Duality of (2,3,5)-distributions and Lagrangian cone structures

Ishikawa, Goo; Kitagawa, Yumiko; Tsuchida, Asahi; Yukuno, Wataru

doi:10.48550/arxiv.1808.00149

Cited by 1 publication

(1 citation statement)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Besides its historical significance and its connection with G 2 , which mediates its relationship with other geometries [7,19,20,21,22,27,28], [29,Section 5], this geometry is significant because of its appearance in simple, nonholonomic kinematic systems [5,6]. It has enjoyed heightened attention in the last decade or so [2,3,11,12,14,15,24,25,30,33,34,35,36,37], owing in part to its realization in the class of parabolic geometries [8,Section 4.3.2], [31,32], a broad family of Cartan geometries for which many powerful results are available.…”

Section: Introductionmentioning

confidence: 99%

Homogeneous Real (2,3,5) Distributions with Isotropy

Willse

Universit²,

Wien³

2019

SIGMA

View full text Add to dashboard Cite

We classify multiply transitive homogeneous real (2, 3, 5) distributions up to local diffeomorphism equivalence.Proposition 2.1 ([18, Section 76]). Let (M, D) be a real (complex) (2, 3, 5) distribution and fix a point u ∈ M . There is a neighborhood U ⊆ M of u, a diffeomorphism (biholomorphism) Ψ : U → V ⊂ J 2,0 (F, F 2 ), and a smooth (complex-analytic) Homogeneous distributions Infinitesimal symmetriesAn infinitesimal symmetry of a (2, 3, 5) distribution (M, D) is a vector field ξ ∈ Γ(D) whose (local) flow preserves D, or equivalently for which L ξ η ∈ Γ(D) for all η ∈ Γ(D). We denote the Lie algebra of infinitesimal symmetries, called the (infinitesimal) symmetry algebra, by aut(D), and we say that (M, D) is infinitesimally homogeneous if aut(D) acts infinitesimally transitively, that is, if {ξ u : ξ ∈ aut(D)} = T u M for all u ∈ M . This article concerns (infinitesimally) multiply transitive homogeneous distributions, that is, infinitesimally homogeneous distributions for which the isotropy subalgebra k u < aut(D) of infinitesimal symmetries vanishing at any u ∈ M is nontrivial, or equivalently, for which dim aut(D) ≥ 6. Algebraic models for homogeneous distributionsFix a homogeneous (2, 3, 5) (real or complex) distribution (M, D) with transitive symmetry algebra h := aut(D), fix a point u ∈ M , and denote by k < h the subalgebra of vector fields in h vanishing at u and by d ⊂ h the subspace d := {ξ ∈ h : ξ u ∈ D u }. Then, k ⊆ d, [k, d] ⊆ d (we call this property k-invariance), and dim(d/k) = 2. The fact that D is a (2, 3, 5) distribution implies the genericity condition d + [d, d] + [d, [d, d]] = h. We call the triple (h, k; d) a (real or complex) algebraic model (of (M, D)).Given an algebraic model, we can reconstruct D up to local equivalence: For any groups H, K with K < H and respectively realizing h, k (for the groups that occur in the classification, we can choose K to be a closed subgroup of H), invoke the canonical identification T id ·K (H/K) ∼ = h/k to take D ⊂ T (H/K) to be the distribution with fibers D h·K = T id ·K L h · (d/k), where L h : H/K → H/K is the map L h : g · K → (hg) · K; by k-invariance this definition is independent of the coset representative h, and by genericity D is an H-invariant (2, 3, 5) distribution. Via the above identification, [D, D] id ·K = (d + [d, d])/k. 1 We declare two algebraic models (h, k; d), (h , k ; d ) to be equivalent iff there is a Lie algebra isomorphism α : h → h satisfying α(k) = k and α(d) = d . Unwinding definitions shows that equivalent algebraic models determine locally equivalent distributions. Multiply transitive homogeneous complex distributionsCartan showed that for all (2, 3, 5) distributions D, dim aut(D) ≤ 14, and that equality holds iff it is locally equivalent to the so-called flat distribution ∆ [9]; his argument applies to both the real and complex settings. We call the corresponding (complex) algebraic model O (see Section 4.1). In this case, aut(D) is isomorphic to the simple complex Lie algebra of type G 2 -we denote it by g 2 (C) -a...

show abstract