# Anticommutative Sets of $SU(N)$ Generators? Anticommutative Analogue to Cartan subalgebra?

I am currently studying $$SU(N)$$ generators in order to find bases that may suit a problem at hand. I am especially interested in getting as large anticommuting sets within a basis as possible.

In $$SU(2)$$ the Pauli matrices provide a nice set of anticommuting operators, however in higher dimensions there are operators that can be neither commuting nor anticommuting.

While trying to grasp abit about the Theory behind $$SU(N)$$. I found out about so called Cartan subalgebras, that seem to be maximally commuting subsets. It seems to be well known that these are exactly of size $$N-1$$ among the $$SU(N)$$ generators, irrespective of the dimension.

Is there a similar result about the anti commutative sets within $$SU(N)$$?

Your question is a bit broad. I assume you are not talking about Lie superalgebras, but, instead, about Lie algebra representations, beyond the fundamental, no? Unlike the Cartan subalgebra of the SU(N) Lie algebra you mentioned, which does not depend on the representation, but only on the architecture of the structure constants $$f_{ijl}$$, the effective d-coefficients of higher Ns for SU(N) do depend on the representation, as I'll illustrate for SU(3), which I assume you must be familiar with.

To start with, the anticommutator of an SU(N) rep always contains the identity, since the square of a generator should not vanish. For N=2, the d-coefficients vanish identically, the reps being real/pseudoreal (more below).

For higher Ns the anticommutator for the fundamental irrep is $$\left[T_a, T_b\right]_+ = \frac{1}{N}\delta_{ab} I_n+ \sum_{c=1}^{N^2 -1}{d_{abc} T_c}$$ for the standard d-coefficients, real from hermiticity like the structure constants, related to anomalies in QFT. So you are asking, which, if any, ''other reps'' of SU(N) have their d-coefficients somehow vanish?

A: The real ones (or pseudoreal ones, effectively real, like all of SU(2), but don't worry about it now). Typically, this amounts to vanishing of the cubic Casimir operator, illustrated here for SU(3), presumably familiar to you.

The fundamental one for SU(3) is the eight Gell-Mann matrices, which are not real, or pure imaginary in our conventions, so the (real) d-s are non-vanishing, the ones you know about. (Well, some ds vanish, corresponding to the SU(2) subgroups of SU(3), so the sets such as $$\lambda_1,\lambda_2,\lambda_3$$, etc... will give you anti-commuting sets equivalent to those of the Pauli matrices.)

If you are confined to NxN reps, fundamentals, there isn't much help you can get. However, if you are futzing systematically with such matrices, the most systematic basis to play with them is the nonhermitian one invented by the great J J Sylvester in 1882, the nonions, sedenions, etc....Clock-and-shift matrices whose multiplication with each other is analytic in their parameters! that should make lots of limits and computations of this type easy.

NB Special thanks to @user196574 who caught a mistake in the original version for higher representations, such as the adjoint. For representations higher than the fundamental, there are extra terms in the anticommutator, beyond the identity and terms proportional to the d-coefficients (which vanish for real representations such as the adjoint). For example, even for SU(2) (!), it is evident that for generators $$T^a_{jk}\propto \epsilon_{ajk}$$ the anticommutator has further nontrivial terms beyond the identity; they are trace-orthogonal to all generators.

• +1, your group theory answers are very illuminating. I am unsure of one thing: You note that the adjoint is real, with $d$ vanishing, which will give "rampant anti commutation." However, won't there be an extra $M^{AB}$ term (like in PSE 221851) spoiling nice anti commutation on the right hand side of your anticommutator as soon as one goes beyond the fundamental representation, since the higher dimensional reps no longer have spanning generators? I feel that would spoil rampant anti commutation for the adjoint rep, unless such an $M^{AB}$ term also vanishes. Commented Jun 20, 2021 at 19:33
• I fear you are right... I shouldn't have assumed that. The true statement is that such an M would be orthogonal to all generators; I was thinking about anomaly coefficients. Correct me if I'm wrong, but for SU(3) and the structure constants of the Gell-Mann matrices, $M^{12}_{31} = -1\neq 0$. Commented Jun 20, 2021 at 20:26
• Up to factors of $2$ in the structure constants definition, using the table on page 2 of scipp.ucsc.edu/~haber/ph251/gellmann17.pdf to avoid algebra mistakes, I have the following: imgur.com/a/6v87KtG, for which $(M^{12})_{12}$ and $(M^{12})_{21}$ are $1$ and nonzero, where $M^{12} = \{T^1_A, T^2_A\}$, so though different, still nonzero. Commented Jun 20, 2021 at 21:01
• Not really... I actually have little experience with those, like anyone else, since they are rarely used... normally, one looks at traces of three generators, etc... Perhaps in the math SE someone might know more... Commented Jun 21, 2021 at 0:12
• I edited my question. In point of fact, SU(2) suffices for a counterexample. For generators proportional to the $\epsilon^{ajk}$, easy to contract, the anticommutator has a piece proportional to the identity, and an extra term M. Commented Jun 21, 2021 at 0:57