On simple restricted modules of Hamiltonian superalgebras with $p$-characters of height 0

TL;DR

This paper investigates simple modules over Hamiltonian superalgebras in prime characteristic, proving simplicity of certain modules with height 0 p-characters, classifying their isomorphism classes, and determining their dimensions.

Contribution

It establishes the simplicity of generalized hi-reduced Kac modules for Hamiltonian superalgebras with height 0 p-characters and classifies their isomorphism classes.

Findings

All generalized hi-reduced Kac modules are simple for height 0 p-characters.

The isomorphism classes of these modules are classified.

The dimensions of these modules are explicitly determined.

Abstract

Let $H(2,1;\underline{t})$ be Hamiltonian superalgebras over $\mathbb{F}$, an algebraically closed field of prime characteristic $p>3$, which are non-restricted simple Lie superalgebras, generally. In this paper, we study generalized $\chi$-reduced simple modules over $H(2,1;\underline{t})$. We proved that all generalized $\chi$-reduced Kac modules of $H(2,1;\underline{t})$ are simple with $p$-characters $\chi$ of height 0. Additionally, the isomorphism classes of these simple $H(2,1;\underline{t})$-modules are classified and their dimensions are determined.