1.2.5 The Dihedral Cone Differential Group $\in 3\mathbb{C}$

We move on to study the de­riv­a­tives for the di­he­dral group $D_{3\mathbb{c}}$ $\in \mathbb{C}^{3 \times 3}$, where there are twenty-four el­e­ments in the set $A^{\mathbb{c}} = \left\{ X_{j,k\ }^{\mathbb{c}}{,Y}_{j,k}^{\mathbb{c}} \right\}:\left( \ \binom{j = \left\{ 1,2,3 \right\}}{k = \left\{ 0,1,2,3 \right\}} \right)$

We keep in mind that the values of $k$ can be con­sid­ered as cyclic, mean­ing $k\ \text{mod}\ 4,$and it should be noted that as the $k$ index in­creas­es from zero, there is a ro­ta­tion by $\frac{i\pi k}{2}$ ra­di­ans that is oc­cur­ring. We start with the ex­po­nen­tial map­ping of the first Lie al­ge­bra $x_{1}X_{1,0}^{\mathbb{c}}$

which has the fol­low­ing dif­fer­en­tial form

We see that the de­riv­a­tive be­haves like the $\left( x_{j}X_{j}^{\mathbb{R}} \right)^{'}$ de­riv­a­tive al­ge­bras, where there is the same switch­ing of the hy­per­bol­ic trigono­met­ric func­tions for the non-axis en­tries, dif­fer­ing given that its com­plex-valued ar­gu­ment leads to a scalar mul­ti­pli­ca­tion of the group by $( - 1)^{\frac{1}{4}}$. $G^{'}\left( {x_{1}X}_{1,0}^{\mathbb{c}} \right)$ can also be rep­re­sent­ed as the ex­po­nen­tial map­ping of its de­riv­a­tive al­ge­bra $\left( {x_{1}X}_{1,0}^{\mathbb{c}} \right)^{'}$, which is the fol­low­ing phase-shift of the ${x_{1}X}_{1,0}^{\mathbb{c}}$ al­ge­bra

where the axis entry is phase-shift­ed by $\frac{i\pi}{4}$ ra­di­ans, the non-axis di­ag­o­nal en­tries by $- \frac{i\pi}{4}$ ra­di­ans, and the non-axis vari­able en­tries by $\frac{i\pi}{2}$ ra­di­ans. Sim­i­lar­ly, the gen­er­al­iza­tion for the $\left( {x_{1}X}_{1,k}^{\mathbb{c}} \right)^{'}$ de­riv­a­tive al­ge­bras can be rep­re­sent­ed as

which leads to the fol­low­ing gen­er­al­iza­tion for the $\left( {x_{j}X}_{j,k}^{\mathbb{c}} \right)^{'}$ group of de­riv­a­tive al­ge­bras

It is also pos­si­ble to find a gen­er­al­iza­tion for the $\left( y_{j}Y_{j,k}^{\mathbb{c}} \right)^{'}$ de­riv­a­tive al­ge­bras. To show this, we take the $\left( y_{2}Y_{2,k}^{\mathbb{c}} \right)^{'}$ ex­am­ple and ex­press it in the di­ag­o­nal­ized form $y_{2}\alpha_{k}Y_{2} = VDV^{- 1}$

for the vari­ables $\alpha_{k}\in \mathbb{C}$, and $y_{2} \in \mathbb{R}^{+}$. As we did in the pre­vi­ous sec­tion, we eval­u­ate op­er­a­tions on the di­ag­o­nal­ized matrix of eigen­val­ues $D$ in the fol­low­ing order: the ex­po­nen­tial map­ping, its de­riv­a­tive, and then the log­a­rithm of the re­sult­ing di­ag­o­nal matrix

After which, the de­com­po­si­tion be­comes

where­by sub­sti­tu­tion with $\alpha_{k} = e^{\left( \frac{i\pi k}{2} \right)}$, the di­ag­o­nal­ized matrix equals

and by as­sum­ing the prin­ci­pal-valued log­a­rithm func­tion, it be­comes

For $V$, the eigen­vec­tor matrix

the prod­uct

yields the gen­er­al­ized de­riv­a­tive al­ge­bra, $\left( {y_{2}Y}_{2,k}^{\mathbb{c}} \right)^{'}$ rep­re­sen­ta­tion

The gen­er­al­ized form above dif­fers from the dif­fer­en­tial al­ge­bra $\left( {y_{2}Y}_{2}^{\mathbb{R}} \right)^{'}$ by the ad­di­tion of the $\ln\left( e^{\left( \frac{i\pi k}{2} \right)} \right)$ func­tions for its di­ag­o­nal el­e­ments, and each $y_{2}$ vari­able is now mul­ti­plied by $e^{\left( \frac{i\pi k}{2} \right)}.$