1.1.20 The $U(3)$ Symmetry Group

We are now able to build on the $S_{3\mathbb{C}}$ sym­me­try group and what we have learned about geo­met­ric sym­me­try rep­re­sen­ta­tion in $\mathbb{C}^{3}.$The bound­aries of the $S_{3\mathbb{C}}$ group are rep­re­sent­ed by three in­ter­con­nect­ed com­plex cir­cles, which act to cir­cum­scribe the equi­lat­er­al tri­an­gle's ver­tices. If we define the dis­tance from the origin as $l_{j},$each vertex of the equi­lat­er­al tri­an­gle will be a unit mod­u­lus from the origin

We can also see that each of the ver­tices are sep­a­rat­ed by edge lengths equal to $\left| \sqrt{2} \right|$, as de­pict­ed in Figure 1.1.10. With the geom­e­try of the com­plex planes set to unit lengths at the bound­aries along the axes, and the two-di­men­sion­al area of the equi­lat­er­al tri­an­gle uni­tar­i­ly set as $\frac{1}{2}\left( \sqrt{2} \bullet \sqrt{2} \right) = 1,$we can better con­sid­er the geom­e­try $\in \mathbb{C}^{3}.$In this way, each com­plex unit circle plane can be rep­re­sent­ed in polar form as

where the mod­u­lus $r$ is omit­ted, since it is equal to 1. For geo­met­ric clar­i­ty, we name the imag­i­nary-valued per­mu­ta­tions rep­re­sen­ta­tive of $\mathbb{C}^{3}$, using $X_{j}$, which is rep­re­sen­ta­tive of tri­an­gle's ver­tices in each com­plex plane, seen as

The real-valued el­e­ments of the group are de­not­ed by $Y_{j}$, with $Y_{1}$ being the iden­ti­ty el­e­ment $\epsilon$

We define an al­ge­bra $\mathfrak{g}$, as the set of el­e­ments

The­o­rem (1.1.101) – The set of per­mu­ta­tions $\mathfrak{g} \in U(3)$ forms a group under com­po­si­tion.

Proof –

• Clo­sure $\rightarrow \ \forall\ X,Y \in \mathfrak{g},\ \ X\ \bullet Y \in \mathfrak{g},$

• As­so­cia­tiv­i­ty $\rightarrow \ \forall\ X,Y,Z \in G,\ \ (\ X\ \bullet Y) \bullet Z = X \bullet (Y \bullet Z),$

• Iden­ti­ty $\rightarrow \ \exists\ e \in \mathfrak{g}:\forall\ a \in G,\ \ a \bullet e = e \bullet a = a,$

• In­verse El­e­ment $\rightarrow \forall\ a \in G\ \exists\ b \in G:\ a \bullet b = b \bullet a = e.$

De­f­i­n­i­tion (1.1.102) – A finite-di­men­sion­al real or com­plex Lie al­ge­bra is a finite-di­men­sion­al real or com­plex vector space $\mathfrak{g}$, to­geth­er with a Lie brack­et $\lbrack X,Y\rbrack$, from $\mathfrak{g} \times \mathfrak{g}$ onto $\mathfrak{g}$, having the fol­low­ing prop­er­ties¹

• Bi­lin­ear­i­ty:

and for ar­bi­trary num­bers $a$ and $b$.

• Skew-sym­met­ric:

• The Jacobi iden­ti­ty holds:

The­o­rem (1.1.103) – The set of per­mu­ta­tions$\ \mathfrak{g} =\left\{ X_{j},Y_{j} \right\}$ forms a finite-di­men­sion­al com­plex Lie al­ge­bra over a com­plex vector space.

