The Gell-Mann matrices, named for Murray Gell-Mann, are one possible representation of the infinitesimal generators of the special unitary group called SU(3).
This group (a real Lie algebra in fact) has dimension eight and therefore it has some set with eight linearly independent generators, which can be written as, with i taking values from 1 to 8.
These Lie Algebra elements obey the commutation relations
where a sum over the index k is implied. The structure constants are completely antisymmetric in the three indices and have values
Any set of Hermitian matrices which obey these relations qualifies. A particular choice of matrices is called a group representation, because any element of SU(3) can be written in the form, where are real numbers and a sum over the index j is implied. Given one representation, another may be obtained by an arbitrary unitary transformation, since that leaves the commutator unchanged.
An important representation involves 3×3 matrices, because the group elements then act on complex vectors with 3 entries, i.e., on the fundamental representation of the group. A particular choice of this representation is
and .
These matrices are traceless, Hermitian, and obey the extra relation . These properties were chosen by Gell-Mann because they then generalize the Pauli matrices for SU(2). They also naturally extend to general SU(n), cf. Generalizations of Pauli matrices.
In this representation, it is clear that the Cartan subalgebra is the set of linear combinations (with real coefficients) of the two matrices and, which commute with each other. There are 3 independent SU(2) subgroups:, and, where the x, y, z must consist of linear combinations of and .
These matrices form a useful model to study the internal rotations in the SU(3) space in the quark model (mixing up of quark colors), and, to a lesser extent, in quantum chromodynamics.