1.6.9 The Infinite Sphere of Tangent Spaces

We are now well versed in the as­so­ci­a­tion that in­te­gra­tion has with ex­po­nen­ti­a­tion and the as­so­ci­a­tion that dif­fer­en­ti­a­tion, the in­verse of in­te­gra­tion, has with the log­a­rithm. It is im­por­tant to re­mem­ber that the in­fi­nite number of tan­gent spaces that reside on the bound­ary of the in­fi­nite sphere of tan­gent spaces are Lie al­ge­bra rep­re­sen­ta­tions, con­tain­ing all the in­for­ma­tion of the whole in­fi­nite sym­me­try group $S_{3\mathbb{C}}^{\infty}$. The geo­met­ric spaces formed by the prod­ucts of the matrix groups of tan­gent space el­e­ments $\in \mathfrak{s}_{3\mathbb{C}}^{\infty}$, result in the ex­ten­sion of the geo­met­ric spaces away from the sin­gu­lar­i­ty based on the mag­ni­tude of the real-valued por­tion in the ex­po­nen­tial vari­able $e^{\theta_{r}^{\lbrack m\rbrack}}$. As $\theta_{r}^{\lbrack m\rbrack} \in \mathbb{C}^{m}$, there is a real por­tion of the Lie al­ge­bra and an imag­i­nary one. We offer the ex­am­ple for $\theta_{r}^{\lbrack m\rbrack}$, with $e^{i\theta_{j}}\in \mathbb{C},\left| e^{i\theta_{j}} \right| = 1$, for an $X_{1}$-type el­e­ment, giving

for $a_{j},b_{j}\in \mathbb{R}\ ,\ z_{j}\in \mathbb{C}$. In this sec­tion, we will be work­ing with the groups pre­sent­ed in the book and we will use the su­per­script from $\theta_{r_{j}}^{\lbrack m\rbrack}$, to insert the di­he­dral group type and per­mu­ta­tion type. Using the $X_{1}$ per­mu­ta­tion type, we can ex­press the Lie al­ge­bras of our di­he­dral groups in terms of their real and imag­i­nary vari­ables

and in the polar form, as

which can be de­com­posed into the prod­uct of its radial mod­u­lus matrix and spher­i­cal matrix group

Note, the prod­uct result will change if the spher­i­cal group is right mul­ti­plied by the mod­u­lus matrix

where the moduli vari­ables have swapped po­si­tions in the bottom-right $2 \times 2$ sub­ma­trix.

The tan­gent space group $\theta_{r_{j}}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack}$ is the result of the ex­po­nen­tial map­ping of its sub­tan­gent form $\ln\left( \theta_{r_{j}}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack} \right)$, and there­fore, the sub­tan­gent form for $Z_{j}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack}$ is found by the matrix log­a­rithm

The matrix ex­po­nen­tial map­ping of the Lie al­ge­bra $Z_{j}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack}$ in­volves the two-fold ex­po­nen­tial map­ping of the sub­tan­gent space $G\left( Z_{j}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack} \right) = \exp\left( \exp\left( \ln\left( Z_{j}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack} \right) \right) \right)$, and is equal to the fol­low­ing form

Con­verse­ly, the sub­tan­gent space is the result of the two-fold log­a­rithm map­ping of the Lie group $\ln(\ln\left( G\left( Z_{j}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack} \right) \right))$. To ex­press the Lie group $G\left( Z_{j}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack} \right)$ in polar form, we al­ge­braical­ly ma­nip­u­late the vari­able $\sqrt{z_{2}z_{3}}$ to its polar equiv­a­lent, as fol­lows

Next, using the con­stant $\frac{z_{3}}{\sqrt{{z_{2}z}_{3}}}$ as an ex­am­ple

and for $r_{3},r_{2} \in \mathbb{R}^{+}$, $\frac{z_{3}}{\sqrt{{z_{2}z}_{3}}}$ has the equiv­a­lent polar form

Given $\theta_{r_{j}}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack} = Z_{j}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack}$, their ex­po­nen­tial map­pings are also equiv­a­lent, $G\left( Z_{j}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack} \right) = G\left( \theta_{r_{j}}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack} \right),$and the polar form of the Lie group $G\left( \theta_{r_{j}}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack} \right)$ is

