Lie bialgebra structures on the twisted Heisenberg–Virasoro algebra

Liu, Dong; Pei, Yufeng; Zhu, Linsheng

doi:10.1016/j.jalgebra.2012.03.009

Cited by 21 publications

(13 citation statements)

References 27 publications

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“…Then Claim 4 follows. This subclaim can be found in Remark 1 of [13]. This subclaim can be proved similar to the proof of Theorem 4.5 (i) of [13].…”

Section: Lemma 33 Every Lie Bialgebra On the Witt Algebra W And Virasupporting

confidence: 61%

“…This subclaim can be found in Remark 1 of [13]. This subclaim can be proved similar to the proof of Theorem 4.5 (i) of [13]. According to Subclaim 1 and Subclaim 2, we know that ϕ s λ (L n ) = ϕ s λ (M n ) = 0 for any n ∈ Z and ϕ s λ ∈ H 1 (L s λ , H s λ /L C s λ ) when s = 1 2 , which implies ϕ s λ (Y 1 2 ) = 0.…”

Section: Lemma 33 Every Lie Bialgebra On the Witt Algebra W And Viramentioning

confidence: 76%

“…The Lie bialgebra structures on the original case L 1 2 0 (with c = 0) and a twisted case L 0 −3 (with c = 0) were respectively investigated by [6] and [3]. In this paper we shall investigate Lie bialgebra structures on L s λ mainly using the techniques introduced recently in [13]. §2.…”

Section: §1 Introductionmentioning

confidence: 99%

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Lie bialgebra structures on the deformative Schrödinger-Virasoro algebras

Zheng

2015

Journal of Mathematical Physics

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“…Then Claim 4 follows. This subclaim can be found in Remark 1 of [13]. This subclaim can be proved similar to the proof of Theorem 4.5 (i) of [13].…”

Section: Lemma 33 Every Lie Bialgebra On the Witt Algebra W And Virasupporting

confidence: 61%

Section: Lemma 33 Every Lie Bialgebra On the Witt Algebra W And Viramentioning

confidence: 76%

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Lie bialgebra structures on the deformative Schrödinger-Virasoro algebras

Zheng

2015

Journal of Mathematical Physics

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“…one can easily verify that the linear map ̺ : L0 → L0 ⊗ L0 defined below is a derivation:̺(L n ) = (nα + γ)I 0 ⊗ I n + (nα † + γ † )I n ⊗ I 0 , ̺(I n ) = βI 0 ⊗ I n + β † I n ⊗ I 0 , n ∈ Z .Denote D the vector space spanned by the such elements ̺ over C . From Theorem 3.2 and Corollary 4.5 in[13], we have following propositions.Proposition 2.3. [13] H 1 (L0, L0 ⊗ L0) = D.…”

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confidence: 93%

Lie superbialgebra structures on the N=2 superconformal Neveu–Schwarz algebra

Dong

Chen

Zhu

2012

Journal of Geometry and Physics

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“…Notice that the center of H is spanned by {C 0 := I 0 , C i | i = 1, 2, 3}. Moreover, the twisted Heisenberg-Virasoro algebra has a triangular decomposition:The twisted Heisenberg-Virasoro algebra is one of the most important Lie algebras both in mathematics and in mathematical physics, whose structure theory has extensively studied (see, e.g., [7,10,20]). A fundamental problem in the representation theory of the twisted Heisenberg-Virasoro algebra is to classify all its irreducible modules.…”

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confidence: 99%

Two classes of non-weight modules over the twisted Heisenberg–Virasoro algebra

Chen

Han

et al. 2018

manuscripta math.

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In the present paper, we construct two classes of non-weight modules Ω(λ, α, β) ⊗ Ind(M ) and M V, Ω(λ, α, β) over the twisted Heisenberg-Virasoro algebra, which are both associated with the modules Ω (λ, α, β). We present the necessary and sufficient conditions under which modules in these two classes are irreducible and isomorphic, and also show that the irreducible modules in these two classes are new. Finally, we construct non-weight modules Ind y,λ (C RS ) and Ind z,λ (C P Q ) over the twisted Heisenberg-Virasoro algebra and then apply the established results to give irreducible conditions for Ind y,λ (C RS ) and Ind z,λ (C P Q ).[L m , I n ] = nI m+n + δ m+n,0 (m 2 + m)C 2 ,Clearly, the subspaces spanned by {I m , C 3 | 0 = m ∈ Z} and by {L m , C 1 | m ∈ Z} are respectively the Heisenberg algebra and the Virasoro algebra. Notice that the center of H is spanned by {C 0 := I 0 , C i | i = 1, 2, 3}. Moreover, the twisted Heisenberg-Virasoro algebra has a triangular decomposition:The twisted Heisenberg-Virasoro algebra is one of the most important Lie algebras both in mathematics and in mathematical physics, whose structure theory has extensively studied (see, e.g., [7,10,20]).A fundamental problem in the representation theory of the twisted Heisenberg-Virasoro algebra is to classify all its irreducible modules. In fact, the theory of weight modules with all weight subspaces being finite dimensional (namely, Harish-Chandra-modules) is welldeveloped. Irreducible weight modules over H with a nontrivial finite dimensional weight subspace were proved to be Harish-Chandra modules [22]. And irreducible Harish-Chandra H-modules were classified in [15], each of which was shown to be either the highest (or lowest) weight module, or the module of intermediate series, consistent with the well-known result for Virasoro algebra [16]. While weight modules with an infinite dimensional weight subspace were also studied (see [6,19]).Non-weight modules constitute the other important ingredients of the representation theory of H, the study of which is definitely necessary and became popular in the last few years. A large class of new non-weight irreducible H-modules were constructed in [3], which includes the highest weight modules and Whittaker modules. Non-weight H-modules whose restriction to the universal enveloping algebra of the degree-0 part (modulo center) are free of rank 1 were studied in [4] (see also [8]). While by twisting the weight modules, the authors [6] obtained a family of new non-weight irreducible H-modules. However, the theory of non-weight H-modules is far more from being well-developed.As a continuation of [6], we still study the representation theory of H in this paper. But we shall be concerned with non-weight H-modules. To be more precisely, we construct two classes of new irreducible non-weight H-modules in the present paper, which are both related to the modules Ω(λ, α, β) (see [4]). It is well known that an important way of constructing modules is to consider the linear tensor product of two modules,...

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Lie bialgebra structures on the twisted Heisenberg–Virasoro algebra

Cited by 21 publications

References 27 publications

Lie bialgebra structures on the deformative Schrödinger-Virasoro algebras

Lie bialgebra structures on the deformative Schrödinger-Virasoro algebras

Lie superbialgebra structures on the N=2 superconformal Neveu–Schwarz algebra

Two classes of non-weight modules over the twisted Heisenberg–Virasoro algebra

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