New simple modules for the $W$-algebra $W(2,2)$

TL;DR

This paper introduces a new class of simple modules for the $W$-algebra $W(2,2)$ by tensoring non-weight modules with simple restricted modules, providing conditions for simplicity and classifying their isomorphism types.

Contribution

It presents a novel construction method for simple modules of $W(2,2)$ using tensor products, expanding the known module categories for this algebra.

Findings

Provided necessary and sufficient conditions for module simplicity

Classified the isomorphism classes of the constructed modules

Demonstrated the novelty of these modules compared to existing literature

Abstract

In this paper, we construct a novel class of simple modules for the $W$-algebra $W(2,2)$. Our approach involves taking tensor products of finitely many non-weight simple modules $\Omega(\lambda,\alpha,h)$ with an arbitrary simple restricted module. We provide a necessary and sufficient condition for these modules to be simple, and subsequently determine their isomorphism classes. Through a comparative analysis with other known simple modules in the literature, we establish that these constructed modules are generically new.