# Category O for Takiff sl_2

## TL;DR

This paper explores the extension of the BGG category O to the Takiff algebra of sl_2, describing its structure and properties, including Gabriel quivers and extension fullness.

## Contribution

It introduces and analyzes a new analogue of category O for Takiff sl_2, detailing its block structure and extension properties.

## Key findings

- Gabriel quivers for the category O analogue are described.
- Extension fullness of one category O analogue is proven.
- Structural properties of the modules are characterized.

## Abstract

We investigate various ways to define an analogue of BGG category $\mathcal{O}$ for the non-semi-simple Takiff extension of the Lie algebra $\mathfrak{sl}_2$. We describe Gabriel quivers for blocks of these analogues of category $\mathcal{O}$ and prove extension fullness of one of them in the category of all modules.