Homomorphisms from ${\rm SL}(2, k)$ to ${\rm SL}(4, k)$ in positive characteristic

TL;DR

This paper classifies and describes homomorphisms from the algebraic group SL(2, k) to SL(4, k) over an algebraically closed field of positive characteristic, including their indecomposable decompositions.

Contribution

It provides a complete classification of homomorphisms from SL(2, k) to SL(4, k) in positive characteristic, detailing their structure and indecomposable components.

Findings

Classification of all homomorphisms from SL(2, k) to SL(4, k)

Description of indecomposable decompositions of these homomorphisms

Insights into the structure of these homomorphisms in positive characteristic

Abstract

Let $k$ be an algebraically closed field of positive characteistic $p$ and let ${\rm SL}(n, k)$ denote the special linear algebraic group of degree $n$ over $k$. In this paper, we describe homomorphisms from ${\rm SL}(2, k)$ to ${\rm SL}(4, k)$. As by-products of this description, we give a classification of homomorphisms from ${\rm SL}(2, k)$ to ${\rm SL}(4, k)$ and describe the indecomposable decompositions of homomorphisms from ${\rm SL}(2, k)$ to ${\rm SL}(4, k)$.