A new construction of vertex algebras and quasi-modules for vertex algebras

Li, Haisheng

doi:10.1016/j.aim.2005.03.008

Cited by 56 publications

(110 citation statements)

References 20 publications

(29 reference statements)

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“…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.…”

Section: Introductionmentioning

confidence: 57%

Some Quantum Vertex Algebras of Zamolodchikov–faddeev Type

Karel

2009

Commun. Contemp. Math.

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Section: Introductionmentioning

confidence: 57%

Some Quantum Vertex Algebras of Zamolodchikov–faddeev Type

Karel

2009

Commun. Contemp. Math.

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“…The proof goes similarly as the proof of [L2,Lemma 3.3]. It is sufficient to consider only homogeneous elements a(z)…”

Section: Proposition 25 Definition 24 Is Independent Of the Choicementioning

confidence: 78%

Higher level vertex operators for $$U_q \left( \widehat{\mathfrak {sl}}_2\right) $$ U q sl ^ 2

Kožić

2017

Sel. Math. New Ser.

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Abstract. We study graded nonlocal q-vertex algebras and we prove that they can be generated by certain sets of vertex operators. As an application, we consider the family of graded nonlocal q-vertex algebras V c,1 , c ≥ 1, associated with the principal subspaces W (cΛ 0 ) of the integrable highest weight U q ( sl 2 )-modules L(cΛ 0 ). Using quantum integrability, we derive combinatorial bases for V c,1 and compute the corresponding character formulae. IntroductionIn their work [LP], J. Lepowsky and M. Primc found the, so-called integrability condition for the affine Kac-Moody Lie algebra sl 2 ,on a level c integrable sl 2 -module. In general, (0.1) holds on an arbitrary level c integrable g-module, when the simple root α is replaced by the maximal root of the untwisted affine Kac-Moody Lie algebra g. In this paper, we continue our research on vertex algebraic structures arising from FrenkelJing operators x ± 1 (z) for U q ( sl 2 ), which was initiated in [Ko3]. So far there were several fruitful approaches to associating vertex algebra-like theories with the various quantum objects, such as quantum affine algebras or Yangians, which resulted in some fundamental results and important constructions (cf. [AB],[B],[EK],[FR],[L1]-[L4]). However, motivated by the role of integrability (0.1) in the representation theory of the affine Kac-Moody Lie algebras, we introduce graded nonlocal q-vertex algebras, certain new structures which are designed to make use of quantum integrability (0.2).2000 Mathematics Subject Classification. 17B37 (Primary), 17B69 (Secondary).

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“…These were introduced by Li in a series of papers [Li06a,Li06,Li06b] under the name (G, φ)-coherent quasi-modules. We will find, as a by-product of this definition of Y W , an iterate formula for such quasi-modules: see Theorem 5.11 and Corollary 6.4.…”

Section: Theorem There Is a Linear Isomorphismmentioning

confidence: 99%

“…Following the terminology introduced in [Li06a,Li06] in the context of vertex algebras (which we recall below in §4.1), we shall sometimes refer to a vertex Lie algebra equipped with an action of Γ satisfying (2.22) as a Γ-vertex Lie algebra.…”

Section: Vertex Lie Algebras and 'Little' Lie Algebrasmentioning

confidence: 99%

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Vertex Lie algebras and cyclotomic coinvariants

Young

2017

Commun. Contemp. Math.

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Abstract. Given a vertex Lie algebra L equipped with an action by automorphisms of a cyclic group Γ, we define spaces of cyclotomic coinvariants over the Riemann sphere. These are quotients of tensor products of smooth modules over 'local' Lie algebras L(L )z i assigned to marked points zi, by the action of a 'global' Lie algebraOn the other hand, the universal enveloping vertex algebra V(L ) of L is itself a vertex Lie algebra with an induced action of Γ. This gives 'big' analogs of the Lie algebras above. From these we construct the space of 'big' cyclotomic coinvariants, i.e. coinvariants with respect to L Γ {z i } (V(L )). We prove that these two definitions of cyclotomic coinvariants in fact coincide, provided the origin is included as a marked point. As a corollary we prove a result on the functoriality of cyclotomic coinvariants which we require for the solution of cyclotomic Gaudin models in [VY14].At the origin, which is fixed by Γ, one must assign a module over the stable subalgebra. This module becomes a V(L )-quasi-module in the sense of Li. As a bi-product we obtain an iterate formula for such quasi-modules.

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A new construction of vertex algebras and quasi-modules for vertex algebras

Abstract: In this paper, a new construction of vertex algebras from more general vertex operators is given and a notion of quasimodule for vertex algebras is introduced and studied.

Cited by 56 publications

References 20 publications

Some Quantum Vertex Algebras of Zamolodchikov–faddeev Type

Some Quantum Vertex Algebras of Zamolodchikov–faddeev Type

Higher level vertex operators for $$U_q \left( \widehat{\mathfrak {sl}}_2\right) $$ U q sl ^ 2

Vertex Lie algebras and cyclotomic coinvariants

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