Quantum conformal algebra with central extension

The q-deformation of the infinite-dimensional n-algebra is investigated. Based on the structure of the q-deformed Virasoro-Witt algebra, we derive a nontrivial q-deformed Virasoro-Witt n-algebra which is nothing but a sh-n-Lie algebra. Furthermore in terms of the pseuddifferential operators on the quantum plane, we construct the (co)sine n-algebra and the q-deformed SDif f (T 2 ) n-algebra. We prove that they are the sh-n-Lie algebras for the case of even n. An explicit physical realization of the (co)sine n-algebra is given.

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“…By means of (16), one can confirm the following q-Jacobi identity [5] satisfied by the q-deformed V-W algebra (14):…”

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confidence: 85%

On q-deformed infinite-dimensional n-algebra

Ding

Jia

et al. 2016

Nuclear Physics B

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“…The investigations of various quantum deformations (or q-deformations) of Lie algebras began a period of rapid expansion in 1980's stimulated by introduction of quantum groups motivated by applications to the quantum Yang-Baxter equation, quantum inverse scattering methods and constructions of the quantum deformations of universal enveloping algebras of semi-simple Lie algebras. Since then several other versions of q-deformed Lie algebras have appeared, especially in physical contexts such as string theory, vertex models in conformal field theory, quantum mechanics and quantum field theory in the context of deformations of infinite-dimensional algebras, primarily the Heisenberg algebras, oscillator algebras and Witt and Virasoro algebras [3,16,17,18,19,21,22,23,28,30,41,42,43]. In these pioneering works it has been in particular descovered that in these q-deformations of Witt and Visaroro algebras and some related algebras, some interesting q-deformations of Jacobi identities, extanding Jacobi identity for Lie algebras, are satisfied.…”

Section: Introductionmentioning

confidence: 99%

Enveloping algebras of color hom-Lie algebras

Armakan¹,

Silvestrov²,

Farhangdoost³

2019

Turk J Math

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In this paper the universal enveloping algebra of color hom-Lie algebras is studied. A construction of the free involutive hom-associative color algebra on a hom-module is described and applied to obtain the universal enveloping algebra of an involutive hom-Lie color algebra. Finally, the construction is applied to obtain the well-known Poincaré-Birkhoff-Witt theorem for Lie algebras to the enveloping algebra of an involutive color hom-Lie algebra. of abstract quasi-Lie algebras and subclasses of quasi-Hom-Lie algebras and Hom-Lie algebras as well as their general colored (graded) counterparts in [25,36,39,58,37]. These generalized Lie algebra structures with (graded) twisted skew-symmetry and twisted Jacobi conditions by linear maps are tailored to encompass within the same algebraic framework such quasi-deformations and discretizations of Lie algebras of vector fields using σ-derivations, describing general descritizations and deformations of derivations with twisted Leibniz rule, and the well-known generalizations of Lie algebras such as color Lie algebras which are the natural generalizations of Lie algebras and Lie superalgebras.Quasi-Lie algebras are non-associative algebras for which the skew-symmetry and the Jacobi identity are twisted by several deforming twisting maps and also the Jacobi identity in quasi-Lie and quasi-Hom-Lie algebras in general contains six twisted triple bracket terms. Hom-Lie algebras is a special class of quasi-Lie algebras with the bilinear product satisfying the non-twisted skew-symmetry property as in Lie algebras, whereas the Jacobi identity contains three terms twisted by a single linear map, reducing to the Jacobi identity for ordinary Lie algebras when the linear twisting map is the identity map. Subsequently, hom-Lie admissible algebras have been considered in [45] where also the hom-associative algebras have been introduced and shown to be hom-Lie admissible natural generalizations of associative algebras corresponding to hom-Lie algebras. In [45], moreover several other interesting classes of hom-Lie admissible algebras generalising some non-associative algebras, as well as examples of finite-dimentional hom-Lie algebras have been described. Since these pioneering works [25,36,39,37,40,45], hom-algebra structures have become a popular area with increasing number of publications in various directions.Hom-Lie algebras, hom-Lie superalgebras and hom-Lie color algebras are important special classes of color (Γ-graded) quasi-Lie algebras introduced first by Larsson and Silvestrov in [39,37]. Hom-Lie algebras and hom-Lie superalgebras have been studied further in different aspects by Makhlouf, Silvestrov, Sheng, Ammar, Yau and other authors [62,61,60,45,48,46,47,68,12,44,69,64,65,66,38,51,52,57,59], and hom-Lie color algebras have been considered for example in [68,14,13,1]. In [4], the constructions of Hom-Lie and quasi-hom Lie algebras based on twisted discretizations of vector fields [25] and Hom-Lie admissible algebras have been extended to Hom-Lie superalgebras, a subclass...

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“…The q-deformations of Witt and Virasoro algebras are obtained when the derivation is replaced by a σ-derivation. It was observed in the pioneering works (See [5]- [8]). Motivated by these examples and their generalization, Hartwig, Larsson and Silvestrov introduced the Hom-Lie algebras when they concerned about the q-deformations of Witt and Virasoro algebras in [4].…”

Section: Introductionmentioning

confidence: 99%

Yetter-Drinfeld modules for weak Hom-Hopf algebras

Guo¹,

Ke²

2017

Filomat

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The aim of this paper is to define and study Yetter-Drinfeld modules over weak Hom-Hopf algebras. We show that the category H WYD H of Yetter-Drinfeld modules with bijective structure maps over weak Hom-Hopf algebras is a rigid category and a braided monoidal category, and obtain a new solution of quantum Hom-Yang-Baxter equation. It turns out that, If H is quasitriangular (respectively, coquasitriangular)weak Hom-Hopf algebras, the category of modules (respectively, comodules) with bijective structure maps over H is a braided monoidal subcategory of the category H WYD H of Yetter-Drinfeld modules over weak Hom-Hopf algebras.

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Quantum conformal algebra with central extension

Cited by 92 publications

References 15 publications

On q-deformed infinite-dimensional n-algebra

On q-deformed infinite-dimensional n-algebra

Enveloping algebras of color hom-Lie algebras

Yetter-Drinfeld modules for weak Hom-Hopf algebras

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