## On free conformal and vertex algebras (1999)

Venue: | J. Algebra |

Citations: | 27 - 8 self |

### BibTeX

@ARTICLE{Roitman99onfree,

author = {Michael Roitman},

title = {On free conformal and vertex algebras},

journal = {J. Algebra},

year = {1999}

}

### OpenURL

### Abstract

Vertex algebras and conformal algebras have recently attracted a lot of attention due to their connections with physics and Moonshine representations of the Monster. See, for example, [6], [10], [15], [17], [19]. In this paper we describe bases of free conformal and free vertex algebras (as introduced in [6], see also [20]). All linear spaces are over a field k of characteristic 0. Throughout this paper Z+ will stand for the set of non-negative integers. In §1 and §2 we give a review of conformal and vertex algebra theory. All statements is these sections are either in [9], [15], [16], [17], [18], [20] or easily follow from results therein. In §3 we investigate free conformal and vertex algebras. 1. Conformal algebras 1.1. Definition of conformal algebras. We first recall some basic definitions and constructions, see [16], [17], [18], [20]. The main object of investigation is defined as follows: Definition 1.1. A Conformal algebra is a linear space C endowed with a linear operator D: C → C and a sequence of bilinear products ○n: C ⊗ C → C, n ∈ Z+, such that for any a, b ∈ C one has (i) (locality) There is a non-negative integer N = N(a, b) such that a ○n b = 0 for any n � N; (ii) D(a ○n b) = (Da) ○n b + a ○n (Db); (iii) (Da) ○n b = −na n−1 b. 1.2. Spaces of power series. Now let us discuss the main motivation for the Definition 1.1. We closely follow [14] and [18]. 1.2.1. Circle products. Let A be an algebra. Consider the space of power series A[[z, z −1]]. We will write series a ∈ A[[z, z −1]] in the form a(z) = ∑ a(n)z −n−1, a(n) ∈ A. n∈Z On A[[z, z−1]] there is an infinite sequence of bilinear products ○n, n ∈ Z+, given by n a ○n b (z) = Resw a(w)b(z)(z − w) ). (1.1) Explicitly, for a pair of series a(z) = ∑ n∈Z a(n)z−n−1 and b(z) = ∑ n∈Z b(n)z−n−1 we have −m−1 a ○n b (z) = a ○n b (m)z, where

### Citations

730 |
Infinite dimensional Lie algebras
- Kac
- 1990
(Show Context)
Citation Context ...b = 0 for any n � N; (ii) D(a ○n b) = (Da) ○n b + a ○n (Db); (iii) (Da) ○n b = −na n−1 b. 1.2. Spaces of power series. Now let us discuss the main motivation for the Definition 1.1. We closely follow =-=[14]-=- and [18]. 1.2.1. Circle products. Let A be an algebra. Consider the space of power series A[[z, z −1 ]]. We will write series a ∈ A[[z, z −1 ]] in the form a(z) = ∑ a(n)z −n−1 , a(n) ∈ A. n∈Z On A[[z... |

421 |
Infinite conformal symmetry in two-dimensional quantum field theory
- Belavin, Polyakov, et al.
- 1984
(Show Context)
Citation Context ...his product is denoted by “ (n)”. 1.2.2. Locality. Next we define a very important property of power series, which makes them form a conformal algebra. Let again A be an algebra. Definition 1.2. (See =-=[1]-=-, [15], [17], [18], [20]) A series a ∈ A[[z, z −1 ]] is called local of order N to b ∈ A[[z, z −1 ]] for some N ∈ Z+ if a(w)b(z)(z − w) N = 0. (1.4) If a is local to b and b is local to a then we say ... |

286 |
Y.-Z.Huang and J.Lepowsky, On axiomatic approaches to vertex operator algebras and modules
- Frenkel
- 1993
(Show Context)
Citation Context ...istic 0. Throughout this paper Z+ will stand for the set of non-negative integers. In §1 and §2 we give a review of conformal and vertex algebra theory. All statements is these sections are either in =-=[9]-=-, [15], [16], [17], [18], [20] or easily follow from results therein. In §3 we investigate free conformal and vertex algebras. 1. Conformal algebras 1.1. Definition of conformal algebras. We first rec... |

270 |
J.Lepowsky and A.Meurman, Vertex operator algebras and the monster
- Frenkel
- 1988
(Show Context)
Citation Context ...0 Sep 1998 Vertex algebras and conformal algebras have recently attracted a lot of attention due to their connections with physics and Moonshine representations of the Monster. See, for example, [6], =-=[10]-=-, [15], [17], [19]. In this paper we describe bases of free conformal and free vertex algebras (as introduced in [6], see also [20]). All linear spaces are over a field k of characteristic 0. Througho... |

