1.1.24 The Cartan Subalgebra of U(3)

As is the case for the sym­met­ric group $S_{3}$, there is a cyclic sub­group in the $\mathfrak{g}$ al­ge­bra formed by its real-valued el­e­ments. By way of the fact that this is the $Z_{3}$ cyclic group, we know that it is abelian, and this sub­group is a Cartan sub­al­ge­bra of $\mathfrak{g}$, which is de­fined as fol­lows.

De­f­i­n­i­tion (1.1.174) – If $\mathfrak{g}$ is a com­plex semi­sim­ple Lie al­ge­bra, then a Cartan sub­al­ge­bra of $\mathfrak{g}$ is a com­plex sub­space $\mathfrak{h}$ of $\mathfrak{g}$, with the fol­low­ing prop­er­ties

¹

De­f­i­n­i­tion (1.1.175) – We define the com­plex sub­space $\mathfrak{h = \{}Y_{1},Y_{2},Y_{3}\}$, as the Cartan Sub­al­ge­bra of $\mathfrak{g}$ and the max­i­mal com­mut­ing sub­al­ge­bra of el­e­ments $\mathfrak{h}$

The­o­rem (1.1.176) – For $Y_{j}$ as a set, the set of per­mu­ta­tions of $\mathfrak{g}$ forms a Cartan sub­al­ge­bra of $\mathfrak{g}.$

Proof –

Note, as stated in De­f­i­n­i­tion (1.1.174), all the el­e­ments of $\mathfrak{h}$ com­mute, where for ex­am­ple $Y_{2}Y_{3} = Y_{1}$, $Y_{3}Y_{2} = Y_{1},$and we have that

Triv­ial­ly, this holds for the iden­ti­ty el­e­ment as well. It is also easy to verify that $Y_{2}$ and $Y_{3}$ are the ad­joint ma­tri­ces of one an­oth­er, $ad_{Y_{2}} = Y_{3},\ \ a{d_{Y}}_{3} = Y_{2}a{d_{Y}}_{1}$, and the iden­ti­ty el­e­ment $Y_{1}$ is its own ad­joint. Fur­ther, we have that all el­e­ments of $Y_{j}\in \mathfrak{h}$ are di­ag­o­nal­iz­able. The $Y_{2}$ and $Y_{3}$ mem­bers of the Cartan sub­al­ge­bra are like the $\lambda_{3}$ and $\lambda_{8}$ Gell-Mann ma­tri­ces of $SU(3)$, which belong to the Cartan sub­al­ge­bra of that group, and in ad­di­tion, $\lambda_{3}$ and $\lambda_{8}$ com­mute with each other.

We are ac­cus­tomed to matrix groups in physics with two-di­men­sion­al sub­groups and their re­spec­tive ro­ta­tions. In the $U(3)$ group, we have two-di­men­sion­al ro­ta­tions for the $X_{j}$ gen­er­a­tors, where the fa­mil­iar ex­am­ple of $e^{{iX}_{1}}$ cal­cu­lates to

noting the pres­ence of the ex­po­nen­tial func­tion $\mathbf{e}^{\mathbf{i}\mathbf{x}_{\mathbf{1}}}$ in the $a_{1,1}$ row-column entry. We should also be aware of the re­flec­tion in the $S_{3}$ group has become a ro­ta­tion in the com­plex $X_{2}/X_{3}$-plane, with a sin­gu­lar $X_{1}$-plane ro­ta­tion for the $x_{1}$ value el­e­ments. The matrix ex­po­nen­tial for the real-valued al­ge­bra yields the ex­po­nen­tial func­tion $\mathbf{e}^{\mathbf{i}\mathbf{x}_{\mathbf{1}}}$ in the $a_{1,1}$ row-column entry and its co­in­cid­ing hy­per­bol­ic trigono­met­ric func­tions

Some­thing unique hap­pens when we ex­po­nen­ti­ate the $Y_{2}$ and $Y_{3}$ al­ge­bra el­e­ments of the Cartan sub­al­ge­bra, a three-way split­ting of the ex­po­nen­tial func­tion occurs, rep­re­sent­ing new trigono­met­ric func­tions as de­scribed by Dennis Morris². When we start to cal­cu­late the first sev­er­al terms of the matrix ex­po­nen­tial of the $Y_{2}$ and $Y_{3}$ al­ge­bra el­e­ments, we see a three-way split­ting pat­tern emerges, which is dis­tinct from the two-way split­ting the sine and cosine func­tions

Seeing the three-way split of the ex­po­nen­tial series in the $e^{Y_{2}}$ and $e^{Y_{3}}$ Lie groups leads to the pro­found con­clu­sion that three-di­men­sion­al ro­ta­tions exist in these groups. Given that a re­flec­tion has been changed to a two-plane ro­ta­tion $\in \mathbb{C}^{3}$, for the ex­po­nen­ti­at­ed $X_{j}$ gen­er­a­tors, we can say that a cyclic ro­ta­tion be­comes a three-di­men­sion­al ro­ta­tion de­fined by the $e^{Y_{2}}$ and $e^{Y_{3}}$ gen­er­a­tors.

