Utility
Notation
The only table of symbols on the site. One meaning per symbol; the macro column is the name in assets/site.js.
| symbol | macro | meaning | in \(\SL_3/\SO_3\) | defined |
|---|---|---|---|---|
| \(\Oc\), \(\Oc_\C\) | \Oc | octonions; \(\Oc\otimes_\R\C\) | \(\R\), \(\C\) | Albert, over \(\C\) |
| \(\Nm\), \(\Tr\) | \Nm, \Tr | \(\Nm(x)=x\bar x\), \(\Tr(x)=x+\bar x\), \(\Oc_\C\to\C\) | \(x^2\), \(2x\) | Albert |
| \(\K\), \(d\) | \K | a normed division algebra, \(d=\dim_\R\K\) | \(\R\), \(1\) | row |
| \(\Jalg\) | \Jalg | Albert algebra, \(\dim_\C 27\) | \(\mathrm{Sym}_3(\C)\) | Albert |
| \(\JK{\K}\) | \JK | \(\mathfrak h_3(\K)\otimes_\R\C\) | \(\JK{\R}=\mathrm{Sym}_3(\C)\) | row |
| \(a,b,c\); \(x,y,z\) | diagonal entries; octonions at \((2,3),(3,1),(1,2)\) | entries of \(S\) | Albert | |
| \(\jo\) | \jo | Jordan product \(\tfrac12(XY+YX)\) | same | Albert |
| \(\one\), \(\tr\) | \one, \tr | identity element; \(a+b+c\) | \(I\), trace | Albert |
| \(\Jo\) | \Jo | traceless elements, \(\dim 26\) | traceless symmetric | Albert |
| \(\det\) | cubic norm \(\Jalg\to\C\) | determinant | Albert | |
| \(\Xe\), \(\Xs{\K}\) | \Xe, \Xs | \(\{\det=1\}\subset\Jalg\), resp. \(\subset\JK{\K}\) | \(\Xs{\R}\) | Albert, \(\SL_3\) |
| \(\Pd{n}\) | \Pd | real positive definite symmetric \(n\times n\), \(\det=1\) (\(n=2,3\)) | \(\Pd{3}=\SL_3(\R)/\SO_3\) | \(\SL_2\), \(\SL_3\) |
| \(\Ball\), \(\Disk\) | \Ball, \Disk | unit ball in \(\R^3\); unit disk in \(\R^2\) | \(\Ball\) | \(\SL_3\), \(\SL_2\) |
| \(\Ell{S}\) | \Ell | \(\{v:v^TS^{-1}v\le1\}\), the ellipse or ellipsoid of \(S\) | \(\Ell{gg^T}=g(\Ball)\) | \(\SL_2\), \(\SL_3\) |
| \(k_\theta\) | rotation of \(\R^2\) by the angle \(\theta\) | \(\SO_2=\{k_\theta\}\) | \(\SL_2\) | |
| \(t,x,y\) | coordinates of a symmetric \(2\times2\) matrix, \(S=\begin{pmatrix}t+x&y\\y&t-x\end{pmatrix}\) | hyperboloid \(t^2-x^2-y^2=1\) | \(\SL_2\) | |
| \(Z_{ij}\) | \(E_{ij}-E_{ji}\), a basis vector of \(\kf\) | same | \(\SL_2\) | |
| \(\mathrm i\) | the complex scalar \(\sqrt{-1}\), commuting with \(e_1,\dots,e_7\) | \(\mathrm i\) | octonions over \(\C\) | |
| \(e_1,\dots,e_7\); \(u_k\) | octonion units (Fano labels); basis of the doubling | none | Fano, doubling | |
| \(\Esix\), \(\Ffour\) | \Esix, \Ffour | \(\det\)-preserving group; \(\Aut(\Jalg)\) | \(\SL_3\), \(\SO_3\) | Albert |
| \(\esix\), \(\ffour\) | \esix, \ffour | their Lie algebras | \(\slf_3\), \(\so_3\) | Prop. |
| \(\Der\), \(\Lop{T}\) | \Der, \Lop | derivations of \(\Jalg\); \(X\mapsto T\jo X\) | \(\so_3\); \(X\mapsto\frac12(TX+XT)\) | Prop. |
| \(\theta\) | the involution, \(\delta+\Lop{T}\mapsto\delta-\Lop{T}\) | \(g\mapsto(g^T)^{-1}\) | Prop., lab | |
| \(\gf=\kf\oplus\pf\) | \gf, \kf, \pf | \(\pm1\) eigenspaces of \(\theta\) | \(\slf_3=\so_3\oplus\{\text{sym.}\}\) | lab |
| \(\iot\) | \iot | \(\Jo\to\pf\), \(T\mapsto2\Lop{T}\) | identity on symmetric \(Y\) | Prop. |
| \(\af\), \(H\), \(h_i\) | \af | Cartan subspace; its element \(\diag(h_1,h_2,h_3)\) | same | roots |
| \(\alpha_{ij}\), \(\mult\) | \mult | restricted root \(H\mapsto h_i-h_j\); multiplicity | \(\mult=1\) | roots |
| \(\gf_\alpha\), \(\kf_\alpha\), \(\pf_\alpha\) | root space; its parts in \(\kf\), \(\pf\) | \(\C E_{ij}\); \(\C Y_{ij}\) | roots | |
| \(Y_{ij}\) | \(E_{ij}+E_{ji}\) | same | roots | |
| \(\mf\), \(\nf\) | \mf, \nf | \(Z_\kf(\af)\); \(\bigoplus_{\alpha>0}\gf_\alpha\) | \(0\); upper triangular nilpotent | roots, spherical |
| \(\Wey\) | \Wey | Weyl group \(\cong S_3\) | same | roots |
| \(\Bor\), \(\bfr\) | \Bor, \bfr | Borel subgroup, its Lie algebra | upper triangular | spherical |
| \(\rk\) | \rk | rank (of a group, or \(\dim\af\) for \(G/K\)) | \(\rk\Xs{\R}=2\) | roots |
| \(\sigma\), \(\varphi\) | cyclic permutation; \(e_k\mapsto e_{k+1}\) entrywise (both in \(\Ffour\)) | \(PSP^T\); — | Albert |
Other macros: \R \C \Ha \SL \SO \Sp \GL \slf \so \diag \Id \Aut for \(\R,\C,\Ha,\SL,\SO,\Sp,\GL,\slf,\so,\diag,\Id,\Aut\).