1.5.8 The Infinite Symmetry Group and the Subtangent Space

In the last sec­tion, we saw that by taking the limit of the min­i­mal el­e­ment in­fin­i­tes­i­mal­ly close to the iden­ti­ty el­e­ment, we were able to smooth the min­i­mal el­e­ment yield­ing a con­tin­u­ous func­tion space that was in­fi­nite­ly dif­fer­en­tiable. All the di­he­dral groups that we have de­scribed in this book have gen­er­al­ized per­mu­ta­tions in six forms, which we arrange as the fol­low­ing two sub­sets $\mathcal{X}_{k}$, $\mathcal{Y}_{k}$:

Up until this point, the matrix en­tries $a,b,c$ were from the fol­low­ing number set $\left\{ \pm 1, \pm i, \pm ( - 1)^{\left( \frac{1}{4} \right)}, \pm ( - 1)^{\left( \frac{3}{4} \right)} \right\}$, having its basis de­ter­mined by the min­i­mal el­e­ment choice $( - 1)^{\left( \frac{1}{4} \right)}$. For a min­i­mal el­e­ment taken as the in­fin­i­tes­i­mal of the $C^{\infty}$ func­tion, de­scribed in the last sec­tion, the matrix en­tries $a,b,c$ can locate on any point on the unit circle for spe­cif­ic values in the in­fi­nite set $U(1)$. We can recast the el­e­ment sub­sets with the $U(1):C^{\infty}$ func­tion in mind, where the set $A^{\infty} = \left\{ \mathcal{X}_{k}^{\infty},\mathcal{Y}_{k}^{\infty} \right\}$ is com­posed of the fol­low­ing sub­sets

We can see from the above that the prod­ucts of any el­e­ments from these sub­sets would yield an­oth­er el­e­ment with all en­tries $\in U(1)$, for $a,b,c$ $\in \mathbb{R}$. All these el­e­ments can be fur­ther con­sid­ered as the in­fi­nite tan­gent space of the $S_{3\mathbb{C}}$ group $\in \mathbb{C}^{3}$, de­not­ed as

and this in­fi­nite tan­gent space is an in­fi­nite Lie al­ge­bra as well. Con­cern­ing this in­fi­nite tan­gent space, we can now define the in­fi­nite sym­me­try group $S_{3\mathbb{C}}^{\infty}$:

De­f­i­n­i­tion (1.5.127) – The uni­tary In­fi­nite Sym­me­try Group $\mathbf{S}_{\mathbf{3}\mathbb{C}}^{\mathbf{\infty}}$ is a group under com­po­si­tion sit­ting in the com­plex geo­met­ric space $\mathbb{C}^{3 \times 3}$, which is com­prised of $C^{\infty}$ func­tions for its in­fin­i­tes­i­mal min­i­mal el­e­ment basis $\in \mathbb{C}^{3},$ yield­ing the in­fi­nite set $A^{\infty}$ of uni­tary el­e­ments in tan­gent space.

The­o­rem (1.5.128) – For $A^{\infty}$as a set, the set of per­mu­ta­tions $\in S_{3\mathbb{C}}^{\infty}$ forms a group under com­po­si­tion

Proof – Ver­i­fi­ca­tion of group axioms

• Clo­sure: All el­e­ments $\in A^{\infty}$are bi­jec­tive, and the com­po­si­tion of any el­e­ments $\phi \circ \psi\ \in A^{\infty}$ is also bi­jec­tive.

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

• Iden­ti­ty: The iden­ti­ty func­tion $\epsilon_{A^{\infty}}$ bi­jec­tive­ly maps all el­e­ments of the set $A^{\infty}$to them­selves $\left( \epsilon_{A^{\infty}}:A^{\infty} \rightarrow A^{\infty} \right)$.

