# Triality, characteristic classes, \(D_4\) and \(G_2\) singularities

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## 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.

## Keywords

Triality Characteristic classes Lie algebra \(G_2\) Singularities Milnor fibration## Mathematics Subject Classification

55R40 17B25 14J17 32S05The Lie algebra \(\mathfrak {so}(8)\) is the first algebra of the series \(D_4,D_5,D_6,\dots \). It is a classical simple algebra but it is also considered exceptional since it is the only one which admits an automorphism of order three. The automorphism of the Lie algebra lifts to an automorphism of the Lie group \({\textit{Spin}}(8)\), the universal cover of \(SO(8)\). The fixed point set of that automorphism is the exceptional group \(G_2\). Cartan [9] has constructed a concrete triality automorphism for the real compact form of \(\mathfrak {so}(8)\). It seems that the approach of E. Cartan to triality was almost forgotten. We think it is worth to recall the original construction. In the “Appendix” we give a formula for the choice of a quadratic form with signature \((4,-4)\). It has the advantage, that the root spaces coincides with the coordinates of the matrix.

We give formulas for the action of triality on the characteristic classes for \({\textit{Spin}}(8)\)-bundles. It is remarkable, that the space spanned by the Euler classes of the natural representation and spin representations \(S^+\), \(S^-\) is a two-dimensional nontrivial representation of \(\mathbb {Z}_3\). In other words the sum of the Euler classes is equal to zero and \(\mathbb {Z}_3\) permutes the Euler classes cyclically. The remaining generators of the ring of characteristic classes can be chosen to be invariant (except the case when the base field is of characteristic three).

The Dynkin diagram \(D_4\) also appears in the singularity theory. The singularity of the type \(D_4\) defined by \(x^3-3xy^2\) admits an automorphism of order three. Moreover this automorphism can be extended to an automorphism of the parameter space of the miniversal deformation. The action of the triality automorphism on the functions on the parameter space is the same as the action on the cohomology of \(H^*({\textit{BSpin}}(8);\mathbb {R})\). Taking the quotient by the cyclic group \(\mathbb {Z}_3\) we obtain a map germ \(\mathbb {C}^2/\mathbb {Z}_3\rightarrow \mathbb {C}\) with a two-dimensional cohomology of the Milnor fiber. In the distinguished basis of vanishing cycles the intersections are described by the Dynkin diagram \(G_2\). We study the geometry and topology of that singularity. Here the domain of the function is singular; it has an isolated singularity of the type \(A_2\), and taking a ramified cover, the function itself becomes the classical singularity of type \(A_2\). Therefore we can say that the singularity \(G_2\) is in some sense built from two singularities \(A_2\). This resembles the picture of the root system of the Lie algebra \(\mathfrak {g}_2\) which contains two copies of the systems \(A_2\) intertwined together.

We wish to put an emphasis on similarities between the theory of Lie algebras and the singularity theory. The formulas for triality in both theories are formally the same, although they describe objects of completely different natures.

## 1 The original approach of Elie Cartan

*Le principe de dualité et la théorie des groupes simples et semi-simples*[9]. In fact the main subject of the article is not duality but a symmetry of order three. After a general introduction motivated by the duality in the matrix group \(GL(n)\)

Moreover \(\mathfrak {so}(8)^\phi \) is the full stabilizer of \(\omega \). It follows that the group of transformations of \(\mathbb {R}^8\) preserving octonionic multiplication coincides with the Lie group associated to \(\mathfrak {so}(8)^\phi \). There are no computations in the Cartan’s paper. We refer the readers who wish to check the formulas to [17]. A general point of view is presented and in [15, §35], but there the explicit form of \(\phi \) is not given. Some forms of triality is given in [21, §24] or [23, §3.3.3].

## 2 Action of triality on maximal torus

Our goal is to describe the triality in a way which does not look like a magical trick. The approach presented here is equivalent, to the Cartan’s work in the complex case. In our construction it will be clear where the formulas come from. The triality automorphism given below has an advantage, that the root spaces coincide with the coordinates of the matrix and these coordinates are permuted by \(\mathbb {Z}_3\).

## 3 Action on rational cohomology of \( BSO (8)\)

**Theorem 1**

**Proposition 1**

The space \(P^7\) is spanned by the Euler class \(e\) and its image with respect to the triality automorphism.

## 4 Cohomology with finite coefficients

For completeness we discuss now the cohomology with finite coefficients, although it will not be used in the remaining part of the paper. The cohomology of \({\textit{BSpin}}(8)\) has only 2-torsion, therefore for \(q\not =2\) the cohomology \(H^*({\textit{BSpin}}(8);\mathbb {F}_q)\) is generated by the same generators as for rational coefficients and the action of the triality is given by the same formula. The only special issue for \(q=3\) is the fact that \(H^{12}({\textit{BSpin}}(8))\) as a representation of \(\mathbb {Z}_3\) is not semisimple. The invariant subspace spanned by the Euler classes \(e\), \(e(S^+)\) and \(e(S^-)\) does not admit any invariant complement. The formula (5) does not make sense for \(\mathbb {F}_3\).

