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Generalized Gell-Mann Matrix


The generalized Gell-Mann matrices are the n^2-1 matrices generating the Lie algebra associated to the special unitary group SU(n), n>=2. As their name suggests, these matrices are intended to generalize both the standard 3×3 Gell-Mann matrices, which generate the Lie algebra associated to SU(3), as well as the 2×2 Pauli matrices which generate the Lie algebra associated to SU(2).

The algorithm for constructing the generalized Gell-Mann matrices is as follows. Throughout, let E_(j,k) denote the matrix with a 1 in the (j,k)th entry and 0 elsewhere. This allows one to define three collections of matrices. The first collection is symmetric:

 lambda_(j,k)^s=E_(k,j)+E_(j,k)
(1)

for 1<=j<k<=n. The second collection is antisymmetric:

 lambda_(j,k)^a=-i(E_(j,k)-E_(k,j))
(2)

for 1<=j<k<=n. The third collection is diagonalDiagonal Matrix:

 lambda_l=sqrt(2/(l(l+1)))(sum_(j=1)^lE_(j,j)-lE_(l+1,l+1))
(3)

for 1<=l<=n-1.

This gives a total of

 1/2n(n-1)+1/2n(n-1)+(n-1)=n^2-1
(4)

generalized Gell-Mann matrices, matching exactly the real-dimension of the Lie algebra associated to SU(n).

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 n=2 and n=3 yield the Pauli and Gell-Mann matrices, respectively.


See also

Antisymmetric Matrix, Bra, Diagonal Matrix, Gell-Mann Matrix, Hermitian Matrix, Ket, Kronecker Delta, Lie Algebra, Lie Group, Matrix Trace, Pauli Matrices, Special Unitary Group, Symmetric Matrix

This entry contributed by Christopher Stover

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References

Bertlmann, R. and Krammer, P. "Bloch Vectors for Qudits." 6 June 2008. http://arxiv.org/abs/0806.1174.

Cite this as:

Stover, Christopher. "Generalized Gell-Mann Matrix." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/GeneralizedGell-MannMatrix.html

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