Automorphism Group And Representation Over Extended Witt Algebra

Let d be a positive integer. Witt algebra is the derivation algebra of the Laurent polynomial ring Ad =C[t1±1,t2±1,…,td±1] in commuting variables, denoted by Wd. The representation theory over Witt algebra has been studied by many scholars. Extended Wiitt algebra is obtained by extendiing the strueture of Witt algebra, denoted by Wd (?) Ad.Eswara Rao first put forward its representation.In 1986, Guangyu Shen defined a class of weight module over Witt algebra, for α ∈Cd,b ∈C and a sld-module V on which the identity matrix acts as the scalar b. Let Fbα(V) = V(?)Ad , then Fbα(V) became a natural Wd-module. Later, these modules as well as the functor Fbα, known as Larsson functor, were also studied by Larsson and Rao. The module Fbα(V) can be realized as module over Wd(?) Ad algebra. In this paper, we use the"twisting technique" to construct a new module Fbα,β(V) and determine the irreducibility of these modules, and our main result is as follows: Whether the weight space is finite or infinite dimensional, when V is an irreducible sld-module and V is not isomorphic to V(ωk) for any k=1,…,d,Fbα,β(V) is an irreducible Wd(?) Ad module.we also determine the automorphism group of Wd(?)Ad. Any automorphism over Wd(?)Ad can be written to the form of composition of several special automorphisms.