Proof – The Lie Al­ge­bra De­f­i­n­i­tion

• Bi­lin­ear­i­ty:$\ \lbrack aX + bY,Z\rbrack = a\lbrack X,Z\rbrack + b\lbrack Y,Z\rbrack$and $\lbrack Z,aX + bY\rbrack = a\lbrack Z,X\rbrack + b\lbrack Z,Y\rbrack\ \forall\ X,Y,Z\ \in$ $\mathfrak{g}$

and for ar­bi­trary num­bers $a\ and\ b$,

• Skew-sym­met­ric:

$\lbrack X,Y\rbrack = - \lbrack Y,X\rbrack\ \forall$ $X,Y \in \mathfrak{g}$,

• The Jacobi Iden­ti­ty holds:

$\left\lbrack X,\lbrack Y,Z\rbrack \right\rbrack + \left\lbrack Y,\lbrack Z,X\rbrack \right\rbrack + \left\lbrack Z,\lbrack X,Y\rbrack \right\rbrack = 0,$ $\forall\ X,Y \in \mathfrak{g}$.

$\square$

Taking the ex­po­nen­tial map­ping of our Lie al­ge­bra el­e­ments $\in \mathfrak{g}$

we can define that a matrix Lie group by ex­po­nen­ti­a­tion is a smooth man­i­fold over $G$.

De­f­i­n­i­tion (1.1.105) – The Uni­tary Group $\mathbf{U}\left( \mathbf{3} \right)$ is a group under com­po­si­tion, sit­ting in the com­plex geo­met­ric space $\mathbb{C}^{3 \times 3}$

with a set of Lie al­ge­bra el­e­ments $\mathfrak{g} =\left\{ X_{j},Y_{j} \right\}$, for angle ar­gu­ments $x_{j},y_{j}\in \mathbb{R}$

The set of ro­ta­tion ma­tri­ces $G = \left\{ e^{X_{j}},e^{Y_{j}} \right\}$, in polar form, can take both $\pm \ \angle\ arguments\in \mathbb{R}$, seen for $e^{X_{j}}$, as

The non-iden­ti­ty cyclic per­mu­ta­tion el­e­ments take on a more com­pli­cat­ed form with ex­po­nen­ti­a­tion. Ef­fec­tive­ly, the ex­po­nen­tial series $e^{ix}$ is split three ways in­stead of the cus­tom­ary two-way splits we know in the trigono­met­ric form by the sine and cosine func­tions. We will study these three-way trigono­met­ric func­tions in more depth in later sec­tions. To be com­plete, we present the Lie groups ${G_{Y}}_{2}$,${G_{Y}}_{3}$, as fol­lows:

We can see the re­sem­blance the $G_{X_{j}}$ ro­ta­tion ma­tri­ces have with the $R_{x},\ R_{y},\ R_{z}$ ma­tri­ces from the $SO(3)$ group, where the latter group has the fol­low­ing gen­er­a­tors

An el­e­ment of $SO(3)$ would be a linear com­bi­na­tion of the above gen­er­a­tors

Ex­po­nen­ti­a­tion of the $G_{j}$ gen­er­a­tors yield the ma­tri­ces

With our $U(3)$ group, in­stead of ro­ta­tions in $\mathbb{R}^{3}$, we have ro­ta­tions that are a linear com­bi­na­tion of three com­plex planes in $\mathbb{C}^{3}$. We use the ex­am­ple of our $X_{1}$ matrix below to com­pare with the $R_{x} \in SO(3)$ matrix above, show­ing the com­plex axis of ro­ta­tion as $\mathbf{e}^{\mathbf{i}\mathbf{X}_{\mathbf{1}}}$, with the other en­tries rep­re­sent­ing a ro­ta­tion in the plane of $X_{2}X_{3} \in \mathbb{C}^{2}$

By taking the com­plex mod­u­lus of $\mathbf{e}^{\mathbf{i}\mathbf{X}_{\mathbf{1}}}$, and using the iso­mor­phism $i \cong \begin{pmatrix} 0 & \mp 1 \\ \pm 1 & 0 \end{pmatrix}$, $\mathbb{R}^{3}$ could be seen as a pro­jec­tive form of $\mathbb{C}^{3}$,

