A contribution to the symmetry classification problem for second-order PDEs in z(x, y)

Hillgarter, Erik

doi:10.1093/imamat/hxh094

Cited by 3 publications

(5 citation statements)

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“…14 which was also noticed in [11,12]. The lifts of nonsolvable Lie algebras are contained in [5], and the case of metric lifts was also considered in [15].…”

Section: Theorem 2 the Following List Contains All Metric Affine And ...mentioning

confidence: 96%

“…Examples areĝ a 10 ,ĝ m 14 ,ĝ a1 14 andĝ a 8 with general α and B = 0. There is in error in the Lie algebra corresponding toĝ a1 14 which was also noticed in [11,12]. The lifts of nonsolvable Lie algebras are contained in [5], and the case of metric lifts was also considered in [15].…”

Section: Theorem 2 the Following List Contains All Metric Affine Anmentioning

confidence: 99%

“…For the primitive Lie algebras of vector fields on the plane, this was completed by Lie [14]. Amaldi continued Lie's work by extending the imprimitive Lie algebras to three-dimensional space [2,3] (see also [11]). Nonsolvable Lie algebras of vector fields on C 3 were classified in [5].…”

Section: Introductionmentioning

confidence: 99%

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Projectable Lie algebras of vector fields in 3D

Schneider

2018

Journal of Geometry and Physics

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Starting with Lie's classification of finite-dimensional transitive Lie algebras of vector fields on C 2 we construct Lie algebras of vector fields on the bundle C 2 × C by lifting the Lie algebras from the base. There are essentially three types of transitive lifts and we compute all of them for the Lie algebras from Lie's classification. The simplest type of lift is encoded by Lie algebra cohomology.

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“…14 which was also noticed in [11,12]. The lifts of nonsolvable Lie algebras are contained in [5], and the case of metric lifts was also considered in [15].…”

Section: Theorem 2 the Following List Contains All Metric Affine And ...mentioning

confidence: 96%

Section: Theorem 2 the Following List Contains All Metric Affine Anmentioning

confidence: 99%

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Projectable Lie algebras of vector fields in 3D

Schneider

2018

Journal of Geometry and Physics

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“…All such systems were computed in the PhD thesis of Hillgarter [12]. The g-action lifted to J 2 (C 2 , C) admits the following three absolute invariants (see p.83 (ip 13 ) and §4.2.1 of [12]):…”

Section: Related Equi-affine Geometrymentioning

confidence: 99%

“…. , i∂ z n+1 is a characteristic 12 (n + 1)-dimensional abelian ideal in hol(M 1 ) and hol(M 2 ). Then M 1 and M 2 are locally biholomorphically equivalent if and only if their bases are locally affinely equivalent.…”

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confidence: 99%

Classification of simply-transitive Levi non-degenerate hypersurfaces in $\mathbb{C}^3$

Doubrov¹,

Merker²,

The³

2020

Preprint

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Holomorphically homogeneous CR real hypersurfaces M 3 ⊂ C 2 were classified by Élie Cartan in 1932. A folklore legend tells that an unpublished manuscript of Cartan also treated the next dimension M 5 ⊂ C 3 (in conjunction with his study of bounded homogeneous domains), but no paper or electronic document currently circulates.In the last 20 years, much progress on this classification problem in C 3 has been made. Fels-Kaup classified the 2-nondegenerate hypersurfaces with constant Levi rank 1, obtaining a list that is entirely tubular. For Levi non-degenerate hypersurfaces, substantial efforts due to Loboda and Doubrov-Medvedev-The settled the multiply-transitive case, and the lists are extensive. Recently, Kossovskiy-Loboda completed the strongly pseudoconvex (Levi-definite) simply-transitive case, and all models found are tubular. Only the simply-transitive case with split Levi signature remained.We perform this last study by employing an independent Lie-algebraic approach. A key new geometric tool is a coordinate-free formula for the fundamental (complexified) quartic tensor, which in particular can be specialized to Lie algebra data. Our final result has a unique (Levi-indefinite) nontubular model, which admits planar equi-affine symmetry.While this article was under preparation, Loboda posted his synthetic memoir arXiv:2006.07835, where he also addressed the simply-transitive Levi-indefinite case. We can hence compare and clarify both lists of simply-transitive models.A striking corollary is that all locally homogeneous real hypersurfaces M 5 ⊂ C 3 are now fully classified by these contributions.

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