Masteries · Jason GlowneyArithmoí Book 1

Masteries

1.1.25 The Anti-Diagonal Symmetry of the Cartan Subalgebra

Jason Glowney

Before moving on, we take a closer look at the abelian sub­al­ge­bra YjY_{j}. The XjX_{j} el­e­ments are sym­met­ric about the main di­ag­o­nal, as all Xj\ X_{j} have skew-sym­me­try (skew-Her­mit­ian). In­spec­tion of the el­e­ments YjY_{j} re­veals sym­me­try about the anti-di­ag­o­nal for each el­e­ment, where the anti-di­ag­o­nal runs from the top right corner of an n×nn \times n matrix to the bottom left, (a1,n,a2,n−1,…,an,1).\left( a_{1,n},a_{2,n - 1},\ldots,a_{n,1} \right).A glance at the pos­i­tive el­e­ments YjY_{j}, with an em­pha­sis in bold along the anti-di­ag­o­nal, shows just this

Yj={(10𝟎0𝟏0𝟎01),(01𝟎0𝟎1𝟏00),(00𝟏1𝟎0𝟎10)}.\begin{array}{r} Y_{j} = \left\{ \begin{pmatrix} 1 & 0 & \mathbf{0} \\ 0 & \mathbf{1} & 0 \\ \mathbf{0} & 0 & 1 \end{pmatrix},\begin{pmatrix} 0 & 1 & \mathbf{0} \\ 0 & \mathbf{0} & 1 \\ \mathbf{1} & 0 & 0 \end{pmatrix},\begin{pmatrix} 0 & 0 & \mathbf{1} \\ 1 & \mathbf{0} & 0 \\ \mathbf{0} & 1 & 0 \end{pmatrix} \right\}. \end{array}
(1.1.193)

Each YjY_{j} el­e­ment above sat­is­fies YjYjT =ϵ,Y_{j}Y_{j}^{T}\ = \epsilon,but does not hold for all yjy_{j} once ex­po­nen­ti­at­ed, given that G(Yj(yj))GT(Yj(yj))≠0G\left( Y_{j}\left( y_{j} \right) \right)G^{T}\left( Y_{j}\left( y_{j} \right) \right) \neq 0, except for when yj=0.y_{j} = 0.To con­sid­er fur­ther the im­pli­ca­tions of sym­me­try about the anti-di­ag­o­nals for these el­e­ments, we define the anti-trans­pose of a square matrix1 A=(Mj,k)1≤j,k≤n∈ℝmA = \left( M_{j,k} \right)_{1 \leq j,k \leq n} \in \mathbb{R}^{m}, as AaTA^{aT},

AaT=JATJ,\begin{array}{r} A^{aT} = JA^{T}J, \end{array}
(1.1.194)

where J=(Jjk)1≤j,k≤nJ = \left( J_{jk} \right)_{1 \leq j,k \leq n} de­notes the matrix for Jjk=1J_{jk} = 1 for j+k=n+1,j + k = n + 1,and for Jjk=0J_{jk} = 0, oth­er­wise. Cal­cu­lat­ing the YjaTY_{j}^{aT} ma­tri­ces from the YℂY^{\mathbb{C}} el­e­ments ∈Aℂ,{\in A}^{\mathbb{C}},in­volves solv­ing the ex­pres­sion

YjaT=(001010100)(YkT)(001010100),\begin{array}{r} Y_{j}^{aT} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}\left( Y_{k}^{T} \right)\begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}, \end{array}
(1.1.195)

which leads to the set of el­e­ments ±YjaT\pm Y_{j}^{aT}

± YjaT={(100010001),(010001100),(001100010),(−1000−1000−1),(0−1000−1−100),(00−1−1000−10)}.\begin{array}{r} \pm \ Y_{j}^{aT} = \left\{ \begin{array}{r} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix},\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix},\begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}, \\ \begin{pmatrix} - 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1 \end{pmatrix},\begin{pmatrix} 0 & - 1 & 0 \\ 0 & 0 & - 1 \\ - 1 & 0 & 0 \end{pmatrix},\begin{pmatrix} 0 & 0 & - 1 \\ - 1 & 0 & 0 \\ 0 & - 1 & 0 \end{pmatrix} \end{array} \right\}. \end{array}
(1.1.196)

Seeing that the anti-trans­pos­es are equal to the el­e­ments being anti-trans­posed, we can state math­e­mat­i­cal­ly that

Yj=YjaT→(Yj−YjaT)=0.\begin{array}{r} Y_{j} = Y_{j}^{aT} \rightarrow \left( {Y_{j} - Y}_{j}^{aT} \right) = 0. \end{array}
(1.1.197)

The Cartan sub­al­ge­bra of U(3)U(3) is a unique sub­group by way of its sym­me­tries and its ap­plic­a­bil­i­ty to being the basis of the strong force in QCD. One might con­sid­er how physics does not depend on ori­en­ta­tion in space, where a pot of water boils at the same tem­per­a­ture no matter which way it faces. This par­al­lels abelian forces like our Cartan sub­al­ge­bra 𝔥⊂U(3)\mathfrak{h} \subset U(3). This abelian sub­group sits in a uni­tary group with the ori­entable char­ac­ter­is­tics of non-abelian XjX_{j} el­e­ments that must be present for three-di­men­sion­al ro­ta­tions to occur.

  1. Goly­shev, Vasily and Stien­stra, Jan, "Fuch­sian Equa­tions of Type DN", https://arxiv.org/abs/math/0701936 2007