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.
Highlights
N δi j δkWhere the indices i, j, k and take on values from 1, 2, . . . , N
There are numerous relations among the generators in the defining and adjoint representations of SU(N)
There are multiple sources for the various identities that will be reviewed in these notes, but there is no single reference that I am aware of that contains all of them
Summary
Where the indices i, j, k and take on values from 1, 2, . . . , N. One can project out the coefficient M0 by taking the trace of eq (7). One can project out the coefficients Ma by multiplying eq (7) by T b and taking the trace of the resulting equation. We can rewrite eq (10) in a more useful form, δi δjk M k = This equation must be true for any arbitrary N × N complex matrix M. It follows that the coefficient of M k in eq (12) must vanish. This yields the identity states in eq (6). If we multiply eq (13) by T c and take the trace of both sides of the resulting equation, the end result is. A more general expression for the trace of four generators (of which eq (14) is a special case) is given in Appendix A
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