Quotient Algebra Partition and Cartan Decomposition for su(N) IV

TL;DR

This paper introduces new partition methods for su(N) Lie algebras using bi-subalgebras, revealing a duality between partitions and enabling diverse Cartan decompositions, applicable to classical and exceptional Lie algebras.

Contribution

It presents novel partition techniques based on nonabelian bi-subalgebras, establishing a duality and universal applicability for Cartan decompositions across Lie algebras.

Findings

Duality between two types of partitions identified

Procedures for merging or detaching co-quotient algebras developed

Universal applicability to classical and exceptional Lie algebras demonstrated

Abstract

Else from the quotient algebra partition considered in the preceding episodes, two kinds of partitions on unitary Lie algebras are created by nonabelian bi-subalgebras. It is of interest that there exists a partition duality between the two kinds of partitions. With an application of an appropriate coset rule, the two partitions return to a quotient algebra partition when the generating bi-subalgebra is abelian. Procedures are proposed to merge or detach a co-quotient algebra, which help deliver type-AIII Cartan decompositions of more varieties. In addition, every Cartan decomposition is obtainable from the quotient algebra partition of the highest rank. Of significance is the universality of the quotient algebra partition to classical and exceptional Lie algebras.