Useful relations among the generators in the defining and adjoint representations of SU(N)

TL;DR

This paper reviews various algebraic relations among generators of SU(N), focusing on the defining and adjoint representations, including those involving the symmetric tensor d_{abc}, with special notes for N=3.

Contribution

It compiles and summarizes many useful relations among SU(N) generators, highlighting those involving the symmetric tensor and special cases for N=3.

Findings

Relations involving the symmetric tensor d_{abc} are summarized.

Special relations for N=3 are highlighted.

The review aids in understanding the structure of SU(N) Lie algebra.

Abstract

There are numerous relations among the generators in the defining and adjoint representations of SU(N). These include Casimir operators, formulae for traces of products of generators, etc. Due to the existence of the completely symmetric tensor $d_{abc}$ that arises in the study of the SU(N) Lie algebra, one can also consider relations that involve the adjoint representation matrix, $(D^a)_{bc}=d_{abc}$. In this review, we summarize many useful relations satisfied by the defining and adjoint representation matrices of SU(N). A few relations special to the case of N=3 are highlighted.