1.4.9 The SU(2) Subgroup of the Symmetric Conic Group

Now that we have de­scribed the sym­met­ric conic group, the cone sub­group and the axis sub­group, we can turn our at­ten­tion to fur­ther ex­plor­ing the sub­groups that lay within. It should be ob­vi­ous that all the di­he­dral groups ex­plored in pre­vi­ous chap­ters are sub­groups of the sym­met­ric conic group, in­clud­ing $S_{3}{/D}_{3},\ D_{3\mathbb{C}},D_{3\mathbb{z}}$. The reader should be think­ing about Cayley's The­o­rem, dis­cussed in Chap­ter 3, and its im­pli­ca­tion that every group is a sub­group of some sym­met­ric group. What might not be ob­vi­ous, while at the same time, re­mark­able is the $SU(2)$ group can be seen as a sub­group of the sym­met­ric conic group $SU(2) \subset S_{3\mathbb{C}}^{\Delta}$. To remind the reader, the $SU(2)$ group's Lie al­ge­bra $\mathfrak{su}(2)$ is rep­re­sent­ed by

gen­er­at­ed by the fol­low­ing ma­tri­ces,

We define two $3 \times 3$ ma­tri­ces, using the no­ta­tion$\ \Delta Y,\Delta X$

which are two per­mu­ta­tions $\in S_{3\mathbb{C}}^{\Delta}$, with de­ter­mi­nants equal to $- 1$. These ma­tri­ces form the $SU(2)$ sub­group by having sim­i­lar prod­ucts re­la­tions stud­ied for the tan­gent spaces of the cone. The re­la­tions are stated by the prod­ucts

Al­though the ma­tri­ces are $3 \times 3$, the en­tries are fixed for $a_{1,1} = 1$, re­main­ing the same value for all prod­ucts in the sub­group. Like the Lie al­ge­bra ma­tri­ces de­scribed using $2 \times 2$ ma­tri­ces, the $3 \times 3$ forms above also have de­ter­mi­nants of $1$. We could have also just cre­at­ed each matrix of the Lie al­ge­bra in­di­vid­u­al­ly but seeing how the prod­ucts of $\Delta Y,\Delta X$ create the sub­group $SU(2)$, this lends itself to a better geo­met­ric per­spec­tive about how the cone num­bers and axis num­bers relate. By as­so­ci­at­ing the vari­ables $a,b,c\in \mathbb{R}$ to the $\mathcal{u}_{j}$ ma­tri­ces, as well as defin­ing $k$ to be a scalar con­stant, we rewrite the ma­tri­ces as the set $\mathcal{u}_{j}^{'}$:

and define $\mathfrak{su}'(2)$ as the group

with de­ter­mi­nants

where these ma­tri­ces have a Eu­clid­ean dis­tance metric, scaled by the con­stant $3k$. By set­ting $k = \frac{1}{3}$, the de­ter­mi­nant re­duces to $a^{2} + b^{2} + c^{2}$, and is the fa­mil­iar de­ter­mi­nant of the $\mathfrak{su}(2)$ group with its $2 \times 2$ ma­tri­ces. As seen in Sec­tion 1.4.2, the chord-re­lat­ed tan­gent spaces of the cone yield­ed the $SU(2)$ group, and these chords can be en­vi­sioned as the en­tries in our $\Delta Y,\Delta X$ ma­tri­ces, high­light­ed in bold, in the fol­low­ing

and as per­mu­ta­tion el­e­ments of the sym­met­ric conic group, they are re­spon­si­ble for these chord re­la­tion­ships. Taking the matrix ex­po­nen­tials of the $\mathcal{u}_{j}^{'}$ ma­tri­ces, we find

which all have de­ter­mi­nants equal to $e^{k}$, rep­re­sent­ing a sym­met­ric scal­ing by way of the ro­ta­tion­al axes. In Figure 1.4.18, we rep­re­sent the ex­am­ple of the $\Delta X$-el­e­ment in re­la­tion to the circle, rep­re­sent­ing a slice of the in­scribed $SU(2)$ sphere group, for the axis of ro­ta­tion of the$\ \Delta X$-el­e­ment, rep­re­sent­ing the min­i­mal slice vector point­ing toward $F$

