Twisted modules for vertex algebras associated with vertex algebroids

Li, Haisheng; Yamskulna, Gaywalee

doi:10.2140/pjm.2007.229.199

Cited by 9 publications

(5 citation statements)

References 21 publications

(29 reference statements)

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“…Now, we recall the definition of vertex algebroid homomorphism (cf. [LiY2]). Let A, A ′ be unital commutative associative algebras, and let B be a vertex A-algebroid, B ′ be a vertex A ′ -algebroid.…”

Section: Vertex Algebroids Associated With Simple Lie Algebrasmentioning

confidence: 99%

“…[GMS]). Also, the classification of graded simple (twisted)modules of vertex algebras associated with vertex algebroid were studied in [LiY1,LiY2] by Li and one of authors of this paper. Recently, Jitjankarn and one of authors of this paper investigated vertex algebroids associated with (semi)simple Leibniz algebras that have the simple Lie algebra sl 2 as their Levi factor ( [JY1]), and constructed an indecomposable non-simple C 2 -cofinite vertex algebra from a certain vertex algebroid such that its Levi factor is isomorphic to sl 2 .…”

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Vertex algebroids and Conformal vertex algebras associated with simple Leibniz algebras

Bui¹,

Yamskulna²

2020

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Section: Vertex Algebroids Associated With Simple Lie Algebrasmentioning

confidence: 99%

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confidence: 99%

Vertex algebroids and Conformal vertex algebras associated with simple Leibniz algebras

Bui¹,

Yamskulna²

2020

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“…It was shown in [GMS] that for a given vertex A-algebroid B, one can construct an N-graded vertex algebra V = ⊕ ∞ n=0 V n such that V 0 = A and V 1 = B. Also, the classification of graded simple twisted and non-twisted modules of vertex algebras associated with vertex algebroids were studied in [LiY1,LiY2] by Li and the last author of this paper.…”

Section: Introductionmentioning

confidence: 99%

On $\mathbb{N}$-graded vertex algebras associated with cyclic Leibniz algebras with small dimensions

Barnes¹,

Martin²,

J.³

et al. 2023

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The main goals for this paper is i) to study of an algebraic structure of Ngraded vertex algebras VB associated to vertex A-algebroids B when B are cyclic non-Lie left Leibniz algebras, and ii) to explore relations between the vertex algebras VB and the rank one Heisenberg vertex operator algebra. To achieve these goals, we first classify vertex A-algebroids B associated to given cyclic non-Lie left Leibniz algebras B. Next, we use the constructed vertex A-algebroids B to create a family of indecomposable non-simple vertex algebras VB. Finally, we use the algebraic structure of the unital commutative associative algebras A that we found to study relations between a certain type of the vertex algebras VB and the vertex operator algebra associated to a rank one Heisenberg algebra.

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“…In [LY1], Li and Yamskulna classify all the N-graded simple modules of those vertex algebras in terms of simple modules for certain Lie algebroids. In [LY2], they construct and classify graded simple twisted modules for those vertex algebras. In addition, they determine the full automorphism groups of those vertex algebras in terms of the automorphism groups of the corresponding vertex algebroids.…”

Section: Introductionmentioning

confidence: 99%

Semi-conformal structure on certain vertex superalgebras associated to vertex superalgebroids

2019

Front. Math. China

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In this paper, we first give the definiton of a vertex superalgebroid. Then we construct a family of vertex superalgebras associated to vertex superalgebroids. As a main result, we find a sufficient and necessary condition that this vertex superalgebras are semi-conformal. In addition, we give an concrete example of this vertex superalgebras and apply our results to this superalgebra. 1 and necessary condition that the associated vertex superalgebras are semi-conformal. In addition, we give an concrete example of such vertex superalgebra and apply our results to this superalgebra. This paper is organized as follows. In Section 2, we review some basic notations, formulas and properties for Lie superalgebras and vertex superalgebras. We give the definitions of 1-truncated conformal superalgebra, vertex superalgebroid and Lie superalgebroid, which are generalization of these algebras in non-super version. We also review tools from [GMS] and [LY1], then we construct an N-graded vertex superalgebra for a given vertex superalgebroid. In Section 3, we recall the definition of semi-conformal vertex superalgebra and find a sufficient and necessary condition that the given vertex superalgebras are semi-conformal. In Section 4, we give an example of this family of vertex superalgebras. PreliminariesWe use the usual symbols Z, Z + and N for the set of integers, the positive integers, and the nonnegative integers respectively.Let M = M0⊕M1 be any superalgebra, i.e., (Z/2Z)-graded algebra. Any element u in M0 (resp. M1) is said to be even (resp. odd). For any homogeneous element u, we define |u| = 0 if u is even, |u| = 1 if u is odd. We define ε u,v = (−1) |u||v| , for any homogeneous elements u, v ∈ M . We note that, the space of endomorphisms of M , denoted by End(M ) is a superalgebra.Throughout this paper, when we write |u| for an element u ∈ M , we will always implicitly assume that u is a homogeneous element. Definition 2.1. A Lie superalgebra is a superalgebra A = A0⊕A1 with multiplication [·, ·] satisfying the following two axioms: for homogeneous a, b, c ∈ A, skew − supersymmetry : [a, b] = −ε a,b [b, a]. super Jacobi identity : [a, [b, c]] = [[a, b], c] + ε a,b [b, [a, c]].Example 2.2. Let A = A0 ⊕ A1 be an associative superalgebra. A becomes a Lie superalgebra with(2.1)Let A be a Lie superalgebra. Then End(A) is an associative superalgebra, and hence it carries a structure of Lie superalgebra by (2.1).Definition 2.4. Let A = A0 ⊕ A1 be an associative superalgebra. An endomorphism D ∈ End(A) s is called a derivation of degree s, if it satisfies the identity D(ab) = D(a)b + (−1) s|a| aD(b), a, b ∈ A.(2.2)Denote by Der(A) s the space of derivations on A of degree s.

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Twisted modules for vertex algebras associated with vertex algebroids

Cited by 9 publications

References 21 publications

Vertex algebroids and Conformal vertex algebras associated with simple Leibniz algebras

Vertex algebroids and Conformal vertex algebras associated with simple Leibniz algebras

On $\mathbb{N}$-graded vertex algebras associated with cyclic Leibniz algebras with small dimensions

Semi-conformal structure on certain vertex superalgebras associated to vertex superalgebroids

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