203 |
The diamond lemma for ring theory
- Bergman
- 1978
(Show Context)
Citation Context ...is an inverse map ϕ−1 : A → Â, and therefore ϕ is an isomorphism. 3.3. The Diamond Lemma. For the future purposes we need a digression on the Diamond Lemma for associative algebras. We closely follow =-=[2]-=-, but use more modern terminology. Let X be some alphabet and K be some commutative ring. Consider the free associative algebra K 〈X〉 of non-commutative polynomials with coefficients in K. Denote by X... |

123 |
On theories with a combinatorial definition of ”equivalence
- Newman
- 1942
(Show Context)
Citation Context ... system is confluent if and only if it is confluent on all the ambiguities, that is, for any ambiguity v ∈ X∗ there is the unique terminal t ∈ K 〈X〉 such that v −→ t. Remark. Statement (a) appears in =-=[21]-=-. A variant of Lemma 3.2 appears in [3] and [4]. It was also known to Shirshov (see [25]). The name “Diamond” is due to the following graphical description of the confluency property, see [21]. Let R ... |

121 |
Vertex algebras, Kac-Moody algebras and the
- Borcherds
(Show Context)
Citation Context ...QA] 10 Sep 1998 Vertex algebras and conformal algebras have recently attracted a lot of attention due to their connections with physics and Moonshine representations of the Monster. See, for example, =-=[6]-=-, [10], [15], [17], [19]. In this paper we describe bases of free conformal and free vertex algebras (as introduced in [6], see also [20]). All linear spaces are over a field k of characteristic 0. Th... |

41 |
Some algorithmic problem for Lie algebras
- Shirshov
(Show Context)
Citation Context ... any ambiguity v ∈ X∗ there is the unique terminal t ∈ K 〈X〉 such that v −→ t. Remark. Statement (a) appears in [21]. A variant of Lemma 3.2 appears in [3] and [4]. It was also known to Shirshov (see =-=[25]-=-). The name “Diamond” is due to the following graphical description of the confluency property, see [21]. Let R be a rewriting system in sense of Definition 3.1 (a), and let “−→” be defined as above. ... |

35 |
Free differential calculus IV, the quotient .groups of the lower central series
- Chen, Fox, et al.
- 1958
(Show Context)
Citation Context ... words in X introduce a (lexicographic) order as follows: if u is a prefix of v then u > v, otherwise u > v whenever for some index i one has ui > vi and uj = vj for all j < i. Definition 3.3. ([25], =-=[7]-=-) A word v ∈ X ∗ is called Lyndon-Shirshov if it is bigger than all its proper suffices. Proposition 3.2. (a) There is a Hall set HLS such that α(HLS) is the set of all Lyndon-Shirshov words and α : T... |

30 |
Imbeddings into simple associative algebras, Algebra i Logika
- Bokut
(Show Context)
Citation Context ...ent on all the ambiguities, that is, for any ambiguity v ∈ X∗ there is the unique terminal t ∈ K 〈X〉 such that v −→ t. Remark. Statement (a) appears in [21]. A variant of Lemma 3.2 appears in [3] and =-=[4]-=-. It was also known to Shirshov (see [25]). The name “Diamond” is due to the following graphical description of the confluency property, see [21]. Let R be a rewriting system in sense of Definition 3.... |

28 | distribution algebras and conformal algebras
- Kac
(Show Context)
Citation Context ...roughout this paper Z+ will stand for the set of non-negative integers. In §1 and §2 we give a review of conformal and vertex algebra theory. All statements is these sections are either in [9], [15], =-=[16]-=-, [17], [18], [20] or easily follow from results therein. In §3 we investigate free conformal and vertex algebras. 1. Conformal algebras 1.1. Definition of conformal algebras. We first recall some bas... |

24 |
Unsolvability of the word problem, and subalgebras of finitely presented Lie algebras
- Bokut
- 1972
(Show Context)
Citation Context ...s confluent on all the ambiguities, that is, for any ambiguity v ∈ X∗ there is the unique terminal t ∈ K 〈X〉 such that v −→ t. Remark. Statement (a) appears in [21]. A variant of Lemma 3.2 appears in =-=[3]-=- and [4]. It was also known to Shirshov (see [25]). The name “Diamond” is due to the following graphical description of the confluency property, see [21]. Let R be a rewriting system in sense of Defin... |

24 |
A basis for free Lie rings and higher commutators in free groups
- Hall
(Show Context)
Citation Context ...bra generated by B with respect to the constant locality N, and L = Coeff C(N). A basis of the Lie algebra L could be obtained by modifying the construction of a Hall basis of a free Lie algebra, see =-=[12]-=-, [23], [24]. Here we review the latter construction. We closely follow [22], except that all the order relations are reversed. As in §3.3, take an alphabet X and a commutative ring K. Let T(X) be the... |