As we saw with the in­scribed spheres of the tan­gent cone, the other it­er­a­tions of the $SU(2)$ group are found the same way. The $SU(2)$ group with op­po­site chi­ral­i­ty can be de­rived from the fol­low­ing el­e­ments of the sym­met­ric conic group

for the matrix $\overline{\Delta X_{2,1}}$, taking the di­ag­o­nal form of a $\Delta Y$-type matrix, de­not­ing the el­e­ment with a bar to show its sub­group re­la­tion­ship. The matrix prod­ucts $\overline{\Delta X_{2,1}}$, $\Delta X_{1,2}$ are found to rep­re­sent the fol­low­ing matrix el­e­ments,

The above ma­tri­ces differ from the prior $\mathcal{u}_{j}$ ma­tri­ces for $\mathcal{u}_{0},\mathcal{u}_{3}$, by having the bottom-right $2 \times 2$ sub­ma­tri­ces flip signs for each entry. The el­e­ments $\mathcal{u}_{1},\mathcal{u}_{2},\mathcal{u}_{3}$ all square to -1, in terms of their bottom-right el­e­ments, which is con­sis­tent with being imag­i­nary number forms. The prod­ucts of the ma­tri­ces are re­versed com­pared to the pre­vi­ous, where

The re­ver­sal of the chi­ral­i­ty occurs be­cause we have swapped the $a_{2,2},a_{3,3}$ el­e­ments $\in \overline{\Delta X_{2,1}}$, from their po­si­tions $\in \Delta X_{2,1}$. In essence, this en­tails going from a coun­ter­clock­wise ori­en­ta­tion in the se­quence of el­e­ments to a clock­wise one.

In Figure 1.4.19, a graph­ic de­picts this chi­ral­i­ty change for the $SU(2)$ group, where the el­e­ments of the matrix $\Delta X_{1,2}$ are in the $X_{1}$-plane, for $A = ( - 1)^{\left( \frac{3}{4} \right)},B = ( - 1)^{\left( \frac{1}{4} \right)}$ . The el­e­ment $A$ is coun­ter­clock­wise from the center point, while the matrix $\overline{\Delta X_{2,1}}$ for $C = - ( - 1)^{\left( \frac{3}{4} \right)},D = ( - 1)^{\left( \frac{1}{4} \right)}$, the el­e­ment $D$ is coun­ter­clock­wise from the center point in the $X_{2}$-plane. To make this clear­er the ma­tri­ces $\Delta X_{2,1},\overline{\Delta X_{2,1}}$ are pre­sent­ed below with the same values for $C$ and $D$

A com­par­i­son of the Lie al­ge­bras of the two sub­groups $\left( \Delta X_{1,1},\Delta X_{3,2} \right),\left( \Delta X_{1,2},\overline{\Delta X_{2,1}} \right)$, shows

where the change in sign for the $a$ el­e­ments on the main di­ag­o­nal become ap­par­ent as the dif­fer­ence be­tween the two,

Gen­er­al­iz­ing the ma­tri­ces that pro­duce the $SU(2)$ sub­group rep­re­sen­ta­tions, de­pends on using a common axis of ro­ta­tion for both the main di­ag­o­nal ma­tri­ces and the ${\Delta X}_{j}$ forms. These can be rep­re­sent­ed by el­e­ments of the sym­met­ric conic group $S_{3\mathbb{C}}^{\Delta}$, as

where the $Im\Delta X_{j,2}$ ma­tri­ces on the left rep­re­sent the fa­mil­iar form of the $X_{j}$-type ma­tri­ces, for $i$, the center of ro­ta­tion in each. The ma­tri­ces $Re\Delta X_{j}X_{k}$ on the right rep­re­sent the re­spec­tive cone num­bers of the or­thog­o­nal plane, with the iden­ti­ty en­tries de­not­ed by the number $1$, co­in­cid­ing with the cen­ters of ro­ta­tion for the $Im\Delta X_{j,2}$ ma­tri­ces. Taking the ex­am­ples $Im\Delta X_{3,1},Re\Delta X_{3}X_{1}$, they form matrix prod­ucts rep­re­sent­ing an $SU(2)$ sub­group with its tan­gent space po­si­tioned at the in­ter­sec­tion of the imag­i­nary $X_{3}$-plane, and the or­thog­o­nal rep­re­sen­ta­tion of the $X_{1}$-plane

