Representations of $asl_2$

TL;DR

This paper explores the representation theory of the Lie antialgebra $asl_2$, revealing its connection to the ghost Casimir element of $osp(1|2)$ and classifying certain infinite-dimensional representations.

Contribution

It establishes the relationship between $asl_2$ representations and the ghost Casimir element, and classifies specific infinite-dimensional weighted representations.

Findings

No non-trivial finite-dimensional representations of $asl_2$

Identification of the connection to the ghost Casimir element

Classification of particular infinite-dimensional weighted representations

Abstract

We study representations of the simple Lie antialgebra $asl_2$ introduced by Ovsienko. We show that representations of $asl_2$ are closely related to the famous ghost Casimir element of the universal enveloping algebra $osp(1|2)$. We prove that $asl_2$ has no non-trivial finite-dimensional representations; we define and classify some particular infinite-dimensional representations that we call weighted representations.