕ is injective. We will prove this theorem in §12. In §7, using Theorem 2, we prove a quantitative version of Dong’s lemma. In §8 we apply this lemma to settle a question raised in =-=[16]-=-: we prove that the locality function of a free conformal algebra has quadratic growth. In §9 we study homogeneous conformal derivations of free vertex algebras. It turns out that a particularly inter... |

2 |
andW(glN) with central charge
- Frenkel, Kac, et al.
- 1995
(Show Context)
Citation Context ...balgebra W+ ⊂ ̂ W spanned by the coefficients pm(n) for n � 0 can be identified with the Lie algebra k 〈 p, t ∣ 〉[−] [t, p] = 1 of differential operators on the disk so that pm(n) = 1 m! pm t n , see =-=[7, 17]-=-. Using (11) and the fact that ε = 1 we calculate that pm(n)v1 = { (−1) m D (m−n) v1 if m � n, 0 otherwise. (18) Therefore, the inner conformal derivations (pm)+ ∈ cder VZ are homogeneous of weight 0 ... |

1 |
Vertex operator realizations of conformal algebras
- Roitman
- 2001
(Show Context)
Citation Context ...reducible lowest weight module over certain conformal algebra ̂ W ⊂ V0, such that the coefficient algebra of ̂ W is a central extension of the Lie algebra of differential operators on the circle, see =-=[7, 11, 17]-=-. Finally, in §10 we find a presentation of lattice vertex algebras in terms of generators and relations, see Theorem 4. It turns out that the required relations are rather minimal. Our proof is compl... |