The generalized Gell-Mann matrices are the 
Hermitian matrices
generating, in the physics convention, the Lie algebra
associated to the special unitary group 
,
.
With the usual matrix commutator, their multiples
by 
form a real vector basis of this Lie
algebra. As their name suggests, these matrices are
intended to generalize both the standard 
Gell-Mann matrices,
which generate, in that convention, the Lie algebra
associated to 
, as well as the 
Pauli matrices which
generate the Lie algebra associated to 
.
The algorithm for constructing the generalized Gell-Mann matrices is as follows. Throughout, let 
denote the matrix with a
1 in the 
th
entry and 0 elsewhere. This allows one to define three collections of matrices.
The first collection is symmetric:
 |
(1)
|
for 
.
The second collection is antisymmetric:
 |
(2)
|
for 
.
The third collection is diagonal:
 |
(3)
|
for 
.
This gives a total of
 |
(4)
|
generalized Gell-Mann matrices, matching exactly the real-dimension of the Lie algebra associated to 
.
Note that the construction can be rephrased using bra-ket notation (Bertlmann and Krammer
2008) as well. In addition, one can easily verify that the cases for 
and 
yield the Pauli matrices
and Gell-Mann matrices, respectively.