In the up­com­ing physics volume in the series, we will elab­o­rate on the math­e­mat­ics putting these func­tions at the heart of color ex­change in quan­tum chro­mo­dy­nam­ics (QCD), and the role they play with light. The Lie al­ge­bras of the $SU(3)$ group have short­com­ings and fail to form a group under com­po­si­tion. More suc­cinct­ly, the eigen­val­ues and eigen­vec­tors of the Cartan gen­er­a­tors $Y_{2},Y_{3}$ can be con­sid­ered the source of color ex­change and the strong force, and these are anal­o­gous to the $T_{3}$ and $T_{8}$ gen­er­a­tors of the Gell-Mann ma­tri­ces, where $T_{A} = \left( \frac{\lambda_{A}}{2} \right).$This treat­ment leads to a sound math­e­mat­i­cal ex­pla­na­tion of why gluons are not seen alone in space­time, given that two-di­men­sion­al ro­ta­tions/trigono­met­ric func­tions com­prise it. As stated in the in­tro­duc­tion, the $SU(3)$ group does not fit the above group cri­te­ria, but the $U(3)$ group does.

We can ex­press the three-way ex­po­nen­tial func­tion splits in more fa­mil­iar trigono­met­ric terms as

and note that if the three series were added to­geth­er, the result would equal $e^{Y_{3}}$. We look more close­ly at the Cartan sub­al­ge­bra set {$\lambda_{3},\ \lambda_{8}\}$ of the Gell-Mann ma­tri­ces $\in SU(3)$, by de­com­pos­ing them to their eigen­val­ues and eigen­vec­tors

Where we see only two eigen­val­ues for the $\lambda_{3}$ gen­er­a­tor, com­pared to three for the $\lambda_{8}$ gen­er­a­tor, eigen­val­ues are scaled by $(1, - 1)$ for the $x_{1},x_{2}$ vari­ables ${\in \lambda}_{3}$, and by $\frac{1}{\sqrt{3}}(1,1, - 2)$ for the $x_{1},x_{2},x_{3}$ vari­ables $\in \lambda_{8}.$The linear prod­uct of $\lambda_{3}$ with a three-vector will make the $x_{3}$ vari­able equal to zero, main­tain­ing the order for the $x_{1},x_{2}$ vari­ables, while $\lambda_{8}$ will keep all the vari­ables intact

The re­sult­ing vec­tors from these $\lambda_{3}\widehat{v}$, $\lambda_{8}\widehat{v}$ matrix-vector prod­ucts have equal two-norm values $\left| \lambda_{3}\widehat{v} \right| = \left| \lambda_{3}\widehat{v} \right|,$when as­sum­ing the mag­ni­tudes of three vector com­po­nents $x_{1},x_{2},x_{3}$ are equiv­a­lent. Com­pare this to the same de­com­po­si­tions for the $Y_{2},Y_{3}$ el­e­ments of the Cartan sub­al­ge­bra $\in U(3),$we see marked sym­me­tries in this form. A sim­i­lar de­com­po­si­tion of the $Y_{2},Y_{3}$ el­e­ments show the same eigen­val­ues for both, with two eigen­vec­tors that differ in the $x_{2},x_{3}$ com­po­nents by a sign in their imag­i­nary por­tions.

Plot­ting the set of eigen­val­ues $\left\{ \lambda_{1},\lambda_{2},\lambda_{3} \right\}$ for the $Y_{2},Y_{3}$ gen­er­a­tors, re­veals the reg­u­lar polyg­o­nal sym­me­try of the $S_{3}/D_{3}$ group's equi­lat­er­al tri­an­gle (Figure 1.1.16).

A sim­i­lar linear prod­uct of the $Y_{2},Y_{3}$ el­e­ments with a three-vector shows cyclic trans­for­ma­tion­al shifts in both pos­i­tive and neg­a­tive di­rec­tions, having a sim­i­lar order of vari­ables in the $Y_{2}$ trans­form, com­pared to the $\lambda_{8}$ trans­form in the Gell-Mann case, except for dif­fer­ent scale fac­tors and the neg­a­tive sign for the $x_{2}$ vari­ables.

The notion of causal­i­ty in matrix groups seems to exist, and for the $U(3)$ group as de­scribed, the Cartan sub­al­ge­bra $Y_{j}$ is caused by the pri­ma­ry $X_{j}$ group el­e­ments, where the $Y_{j}$ el­e­ments alone cannot yield any el­e­ment of the $X_{j}$ group. But the $Y_{j}$ are nec­es­sary for the $X_{j}$ group el­e­ments to ex­change with one an­oth­er in terms of the al­ge­bras, as demon­strat­ed below

1. Hall, Brian C., "Lie Groups, Lie Al­ge­bras, and Rep­re­sen­ta­tions", Grad­u­ate Texts in Math­e­mat­ics, Springer, 2015
2. Morris, Dennis, "Com­plex Num­bers the Higher Di­men­sion­al Forms - 2^nd Edi­tion", Abane & Right, 2015