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Register a new userNew families of irreducible weight modules over $mathfrak{sl}_{3}$
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Let $n>1$ be an integer, $alphain{mathbb C}^n$, $bin{mathbb C}$, and $V$ a $mathfrak{gl}_n$-module. We define a class of weight modules $F^alpha_{b}(V)$ over $sl_{n+1}$ using the restriction of modules of tensor fields over the Lie algebra of vector fields on $n$-dimensional torus. In this paper we consider the case $n=2$ and prove the irreducibility of such 5-parameter $mathfrak{sl}_{3}$-modules $F^alpha_{b}(V)$ generically. All such modules have infinite dimensional weight spaces and lie outside of the category of Gelfand-Tsetlin modules. Hence, this construction yields new families of irreducible $mathfrak{sl}_{3}$-modules.
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For an irreducible module $P$ over the Weyl algebra $mathcal{K}_n^+$ (resp. $mathcal{K}_n$) and an irreducible module $M$ over the general liner Lie algebra $mathfrak{gl}_n$, using Shens monomorphism, we make $Potimes M$ into a module over the Witt algebra $W_n^+$ (resp. over $W_n$). We obtain the necessary and sufficient conditions for $Potimes M$ to be an irreducible module over $W_n^+$ (resp. $W_n$), and determine all submodules of $Potimes M$ when it is reducible. Thus we have constructed a large family of irreducible weight modules with many different weight supports and many irreducible non-weight modules over $W_n^+$ and $W_n$.
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