Simple non-weight modules over Lie superalgebras of Block type

TL;DR

This paper constructs and classifies non-weight modules over Lie superalgebras of Block type, providing conditions for simplicity and determining isomorphism classes, thus extending existing results in the field.

Contribution

It introduces new classes of non-weight modules over Lie superalgebras of Block type, with a complete classification and simplicity criteria.

Findings

Constructed free modules over Ramond-Block and Neveu-Schwarz-Block algebras.

Provided necessary and sufficient conditions for module simplicity.

Determined isomorphism classes of the modules.

Abstract

In this paper, a family of non-weight modules over Lie superalgebras $S(q)$ of Block type are studied. Free $U(\eta)$-modules of rank $1$ over Ramond-Block algebras and free $U(\mathfrak{h})$-modules of rank $2$ over Neveu-Schwarz-Block algebras are constructed and classified. Moreover, the sufficient and necessary conditions for such modules to be simple are presented, and their isomorphism classes are also determined. The results cover some existing results.