1.2.3 The Dihedral Differential Forms $\in 3\mathbb{C}$

This sec­tion con­sid­ers the de­riv­a­tives of matrix gen­er­a­tors and the re­la­tion­ship with the re­sult­ing Lie al­ge­bras that un­der­ly the dif­fer­en­tial func­tion. The sub­ject matter is of par­tic­u­lar im­por­tance in math­e­mat­i­cal physics. We start the sec­tion by con­sid­er­ing the el­e­ment-by-el­e­ment de­riv­a­tive of the $G\left( {x_{1}X}_{1}^{\mathbb{C}} \right)$ matrix

where the con­tin­u­ous nature of $G\left( {x_{1}X}_{1}^{\mathbb{C}} \right)$ en­sures the de­riv­a­tive's ex­is­tence. This dif­fer­en­tial form is anal­o­gous to the ve­loc­i­ty of the $G\left( {x_{1}X}_{1}^{\mathbb{C}} \right)$ group. When taken el­e­ment-by-el­e­ment, there are five dis­tinct ve­loc­i­ty vector func­tions in the result, which to­geth­er sit in $\mathbb{C}^{3}$ space. The $D_{3\mathbb{C}}$ set of matrix el­e­ments $A_{\mathbb{C}}$ is com­posed of the fol­low­ing twelve per­mu­ta­tion el­e­ments

for which the $k = \left\{ 0,1 \right\}.$Pro­ceed­ing, we dif­fer­en­ti­ate $G^{'}\left( {x_{1}X}_{1}^{\mathbb{C}} \right)$ to find the dif­fer­en­tial form anal­o­gous to the ac­cel­er­a­tion of the $G\left( {x_{1}X}_{1}^{\mathbb{C}} \right)$ group,

which is also com­prised of five dis­tinct ac­cel­er­a­tion vector func­tions $\in \mathbb{C}^{3}$ space. For $G^{'}\left( {x_{1}X}_{1}^{\mathbb{C}} \right),$we look to find the Lie al­ge­bra that cap­tures the essence of the group, which we denote $\left( {x_{1}X}_{1}^{\mathbb{C}} \right)'$, and then con­sid­er the $\left( {x_{1}X}_{1}^{\mathbb{C}} \right)^{'}$ de­riv­a­tive al­ge­bras as a whole $\in \left( \mathfrak{g}^{\mathbb{C}} \right)^{'}$. To do this, we first note a phase shift of the $x_{1}$ ar­gu­ment by $\frac{i\pi}{2}$ ra­di­ans will bring an ini­tial ex­po­nen­tial func­tion cen­tered at the iden­ti­ty $e^{(0)} = 1$, to the func­tion $e^{\left( \frac{i\pi}{2} \right)} = i.$In the frame­work of phase-shift­ing lies the key to how the Lie al­ge­bra changes by the dif­fer­en­ti­a­tion of the co­in­cid­ing Lie group. For in­stance, the de­riv­a­tive of a group $G = e^{iXt}$

can be put into a form that is more con­sis­tent with the phase-shift­ing con­cept, as

where $G'$is phase-shift­ed by $\frac{i\pi}{2}$ com­pared to $G.$By ap­ply­ing a sim­i­lar phase-shift for the Lie al­ge­bra rep­re­sent­ing $G^{'}\left( {x_{1}X}_{1}^{\mathbb{C}} \right)$, we find

whose ex­po­nen­tial map­ping is the de­sired group de­riv­a­tive $G^{'}\left( {x_{1}X}_{1}^{\mathbb{C}} \right)$

A sim­i­lar phase-shift­ing of the other Lie al­ge­bras of the group, $\left( X_{j}^{\mathbb{C}} \right)^{'} \in \left( \mathfrak{g}^{\mathbb{C}} \right)^{'}$, leads to the fol­low­ing dif­fer­en­tial al­ge­bras

where for brevi­ty, the $\pm$ sign is used to rep­re­sent the six al­ge­bras in the sub­group. The link be­tween the phase-shift and the matrix log­a­rithm should be an ob­vi­ous one, and for the $\left( {x_{1}X}_{1}^{\mathbb{C}} \right)^{'}$ al­ge­bra, this is seen by cen­ter­ing the vari­able at $x_{k} = 1$, then eval­u­at­ing the log­a­rithm of the el­e­ments

where $\phi_{\lbrack 1\rbrack}$ de­notes the amount of phase-shift oc­cur­ring for the first de­riv­a­tive of a par­tic­u­lar el­e­ment $\in \left( {x_{1}X}_{1}^{\mathbb{C}} \right)^{'}$. For higher-order de­riv­a­tives of the $G\left( {x_{1}X}_{1}^{\mathbb{C}} \right)$ Lie groups, the phase shifts will equal

which are simply mul­ti­ples of the phase shift of the first-order de­riv­a­tive al­ge­bra $\left( {x_{1}X}_{1}^{\mathbb{C}} \right)^{'}$. There­fore, the $n^{th}$ de­riv­a­tive of the con­tin­u­ous Lie groups $G\left( {x_{1}X}_{1}^{\mathbb{C}} \right)$ will have the de­riv­a­tive al­ge­bra forms