The matrix above rep­re­sents the ex­po­nen­tial map­ping of the $\theta_{r_{j}}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack}$ tan­gent space, and given that the matrix de­ter­mi­nant re­lates to the volume of a region, we cal­cu­late the de­ter­mi­nant of the G$\left( \theta_{r_{j}}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack} \right)$ group

where the de­ter­mi­nate equals the axis value $e^{r_{1}e^{i\theta_{1}}},$which is equiv­a­lent to $e^{z_{1}}$, given that ${r_{1}e^{i\theta_{1}} = z}_{1}$. When $r_{1} = 1$, the de­ter­mi­nant will be equal to $e^{e^{i\theta_{1}}}$, which in­di­cates the de­ter­mi­nant will vary de­pend­ing on the angle value $\theta_{1}$. So, real-valued peaks are equal to the con­stant $e$, oc­cur­ring when $\theta_{1} = 2\pi k$, for $k\in \mathbb{Z}$. In turn, real-valued troughs occur at $e^{- 1} \approx 0.3679$, when $\theta_{1} = \pi k$, for $k\in \mathbb{Z}$. In a beau­ti­ful and anal­o­gous manner, imag­i­nary-valued peaks locate at $e^{i} \approx (0.5403 + 0.8415i)$, oc­cur­ring when $\theta_{1} = \left( \frac{\pi}{2} + 2\pi k \right)$, for $k\in \mathbb{Z}$, with troughs at $e^{- t} \approx (0.5403 - 0.8415i)$, when $\theta_{1} = \left( \frac{3\pi}{2} + 2\pi k \right)$, for $k\in \mathbb{Z}$. To better vi­su­al­ize this, we plot the de­ter­mi­nant wave­forms for $G\left( \theta_{1}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack} \right)$ in Figure 1.6.24.

It is im­por­tant to notice that the peaks and troughs for the real and imag­i­nary wave­forms are equal to the iden­ti­ty when con­sid­ered as prod­ucts, since $e^{1}e^{- 1} = \epsilon$ and $e^{i}e^{- i} = \epsilon$, and the fre­quen­cy of the imag­i­nary wave­form is double that of the real wave­form, given the cosine func­tion is an even func­tion, while the sine func­tion is odd. To de­lin­eate be­tween the real and imag­i­nary com­po­nents of $G\left( \theta_{r_{j}}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack} \right)$, we can use the fol­low­ing hy­per­bol­ic iden­ti­ties

and sub­sti­tute $r = \sqrt{r_{2}r_{3}}$ , $e^{i\theta} = e^{i\left( \theta_{2} + \theta_{3} \right)/2},$to ex­press the matrix group in the matrix figure below.

$$\begin{array}{r} \begin{pmatrix} e^{r_{1}e^{i\theta_{1}}} & 0 & 0 \\ 0 & \left( \begin{array}{r} {\cos\left( \sqrt{r_{2}r_{3}}\sin\left( \frac{\theta_{2}}{2} + \frac{\theta_{3}}{2} \right) \right)\cosh}\left( \sqrt{r_{2}r_{3}}\cos\left( \frac{\theta_{2}}{2} + \frac{\theta_{3}}{2} \right) \right) \\ + \ isin\left( \sqrt{r_{2}r_{3}}\sin\left( \frac{\theta_{2}}{2} + \frac{\theta_{3}}{2} \right) \right)\sinh\left( \sqrt{r_{2}r_{3}}\cos\left( \frac{\theta_{2}}{2} + \frac{\theta_{3}}{2} \right) \right) \end{array} \right) & \sqrt{\frac{r_{2}}{r_{3}}}\left( e^{i\left( \frac{\theta_{2}}{2} - \frac{\theta_{3}}{2} \right)} \right)\left( \begin{array}{r} {\cos\left( \sqrt{r_{2}r_{3}}\sin\left( \frac{\theta_{2}}{2} + \frac{\theta_{3}}{2} \right) \right)\sinh}\left( \sqrt{r_{2}r_{3}}\cos\left( \frac{\theta_{2}}{2} + \frac{\theta_{3}}{2} \right) \right) \\ + \ isin\left( \sqrt{r_{2}r_{3}}\sin\left( \frac{\theta_{2}}{2} + \frac{\theta_{3}}{2} \right) \right)\cosh\left( \sqrt{r_{2}r_{3}}\cos\left( \frac{\theta_{2}}{2} + \frac{\theta_{3}}{2} \right) \right) \end{array} \right) \\ 0 & \sqrt{\frac{r_{3}}{r_{2}}}\left( e^{i\left( \frac{\theta_{3}}{2} - \frac{\theta_{2}}{2} \right)} \right)\left( \begin{array}{r} {\cos\left( \sqrt{r_{2}r_{3}}\sin\left( \frac{\theta_{2}}{2} + \frac{\theta_{3}}{2} \right) \right)\sinh}\left( \sqrt{r_{2}r_{3}}\cos\left( \frac{\theta_{2}}{2} + \frac{\theta_{3}}{2} \right) \right) \\ + \ isin\left( \sqrt{r_{2}r_{3}}\sin\left( \frac{\theta_{2}}{2} + \frac{\theta_{3}}{2} \right) \right)\cosh\left( \sqrt{r_{2}r_{3}}\cos\left( \frac{\theta_{2}}{2} + \frac{\theta_{3}}{2} \right) \right) \end{array} \right) & \left( \begin{array}{r} {\cos\left( \sqrt{r_{2}r_{3}}\sin\left( \frac{\theta_{2}}{2} + \frac{\theta_{3}}{2} \right) \right)\cosh}\left( \sqrt{r_{2}r_{3}}\cos\left( \frac{\theta_{2}}{2} + \frac{\theta_{3}}{2} \right) \right) \\ + \ isin\left( \sqrt{r_{2}r_{3}}\sin\left( \frac{\theta_{2}}{2} + \frac{\theta_{3}}{2} \right) \right)\sinh\left( \sqrt{r_{2}r_{3}}\cos\left( \frac{\theta_{2}}{2} + \frac{\theta_{3}}{2} \right) \right) \end{array} \right) \end{pmatrix} \end{array}$$

