Triality, characteristic classes, D4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_4$$\end{document} and G2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_2$$\end{document} singularities

We recall the construction of triality automorphism of so(8)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {so}(8)$$\end{document} given by E. Cartan and we give a matrix representation for the real form so(4,4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {so}(4,4)$$\end{document}. We compute the induced results on the characteristic classes. Paralelly we study the triality automorphism of the singularity D4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_4$$\end{document} (in Arnolds classification of smooth functions) and its miniversal deformation. The similarity with Lie theory leads us to a definition of G2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_2$$\end{document} singularity.

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 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 Z 3 . In other words the sum of the Euler classes is equal to zero and 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 − 3x y 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 * (BSpin (8); R). Taking the quotient by the cyclic group Z 3 we obtain a map germ C 2 /Z 3 → 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 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.

The original approach of Elie Cartan
In 1925 Cartan published a paper under the title 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 G L(n) Cartan introduced a very concrete automorphism of the Lie algebra so(8). The construction of the transformation is the following: Let (a i, j ) 0≤i, j≤7 ∈ so(8) be an antisymmetric matrix. For each i ∈ {1, 2, . . . , 7} the quadruple is transformed linearly by the matrix Here the index i is understood modulo 7. The value i = 0 plays a special role and it is excluded from the cycle 1, 2, . . . 7. The resulting self-map φ : for a, b, c ∈ I m(O). We note that ω has all coefficient equal to one, therefore this definition of octonionic multiplication cannot agree with the most common definition via Fano plane, see e.g. [5]. To repair the discrepancy it is enough to change the sign of the basis element e 4 . Then the multiplication is given by the rule encoded in the picture: Moreover so(8) φ is the full stabilizer of ω. It follows that the group of transformations of R 8 preserving octonionic multiplication coincides with the Lie group associated to so(8) φ . 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 φ is not given. Some forms of triality is given in [21, §24] or [23, §3.3.3].
It is a pity that this original very simple point of view on triality and the definition of the group G 2 as the fixed set is not wide spread in the literature. Usually there are discussed mainly the spinor representations an the nondegenerate map permuted by Z 3 cyclically, [1], [10, §20.3], [15], [24]. We will construct another matrix representation of triality for so (4,4) in the "Appendix".

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 Z 3 .
Working with the complex coefficients we choose a basis in C 8 (as in [10, §19]) in which the quadratic form is equal to For real coefficients this means that we deal with SO (4,4). The maximal torus of SO (4,4) consists of the diagonal matrices diag(e t 1 , e t 2 , e t 3 , e t 4 , e −t 4 , e −t 2 , e −t 3 , e −t 1 ).
The following weights form the root system of so (4,4): where L i (t 1 , t 2 , t 3 , t 4 ) = t i . Choosing the Borel subgroup as the upper triangular matrices we obtain the Dynkin diagram of simple roots: The triality automorphism rotates the diagram anti-clockwise: and fixes the root L 2 − L 3 . In the basis consisting of the weights L i the triality automorphism is given by the remarkable matrix:

Action on rational cohomology of BSO(8)
The rational cohomology ring of the classifying space of SO (8) is well known, see e.g. [20,Theorem 15.9] where p i 's are the Pontryagin classes and e is the Euler class of the tautological bundle.
We identify the cohomology of the classifying space with the invariants of the Weyl group acting on the polynomials on t where t is the Lie algebra of the maximal torus (Cartan subalgebra), [7, Proposition 27.1]. We have where σ i is the elementary symmetric function.

Theorem 1 Let φ be the triality automorphism acting on the cohomology of
Then: The proof is a direct computation in linear algebra. Alternatively one can ask what are the Pontryagin and Euler classes of the spin bundle S + associated to the universal principal bundle over BSpin (8). To know that, one has to compute the weights of the original representation and apply the corresponding W -symmetric function. For example the triality automorphism sends the Euler class of the standard representation to the Euler class of the spin S + representation. Its Euler class is product of weights (3) see [10, §20]. The result should be developed in the basis consisting of e, p 2 1 and p 2 . We will chose another set of generators, which is better adapted to homotopy theory. For any connected Lie group G the cohomology H * (G; Q) is a Hopf algebra. By [18] it has to be of the form P • , where P • is the graded space of primitive elements of the Hopf algebra.
identifies the primitive generators of H * (G; Q) with the generators of are preserved by automorphisms of G, and therefore they form a distinguished subspaces of generators. For G = SO (8) or Spin(8) the dimensions of the spaces P i are equal to 1 for i = 3, 11 and dim P 7 = 2. Therefore the action of the triality automorphism is nontrivial only on P 7 . We identify the space P 7 with a subspace of H 8 (B SO (8); Q) P 7 ⊕ S 2 P 3 .