24 | Commutative quantum operator algebras - LIAN, ZUCKERMAN - 1995 |

17 |
On free Lie rings
- Shirshov
(Show Context)
Citation Context ...nerated by B with respect to the constant locality N, and L = Coeff C(N). A basis of the Lie algebra L could be obtained by modifying the construction of a Hall basis of a free Lie algebra, see [12], =-=[23]-=-, [24]. Here we review the latter construction. We closely follow [22], except that all the order relations are reversed. As in §3.3, take an alphabet X and a commutative ring K. Let T(X) be the set o... |

14 |
Free Lie algebras, volume 7 of London Mathematical Society Monographs. New Series
- Reutenauer
- 1993
(Show Context)
Citation Context ...N). A basis of the Lie algebra L could be obtained by modifying the construction of a Hall basis of a free Lie algebra, see [12], [23], [24]. Here we review the latter construction. We closely follow =-=[22]-=-, except that all the order relations are reversed. As in §3.3, take an alphabet X and a commutative ring K. Let T(X) be the set of all binary trees with leaves from X. For typographical reasons we wi... |

10 |
Universal algebra, volume 6 of Mathematics and its Applications
- Cohn
- 1981
(Show Context)
Citation Context ...ith the identities (1.7). This is indeed the case: D ( (Da)(n) + na(n − 1) ) = −n ( (Da)(n − 1) + (n − 1)a(n − 2) ) . n 1.6. Varieties of conformal algebras. Consider now a variety of algebras A (see =-=[8]-=-, [13]). Definition 1.3. A conformal algebra C is a A-conformal algebra if Coeff C lies in the variety A. The identities in A-conformal algebras are all the circle-products identities R such that for ... |

7 |
Grobner]Shirshov bases for relations of a Lie algebraÈ and its enveloping algebra, preprint
- Bokut, Malcolmson
- 1998
(Show Context)
Citation Context ...ery term v that appears during this process must satisfy properties 1 and 2 and there are only finitely many such terms. Remark. Altrernatively we could use the theorem of L. Bokut’ and P. Malcolmson =-=[5]-=-. As in (b) of Theorem 3.1, we deduce that all the elements of ϕ(λ(Hterm)) containing only symbols from X0 form a basis of L+. Note that we have an algorithm for building a basis of the free Lie confo... |

5 |
Vertex operator algebras associated to repersentations of affine and Virasoro algebras
- Frenkel, Zhu
- 1992
(Show Context)
Citation Context ...(M). Then ψ(C) ⊂ F(M) consists of pairwise local fields, and by Dong’s Lemma 1.2, ψ(C) together with 1 - ∈ F(M) generates a vertex algebra SM ⊂ F(M). The following proposition is well-known, see e.g. =-=[11]-=-. Proposition 2.2. (a) The vertex algebra S = SM has a structure of a highest weight module over L with the highest weight vector 1 - . The action is given by a(n)β = ψ(a) ○n β, Moreover this action a... |

3 |
On the bases of a free Lie algebras, Algebra i Logika
- Shirshov
(Show Context)
Citation Context ...d by B with respect to the constant locality N, and L = Coeff C(N). A basis of the Lie algebra L could be obtained by modifying the construction of a Hall basis of a free Lie algebra, see [12], [23], =-=[24]-=-. Here we review the latter construction. We closely follow [22], except that all the order relations are reversed. As in §3.3, take an alphabet X and a commutative ring K. Let T(X) be the set of all ... |

1 |
The idea of locality. Wigner medal acceptance speech
- Kac
- 1996
(Show Context)
Citation Context ...1998 Vertex algebras and conformal algebras have recently attracted a lot of attention due to their connections with physics and Moonshine representations of the Monster. See, for example, [6], [10], =-=[15]-=-, [17], [19]. In this paper we describe bases of free conformal and free vertex algebras (as introduced in [6], see also [20]). All linear spaces are over a field k of characteristic 0. Throughout thi... |

1 |
Vertex Algebras for Beginners, volume 10
- Kac
- 1997
(Show Context)
Citation Context ...ertex algebras and conformal algebras have recently attracted a lot of attention due to their connections with physics and Moonshine representations of the Monster. See, for example, [6], [10], [15], =-=[17]-=-, [19]. In this paper we describe bases of free conformal and free vertex algebras (as introduced in [6], see also [20]). All linear spaces are over a field k of characteristic 0. Throughout this pape... |

1 | Moonshine cohomology
- Lian, Zuckerman
- 1995
(Show Context)
Citation Context ...algebras and conformal algebras have recently attracted a lot of attention due to their connections with physics and Moonshine representations of the Monster. See, for example, [6], [10], [15], [17], =-=[19]-=-. In this paper we describe bases of free conformal and free vertex algebras (as introduced in [6], see also [20]). All linear spaces are over a field k of characteristic 0. Throughout this paper Z+ w... |