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

TL;DR

This paper classifies irreducible finite-dimensional modules over a specific Lie algebra of vector fields on a complex surface, revealing their structure and providing tensor product realizations for jet modules.

Contribution

It introduces a new approach to classify irreducible jet modules for the vector field Lie algebra on a complex surface, linking them to -modules and tensor product constructions.

Findings

Irreducible finite-dimensional modules are isomorphic to -modules.

The algebra decomposes into a Weyl algebra tensor a Lie algebra L.

Tensor product realizations of jet modules are constructed.

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 $\theta:L \rightarrow \mathfrak{m}_{1,0}\Delta$, 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.