In the $G\left( \theta_{r_{j}}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack} \right)$ matrix above, it be­comes ap­par­ent that the group is a mix of hy­per­bol­ic and circle trigono­met­ric func­tions, and the group can be con­sid­ered as a prod­uct of two dis­tinct matrix group types, the ex­po­nen­tial com­plex cir­cu­lar form

and the hy­per­bol­ic ex­po­nen­tial form

whose matrix prod­uct is equal to $G\left( \theta_{r_{j}}^{3,X_{1}} \right)$

When the angle vari­ables are equal, $\theta_{2} = \theta_{3} = \theta$, the matrix groups are as­sured to fall into the dis­tinct imag­i­nary and real valued forms of the com­plex sphere and the hy­per­bo­la

In these forms, the $cos,\ cosh$ en­tries are on the main di­ag­o­nals of the matrix groups, while the $sin,sinh$ func­tions locate on the off di­ag­o­nals of the bottom-right $2 \times 2$ sub­ma­tri­ces.

We return to $G\left( \theta_{r_{j}}^{\left\lbrack D_{3\mathbb{z}}\left( {\ X}_{1} \right) \right\rbrack} \right)$, set­ting, $r_{j} = 1$, $a_{1} = 0$, and for $z_{1} = \left( a_{1} + ib \right),$the group be­comes

and plot each func­tion type in the ro­ta­tor sub­ma­trix in Figure 1.6.26, to better vi­su­al­ize their wave­forms.

The plot­ted wave­forms above reveal the real-valued func­tion, $\cos\left( \sin(\theta) \right)$, os­cil­lates be­tween troughs for $\theta$ values

and peaks

for $k \in \mathbb{Z}^{+}$. The func­tion ${isin}\left( \sin(\theta) \right)$, os­cil­lates in the plane be­tween troughs for $\theta$ values

and peaks

Note­wor­thy in the plots, the wave­form fre­quen­cy for $\cos\left( \sin(\theta) \right)$ dic­tates two peaks on each in­ter­val, $\theta = (0,2\pi\rbrack$, while the ${isin}\left( \sin(\theta) \right)$ wave­form's fre­quen­cy is half of that, ev­i­denced by its one peak on the same in­ter­val $\theta = (0,2\pi\rbrack$.

For the case of the hy­per­bol­ic group form plot, Figure 1.6.27, we can see the real-valued func­tion $\cosh\left( \cos(\theta) \right),$os­cil­lates be­tween troughs values of $1$, for $\theta$, equal to

and peaks for ar­gu­ments equal to $\cos{(\theta) = \pm 1}$

The func­tion $\sinh\left( \cos(\theta) \right)$, also os­cil­lates at half the fre­quen­cy of the $\cosh\left( \cos(\theta) \right)$ func­tion, with troughs oc­cur­ring at $\theta$ values