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

$\square$

When con­sid­er­ing the in­fi­nite tan­gent space of el­e­ments and how the el­e­ments are com­posed of $C^{\infty}$, we can math­e­mat­i­cal­ly con­sid­er the ex­is­tence of a sub­tan­gent space to the tan­gent space, de­fined as:

De­f­i­n­i­tion (1.5.129) – The Sub­tan­gent Space $\mathcal{T}\left( \mathfrak{s}_{\mathbf{3}\mathbb{C}}^{\mathbf{\infty}} \right)$ is the matrix log­a­rithm of the in­fi­nite tan­gent space, $\mathfrak{T}\left( S_{3\mathbb{C}}^{\infty} \right)$,

In the de­f­i­n­i­tion above, the sub­tan­gent space, when paired down, op­er­ates as the matrix log­a­rithm of the Lie al­ge­bra el­e­ments $\in \mathfrak{s}_{3\mathbb{C}}^{\infty}$, where­as the tan­gent space op­er­ates as the matrix log­a­rithm of the Lie group $S_{3\mathbb{C}}^{\infty}.$It turns out that the sub­tan­gent space is very con­sis­tent with what we al­ready know about Lie al­ge­bras in gen­er­al, where many of the same prop­er­ties hold in the sub­tan­gent space, and this opens up many pos­si­bil­i­ties about how we may go about con­sid­er­ing group in­ter­re­la­tion­ships. We start by first defin­ing the iden­ti­ty group as the $S_{3}^{\epsilon}$ sym­me­try group:

De­f­i­n­i­tion (1.5.130) – The Iden­ti­ty Group $\mathbf{S}_{\mathbf{3}}^{\mathbf{\epsilon}}$, of the in­fin­i­ty sym­me­try group $S_{3\mathbb{C}}^{\infty}$ is the set of el­e­ments $A^{\epsilon},$ com­pris­ing the $S_{3}^{\epsilon}$ sym­me­try group

We take the tan­gent space $\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$ to be the fol­low­ing set of in­fin­i­ty el­e­ments

to define the matrix log­a­rithm $\ln\left( \mathfrak{T}\left( {u \times \mathfrak{s}}_{3\mathbb{C}}^{\infty} \right) \right)$, as $\mathcal{T}\left( \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$

then state the fol­low­ing the­o­rem:

The­o­rem (1.5.133) – The dif­fer­en­tial sub­tan­gent space of $S_{3\mathbb{C}}^{\infty}$, de­not­ed by $\mathcal{T}\left( \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$, con­sists of the com­plex $3 \times 3$ ma­tri­ces which rep­re­sent the dif­fer­en­tial angle space of the tan­gent space, for all $\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$, as equal to an el­e­ment of the iden­ti­ty group $S_{3}^{\epsilon}$, for $u = 0$, and the de­riv­a­tive of the tan­gent space $u\mathfrak{T}\left( \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$, equals the sub­tan­gent space$\ \mathcal{T}\left( \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$, in the neigh­bor­hood of the iden­ti­ty group and its el­e­ments $S_{3}^{\epsilon}$ , for $u = 0$.

Proof – The el­e­ments of the tan­gent space $\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$ have three matrix en­tries in the form $e^{u\left( i\theta_{k} \right)}$, re­sult­ing from the ex­po­nen­ti­a­tion of the in­di­vid­ual matrix en­tries and not the matrix ex­po­nen­tial. As such, the matrix dif­fer­en­tial will take the form com­prised of el­e­ments $i\theta_{k}$, for $u = 0,$ rep­re­sent­ing the angle ar­gu­ments for each ex­po­nen­tial entry $\in \mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$. These angle ar­gu­ments, arranged in their re­spec­tive ma­tri­ces, have been de­fined as the sub­tan­gent space $\mathcal{T}\left( \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$, and there­fore, $\frac{d}{d\mathcal{u}}\left( \left. \ \mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right) \right|_{u = 0} \right) = \mathcal{T}\left( \mathfrak{s}_{3\mathbb{C}}^{\infty} \right).$ For any ex­po­nen­tial entry in a tan­gent space matrix of the form $e^{u\left( i\theta_{k} \right)}$, for $u = 0$, we find that $e^{u\left( i\theta_{k} \right)} = 1$, and all ma­tri­ces take the form of an el­e­ment in the iden­ti­ty group $S_{3}^{\epsilon}$.

