Generalised holonomies and K(E9)

The involutory subalgebra K($$ \mathfrak{e} $$ e 9) of the affine Kac-Moody algebra $$ \mathfrak{e} $$ e 9 was recently shown to admit an infinite sequence of unfaithful representations of ever increasing dimensions [1]. We revisit these representations and describe their associated ideals in more detail, with particular emphasis on two chiral versions that can be constructed for each such representation. For every such unfaithful representation we show that the action of K($$ \mathfrak{e} $$ e 9) decomposes into a direct sum of two mutually commuting (‘chiral’ and ‘anti-chiral’) parabolic algebras with Levi subalgebra $$ \mathfrak{so} $$ so (16)+ ⊕ $$ \mathfrak{so} $$ so (16)−. We also spell out the consistency conditions for uplifting such representations to unfaithful representations of K($$ \mathfrak{e} $$ e 10). From these results it is evident that the holonomy groups so far discussed in the literature are mere shadows (in a Platonic sense) of a much larger structure.

Transverse Kähler holonomy in Sasaki Geometry and S-Stability

We study the transverse Kähler holonomy groups on Sasaki manifolds (M, S) and their stability properties under transverse holomorphic deformations of the characteristic foliation by the Reeb vector field. In particular, we prove that when the first Betti number b 1(M) and the basic Hodge number h 0,2 B(S) vanish, then S is stable under deformations of the transverse Kähler flow. In addition we show that an irreducible transverse hyperkähler Sasakian structure is S-unstable, whereas, an irreducible transverse Calabi-Yau Sasakian structure is S-stable when dim M ≥ 7. Finally, we prove that the standard Sasaki join operation (transverse holonomy U(n 1) × U(n 2)) as well as the fiber join operation preserve S-stability.

Almost All Finsler Metrics have Infinite Dimensional Holonomy Group

We show that the set of Finsler metrics on a manifold contains an open everywhere dense subset of Finsler metrics with infinite-dimensional holonomy groups.

Irreducible holonomy groups and first integrals for holomorphic foliations

We study groups of germs of complex diffeomorphisms having a property called irreducibility. The notion is motivated by the similar property of the fundamental group of the complement of an irreducible hypersurface in the complex projective space. Natural examples of such groups of germ maps are given by holonomy groups and monodromy groups of integrable systems (foliations) under certain conditions. We prove some finiteness results for these groups extending previous results in [D. Cerveau and F. Loray, Un théorème de Frobenius singulier via l’arithmétique élémentaire, J. Number Theory 68 1998, 2, 217–228]. Applications are given to the framework of germs of holomorphic foliations. We prove the existence of first integrals under certain irreducibility or more general conditions on the tangent cone of the foliation after a punctual blow-up.

Generalization of Rubik’s cube group to n-dimensional case

In this paper, Rubik’s cube group in [Formula: see text]-dimensional case is described. In that case, some interesting effects arise, which is related with angular rotations. For [Formula: see text], the group [Formula: see text] arises, which can be realized as a factor group of [Formula: see text] by commutation [Formula: see text]. For [Formula: see text] there are no such effects, because corresponding factor is trivial. Rubik’s cube is an important model to study effects of groups related with, in particular, holonomy groups from differential geometry.