# Bases for local Weyl modules for the hyper and truncated current $\mathfrak{sl}_2$-algebras

## TL;DR

This paper constructs explicit linear bases for local Weyl modules of hyper and truncated current algebras associated with rak{sl}_2, using Grf6bner-Shirshov bases, providing new insights especially in positive characteristic.

## Contribution

It offers the first characteristic-free basis construction for hyper current rak{sl}_2$-algebras and presents a novel approach to existing bases in characteristic zero.

## Key findings

- First characteristic-free basis construction for hyper current rak{sl}_2$-algebras.
- New bases with specific properties for truncated current rak{sl}_2$-modules.
- Alternative construction of known bases in characteristic zero.

## Abstract

We use the theory of Gr\"obner-Shirshov bases for ideals to construct linear bases for graded local Weyl modules for the (hyper) current and the truncated current algebras associated to the finite-dimensional complex simple Lie algebra $\mathfrak{sl}_2$. The main result is a characteristic-free construction of bases for this important family of modules for the hyper current $\mathfrak{sl}_2$-algebra. In the positive characteristic setting this work represents the first construction in the literature. In the characteristic zero setting, the method provides a different construction of the Chari-Pressley (also Kus-Littelmann) bases and the Chari-Venkatesh bases for local Weyl modules for the current $\mathfrak{sl}_2$-algebra. Our construction allows us to obtain new bases for the local Weyl modules for truncated current $\mathfrak{sl}_2$-algebras with very particular properties.