$\square$

Mind­ful as we con­tin­ue, stated in the the­o­rem is that the sub­tan­gent space basis is made up of the space of angles from the group's tan­gent space. Ex­pand­ing the de­f­i­n­i­tion of how the iden­ti­ty is a pro­found con­cept, help­ing to link pre­vi­ous­ly dis­parate groups to­geth­er. Pre­vi­ous­ly we con­sid­ered the iden­ti­ty el­e­ment of our di­he­dral groups, as

and in the con­text of the sub­tan­gent space, we can see that the iden­ti­ty el­e­ment $\epsilon$ is just an el­e­ment in the $S_{3}^{\epsilon}$ iden­ti­ty group. We may con­sid­er our orig­i­nal iden­ti­ty $\epsilon$, as the iden­ti­ty of an iden­ti­ty group, where the con­cept of nest­ing iden­ti­ties nec­es­sar­i­ly arises, which leads to the fol­low­ing de­f­i­n­i­tion:

De­f­i­n­i­tion (1.5.135) – The Iden­ti­ty Group $\mathbf{S}_{\mathbf{3}}^{\mathbf{\epsilon}}$ of the sub­tan­gent space $\mathcal{T}\left( \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$ is the group of el­e­ments from the de­fined tan­gent space $\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$

re­al­ized for $u = 0$

for $S_{3}^{\epsilon}$, equal to $D_{3}/S_{3}$, and is there­fore a group under com­po­si­tion as proven pre­vi­ous­ly.

As de­fined, the dif­fer­en­tial sub­tan­gent space is the ar­gu­ment com­po­si­tion of the angle space. From the log­a­rith­mic stand­point, these angle space ar­gu­ments would rep­re­sent the el­e­ment-by-el­e­ment prin­ci­pal log­a­rithms of the tan­gent space for $a,b,c \in \mathbb{R}^{+}$, and not the matrix log­a­rithm func­tion. From the per­spec­tive of the taking the log­a­rithm of in­di­vid­ual en­tries in the matrix el­e­ments $\in \mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$, neg­a­tive in­fini­ties are the result for the zero-valued en­tries. An ex­am­ple of­fered is the el­e­ment-by-el­e­ment prin­ci­pal log­a­rithm of $\mathcal{X}_{1}^{\infty}\in \mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$, which we define using the no­ta­tion ${\ln_{element}(\mathcal{X}}_{1}^{\infty})$

We know that these neg­a­tive in­fini­ties arise as so­lu­tions for $x$ in the equa­tion $e^{x} = 0$. In Figure 1.5.1, we plot the func­tion, $e^{x}$ for $x\in \mathbb{R}$, where we see the iden­ti­ty is the point where the func­tion cross­es the ver­ti­cal axis, and the func­tion goes to $+ \infty$, as $x \rightarrow \infty$, while for $x \rightarrow - \infty$, the $e^{x}$ func­tion goes to zero $e^{- \infty} = 0.$

Like the prin­ci­pal el­e­ment-by-el­e­ment log­a­rithm, the dif­fer­en­tial sub­tan­gent al­ge­bra is trans­formed to the al­ge­bras of $\mathfrak{T}\left( \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$ tan­gent space by the ex­po­nen­tial func­tion, and not the matrix ex­po­nen­tial. Since the dif­fer­en­tial sub­tan­gent space is de­fined as a de­riv­a­tive with re­spect to $u$ for $\frac{d}{d\mathcal{u}}\left( \left. \ \mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right) \right|_{u = 0} \right),$it is con­sis­tent with the de­f­i­n­i­tion for the tan­gent space in Lie theory, dif­fer­ing from the matrix log­a­rithm op­er­a­tion, which de­fines the tan­gent space $\mathfrak{T}\left( \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$. We demon­strate this with an ex­am­ple using the el­e­ment $\mathcal{X}_{1}^{\infty}$ $\in \mathfrak{T}\left( \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$

and cal­cu­late its matrix log­a­rithm $\ln\left( \mathcal{X}_{1}^{\infty} \right)$

and by as­sum­ing a prin­ci­pal valued matrix log­a­rithm for $a,b,c \in \mathbb{R}^{+}$, this sim­pli­fies to

A sim­i­lar ap­proach ap­plied for the other el­e­ment $\mathcal{X}_{1}^{\infty}$ $\in \mathfrak{T}\left( \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$, re­sults in the $\mathcal{X}_{k}^{\infty}$ subset of el­e­ments based upon the prin­ci­pal matrix log­a­rithms for $a,b,c \in \mathbb{R}^{+}$,