Proposition 1
The space P 7 is spanned by the Euler class e and its image with respect to the triality automorphism.
The orbit of the Euler class e consists of e and the Euler classes of the spin representations S + , S − . The space spanned by them is invariant. We will show that it is of dimension two, in fact Having the expression (3), and the formula for the Euler class e(S − ) we check directly that and the result follows.
In addition it is not hard to compute that the invariant subspace of the cohomology H 12 (B SO (8); Q) is spanned by

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 BSpin (8) has only 2-torsion, therefore for q = 2 the cohomology H * (BSpin (8); 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 (BSpin (8)) as a representation of 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 F 3 . The case q = 2 is very different. The cohomology of BSpin(n) with coefficients in F 2 was computed by Quillen [22,Theorem 6.5]. For n = 8 we have where w i are the Stiefel-Whitney classes of the universal bundle and w + 8 is the class of the spinor bundle. The ideal J is generated by w 2 and the results of Steenrod operation: Sq 1 (w 2 ) and [Sq 2 , Sq 1 ](w 2 ). In general we have and since here w 1 = 0 we find that

Proposition 2
The action of the triality on H * (BSpin (8); F 2 ) is the following: φ fixes w 4 , w 6 and w 7 and The triality has to fix w 4 , w 6 and w 7 since the cohomology at that gradations is one dimensional. Moreover the triality permutes cyclically the representations: the natural one, S + and S − , hence φ(w + 8 ) = w 8 (S − ). It remains to show that w 8 (S − ) = w 8 +w + 8 . By Proposition 1 we have e + e(S + ) + e(S − ) = 0 in rational cohomology. The group H 8 (BSpin (8); F 2 ) is spanned by w 8 , w + 8 and w 2 4 . The top Stiefel-Whitney classes are the reductions modulo 2 of the integral Euler classes e, e(S + ) and e(S − ). Also the class w 4 is the reduction modulo 2 of the integral Pontryagin class p 1 . Therefore the relation (6) has to hold also in F 2 cohomology.
We will not discuss here the cohomology with integral coefficients, their generators are to be found in [6].

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 C n → C are indexed by the Dynkin diagrams A μ for μ ≥ 1, D μ for μ ≥ 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 singularity D 4 is given by the formula We have chosen the real form D − 4 since the triality does not act on the real form D + 4 defined by x 3 + 3x y 2 . We will observe how triality acts on the spaces related to that singularity. The link of the singularity consists of three circle in S 3 linked with each other. They bound a surface, which is homeomorphic to the Milnor fiber: 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 • . The rotation of the (x, y) plane by that angle preserves the function. Denote this rotation by φ 0 . The map φ 0 of R 2 (or 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 − x y 2 as the germ representing the singularity, then the formula for φ 0 and φ does not involve irrational coefficients like cos(120 • ) and sin(120 • ).
The Milnor fiber M ε for ε << 1 is described by the equation It is an elliptic curve with three discs removed. We can forget the inequality (since f is homogeneous) and we identify the Milnor fibre with the plane cubic curve. The automorphism φ 0 preserves M ε permuting cyclically the removed discs. Let The function F is clearly invariant since φ 0 preserves the scalar product (b, c), (x, y) . 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 , as desired in the definition of miniversal deformation.
Let us take a φ-invariant morsifications of f We easily compute that f a has the critical points at The first critical value is −4a 3 (minimum), the remaining three values are equal to 0 (saddle points). It follows that the vanishing cycles associated to p A , p B and p C are perpendicular (see [25,§4,Theorem 4.26]) they correspond to the roots A, B and C.
The map φ 0 rotates them cyclically. Therefore φ 0 realizes the triality automorphism of the root system D 4 . Topologically the system of vanishing cycles is homeomorphic to the following configuration: Note that the action of φ coincides with the action on the distinguished generators of H * (BSpin (8); R) found in Proposition 1. The singularity D 4 is homogeneous, and the coefficients of the miniversal deformation can be given the gradations so that the whole function F is quasihomogeneous. We obtain

