Irreducible Jet modules for the vector field Lie algebra on ${\mathbb{S}}^1\times \mathbb{C}$

Abstract

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 $A\#U(\mathfrak{g})\cong \mathcal{D}\otimes U(L)$, 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 $\theta :L\to {\mathfrak{m}}_{1,0}\Delta$, 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.

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• Classification of simple bounded weight modules of the Lie algebra of vector fields on $\C^n$
• Simple weight modules with finite-dimensional weight spaces over Witt superalgebras
• Quantization and injective submodules of differential operator modules
• $\mathcal{A}\mathcal{V}$ modules of finite type on affine space