We can see the $Im\Delta X_{3,1},Re\Delta X_{3}X_{1}$ prod­ucts have the fa­mil­iar $SU(2)$ sub­group form, dif­fer­ing in the po­si­tion for the axis of ro­ta­tion $a_{3,3}$ for the $\mathcal{u}_{j}$ ma­tri­ces. From the Lie al­ge­bra stand­point, this sub­group can be de­fined as

These arrange­ments have a nice geo­met­ric in­tu­ition, which un­der­scores that we can rotate the $Im\Delta X_{3,2},Re\Delta X_{3}X_{1}$ el­e­ments to arrive at the other chiral forms of $SU(2)$, which is what was oc­cur­ring for the $\Delta X_{1,2},\overline{\Delta X_{2,1}}$ ex­am­ple. To demon­strate, using $Im\Delta X_{3,2}$, $Re\Delta X_{3}X_{1}$, we rotate both ma­tri­ces by rais­ing them to the odd power of three

seeing that $Im\Delta X_{3,2}$ has ro­tat­ed to its an­tipo­dal point on the sphere of tan­gent spaces, and $Re\Delta X_{3}X_{1}$ has mapped to its neg­a­tive com­plex con­ju­gate, $Re\Delta X_{3}X_{1} \rightarrow - \overline{Re\Delta X_{3}X_{1}}$. The ma­tri­ces of the re­sult­ing $SU(2)$ group become

show­ing that the prod­uct $\left( Im\Delta X_{3,2} \right)^{3} \times \left( Re\Delta X_{3}X_{1} \right)^{3}$ now maps to the $\mathcal{u}_{2}$ matrix. Fur­ther, $\left( Re\Delta X_{3}X_{1} \right)^{3} \times \left( Im\Delta X_{3,2} \right)^{3}$ now maps to $\mathcal{u}_{1}$, in­stead of the pre­vi­ous $\mathcal{u}_{2}$. The over­all effect is to re­verse the sub­group's chi­ral­i­ty, lead­ing to the chiral Lie al­ge­bra

The essence of the trans­for­ma­tion maps $Im\Delta X_{3,2}$ to its neg­a­tive and an­tipo­dal po­si­tion, where $Re\Delta X_{3}X_{1}$ be­comes its neg­a­tive com­plex con­ju­gate, and there­fore, we can rep­re­sent the trans­for­ma­tion as

In Figure 1.4.20, we show a geo­met­ric ren­der­ing of this trans­for­ma­tion, where the ma­tri­ces $Im\Delta X_{3,2},Re\Delta X_{3}X_{1}$ trans­form for co­in­cid­ing points $A,B,C,D \rightarrow A^{'},B^{'},C^{'},D'$, as fol­lows:

The point to not over­look, is that $Im\Delta X_{3,2}$ ro­tates in the $X_{3}$-plane, while $Re\Delta X_{3}X_{1}$ trans­forms in the $X_{1}$-plane.

The final thing we con­sid­er, are the ex­po­nen­tial map­pings of the $SU(2)$ sub­group al­ge­bras as forms based $\in \mathbb{C}^{3}$. It turns out that since the axis prop­er­ties of the Eu­clid­ean dis­tance metric re­mains intact, the finite group re­la­tion­ship re­mains intact, seen by using de­not­ing the $\mathfrak{su}'(2)$ Lie al­ge­bra, as

then pro­ceed­ing to find the matrix ex­po­nen­tial $\exp\left( \mathfrak{s}\mathfrak{u}^{'}(2) \right)$, re­sult­ing in the Lie group rep­re­sen­ta­tion

The group above shares the same result for its bottom-right $2 \times 2$ sub-matrix, with the matrix ex­po­nen­ti­a­tion of the Lie al­ge­bra $\mathfrak{su}(2)$. In ad­di­tion, the $e^{\left( x_{1} + x_{2} + x_{3} \right)}$ func­tion lo­cat­ed in the $a_{1,1}$ entry acts to scale the axis of ro­ta­tion for the group.