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

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.


Introduction
The SU(N) Lie group and its Lie algebra are ubiquitous in theoretical physics. Numerous relations among the generators in the defining and adjoint representations of SU(N) are often useful in a variety of physics applicaitons. Many of these relations are well known and others are more obscure. There are multiple sources for the various identities that will be reviewed in this note, but there is no single reference that I am aware of that contains all of them. For my own benefit, as well as for the benefit of others, I have collected many of the relevant identities and assembled them in this short review. In a few instances, I found typographical errors in some of the original sources that I was able to correct.
Recently, two authors contacted me concerning these notes, which they apparently found quite useful [1,2]. They urged me to post this review to the arXiv. I have decided to follow their advice. I hope that anyone who consults this work will find it useful. Please excuse me for not including a more comprehensive list of references. The references included at the end of this review focus mainly on the sources that I used to obtain all the formulae displayed below.

The defining representation of the SU(N ) Lie algebra
In these notes, I will provide some useful relations involving the generators of the SU(N) Lie algebra, henceforth denoted by su(N). We employ the physicist's convention, where the N 2 − 1 generators in the defining representation of su(N), denoted by T a , serve as a basis for the set of traceless hermitian N × N matrices. The generators satisfy the commutation relations, In particular Tr T a = 0 .
We employ the following normalization convention for the generators in the defining representation of su(N), In this convention, the f abc are totally antisymmetric with respect to the interchange of any pair of its indices. Consider a d-dimensional irreducible representation, R a of the generators of su(N). The quadratic Casimir operator, C 2 ≡ R a R a , commutes with all the su(N) generators. 1 Hence in light of Schur's lemma, C 2 is proportional to the d × d identity matrix. In particular, the quadratic Casimir operator in the defining representation of su(N) is given by where 1 is the N × N identity matrix. To evaluate C F , we take the trace of eq. (4) and make use of Tr 1 = N. Summing over a, we note that δ aa = N 2 − 1. Using the normalization of the generators specified in eq. (3), it follows that 1 Next we quote an important identity involving the su(N) generators in the defining representation, where the indices i, j, k and ℓ take on values from 1, 2, . . . , N. To derive eq. (6), we first note that any N × N complex matrix M can be written as a complex linear combination of the N × N identity matrix and the T a , This can be regarded as a completeness relation on the vector space of complex N × N matrices. One can project out the coefficient M 0 by taking the trace of eq. (7). Likewise, one can project out the coefficients M a by multiplying eq. (7) by T b and then taking the trace of the resulting equation. Using eqs. (2) and (3), it follows that 1 It is straightforward to show that C 2 commutes with all the generators of su(N ). In particular, using the commutation relations, due to the antisymmetry of f abc under the interchange of any pair of indices. 2 In the older literature, the defining representation is (inaccurately) called the fundamental representation. It is for this reason that the Casimir operator in the defining representation is often denoted by C F . Inserting these results back into eq. (7) yields The matrix elements of eq. (9) are therefore where the sum over repeated indices is implicit. We can rewrite eq. (10) in a more useful form, It follows that This equation must be true for any arbitrary N × N complex matrix M. It follows that the coefficient of M ℓk in eq. (12) must vanish. This yields the identity states in eq. (6). The proof is complete. Many important identities can be obtained from eq. (6). For example, multiplying eq. (6) by T b jk and summing over j and k yields after employing eq. (2). If we now multiply eq. (13) by T c and take the trace of both sides of the resulting equation, then the end result is after using eq. (3). A more general expression for the trace of four generators (of which eq. (14) is a special case) is given in Appendix A.

Introducing the symmetric third rank tensor d abc
In su(N), one can also define a totally symmetric third rank tensor called d abc via the relation, where 1 is the N × N identity matrix. Combining eqs. (1) and (15) yields the following anticommutation relation, Using eqs. (3) and (16), one obtains an explicit expression, which can be taken as the definition of the d abc . It then follows that d aac = 0 (where a sum over the repeated index a is implicit). Indeed, since d abc is a totally symmetric tensor, it follows that d aca = d caa = 0. The case of su(2) provides the simplest example. In this case, we identify T a = 1 2 σ a , where the σ a (for a = 1, 2, 3) are the well-known Pauli matrices, and f abc = ǫ abc are the components of the Levi-Civita tensor. It is a simple matter to check that in the case of su(2), we have d abc = 0. In contrast, the d abc are generally non-zero for N ≥ 3.
Consider the trace identity obtained by multiplying eq. (15) by T c and taking the trace. In light of eqs. (2) and (3), It then follows that In obtaining eqs. (19) and (20), we used the fact that d abc is symmetric and f abc is antisymmetric under the interchange of any pair of indices, which implies that To evaluate the products f abc f abd and d abc d abd , we proceed as follows. Using eqs. (1) and (16), The traces are easily evaluated using eqs. (3)- (5) and (14), and we end up with Comparing eqs. (24) and (25) with eqs. (19) and (20), we conclude that, 3 Consider a d-dimensional irreducible representation, R a of the generators of su(N). The cubic Casimir operator C 3 ≡ d abc R a R b R c , commutes with all the su(N) generators. Hence in light of Schur's lemma, C 3 is proportional to the d × d identity matrix. In particular, the cubic Casimir operator in the defining representation of su(N) is given by To evaluate C 3F , we multiply eq. (15) d abd to obtain after using eqs. (26) and (27). Multiplying the above result by T c and employing eq. (4) yields Hence, using eqs. (5) and (28), we obtain For completeness, we note the following result that resembles eq. (29), iNT c , after employing eq. (24). Hence, in light of eqs. (4) and (5) it follows that where C 2 ≡ R a R a is the quadratic Casimir operator in representation R. Hence, f abc R a R b R c is proportional to C 2 and thus is not an independent Casimir operator. 4

