The generalized Gell-Mann matrices are the matrices generating the Lie algebra associated to the special unitary group , . As their name suggests, these matrices are intended to generalize both the standard Gell-Mann matrices, which generate 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)
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for . The second collection is antisymmetric:
(2)
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for . The third collection is diagonalDiagonal Matrix:
(3)
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for .
This gives a total of
(4)
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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 and Gell-Mann matrices, respectively.