**Proposition 2**

We will not discuss here the cohomology with integral coefficients, their generators are to be found in [6].

## 5 Triality of the singularity \(D_4\)

Triality phenomenon seem to attract recently mathematicians working on singularity theory. In a preprint [14] the triality was related to the study of integral curves. We will discuss here only very basic and obvious appearance of triality in singularities of scalar functions. The simple singularities of germs of holomorphic functions \(\mathbb {C}^n\rightarrow \mathbb {C}\) are indexed by the Dynkin diagrams \(A_\mu \) for \(\mu \ge 1\), \(D_\mu \) for \(\mu \ge 4\), \(E_6\), \(E_7\) and \(E_8\). The Dynkin diagram \(D_4\) describes the intersection form in the homology of the Milnor fiber in the distinguished basis corresponding to the basic vanishing cycles (defined by a choice of paths joining the singular values of a morsification with a regular one as in [4, Ch. 2] or [25, §4]).

The link and the Milnor fibre ^{1} contained in \(S^3\)

The real part of the zero set of the function \(f\) is the union of three lines intersecting at the angle 120\(^{\circ }\). The rotation of the \((x,y)\) plane by that angle preserves the function. Denote this rotation by \(\phi _0\). The map \(\phi _0\) of \(\mathbb {R}^2\) (or \(\mathbb {C}^2\)) is determined by the angles at which the lines intersect (up to a cubic root of unity in the complex case). We remark, that if one takes \(x^3-xy^2 \) as the germ representing the singularity, then the formula for \(\phi _0\) and \(\phi \) does not involve irrational coefficients like \(\cos (120^{\circ })\) and \(\sin (120^{\circ })\).

First let us find a deformation which is invariant with respect to the rotation \(\phi _0\).

**Proposition 3**

The function \(F\) is clearly invariant since \(\phi _0\) preserves the scalar product \(\langle (b,c),(x,y)\rangle \). The choice of the quadratic term is forced by the invariance condition. The functions \(1\), \(x\), \(y\) and \(x^2+y^2\) form a basis of the local algebra \(\mathbb {C}[x,y]/(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y})\), as desired in the definition of miniversal deformation.

**Theorem 2**

## 6 Singularity \(G_2\)?

The Dynkin diagram \(G_2\) does not appear in the original Arnold’s classification of simple singularities as well as the series \(B_\mu \), \(C_\mu \) and the exceptional \(F_4\). The other diagrams appear in [2] (see also references in [12]), while \(G_2\) is only mentioned in remark at the end of §\(9\). The series \(B_\mu \) and \(C_\mu \) arise as diagrams for the singularities with boundary condition. We will show how the diagram \(G_2\) appears for singularities with \(\mathbb {Z}_3\) symmetry.

It is worth to continue the analogy with the world of Lie algebras. Here the situation is dual, instead of taking the fixed points we divide by the \(\mathbb {Z}/3\) action. The function \(f\) factors to the quotient \(\bar{f}: \mathbb {C}^2/\mathbb {Z}_3\rightarrow \mathbb {C}\). The quotient space is not smooth, but it has mild singularities, an isolated du Val singularity of the type \(A_2\). The new Milnor fiber \(\overline{M}_\varepsilon \) is the quotient of the original Milnor fiber \(M_\varepsilon \). The quotient map is an unbranched cover. Therefore the \(\overline{M}_\varepsilon \) is homeomorphic to an elliptic curve with one disc removed. The cohomology \(H^1(\overline{M}_\varepsilon ;\mathbb {Q})\) is generated by two vanishing cycles corresponding to the singular values of the invariant morsification (7). The vanishing cycle corresponding to the value \(0\) is the usual one. The value \(-4 a^3\) corresponds to the vanishing cycle shrieked to the singular point of the domain. It is reasonable to treat the quotient space as \(\mathbb {C}^2/\mathbb {Z}_3\) as a stack. Here it simply means that we consider the singularity \(D_4\) together with the \(\mathbb {Z}_3\)-symmetry, as it was done in [11] for unitary reflection groups. We compute the intersection number of a pair of cycles taking their inverse images in the cover and dividing the result by the order of the cover.

## Footnotes

- 1.
For the purpose of the picture we have taken the original definition of [19] according to which the Milnor fiber is the surface in \(S^3\) defined by \({\textit{arg}}(f)={\textit{const}}\). Its boundary is equal to the link of the singularity. The picture of the Milnor fiber is obtained by a stereographic projection from \(S^3\) to \(\mathbb {R}^3\).

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