For a commutative algebra $A$ over $mathbb{C}$,denote $mathfrak{g}=text{Der}(A)$. A module over the smash product $A# U(mathfrak{g})$ is called a jet $mathfrak{g}$-module, where $U(mathfrak{g})$ is the universal enveloping algebra of $mathfrak{g}$.In
the present paper, we study jet modules in the case of $A=mathbb{C}[t_1^{pm 1},t_2]$.We show that $A#U(mathfrak{g})congmathcal{D}otimes U(L)$, where $mathcal{D}$ is the Weyl algebra $mathbb{C}[t_1^{pm 1},t_2, frac{partial}{partial t_1},frac{partial}{partial t_2}]$, and $L$ is a Lie subalgebra of $A# U(mathfrak{g})$ called the jet Lie algebra corresponding to $mathfrak{g}$.Using a Lie algebra isomorphism $theta:L rightarrow mathfrak{m}_{1,0}Delta$, where $mathfrak{m}_{1,0}Delta$ is the subalgebra of vector fields vanishing at the point $(1,0)$, we show that any irreducible finite dimensional $L$-module is isomorphic to an irreducible $mathfrak{gl}_2$-module. As an application, we give tensor product realizations of irreducible jet modules over $mathfrak{g}$ with uniformly bounded weight spaces.
In this paper, we study weight representations over the Schr{o}dinger Lie algebra $mathfrak{s}_n$ for any positive integer $n$. It turns out that the algebra $mathfrak{s}_n$ can be realized by polynomial differential operators. Using this realization
, we give a complete classification of irreducible weight $mathfrak{s}_n$-modules with finite dimensional weight spaces for any $n$. All such modules can be clearly characterized by the tensor product of $mathfrak{so}_n$-modules, $mathfrak{sl}_2$-modules and modules over the Weyl algebra.
The rank $n$ symplectic oscillator Lie algebra $mathfrak{g}_n$ is the semidirect product of the symplectic Lie algebra $mathfrak{sp}_{2n}$ and the Heisenberg Lie algebra $H_n$. In this paper, we study weight modules with finite dimensional weight spa
ces over $mathfrak{g}_n$. When $dot z eq 0$, it is shown that there is an equivalence between the full subcategory $mathcal{O}_{mathfrak{g}_n}[dot z]$ of the BGG category $mathcal{O}_{mathfrak{g}_n}$ for $mathfrak{g}_n$ and the BGG category $mathcal{O}_{mathfrak{sp}_{2n}}$ for $mathfrak{sp}_{2n}$. Then using the technique of localization and the structure of generalized highest weight modules, we also give the classification of simple weight modules over $mathfrak{g}_n$ with finite-dimensional weight spaces.
In this paper, we study the BGG category $mathcal{O}$ for the quantum Schr{o}dinger algebra $U_q(mathfrak{s})$, where $q$ is a nonzero complex number which is not a root of unity. If the central charge $dot z eq 0$, using the module $B_{dot z}$ over
the quantum Weyl algebra $H_q$, we show that there is an equivalence between the full subcategory $mathcal{O}[dot z]$ consisting of modules with the central charge $dot z$ and the BGG category $mathcal{O}^{(mathfrak{sl}_2)}$ for the quantum group $U_q(mathfrak{sl}_2)$. In the case that $dot z=0$, we study the subcategory $mathcal{A}$ consisting of finite dimensional $U_q(mathfrak{s})$-modules of type $1$ with zero action of $Z$. Motivated by the ideas in cite{DLMZ, Mak}, we directly construct an equivalent functor from $mathcal{A}$ to the category of finite dimensional representations of an infinite quiver with some quadratic relations. As a corollary, we show that the category of finite dimensional $U_q(mathfrak{s})$-modules is wild.
In 2006, Gao and Zeng cite{GZ} gave the free field realizations of highest weight modules over a class of extended affine Lie algebras. In the present paper, applying the technique of localization to those free field realizations, we construct a clas
s of new weight modules over the extended affine Lie algebras. We give necessary and sufficient conditions for these modules to be irreducible. In this way, we construct free field realizations for a class of simple weight modules with infinite weight multiplicities over the extended affine Lie algebras.
Let $dge1$ be an integer, $W_d$ and $mathcal{K}_d$ be the Witt algebra and the weyl algebra over the Laurent polynomial algebra $A_d=mathbb{C} [x_1^{pm1}, x_2^{pm1}, ..., x_d^{pm1}]$, respectively. For any $mathfrak{gl}_d$-module $M$ and any admissib
le module $P$ over the extended Witt algebra $widetilde W_d$, we define a $W_d$-module structure on the tensor product $Potimes M$. We prove in this paper that any simple $W_d$-module that is finitely generated over the cartan subalgebra is a quotient module of the $W_d$-module $P otimes M$ for a finite dimensional simple $mathfrak{gl}_d$-module $M$ and a simple $mathcal{K}_d$-module $P$ that are finitely generated over the cartan subalgebra. We also characterize all simple $mathcal{K}_d$-modules and all simple admissible $widetilde W_d$-modules that are finitely generated over the cartan subalgebra.
In the present paper, using the technique of localization, we determine the center of the quantum Schr{o}dinger algebra $S_q$ and classify simple modules with finite-dimensional weight spaces over $S_q$, when $q$ is not a root of unity. It turns out
that there are four classes of such modules: dense $U_q(mathfrak{sl}_2)$-modules, highest weight modules, lowest weight modules, and twisted modules of highest weight modules.
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.
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 a
lgebra $W_n^+$ (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 $W_n$.
In this paper, we studied the jet modules for the centerless Virasoro-like algebra which is the Lie algebra of the Lie group of the area-preserving diffeomorphisms of a $2$-torus. The jet modules are certain natural modules over the Lie algebra of se
mi-direct product of the centerless Virasoro-like algebra and the Laurent polynomial algebra in two variables. We reduce the irreducible jet modules to the finite-dimensional irreducible modules over some infinite-dimensional Lie algebra and then characterize the irreducible jet modules with irreducible finite dimensional modules over $mathfrak{sl}_2$. To determine the indecomposable jet modules, we use the technique of polynomial modules in the sense of cite{BB, BZ}. Consequently, indecomposable jet modules are described using modules over the algebra $BB_+$, which is the positive part of a Block type algebra studied first by cite{DZ} and recently by cite{IM, I}).