and peaks

When con­sid­er­ing the math­e­mat­i­cal unit sin­gu­lar­i­ty, with an on­to­log­i­cal unit radial dis­tance from its center to its bound­ary, we find the com­plex cir­cu­lar func­tions $\cos\left( \sin(\theta) \right)$, ${isin}\left( \sin(\theta) \right)$ have wave­forms be­neath the bound­ary's sur­face in com­plex space. For the hy­per­bol­ic func­tion forms, we find that the func­tion $\cosh\left( \cos(\theta) \right)$ lays en­tire­ly above the bound­ary, except for just touch­ing its sur­face at its troughs, while the ma­jor­i­ty of the $\sinh\left( \cos(\theta) \right)$ func­tion's wave­form is inside the bound­ary, except in the neigh­bor­hood of the caps of its peaks and troughs, re­spec­tive­ly, start­ing at ${cos^{- 1}}\left( \sinh^{- 1}(1) \right)$ and ${cos^{- 1}}\left( \sinh^{- 1}( - 1) \right)$.

Our treat­ment of the Lie al­ge­bra $\theta_{r_{j}}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack}$, which lays on the sin­gu­lar­i­ty's bound­ary, has not yet in­clud­ed a vari­able for its ex­po­nen­tial map­ping form. To remedy this, we define the vari­able $t_{j}\in \mathbb{R}$, and the Lie al­ge­bra $t_{j}\theta_{r_{j}}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack}$, as

with the com­plex form $t_{j}Z_{j}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack}$

and the matrix ex­po­nen­tial map­ping trans­forms these Lie al­ge­bras to the forms

The two vari­ables in the matrix groups above should be viewed as the means for which the sub­tan­gent space $\left( \ln\left( \theta_{r}^{\lbrack m\rbrack} \right) \right)$ in­ter­acts with the sphere of tan­gent spaces $\left( t\theta_{r}^{\lbrack m\rbrack} \right)$, and in turn, shows how the sphere of tan­gent spaces in­ter­acts with the ex­po­nen­tial­ly ex­tend­ed group spaces $\left( G\left( t\theta_{r}^{\lbrack m\rbrack} \right) \right)$. To better un­der­stand the rea­son­ing behind this as­ser­tion, we return to the sub­tan­gent space ex­am­ple $\ln\left( Z_{j}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack} \right)$

which using polar form vari­ables, is

Eval­u­at­ing the log­a­rithm for each entry, the sub­tan­gent space is equiv­a­lent to

a result, with com­plex matrix en­tries, ex­pressed in their real and imag­i­nary por­tions as fol­lows

and in con­junc­tion, the two lower en­tries of the main di­ag­o­nal

With these func­tions in the $\mathcal{z}_{j}$ form, it en­ables the ex­pres­sion of the sub­tan­gent space $\ln\left( \left( X_{1}\left( z_{j} \right) \right) \right)$, in its polar form

where each entry rep­re­sents the fol­low­ing

for $\mathcal{r}_{j}$, the radial moduli, $\Theta_{j}$ the an­gu­lar space vari­ables, and these are anal­o­gous to the vari­ables $r_{j}$, $\theta_{j}$. An ex­po­nen­ti­at­ed and ex­tend­ed space of the form $e^{re^{i\theta}}$, there­fore, has the $r$-vari­able used in the typ­i­cal physics treat­ments of Lie theory, where ro­ta­tions and or boosts can be per­formed in the space, but the $\theta$-vari­able rep­re­sents the angle space, and the geom­e­try of the space hinges on its given value. The move­ment in the angle space re­lat­ed to $\theta$, occurs on the sphere of tan­gent spaces, and this rep­re­sents a cru­cial con­cept. To help ensure the reader grasps this in­tu­ition, we con­sid­er the matrix $G\left( t_{j}\theta_{r_{j}}^{3,X_{1}} \right)$, as

and con­sid­er this group for all $r_{j} = 1,$and all $\theta_{j} = \frac{\pi}{2}$ , find­ing that $G\left( t_{j}{\theta\left( \frac{\pi}{2} \right)}_{1}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack} \right)$, be­comes

The ex­am­ple for the chosen vari­ables, re­sults in a$\ e^{\left( \frac{i\pi}{2} \right)}$ ro­ta­tion of the group in tan­gent space, to the equiv­a­lent matrix result

and given the iden­ti­ties, $\sinh{(i\theta) = isin(\theta),}\cosh{(i\theta) = \cos(\theta),}$ we find $G\left( t_{j}{\theta\left( \frac{\pi}{2} \right)}_{1}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack} \right)$ equals

By as­sum­ing the $t_{j}$ vari­ables are all equal, we find $G\left( t{\theta\left( \frac{\pi}{2} \right)}_{1}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack} \right)$ equals the fa­mil­iar $U(3)$ group gen­er­a­tor, $G\left( iX_{1} \right)$,

