1.2.4 The Dihedral Differential Group $\in 3\mathbb{R}$

We con­tin­ue with dif­fer­en­tial forms for the $D_{3\mathbb{R}}$ group, which has a per­mu­ta­tion set $A^{\mathbb{R}} \in \mathbb{R}^{3}$, com­prised of the six fol­low­ing ma­tri­ces

The $G\left( {x_{1}X}_{1}^{\mathbb{R}} \right)$ group is the result of taking the ex­po­nen­tial map­ping of the ${x_{1}X}_{1}^{\mathbb{R}}$ al­ge­bra,

which com­pris­es the ex­po­nen­tial func­tion for its axis entry and the hy­per­bol­ic trigono­met­ric func­tions for the non-axis matrix en­tries. The matrix de­riv­a­tive of the $G\left( {x_{1}X}_{1}^{\mathbb{R}} \right)$ group is

for which the $\cosh\left( x_{1} \right)$ and $\sinh\left( x_{1} \right)$ terms have switched. The reason for this switch is due to the pres­ence of the hy­per­bol­ic

trigono­met­ric func­tions $\in e^{x_{1}}$, and eval­u­at­ing the de­riv­a­tive of the ex­pand­ed trigono­met­ric form $e^{x_{1}}$, we find

To com­pare the forms $G\left( {x_{1}X}_{1}^{\mathbb{R}} \right)$ and $G^{'}\left( {x_{1}X}_{1}^{\mathbb{R}} \right)$, we set $x_{1} = 0$, and eval­u­ate the fol­low­ing matrix-vector prod­ucts

find­ing that $G\left( {x_{1}X}_{1}^{\mathbb{R}} \right)$ acts as an iden­ti­ty, leav­ing the vector un­changed, while $G^{'}\left( {x_{1}X}_{1}^{\mathbb{R}} \right)$ re­flects in the $X_{2}/X_{3}$-plane. With the above cal­cu­la­tion, we get the first clues sur­round­ing the nature of the de­riv­a­tive $D_{3\mathbb{R}}$, namely, a swap­ping of the hy­per­bol­ic trigono­met­ric func­tions for the $X_{j}$ el­e­ments of the group. This means that each $\cosh\left( x_{k} \right)$ el­e­ment ro­tates in $\mathbb{C}^{3}$ space by $\frac{i\pi}{2}$ ra­di­ans in the coun­ter­clock­wise di­rec­tion, ar­riv­ing to ori­en­tate itself with what was pre­vi­ous­ly the $\sinh\left( x_{k} \right)$ coun­ter­part. For the $\sinh\left( x_{k} \right)$ func­tions, they rotate$\ \frac{- i\pi}{2}$ ra­di­ans for a clock­wise ro­ta­tion in $\mathbb{C}^{3}$ space, to arrive at the ori­en­ta­tion of the orig­i­nal $\cosh\left( x_{k} \right)$ coun­ter­part.

In Fig­ures 1.2.3-4, we see how the ver­tices are arranged for each iden­ti­ty in re­la­tion­ship to one an­oth­er in $\mathbb{C}^{3}$ space. For the $\left( x_{j}X_{j}^{\mathbb{R}} \right)^{'}$ el­e­ments, we find com­plex phase shifts in ra­di­ans for the fol­low­ing Lie al­ge­bra el­e­ments $\in \left( \mathfrak{g}^{\mathbb{R}} \right)^{'}$

and once ex­po­nen­ti­at­ed, the $e^{\left( x_{j}\left( X_{j}^{\mathbb{R}} \right)^{'}\ \right)}$ gen­er­a­tor­sare the fol­low­ing

