# Orthogonal decompositions of classical Lie algebras over finite commutative rings

## TL;DR

This paper investigates the conditions under which classical Lie algebras over finite commutative rings can be orthogonally decomposed, focusing on special linear, symplectic, and orthogonal Lie algebras.

## Contribution

It provides a necessary condition for orthogonal decompositions of these Lie algebras over finite commutative rings, extending understanding of their structural properties.

## Key findings

- Necessary condition for orthogonal decomposition of special linear Lie algebra
- Analysis of orthogonal decompositions of symplectic Lie algebra
- Study of orthogonal decompositions of special orthogonal Lie algebra

## Abstract

Let $R$ be a finite commutative ring with identity. In this paper, we give a necessary condition for the existence of an orthogonal decomposition of the special linear Lie algebra over $R$. Additionally, we study orthogonal decompositions of the symplectic Lie algebra and the special orthogonal Lie algebra over $R$.