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

TL;DR

This paper extends quotient algebra partitions of su(N) to higher ranks, enabling systematic Cartan decompositions of types AI, AII, and AIII, and introduces the concept of s-rotation for computational universality.

Contribution

It introduces higher-rank quotient algebra partitions and systematic methods for Cartan decompositions, along with the novel s-rotation transformation for universal computation.

Findings

Extended quotient algebra partitions to higher ranks.

Achieved systematic Cartan decompositions of types AI, AII, and AIII.

Demonstrated computational universality using s-rotation transformations.

Abstract

In the 3rd episode of the serial exposition, quotient algebra partitions of rank zero earlier introduced undergo further partitions generated by bi-subalgebras of higher ranks. The refined versions of quotient algebra partitions admit not only Cartan decompositions of type AI but also decompositions of types AII and AIII, resorting to systematic applications of the operation tri-addition. Details of quotient algebra partitions of higher ranks are extensively examined in this longest episode of the serial. Furthermore, the computational universality is attained taking advantage of a special form of transformation called the s-rotation. The structure of quotient algebra partition is preserved under mappings composed of spinor-to-spinor s-rotations.