The $\left( x_{j}X_{j}^{\mathbb{R}} \right)^{'}$ de­riv­a­tive al­ge­bras are a de­par­ture from the last sec­tion's $\left( x_{j}X_{j}^{\mathbb{C}} \right)^{'}$, given the al­ge­bras have gone from three non-zero en­tries to five. From 1.2.68, the de­riv­a­tive al­ge­bras $\left( x_{j}X_{j}^{\mathbb{R}} \right)^{'}$ have phase shifts of $\frac{i\pi}{2}$ ra­di­ans for the non-axis di­ag­o­nal el­e­ments, and $- \frac{i\pi}{2}$ ra­di­ans for the non-axis en­tries with $x_{j}$ vari­ables. In Figure 1.2.5, we show the phase-shift­ing ro­tates the non-axis hy­per­bol­ic func­tions $\in G\left( {x_{1}X}_{1}^{\mathbb{R}} \right)$, to the that of the non-axis hy­per­bol­ic func­tions $\in G^{'}\left( {x_{1}X}_{1}^{\mathbb{R}} \right)$. The phase-shift­ing be­comes a ro­ta­tion once the al­ge­bra is ex­po­nen­ti­at­ed, seen as the arcs that rotate by $\frac{i\pi}{2}$ ra­di­ans for $\cosh\left( x_{1} \right)G\left( {x_{1}X}_{1}^{\mathbb{R}} \right) \rightarrow \cosh\left( x_{1} \right)G'\left( {x_{1}X}_{1}^{\mathbb{R}} \right)$, and by $- \frac{i\pi}{2}$ ra­di­ans for $\sinh\left( x_{1} \right)G\left( {x_{1}X}_{1}^{\mathbb{R}} \right) \rightarrow \sinh\left( x_{1} \right)G'\left( {x_{1}X}_{1}^{\mathbb{R}} \right)$.

func­tions (black), and that of its de­riv­a­tive group's, $G'\left( {x_{j}X}_{j}^{\mathbb{R}} \right)$, (red).

For unit hy­per­bo­lae $\in \mathbb{R}^{2}$, those ori­ent­ed along the hor­i­zon­tal axis will have the equa­tion

which has the fol­low­ing hy­per­bol­ic trigono­met­ric func­tion rep­re­sen­ta­tion

The equa­tion for a ver­ti­cal­ly ori­ent­ed unit hy­per­bo­la is

which has the hy­per­bol­ic trigono­met­ric func­tion rep­re­sen­ta­tion

Using the fol­low­ing $2 \times 2$ matrix com­posed of hy­per­bol­ic trigono­met­ric func­tions, we find its de­ter­mi­nant is a match with the first hy­per­bol­ic iden­ti­ty

and the de­riv­a­tive de­ter­mi­nant match­es the second hy­per­bol­ic iden­ti­ty

When taking the de­ter­mi­nants of the ex­po­nen­tial de­riv­a­tives $e^{\left( x_{j}X_{j}^{\mathbb{R}} \right)^{'}\ }$, we find a sim­i­lar re­la­tion­ship to the ver­ti­cal unit hy­per­bo­la with neg­a­tive de­ter­mi­nants, where­by taking the result for $e^{\left( x_{1}X_{1}^{\mathbb{R}} \right)^{'}\ }$

we find that the neg­a­tive de­ter­mi­nant is scaled by the mag­ni­tude of the axis com­po­nent, $e^{x_{1}}.$

The key in­sight re­vealed by the $\left( x_{j}X_{j}^{\mathbb{R}} \right)^{'}$ de­riv­a­tive al­ge­bras is that the $G\left( {x_{j}X}_{j}^{\mathbb{R}} \right)$ groups sit in $\mathbb{C}^{3}$ space, ev­i­denced by the imag­i­nary phase shift values in its Lie al­ge­bras. To argue that these do not sit in $\mathbb{C}^{3}$ space would be to accept the non-ex­is­tence of their de­riv­a­tives. The $\left( {y_{j}Y}_{j}^{\mathbb{R}} \right)^{'}$ are the same as the $\left( {y_{j}Y}_{j}^{\mathbb{C}} \right)^{'}$ al­ge­bras, with the ex­cep­tion that they are solely pos­i­tive in value, and for the sake of brevi­ty, the reader may go back to ref­er­ence Eq. 1.2.61.