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 μ , C μ 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 μ and C μ arise as diagrams for the singularities with boundary condition. We will show how the diagram G 2 appears for singularities with 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 Z/3 action. The function f factors to the quotientf : C 2 /Z 3 → 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 M ε is the quotient of the original Milnor fiber M ε . The quotient map is an unbranched cover. Therefore the M ε is homeomorphic to an elliptic curve with one disc removed. The cohomology H 1 (M ε ; 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 −4a 3 corresponds to the vanishing cycle shrieked to the singular point of the domain. It is reasonable to treat the quotient space as C 2 /Z 3 as a stack. Here it simply means that we consider the singularity D 4 together with the 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.
To see what is the associated diagram of that singularity we pass to an odd dimension, adding a nondegenerate quadratic form, e.g.
Then the self-intersection of the cycle corresponding to [ p A ] is −2, and the self intersection of the cycle corresponding to [ p Y ] is equal to − 2 3 . The intersection diagram is exactly G 2 : We will give more precise description of the Milnor fiber off . It is contained in the quotient space C 2 /Z 3 which has the ring of algebraic functions isomorphic to by setting u = x 3 − 3x y 2 , v = y 3 − 3x 2 y and w = x 2 + y 2 . The functionf is equal to u. We see that M is described in C 3 by the equations Hence it is isomorphic to the plane cubic All the fibers are isomorphic as algebraic curves. Topologically, the classical Milnor fibration descends to a fibration from the complement of a circle in the lens space with the fiber homeomorphic to the topological 2-torus with one point removed.
Let g(s, t) = s 3 − t 2 be a germ of A 2 singularity. Note that we have a commutative diagram Therefore the germ off is induced from g by a double covering of C. It is not coincidence. The root system of the Lie algebra g 2 contains the root system of the algebra of the type A 2 . The miniversal deformation of G 2 germ is understood as a slice in the space of jets of invariant function, which is transverse to the orbit of the function f . It is clearly equal to The associated family of fibers is a nontrivial family of plane cubic curves We conclude with the remark that an analogous construction can be applied to the singularity of n lines intersecting at one point which is symmetric with respect to the group Z n acting by the rotations with the angles 2kπ n . The resulting quotient Milnor fiber is isomorphic to a complex curve of genus n−1 2 with one disc removed, when n is odd. For n even the quotient Milnor fiber is a complex curve of genus n−2 2 with two discs removed. That construction for n > 3 seem not to have a counterpart in the realm of Lie algebras.
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Triality in so(4, 4)
In Sect. 2 we have given a formula for triality automorphism acting on the dual t * of the Cartan subalgebra of so (4,4). It does not preserve the lattice corresponding to the group SO(4, 4) but it preserves the lattice spanned by L i 's and 1 2 (L 1 + L 2 + L 3 + L 4 ), which corresponds to Spin (4,4), the cover of SO (4,4). We list below the set of positive roots: The boxed roots are fixed by triality. It can be easily seen when we express the roots in the basis of simple roots: Here for example ABC2Y denotes the root A + B + C + 2Y . Dividing roots into orbits of the triality automorphism we see that we have the fixed roots Y , ABCY and ABC2Y . Three free orbits are generated by A, AY and ABY .
From general theory it follows that the triality automorphism of weights lifts to a self-map of the Lie algebra so (4,4). But a priori it is not clear that one can find such a lift of order three. Not every lift satisfies φ • φ • φ = I d. The choice of signs is not obvious and demands a careful check. All the calculations can be found in [17]. The elements of so(4, 4) for our quadratic form defined by the matrix with 1's on the antidiagonal are the matrices m i j 1≤i, j≤8 which are antisymmetric with respect to the reflection in the antidiagonal. Such a matrix is transformed by the triality automorphism to the following one ⎛ We remark that this is an unique automorphism with real coefficients extending the selfmap of the maximal torus. We would like to stress, that both: the Cartan construction of triality for so (8) and the triality for so(4, 4) presented here works for any ring in which 2 is invertible. On the other hand we note that the remaining real forms of the orthogonal algebra so(k, k − 8) for k = 1, 2, 3, 5, 6, 7 do not admit any triality automorphism with real coefficients. Already on the level of t * we obtain the rotation matrices with imaginary coefficients.

Noncompact version of G 2
The algebra fixed by φ consist of matrices of the form Below we formulate a fundamental fact (known to the specialists) to which we could not find a reference except the paper of [9] for so(8) form or without a explicit matrix realization in [21, §24], [23, §3.3.3], [15,Prop 35.9].

Theorem 3
The fixed points of the triality automorphism is a Lie algebra of the type g 2 .