Contragredient Lie algebras and Lie algebras associated with a standard pentad

TL;DR

This paper introduces standard pentads of Cartan type, generalizing Cartan subalgebras, and explores their associated graded Lie algebras, revealing connections with contragredient Lie algebras.

Contribution

It constructs a new class of standard pentads of Cartan type and analyzes the structure of their related graded Lie algebras, linking them to contragredient Lie algebras.

Findings

Defined pentads of Cartan type using two integers and three matrices

Established the relationship between these Lie algebras and contragredient Lie algebras

Provided a framework for studying graded Lie algebras via standard pentads

Abstract

We will construct standard pentads which are analogues of Cartan subalgebras, and moreover, we will study graded Lie algebras corresponding to these standard pentads. We call such pentads pentads of Cartan type and describe them by two positive integers and three matrices. The structures of their corresponding Lie algebras are related with contragredient Lie algebras.