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

### Submission summary

 As Contributors: Howard Haber Arxiv Link: https://arxiv.org/abs/1912.13302v1 (pdf) Date accepted: 2021-01-08 Date submitted: 2020-09-10 09:02 Submitted by: Haber, Howard Submitted to: SciPost Physics Lecture Notes Academic field: Physics Specialties: High-Energy Physics - Phenomenology Mathematical Physics Approach: Theoretical

### 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.

Published as SciPost Phys. Lect. Notes 21 (2021)

### Submission & Refereeing History

Submission 1912.13302v1 on 10 September 2020

## Reports on this Submission

### Anonymous Report 1 on 2020-12-28 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1912.13302v1, delivered 2020-12-28, doi: 10.21468/SciPost.Report.2342

### Strengths

1-The writer is clear in their goal and presentation of their work
2-Many identities that require long derivation are presented in an easy to understand and compact manner
3-Identities presented in this work are easily applied to problems in physics. In particularly, trace identities which appear when working with chiral Lagrangians in quantum field theory come to mind.

### Weaknesses

1-There do not appear to be any new results in this work. The point of this work is to collect results in the literature and present them in a succinct way.

### Report

Assuming that this work is correct in its assertion that there is no single reference compiling the identities, I find that this work meets the criteria for publishing in SciPost Physics Lecture Notes. The work is clear and a useful resource for anyone working with the generators of SU(N). I was able to reproduce all equations just by following the work.

### Requested changes

I have no requested changes.

• validity: high
• significance: low
• originality: ok
• clarity: good
• formatting: good
• grammar: good