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Let d ≥ 1 be an integer and let 풲 d be the Witt algebra. For any admissible 풲 d -module P and any 픤 ⁢ 픩 d -module V, one can form a 풲 d -module ℱ ⁢ (P , V) , which as a vector space is P ⊗ V . Since 풲 d has a natural subalgebra isomorphic to 픰 ⁢ 픩 d + 1 , we can view ℱ ⁢ (P , V) as an 픰 ⁢ 픩 d + 1 -module. Taking P = Ω ⁢ (흀) , the rank-1 U ⁢ (픥) -free 풲 d -module, and V = V ⁢ (퐚 , b) , the simple cuspidal module over 픤 ⁢ 픩 d , we get the special 픰 ⁢ 픩 d + 1 -modules ℱ ⁢ (흀 ; 퐚 , b) = ℱ ⁢ (Ω ⁢ (흀) , V ⁢ (퐚 , b)) which are U ⁢ (픥) -free modules of infinite rank. We determine the necessary and sufficient condition for the 픰 ⁢ 픩 d + 1 -module ℱ ⁢ (흀 ; 퐚 , b) to be simple, and for the non-simple case we construct their proper submodules explicitly. At last, using the above results, we deduce an explicit simplicity criterion for the generalized Verma modules induced from V ⁢ (퐚 , b) and obtain a family of simple affine modules from ℱ ⁢ (흀 ; 퐚 , b) , which can be viewed as the non-weight version of loop modules. [ABSTRACT FROM AUTHOR]

The concept of -differential operators is a natural generalization of differential operators and difference operators. In this paper, we determine the -differential Lie algebraic structure on the Witt algebra and the Virasoro algebra for invertible. Then we consider several families of modules over the Virasoro algebra with explicit module actions and determine the -differential module structures on them. [ABSTRACT FROM AUTHOR]

We construct a class of modules over the affine Lie algebra sl ˆ 2 from the modules over the degree-2 Weyl algebra K 2 and modules over the 2-dimensional solvable Lie algebra b. We determine the irreducibility and isomorphisms of these modules with examples given by Bavula's construction of modules for generalized Weyl algebras in [5]. Finally, we show these irreducible sl ˆ 2 -modules are generally new. [ABSTRACT FROM AUTHOR]

In Tang and Bai [Math Nachr. 2012;285:922–935] classified a class of non-graded left-symmetric algebraic structures on the Witt algebra under a certain rational condition. In this note, we show that this rational condition is not necessary. This leads to a more elegant classification of the left-symmetric algebraic structures and Novikov algebraic structures on the Witt algebra. [ABSTRACT FROM AUTHOR]

In this paper, we study Whittaker modules over the loop Virasoro algebra relative to some total order. We give a description of all Whittaker vectors for the universal Whittaker modules. We also show that any universal Whittaker module admits a unique simple quotient modules except for a special case. [ABSTRACT FROM AUTHOR]

Truncated Virasoro algebras can be viewed as quotient algebras of the loop-Virasoro algebra. In this paper, we prove that any weight module M with a finite dimensional weight space over any truncated Virasoro algebra L(f)=Virx(C[t±1]/fC[t±1]) has a Harish-Chandra submodule, where Vir is the Virasoro algebra and f is a nonzero and not invertible Laurent polynomial in C[t±1]. More precisely, if M has a nontrivial upper bounded Vir-submodule, then it has a highest weight L(f)-submodule; if M has a nontrivial lower bounded Vir-submodule, then it has a lowest weight L(f)-submodule; if M is nontrivial over L(f) and does not contain any nontrivial upper bounded nor any nontrivial lower bounded Vir-submodules, then it has a nontrivial Harish-Chandra L(f)-submodule S such that dimSλ=1 for all λ ∈ supp(S)\{0}. As a corollary, any irreducible weight module with a finite dimensional weight space over L(f) is a Harish-Chandra module. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of Z d -graded modules, which are constructed using Shen–Larsson's functor from sl d -modules V , for the Lie algebras of divergence zero vector fields on tori and quantum tori. We determine the irreducibility of these modules for finite-dimensional or infinite-dimensional V using a unified method. In particular, these modules provide new irreducible weight modules with infinite-dimensional weight spaces for the corresponding algebras. [ABSTRACT FROM AUTHOR]

In this paper, we construct and study a new class of modules over the Block algebra B (q) , where q is a nonzero complex number. We determine the irreducibilities of these modules and the isomorphisms among them. We also show that these modules exhaust all U (h) -free B (q) -modules of rank-1. [ABSTRACT FROM AUTHOR]