For the $\mathcal{Y}_{k}^{\infty}$ in­fin­i­ty el­e­ments the prin­ci­pal matrix log­a­rithm of $\mathcal{Y}_{k}^{\infty}$ is straight­for­ward

For the eval­u­a­tion of the in­fin­i­ty el­e­ments $\mathcal{Y}_{2}^{\infty},\mathcal{Y}_{3}^{\infty}$, we apply the eigen­de­com­po­si­tion for $\mathcal{Y}_{2}^{\infty}$

find­ing the eigen­val­ue, eigen­vec­tor ma­tri­ces, and in­verse eigen­vec­tor ma­tri­ces$\ D,V,V^{- 1}$

The prin­ci­pal matrix log­a­rithm $\ln(D)$ is then easily cal­cu­lat­ed

with which we de­ter­mine for$\ a,b,c \in \mathbb{R}^{+}$, the prod­uct $\mathcal{Y}_{2}^{\infty} = VDV^{- 1}$

The sim­i­lar prin­ci­ple matrix log­a­rithm $\ln\left( \mathcal{Y}_{3}^{\infty} \right)$, using the eigen­de­com­po­si­tion, would then round out the $\mathcal{Y}_{k}^{\infty}$ in­fin­i­ty matrix log­a­rithm subset, hold­ing for $a,b,c \in \mathbb{R}^{+}$

With the matrix log­a­rithm sub­tan­gent space forms de­ter­mined, we state the fol­low­ing the­o­rem:

The­o­rem (1.5.147) – The log­a­rith­mic sub­tan­gent space of $S_{3\mathbb{C}}^{\infty}$, de­not­ed $\mathcal{T}_{\ln}\ \left( \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$, is com­prised of com­plex $3 \times 3$ ma­tri­ces which rep­re­sent the angle func­tion space of the tan­gent space for all $\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$, equal to an el­e­ment of the iden­ti­ty group $S_{3}^{\epsilon}$ for $u = 0$. The prin­ci­pal log­a­rithm of the tan­gent space $\mathfrak{T}\left( \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$, for angle ar­gu­ments $\in \mathbb{R}$, equals the log­a­rith­mic sub­tan­gent space$\ \mathcal{T}_{\ln}\left( \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$, in the neigh­bor­hood of the iden­ti­ty group and its el­e­ments $S_{3}^{\epsilon}$, for $u = 0$.

Proof – The el­e­ments of the tan­gent space $\mathfrak{T}\left( \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$ have three matrix en­tries $e^{\left( i\theta_{k} \right)}$, re­sult­ing from the matrix ex­po­nen­tial. As such, the matrix log­a­rithm will take the form of al­ge­bra­ic func­tion groups$,$ which rep­re­sent the matrix log­a­rithm of the angle ar­gu­ments of the tan­gent space ma­tri­ces $\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$. These angle func­tions arranged in their re­spec­tive ma­tri­ces are de­fined as the log­a­rith­mic sub­tan­gent space $\mathcal{T}_{\ln}\left( \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$ , and there­fore, $\ln\left( \mathfrak{T}\left( \mathfrak{s}_{3\mathbb{C}}^{\infty} \right) \right) = \mathcal{T}_{\ln}\left( \mathfrak{s}_{3\mathbb{C}}^{\infty} \right).$ Taking any matrix group ${\mathcal{X}_{k}^{\infty},\mathcal{Y}}_{k}^{\infty}$, for all angle vari­ables $\theta_{k} = 0$, it holds $\forall\ e^{\ln\left( {\mathcal{X}_{k}^{\infty},\mathcal{Y}}_{k}^{\infty} \right)}$ that they take the form of a el­e­ment in the iden­ti­ty group $S_{3}^{\epsilon}$.

$\square$

Math­e­mat­i­cal­ly, we have seen that a con­sis­tent sub­tan­gent space exists and is log­a­rith­mi­cal­ly re­lat­ed to the tan­gent space lo­cat­ed on the sphere of tan­gent spaces. The de­fined dif­fer­en­tial sub­tan­gent space, presents an arrange­ment com­prised of finite angle el­e­ments, taking form as a result of the zero-point $e^{- \infty}$. Where­as the de­fined log­a­rith­mic sub­tan­gent space is a well-de­fined angle space func­tion, ori­ent­ed at a math­e­mat­i­cal cross­roads where in­fini­ties take finite forms.