The sim­i­lar­i­ties be­tween our ex­po­nen­ti­at­ed $X_{j}$ gen­er­a­tors and that of $SO(3)$) begets the ques­tion, is the three-di­men­sion­al space we are all fa­mil­iar with ac­tu­al­ly $\mathbb{C}^{3}$? We would argue that it is indeed. The notion of need­ing a real-axis to rotate about in space is not nec­es­sary. To show this, we call at­ten­tion to $U(1)$, and its real form $SO(2)$, which have no fixed real axes but rotate in two-di­men­sion­al space seam­less­ly. In the $U(3)$ group, the $X_{j}$ gen­er­a­tor ro­ta­tions $\in \mathbb{C}^{3}$ do not change the com­plex plane that the axis func­tion of the $X_{j}$ group sits in. This occurs while an inter-planar ro­ta­tion occurs with the ar­gu­ment for $X_{j}$ in the $X_{k}X_{l}$ sub­space. Look­ing at Eq. 1.1.116 above, $\mathbf{e}^{\mathbf{i}\mathbf{X}_{\mathbf{1}}}$ is the fixed two-di­men­sion­al "axis" $\in \mathbb{C},$while the pri­ma­ry ro­ta­tion occurs in the $\mathbb{C}^{2}$-plane, $X_{2}{/X}_{3}.$The rep­re­sen­ta­tion in the polar form of the whole group can be cal­cu­lat­ed by taking ad­van­tage of the fact that the al­ge­bra $\mathfrak{g}$ is di­ag­o­nal­iz­able

The al­ge­bra set $\mathfrak{g} =\left\{ X_{j},Y_{j} \right\}$ can be put into matrix form as

In the eigen­de­com­po­si­tion $\mathfrak{g} =Y_{1}\left( VDV^{- 1} \right)$, the group iden­ti­ty $Y_{1}$ will simply scale the $VDV^{- 1}$ ma­tri­ces uni­form­ly by $e^{1}$, through the ex­po­nen­tial map­ping. The eigen­val­ue and eigen­vec­tor ma­tri­ces $D,V$ have the fol­low­ing repere­sen­ta­tions

for $\mathcal{R} =\left( y_{2} + y_{3} \right)$, $\ \ \gamma = \left( x_{1} + x_{2} + x_{3} \right),\ \ \mathbf{\lambda} = \ \sqrt{\begin{array}{r} - 3y_{2}^{2} + 6y_{2}y_{3} - 3y_{3}^{2} + 4x_{1}^{2} \\ - 4x_{1}x_{2} - 4x_{1}x_{3} + 4x_{2}^{2} - 4x_{2}x_{3} + 4x_{3}^{2} \end{array}}.$

We can take ad­van­tage of the di­ag­o­nal­ized form to cal­cu­late the polar form of the entire group

find­ing the matrix rep­re­sen­ta­tion $G = e^{y_{1}}\left( Ve^{D}V^{- 1} \right)$, as