We recall the $y_{j}Y_{j}^{\mathbb{C}}$ el­e­ments that form a sub­group $\in \mathfrak{g}^{\mathbb{C}}$

and we con­sid­er de­riv­a­tives of their ex­po­nen­tial map­pings using the di­ag­o­nal­iza­tion tech­nique

Taking the $Y_{2}^{\mathbb{C}}$ al­ge­bra, this can be ex­pressed as the fol­low­ing prod­uct of nor­mal­ized eigen­vec­tors and di­ag­o­nal matrix of eigen­val­ues

which sim­pli­fies the abil­i­ty to cal­cu­late the ex­po­nen­tial map­ping of $Y_{2}^{\mathbb{C}}$

where only the di­ag­o­nal eigen­val­ues need to be ex­po­nen­ti­at­ed. Like what was found in the last sec­tion for $\exp\left( - {y_{2}Y}_{2}^{\mathbb{C}} \right)$, the matrix group $G\left( {y_{2}Y}_{2}^{\mathbb{C}} \right)$ is com­posed of a sim­i­lar arrange­ment of cyclic trigono­met­ric func­tions

and using this eigen­vec­tor/eigen­val­ue matrix prod­uct de­com­po­si­tion, more in­tu­itive ex­pres­sions for each come in the form of the ex­po­nen­tial func­tion com­po­si­tions

Above, each func­tion can be seen as a sum of the ex­po­nen­tial map­pings of the eigen­val­ues of ${y_{2}Y}_{2}^{\mathbb{C}}$, with phase-shifts of $- \frac{2i\pi}{3}$ or $\frac{2i\pi}{3}$ for each func­tion's second two ex­po­nen­tial func­tions. The di­ag­o­nal­iza­tion tech­nique also en­ables us to sim­pli­fy the ap­proach to find­ing matrix-scalar de­riv­a­tives. For the $G\left( {y_{2}Y}_{2}^{\mathbb{C}} \right)$ group, we can eval­u­ate the de­riv­a­tive by way of the dif­fer­en­tial of the di­ag­o­nal­ized por­tion

and find for the pos­i­tive­ly-valued ${y_{2}Y}_{2}^{\mathbb{C}}$

which, in terms of phase-shift­ing, is

Taking the log­a­rithm of a phase-shift­ed di­ag­o­nal matrix is straight­for­ward

and the first-order de­riv­a­tive al­ge­bra $\left( Y_{2}^{\mathbb{C}} \right)^{'}$ is de­ter­mined to be

where the sum of non-vari­able terms equals zero

The second de­riv­a­tive for $G\left( {y_{2}Y}_{2}^{\mathbb{C}} \right)$ is de­ter­mined by

and is equiv­a­lent to the matrix ex­po­nen­tial map­ping of the second-order de­riv­a­tive al­ge­bra $\left( {y_{2}Y}_{2}^{\mathbb{C}} \right)^{''}$. A sim­i­lar process for $\left( {y_{2}Y}_{2}^{\mathbb{C}} \right)^{''}$ yields

and by eval­u­at­ing the third de­riv­a­tive for $G\left( {y_{2}Y}_{2}^{\mathbb{C}} \right)$, we find the Lie al­ge­bra re­turns to the orig­i­nal $Y_{2}^{\mathbb{C}}$ form

The ex­po­nen­tial map­ping of the ${y_{2}Y}_{2}^{\mathbb{C}}$ and ${y_{3}Y}_{3}^{\mathbb{C}}$ al­ge­bras result in odd splits of the ex­po­nen­tial func­tion for its trigono­met­ric func­tions, and three dis­tinct de­riv­a­tive al­ge­bras in ${y_{2}Y}_{2}^{\mathbb{C}},\left( {y_{2}Y}_{2}^{\mathbb{C}} \right)^{'},\left( {y_{2}Y}_{2}^{\mathbb{C}} \right)^{''}.$As a sub­group $\in \left( \mathfrak{g}^{\mathbb{C}} \right)^{'}$, the $\left( {y_{j}Y}_{j}^{\mathbb{C}} \right)^{'}$ dif­fer­en­tial al­ge­bras are the fol­low­ing, noting that the result can be both pos­i­tive or neg­a­tive as this hap­pens in the com­plex domain,