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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]

In this paper, using Larsson’s functor with irreducible𝔰𝔩2-modulesV, we construct a class ofℤ2-graded modules for the Virasoro-like algebra and itsq-analogs. We determine the irreducibility of these modules for finite-dimensional or infinite-dimensionalVusing a unified method. In particular, these modules provide new irreducible weight modules with infinite-dimensional weight spaces for the corresponding algebras. [ABSTRACT FROM PUBLISHER]

In this article, we study Whittaker modules over generalized Virasoro algebras relative to some total order. We give the description of any Whittaker module and its Whittaker vectors. In particular, we determine all simple Whittaker modules, which provide the first known examples of simple nonweight modules over generalized Virasoro algebras. On the other hand, we also show that any Whittaker module over any generalized Virasoro algebra can be induced from a Whittaker module over some Virasoro subalgebra. [ABSTRACT FROM PUBLISHER]