The Lie conformal algebra of loop Virasoro algebra, denoted by C W , is introduced in this paper. Explicitly, C W is a Lie conformal algebra withThen conformal derivations of C W are determined. Finally, rank one conformal modules and Z-graded free intermediate series modules over C W are classified.

In this article, we study the structure theory of a class of generalized loop Virasoro algebras. In particular, the derivation algebras, the automorphism groups, and the second cohomology groups of generalized loop centerless Virasoro algebras are determined.

For any complex parameters a, b, let [Formula: see text] 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 [Formula: see text] conformal algebra for some a, b and its conformal module of rank one.

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

Lie bialgebra structures on Lie algebras of generalized Weyl type are studied. They are shown to be triangular coboundary.

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,...

The first cohomology group of a generalized loop Virasoro algebra with coefficients in the tensor product of its adjoint module is shown to be trivial. The result is applied to prove that Lie bialgebra structures on generalized loop Virasoro algebras are coboundary triangular. We then generalize the results to generalized map Virasoro algebras.

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