Calculate representations of $SU(N)$ generators

Question

I would like to calculate explicit representations of certain $SU(N)$ generators. I have written some code, but either it is extremely slow, or something just doesn't work. Either way, I would be very happy for some input!

Back to topic: The generators are hermitian, traceless $d\times d$ matrices $t^a$ ($a=1,\ldots,N^2-1$), $$ t^a = (t^a)^\dagger = ((t^a)^T)^*,\qquad \text{Tr}[t^a]=0 $$ where $d$ depends on the representation. They should also fulfill their algebra relation $$ [t^a,t^b]:=t^a t^b - t^b t^a = \text{i} \sum\limits_{c=1}^{N^2-1}f^{abc} t^c $$ where $f^{abc}$ are known constants ("structure constants"). My code looks like this:

(* Specify SU(n) *)
n = 2; 
(* rep(resentation) will determine dim(ension) *)
rep = "fund"; 
(* range for index a *)
dimSU = n^2 - 1; 
(* Define structure constants *)
f[a_, b_, c_] := Switch[n, 2, LeviCivitaTensor[3][[a]][[b]][[c]]]
(* Define dim depending on rep *)
dim := Switch[rep, "trivial", 1, "fund", 2, "adj", 3, 1, 1, 2, 2, 3, 3]
(* Define matrices t^a with indices {ij} *)
t[a_, i_, j_] := Table[
       tt[a, i, j]    (* They are the variables I want to work with *)
       , {ii, 1, dim}
      , {jj, 1, dim}
     , {aa, 1, dimSU}][[a]][[i]][[j]]
(* Print t as a list *)
tMatrix[a_] := Table[t[a, i, j], {i, 1, dim}, {j, 1, dim}]
(* Print t as a matrix *)
tMatrixForm[a_] := 
 Table[t[a, i, j], {i, 1, dim}, {j, 1, dim}] // MatrixForm

Now I implement the equations that they have to fulfill:

requireTraceless = 
  Table[Sum[tt[a, i, i], {i, 1, dim}] == 0, {a, 1, dimSU}];
requireHermitean = Table[
     tt[a, i, j] == Conjugate[tt[a, j, i]]
     , {i, 1, j}
    , {j, 1, dim}
   , {a, 1, dimSU}];
requireLie = Table[
     Sum[t[a, i, j] t[Mod[a, dimSU] + 1, j, k], {j, 1, dim}] - 
       Sum[t[Mod[a, dimSU] + 1, i, j] t[a, j, k], {j, 1, dim}] == 
      I f[a, Mod[a, dimSU] + 1, Mod[a + 1, dimSU] + 1] t[
        Mod[a + 1, dimSU] + 1, i, k]
     , {i, 1, dim}
    , {k, 1, dim}
   , {a, 1, dimSU}];

Finally I collect my equations and variables.

varList = 
  Table[tt[a, i, j], {i, 1, dim}, {j, 1, dim}, {a, 1, 
     dimSU}] // Flatten;
eqList = {requireTraceless, requireHermitean, requireLie} // Flatten;

And try to solve.

Solve[eqList, varList]

For $SU(2)$ with dim=2, this should yield the Pauli matrices.

I have found a related question, although it only covers the case $N=2$. Also, I would like to learn how/why my code is so slow.

edit: included info from comments

Answer 1

Actually, the idea of the structure constants is not to prescribe them and then to find a suitable basis. It is vice versa: Usually, one prescribes a basis satisfying some niceness requirements and computes the structure constants from them. Oftentimes, these requirements are sparsity and orthonormality with respect to some inner product.

Sparse matrices are best set up with SparseArray. Here, I construct a sparse basis of $\mathfrak{su}(3)$ which is orthonormal with respect to the Frobenius inner product. Of course, the method works for all dimension.

n = 3;
a = 1/Sqrt[2] Flatten[Table[SparseArray[{{i, j} -> I, {j, i} -> I}, {n, n}], {i, 1, n}, {j, i + 1, n}], 1];
b = 1/Sqrt[2] Flatten[Table[SparseArray[{{i, j} -> -1, {j, i} -> 1}, {n, n}], {i, 1, n}, {j, i + 1, n}], 1];
c = DiagonalMatrix@*SparseArray /@ Orthogonalize[Table[SparseArray[{{i} -> I, {i + 1} -> -I}, {n}], {i, 1, n - 1}]];
basis = Join[a, b, c];
MatrixForm /@ basis

Up to ordering, scaling, and a multiplication with the imaginary unit, these are the Gell-Mann matrices.

Testing if we really have elements of $\mathfrak{su}(3)$:

Max[Abs[Tr /@ basis]]
Max[Abs[basis + ConjugateTranspose /@ basis]]

(* 0 *)
(* 0 *)

Testing the orthonormality:

innerprod = {x, y} \[Function] Tr[x.ConjugateTranspose[y]];
Outer[innerprod, basis, basis, 1] == IdentityMatrix[n (n - 1) + (n - 1)]
(* True *)

Orthogonality is a nice feature since the structure constants can now be computed with

SparseArray@ Outer[{x, y, z} \[Function] innerprod[x.y - y.x, z], basis, basis, basis, 1]