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In this article, we study extensions of conformal modules over Lie conformal algebras W (a , b , r) with three parameters a, b, r ∈ C , which contain the Virasoro Lie conformal algebra Vir, W (b) and W (a , b) as subalgebras. We obtain the existence conditions of extensions of nontrivial conformal modules over W (a , b , r) and give a complete classification of extensions of finite irreducible conformal modules of W (a , b , r) . [ABSTRACT FROM AUTHOR]

In this paper, by studying the maximal good subspaces, we determine the dual Lie coalgebras of the centerless twisted Heisenberg–Virasoro algebra. Based on this, we construct the dual Lie bialgebras structures of the twisted Heisenberg–Virasoro type. As by-products, four new infinite dimensional Lie algebras are obtained. [ABSTRACT FROM AUTHOR]

In this paper, all Lie bialgebra structures on the derivation Lie algebra W over a rank d ≥ 3 quantum torus associated to q are considered, where q is a d × d matrix with all the entries being roots of unity. They are shown to be triangular coboundary. As a byproduct, it is also proved that the first cohomology group H1 (W, W ⊗ W) is trivial. [ABSTRACT FROM AUTHOR]

Recently, Lie bialgebra structures of the derivation Lie algebra over quantum torus were determined. In this paper, we use the general method of quantization by a Drinfel'd twist to quantize these algebras with their Lie bialgebra structures and present the explicit formulae. [ABSTRACT FROM AUTHOR]

In a previous paper by the authors, we obtain the first example of a finitely freely generated simple Z -graded Lie conformal algebra of linear growth that cannot be embedded into any general Lie conformal algebra. In this paper, we obtain, as a byproduct, another class of such Lie conformal algebras by classifying Z -graded simple Lie conformal algebras G = ⊕ i = − 1 ∞ G i satisfying the following, (1) G 0 ≅ Vir , the Virasoro conformal algebra; (2) Each G i for i ≥ − 1 is a Vir-module of rank one. These algebras include some Lie conformal algebras of Block type. [ABSTRACT FROM AUTHOR]

Let W (Γ) be a class of not-finitely graded Lie algebras related to generalized Virasoro algebras with basis { L α , i , C   |   α ∈ Γ ,   i ∈ Z + } , which satisfies relations [ L α , i , L β , j ] = (β - α) L α + β , i + j + (j - i) L α + β , i + j + 1 + δ α + β , 0 δ i + j , 0 α 3 - α 12 C and [ C , L α , i ] = 0. In this paper, W (Γ) -modules of the intermediate series satisfying some conditions are constructed and classified. We also obtain modules of the intermediate series over the related Lie superalgebra. [ABSTRACT FROM AUTHOR]

In this paper, the structures of dual Lie bialgebras of twisted Schrödinger-Virasoro type are investigated. By studying the maximal good subspaces, we determine the dual Lie coalgebras of the twisted Schrödinger-Virasoro algebras. Then based on this, we construct the dual Lie bialgebra structures of this type. As by-products, four new infinite dimensional Lie algebras are obtained. [ABSTRACT FROM AUTHOR]

Let be the Lie algebra of Block type with basis {Li,j | i,j ∈ ℤ} and relations [Li,j, Lk,l] = ((j + 1)k - (l + 1)i)Li+k,j+l. Since is ℤ2-graded, it is natural to study ℤ2-graded modules over . In this paper, ℤ2-graded -modules of the intermediate series are classified. [ABSTRACT FROM AUTHOR]

For any complex parameters a, b, let be the Lie algebra with basis {Li,Hi | i ∈ ℤ} and relations [Li,Lj]=(j-i)Li+j, [Li,Hj]=(a+j+bi)Hi+j and [Hi,Hj]=0. In this paper, we construct the conformal algebra for some a, b and its conformal module of rank one. [ABSTRACT FROM AUTHOR]

Let L be a Lie algebra of Block type over ℂ with basis {Lα,i | α,i ∈ ℤ} and brackets [Lα,i,Lβ,j]=(β(i+1)-α(j+1)) Lα+β,i+j. In this paper, we first construct a formal distribution Lie algebra of L. Then we decide its conformal algebra B with ℂ[∂]-basis {Lα(w) | α ∈ ℤ} and λ-brackets [Lα(w)λ Lβ(w)]= (α∂+(α+β)λ) Lα+β(w). Finally, we give a classification of free intermediate series B-modules. [ABSTRACT FROM AUTHOR]

The first cohomology group of generalized loop Virasoro algebras with coefficients in the tensor product of its adjoint module is shown to be trivial. The result is used to prove that Lie bialgebra structures on generalized loop Virasoro algebras are coboundary triangular. The authors generalize the results to generalized map Virasoro algebras. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a class of Z -graded conformal algebras which is induced by Heisenberg-Virasoro conformal algebra. More precisely, we classify Z -graded conformal algebras L = ⊕ i = − 1 ∞ L i satisfying the following conditions, (C1) L 0 is the Heisenberg-Virasoro conformal algebra; (C2) Each L i for i ∈ Z ≥ − 1 ⁎ is an L 0 -module of rank one; (C3) [ X − 1 λ X i ] ≠ 0 for i ≥ 0 , where X i is any one of C [ ∂ ] -generators of L i for i ∈ Z ≥ − 1. Further, we prove that all finite nontrivial irreducible modules of these algebras under some special conditions are free of rank one as a C [ ∂ ] -module. The conformal derivations of this class of graded Lie conformal algebras are also determined. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a filtered Lie conformal algebra whose associated graded conformal algebra is isomorphic to that of general conformal algebra . In this paper, we prove that or (the associated graded conformal algebra of ), by making use of some results on the second cohomology groups of the conformal algebra with coefficients in its module of rank 1, where is the semi-direct sum of the Virasoro conformal algebra Vir with its module . Furthermore, we prove that does not have a nontrivial representation on a finite -module, this provides an example of a finitely freely generated simple Lie conformal algebra of linear growth that cannot be embedded into the general conformal algebra for any N. [Copyright &y& Elsevier]

Lie bialgebra structures on Lie algebras of generalized Weyl type are studied. They are shown to be triangular coboundary. [ABSTRACT FROM AUTHOR]