A question on an set of 8 matrices related to the SU(3) generators

Question

SU(2) and SU(3) differ quite a bit.

The Lie algebra of SU(2) formed by the three generators $g_n$ is the same as the algebra formed by the SU(2) matrices/elements $F_n=e^{\pi \cdot i \cdot g_n / 2}$. In fact, the $F_n$ are just given by $F_n=i \cdot g_n$. Speaking sloppily, the $F_n$ are "finite" elements of SU(2), the $g_n$ are "infinitesimal" "in" SU(2).

Instead, for SU(3), the set $F_m$ and its algebraic relations differ from that of the eight generators $g_m$. The $F_m$ have no simple relation to the $g_m$. For example, in the usual representation, 7 of the 8 generators $g_m$ have a diagonal element that is zero; in the same representation, the $F_m$ have a 1 in that same position.

Is the set of the eight finite SU(3) elements $F_m=e^{\pi \cdot i \cdot g_m / 2}$ $-$ defined by the eight SU(3) generators $g_m$ $-$ an object of study or of teaching? Is there a simple way to think about this set? (Is it an algebra?) Is there a simple multiplication or commutation or other table for it? Is the set $F_m$, thus the eight finite elements generated by the infinitesimal generators, used in any physical or other setting? Does the set have a name?

Thank you for any help whatsoever.

Answer 1

To make the discussion easier, here are the Gell-Man matrices $g_n$:

$g_1=\left ( \begin{array}{ccc} 0 & 1 & 0\\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right )$, $g_2=\left ( \begin{array}{ccc} 0 & -i & 0\\ i & 0 & 0 \\ 0 & 0 & 0 \end{array} \right )$, $g_3=\left ( \begin{array}{ccc} -1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array} \right )$, $g_4=\left ( \begin{array}{ccc} 0 & 0 & 1\\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{array} \right )$, $g_5=\left ( \begin{array}{ccc} 0 & 0 & -i\\ 0 & 0 & 0 \\ i & 0 & 0 \end{array} \right )$, $g_6=\left ( \begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array} \right )$, $g_7=\left ( \begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & -i \\ 0 & i & 0 \end{array} \right )$, $g_8=\left ( \begin{array}{ccc} 1/\sqrt{3} & 0 & 0\\ 0 & 1/\sqrt{3} & 0 \\ 0 & 0 & -2/\sqrt{3} \end{array} \right )$,

And these are the first matrices $F_n$:

$F_1=\left ( \begin{array}{ccc} 0 & i & 0\\ i & 0 & 0 \\ 0 & 0 & 1\end{array} \right )$, $F_2=\left ( \begin{array}{ccc} 0 & 1 & 0\\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right )$, $F_3=\left ( \begin{array}{ccc} i & 0 & 0\\ 0 & -i & 0 \\ 0 & 0 & 1 \end{array} \right )$, $F_4=\left ( \begin{array}{ccc} 0 & 0 & i\\ 0 & 1 & 0 \\ i & 0 & 0 \end{array} \right )$, $F_5=\left ( \begin{array}{ccc} 0 & 0 & 1\\ 0 & 1 & 0 \\ -1 & 0 & 0 \end{array} \right )$, $F_6=\left ( \begin{array}{ccc} 1 & 0 & 0\\ 0 & 0 & i \\ 0 & i & 0 \end{array} \right )$, $F_7=\left ( \begin{array}{ccc} 1 & 0 & 0\\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{array} \right )$, $ F_8$ is less simple.

Answer 2

Finite $SU(3)$ transformations, in much more broader context, are considered in https://projecteuclid.org/euclid.cmp/1103841153 (On Gell-Mann's λ-Matrices, d- and f-tensors, octets, and parametrizations of SU (3), by A. J. MacFarlane, A. Sudbery, and P. H. Weisz), http://scitation.aip.org/content/aip/journal/jmp/12/4/10.1063/1.1665634 (Finite Transformations in Various Representations of SU(3), by S.P. Rosen) and in Douglas Holland's dissertation "Finite Transformations of SU(3)": http://arizona.openrepository.com/arizona/bitstream/10150/285022/1/azu_td_6904047_sip1_m.pdf

As far as I know, $\{F_m\}$ subset of $SU(3)$ has no particular significance. However a subset of $SU(3)$ generated by finite transformations $$U(\omega)=\exp{\left(i\frac{\omega^a}{2}\lambda_a\right)}$$ with such eight dimensional vectors $\omega^a$ that satisfy $$\omega^a\omega^a=1,\;\;\;d_{abc}\omega^a\omega^b\omega^c=\frac{1}{\sqrt{3}},$$ does have a significance for the infrared region of QCD, see http://arxiv.org/abs/hep-th/9903060 (Special properties of a submanifold of $su(3)$, by R. Parthasarathy).