$$\begin{array}{r} e^{y_{1}}\begin{pmatrix} \frac{e^{\left( \gamma+ \mathcal{R} \right)}}{3} + e^{- \frac{\mathcal{R}}{2}}\left( \frac{2\cosh\left( \frac{\mathbf{\lambda}}{2} \right)}{3} - \frac{i\frac{\partial\mathbf{\lambda}}{\partial X_{1}}\sinh\left( \frac{\mathbf{\lambda}}{2} \right)}{3} \right) & \frac{e^{\left( \gamma+ \mathcal{R} \right)}}{3} - e^{- \frac{\mathcal{R}}{2}}\left( \frac{\cosh\left( \frac{\mathbf{\lambda}}{2} \right)}{3} + \frac{\frac{\partial\mathbf{\lambda}}{\partial Y_{2}} + i\frac{\partial\mathbf{\lambda}}{\partial X_{3}}\sinh\left( \frac{\mathbf{\lambda}}{2} \right)}{3} \right) & \frac{e^{\left( \gamma+ \mathcal{R} \right)}}{3} - e^{- \frac{\mathcal{R}}{2}}\left( \frac{\cosh\left( \frac{\mathbf{\lambda}}{2} \right)}{3} + \frac{- \frac{\partial\mathbf{\lambda}}{\partial Y_{3}} - i\frac{\partial\mathbf{\lambda}}{\partial X_{2}}\sinh\left( \frac{\mathbf{\lambda}}{2} \right)}{3} \right) \\ \frac{e^{\left( \gamma+ \mathcal{R} \right)}}{3} - e^{- \frac{\mathcal{R}}{2}}\left( \frac{\cosh\left( \frac{\mathbf{\lambda}}{2} \right)}{3} + \frac{- \frac{\partial\mathbf{\lambda}}{\partial Y_{3}} - i\frac{\partial\mathbf{\lambda}}{\partial X_{3}}\sinh\left( \frac{\mathbf{\lambda}}{2} \right)}{3} \right) & \frac{e^{\left( \gamma+ \mathcal{R} \right)}}{3} + e^{- \frac{\mathcal{R}}{2}}\left( \frac{2\cosh\left( \frac{\mathbf{\lambda}}{2} \right)}{3} - \frac{i\frac{\partial\mathbf{\lambda}}{\partial X_{2}}\sinh\left( \frac{\mathbf{\lambda}}{2} \right)}{3} \right) & \frac{e^{\left( \gamma+ \mathcal{R} \right)}}{3} - e^{- \frac{\mathcal{R}}{2}}\left( \frac{\cosh\left( \frac{\mathbf{\lambda}}{2} \right)}{3} + \frac{\frac{\partial\mathbf{\lambda}}{\partial Y_{2}} + i\frac{\partial\mathbf{\lambda}}{\partial X_{1}}\sinh\left( \frac{\mathbf{\lambda}}{2} \right)}{3} \right) \\ \frac{e^{\left( \gamma+ \mathcal{R} \right)}}{3} - e^{- \frac{\mathcal{R}}{2}}\left( \frac{\cosh\left( \frac{\mathbf{\lambda}}{2} \right)}{3} + \frac{\frac{\partial\mathbf{\lambda}}{\partial Y_{2}} + i\frac{\partial\mathbf{\lambda}}{\partial X_{2}}\sinh\left( \frac{\mathbf{\lambda}}{2} \right)}{3} \right) & \frac{e^{\left( \gamma+ \mathcal{R} \right)}}{3} - e^{- \frac{\mathcal{R}}{2}}\left( \frac{\cosh\left( \frac{\mathbf{\lambda}}{2} \right)}{3} + \frac{- \frac{\partial\mathbf{\lambda}}{\partial Y_{3}} - i\frac{\partial\mathbf{\lambda}}{\partial X_{1}}\sinh\left( \frac{\mathbf{\lambda}}{2} \right)}{3} \right) & \frac{e^{\left( \gamma+ \mathcal{R} \right)}}{3} + e^{- \frac{\mathcal{R}}{2}}\left( \frac{2\cosh\left( \frac{\mathbf{\lambda}}{2} \right)}{3} - \frac{i\frac{\partial\mathbf{\lambda}}{\partial X_{3}}\sinh\left( \frac{\mathbf{\lambda}}{2} \right)}{3} \right) \end{pmatrix} \end{array}$$

for par­tial de­riv­a­tives $\frac{\partial\mathbf{\lambda}}{\partial\mathbf{X}_{\mathbf{j}}}$, rep­re­sent­ed by

It is also noted that for the polar form of the group $G,$we have

and

In the $U(3)$ group, we have ex­tend­ed the sym­me­try group $S_{3}$ to the com­plex field of $\mathbb{C}^{3}$. The group's bound­ary, by its uni­tary char­ac­ter­is­tic, forms a sphere $\in \mathbb{C}^{3}$ with a uni­form mod­u­lus of one. It has six di­men­sions, three real and three imag­i­nary, which fits the sym­me­try a per­fect sphere might embody.

1. Hall, Brian C. – "Lie Groups, Lie Al­ge­bras, and Rep­re­sen­ta­tions", Grad­u­ate Texts in Math­e­mat­ics, Springer, 2015