Classical orthogonal decomposition of a modular $\mathfrak{sl}_n$

TL;DR

This paper extends the classical orthogonal decomposition problem of Lie algebras to modular Lie algebras over fields of prime characteristic, constructing such decompositions for $rak{sl}_n$ and analyzing special cases.

Contribution

It introduces a new concept of classical orthogonal decomposition for modular Lie algebras and constructs these decompositions for $rak{sl}_n$ under specific conditions.

Findings

Constructed orthogonal decompositions for $rak{sl}_n$ in prime characteristic

Provided detailed analysis for cases $n=2$ and $n=3$

Extended classical decomposition theory to modular Lie algebras

Abstract

An orthogonal decomposition problem of Lie algebras over the complex numbers has been studied since the 1980s. It has many applications and relations to other areas of mathematics and sciences. In this paper, we consider this decomposition problem over a field of prime characteristic. We define a classical orthogonal decomposition of a modular Lie algebra and construct it for $\mathfrak{sl}_n$ under certain sufficient conditions. Additionally, we provide more detailed analysis of the problem when $n = 2$ and $3$.