Matrices of the adjoint representation of SU(N )
We now introduce the generators of su(N) in the adjoint representation, which will be henceforth denoted by F a . The F a are (N 2 − 1) × (N 2 − 1) antisymmetric matrices, since the dimension of the adjoint representation is equal to the number of generators of su(N). Explicitly, the matrix elements of the adjoint representation generators are determined by the structure constants, It is also convenient to define a set of (N 2 − 1) × (N 2 − 1) traceless symmetric matrices where the d abc is defined by eq. (17). Since d abb = 0 it follows that Tr D a = 0. The properties of the F a and D a matrices have been examined in Refs. [3,4]. The F a satisfy the commutation relations of the su(N) generators, which is equivalent to the Jacobi identity, Likewise, there is a second commutation relation of interest, which is equivalent to the two identities, The relations, are also noteworthy. Combining eqs. (36) and (39) yields, The expression for the commutator [D a , D b ] is more complicated, which is equivalent to the identity, Interchanging b ↔ c and subtracting, the resulting expression can be rewritten as Eq. (43) is equivalent to the identity, The quadratic Casimir operator in the adjoint representation is and I is the (N 2 − 1) × (N 2 − 1) identity matrix, which is equivalent to eq. (26). Two other similar expressions of interest are which are equivalent to eqs. (27) and (21), respectively. Using the above results, we can derive additional identities of interest. For example, It then follows that For completeness, we quote the analogous identities with f abc replaced by d abc . These identities are proved in Appendix B of these notes.
It then follows that Note that eq. (57) implies that the cubic Casimir operator in the adjoint representation vanishes, i.e., d abc F a F b F c = 0.
Tr F a = Tr D a = 0 , Additional identities involving traces of four generators can also be derived. Ref. [6] provides the following results, 5 Alternative expressions for eqs. (66)-(70) are given in Appendix C [5]. As a check of eq. (65), let us set a = c and sum over a. After employing eqs. (26) and (27) and relabeling d by c, we obtain Alternatively, one can obtain the above result directly by using eqs. (26), (45), (62) and (63) to compute which confirms the result of eq. (71). Similarly, the results of eqs. (66)-(70) can also be checked by multiplication by either a Kronecker delta, f abc or d abc and then employing the trace formulae previously derived. Various applications of the identities given in this section can be found in a paper by Roger Cutler and Dennis Sivers [7]. Indeed, many of these identities are also reproduced in Appendix B of Ref. [7], after correcting the latter for some obvious typographical errors. The identities provided in these notes are sufficient to work out the color factors for scattering process involving quarks and gluons. Although the color factors should be computed for the case of N = 3, it is useful to first evaluate the color factors for an SU(N) gauge theory, since these results allow one to identify sets of independent color factors that arise for a given process.

Two additional identities for N = 3
Two additional identities, which were first presented in Ref. [8], are special to the case of N = 3 and do not generalize to arbitrary N. These identities can be derived from the characteristic equation of a general element of the su(3) Lie algebra [4,8], These two identities can be rewritten as Combining eqs. (34) and (73) then yields, Likewise, combining eqs. (41) and (74) yields, Note that the sum of eqs. The trace of a product of four generators in the defining representation also involves the symmetric tensor d abc introduced in Section 2. Applying eq. (15) twice, and taking the trace with the help of eq. (3) yields It is convenient to employ eqs. (38) and (42) of Section 3 to produce a more symmetric version,  63) and (64), after using eqs. (32) and (33). Multiplying eq. (40) on the left by F e and taking a trace yields in light of eqs. (61) and (62). Likewise, multiplying eq. (40) on the right by D e and taking a trace yields Multiplying eq. (43) on the right by (D f ) de and taking the trace (by setting c = e and summing over e) yields, Finally, we use the result of eq. (83) to obtain APPENDIX C: Traces of adjoint representation matrices revisited The traces of products of four matrices (either F a or D a ) in the adjoint representation are given in eqs. (65)-(70). It is sometime convenient to eliminate the product f ade f bce in favor of δ ab and d abc , etc., by using eq. (42). The following results were obtained in Ref. [5], Note that the second equation above is consistent with eq. (66) in light of eq. (36), and the fifth equation above is consistent with eq. (69) in light of eq. (37).