Generalized Gell-Mann matrices are the
2
n
-1
matrices that form the Lie algebra of the special unitary group SU(n) for n ≥ 2. They extend the 3×3 Gell-Mann matrices for SU(3) and the 2×2 Pauli matrices for SU(2). In this short essay, I explored how to construct them using Wolfram quantum framework, since Gell-Mann matrices are used as a basis. Then I explored some their properties.
https://en.wikipedia.org/wiki/Gell-Mann_matrices​
​https://mathworld.wolfram.com/GeneralizedGell-MannMatrix.html
In[]:=
PacletInstall[CloudObject["https://wolfr.am/DevWQCF"],ForceVersionInstall->True]​​<<Wolfram`QuantumFramework`
Out[]=
PacletObject
Name: Wolfram/QuantumFramework
Version: 1.5.1

Set the dimension
In[]:=
n=3;
Find generalized Gell-Mann matrices as the basis (drop identity):
In[]:=
λn=Rest@Flatten[ArrayReshape[Transpose@QuantumBasis["GellMann"[n]]["Matrix"],{n,n,n,n}],1];
number of matrices:
In[]:=
d=n^2-1;
Show them:
In[]:=
MatrixForm/@λn
Out[]=

0
1
0
1
0
0
0
0
0
,
0
-
0

0
0
0
0
0
,
1
0
0
0
-1
0
0
0
0
,
0
0
1
0
0
0
1
0
0
,
0
0
-
0
0
0

0
0
,
0
0
0
0
0
1
0
1
0
,
0
0
0
0
0
-
0

0
,
1
3
0
0
0
1
3
0
0
0
-
2
3

Find the structure constants:
f
abc
=
1
4
Tr[[
λ
a
,
λ
b
]
λ
c
]
In[]:=
fabc=SparseArrayThreadTuples[Range[d],3]->
1
4
Tr[(#1.#2-#2.#1).#3]&@@@Tuples[λn,3],{d,d,d};
d
abc
=
1
4
Tr[{
λ
a
,
λ
b
}
λ
c
]
In[]:=
dabc=SparseArrayThreadTuples[Range[d],3]->
1
4
Tr[(#1.#2+#2.#1).#3]&@@@Tuples[λn,3],{d,d,d};
Check some symmetry feature and some properties of those tensors:
All of them are Hermitian
In[]:=
And@@(HermitianMatrixQ/@λn)
All of them are traceless
In[]:=
Tr/@λn
Out[]=
{0,0,0,0,0,0,0,0}
They form an orthonormal set with respect to the Hilbert-Schmidt inner product, ie
Tr
λ
i
λ
j
=2
δ
ij
In[]:=
Outer[Tr[#1.#2]/2&,λn,λn,1]==IdentityMatrix[d]//FullSimplify
Out[]=
True
Symmetric and antisymmetric features of structure constants:
In[]:=
fabc==-Transpose[fabc]//FullSimplify​​dabc==Transpose[dabc]//FullSimplify
Out[]=
True
Out[]=
True

λ
i
,
λ
j
=2
∑
k
f
ijk
λ
k
In[]:=
Apply[And]@Flatten@Table[DeleteDuplicates@Flatten@Chop@FullSimplify@Normal[(λn[[a]].λn[[b]]-λn[[b]].λn[[a]])-(2Sum[fabc[[a,b,c]]λn[[c]],{c,Range[d]}])]=={0},{a,Range[d]},{b,Range[d]},{c,Range[d]}]
Out[]=
True

λ
i
,
λ
j
=
4
n
δ
ij
+2
∑
k
d
ijk
λ
k
In[]:=
Apply[And]@Flatten@TableDeleteDuplicates@Flatten@Chop@FullSimplify@Normal(λn[[a]].λn[[b]]+λn[[b]].λn[[a]])-2Sum[dabc[[a,b,c]]λn[[c]],{c,Range[d]}]+
4
n
KroneckerDelta[a,b]IdentityMatrix[n]=={0},{a,Range[d]},{b,Range[d]},{c,Range[d]}
Out[]=
True
Anti-Hermitian for
-
λ
i
In[]:=
And@@(AntihermitianMatrixQ/@(-λn))
Out[]=
True
Show Gell-Mann matrices in other dimensions:
In[]:=
With[{n=4},MatrixForm/@Rest@Flatten[ArrayReshape[Transpose@QuantumBasis["GellMann"[n]]["Matrix"],{n,n,n,n}],1]]
Out[]=
In[]:=
With[{n=5},MatrixForm/@Rest@Flatten[ArrayReshape[Transpose@QuantumBasis["GellMann"[n]]["Matrix"],{n,n,n,n}],1]]
Out[]=

CITE THIS NOTEBOOK

Generalized Gell-Mann matrices: tensor analysis, and algebraic properties​
by Mohammad Bahrami​
Wolfram Community, STAFF PICKS, April 3, 2025
​https://community.wolfram.com/groups/-/m/t/3439352