Abstract
We recall the construction of triality automorphism of \(\mathfrak {so}(8)\) given by E. Cartan and we give a matrix representation for the real form \(\mathfrak {so}(4,4)\). We compute the induced results on the characteristic classes. Paralelly we study the triality automorphism of the singularity \(D_4\) (in Arnolds classification of smooth functions) and its miniversal deformation. The similarity with Lie theory leads us to a definition of \(G_2\) singularity.
Highlights
The Dynkin diagram D4 appears in the singularity theory
This automorphism can be extended to an automorphism of the parameter space of the miniversal deformation
In the distinguished basis of vanishing cycles the intersections are described by the Dynkin diagram G2
Summary
In 1925 Cartan published a paper under the title Le principe de dualité et la théorie des groupes simples et semi-simples [9]. Ω = e1∗26 + e1∗34 + e1∗57 + e2∗37 + e2∗45 + e3∗56 + e4∗67 , where ei∗jk = ei∗ ∧ e∗j ∧ ek∗ This particular form defines the imaginary part of the octonionic multiplication in R8. The multiplication is given by the rule encoded in the picture: so(8)φ is the full stabilizer of ω. It follows that the group of transformations of R8 preserving octonionic multiplication coincides with the Lie group associated to so(8)φ. Permuted by Z3 cyclically, [1], [10, §20.3], [15], [24] We will construct another matrix representation of triality for so(4, 4) in the “Appendix”
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