Angle space ro­ta­tions that occur on the sin­gu­lar­i­ty's bound­ary can be rep­re­sent­ed in mul­ti­ple ways, shown first, by using the $X_{1}$-type matrix

which serves to rotate each polar vari­able in the Lie al­ge­bra $\theta_{r_{j}}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack}$. The second ex­am­ple is more straight­for­ward, where we di­ag­o­nal­ize the vector $\widehat{v} = \left( e^{i\rho_{1}},e^{i\rho_{2}},e^{i\rho_{3}} \right)$, for $\rho_{j}\in \mathbb{R}$. The prod­uct of this vector with the Lie al­ge­bra $\theta_{r_{j}}^{\left\lbrack D_{3\mathbb{z}}\left( X_{1} \right) \right\rbrack}$, re­sults in solely ro­tat­ing in the angle space, with the radial moduli re­main­ing un­changed, seen as

To gain fur­ther in­tu­ition on how a ro­ta­tion in the angle space af­fects geom­e­try, we set $\theta_{j} = \frac{\pi}{2},\ r_{j} = r,$and define a vector, $\widehat{v} = \left( e^{\left( - \frac{i\pi}{2} \right)},e^{\left( - \frac{i\pi}{2} \right)},e^{\left( - \frac{i\pi}{2} \right)} \right),$to eval­u­ate

find­ing the ro­ta­tion in the angle space has changed the geom­e­try from the imag­i­nary-valued $X_{1}$-type Lie al­ge­bra, ${i \times |D}_{3\mathbb{z}}|$, to a real-valued $X_{1}$-type hy­per­bol­ic Lie al­ge­bra, $|D_{3\mathbb{z}}|$.

The con­tent of this chap­ter, and in par­tic­u­lar this sec­tion, leads to the fol­low­ing de­f­i­n­i­tion of The Tan­gent Space Com­plex Sym­me­try Group.

De­f­i­n­i­tion (1.6.304) – We define The Tan­gent Space Com­plex Sym­me­try Group $\mathbf{\theta}_{\mathbf{r}_{\mathbf{j}}}^{\left\lbrack \mathbf{D}_{\mathbf{3}\mathbb{z}}\left( \mathbf{X}_{\mathbf{k}}\mathbf{,}\mathbf{Y}_{\mathbf{k}} \right) \right\rbrack} \in \mathbb{C}^{3}$, as a group under com­po­si­tion, com­prised of $C^{\infty}$ func­tions for its in­fin­i­tes­i­mal min­i­mal el­e­ment basis $\in \mathbb{C}^{3},$ yield­ing the in­fi­nite set ${\theta_{r}^{\lbrack m\rbrack}}^{\infty}$ of uni­tary el­e­ments in tan­gent space, for $r_{j}$=1, and non-unit radial lengths $r_{j} \neq 1$, for the form ex­tend­ed or con­tract­ed in $\mathbb{C}^{3}$-space from the uni­tary state

con­tain­ing el­e­ment sets,

The­o­rem (1.6.305) – For ${\theta_{r}^{\lbrack m\rbrack}}^{\infty}$as a set, the set of per­mu­ta­tions $\theta_{r_{j}}^{\left\lbrack D_{3\mathbb{z}}\left( X_{k},Y_{k} \right) \right\rbrack}$ forms a group under com­po­si­tion.

Proof – Ver­i­fi­ca­tion of group axioms

• Clo­sure: All el­e­ments $\in {\theta_{r}^{\lbrack m\rbrack}}^{\infty}$are bi­jec­tive, and the com­po­si­tion of el­e­ments $\phi \circ \psi\ \in {\theta_{r}^{\lbrack m\rbrack}}^{\infty}$ is also bi­jec­tive.

• As­so­cia­tiv­i­ty: Per­mu­ta­tions are func­tions, and their com­po­si­tions are as­so­cia­tive.

• Iden­ti­ty: The iden­ti­ty func­tion $\epsilon_{\left( Y_{1} \right)}$ bi­jec­tive­ly maps all el­e­ments in the set ${\theta_{r}^{\lbrack m\rbrack}}^{\infty}$to them­selves $\left( \epsilon_{\left( Y_{1} \right)}:{\theta_{r}^{\lbrack m\rbrack}}^{\infty} \rightarrow {\theta_{r}^{\lbrack m\rbrack}}^{\infty} \right)$.

• In­verse: $\phi$ is a per­mu­ta­tion and is thus bi­jec­tive, and there exists a func­tion $\phi^{- 1}$ that is also bi­jec­tive.

$\square$