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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 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, 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]