Pith. sign in

REVIEW 3 major objections 7 minor 47 references

On Homogeneous CR Manifolds of Arbitrary Order of Levi Nondegeneracy

T0 review · 3 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read For every $k \ge 1$, the paper constructs a homogeneous CR hypersurface that is $k$-nondegenerate and gives its local model equation $\operatorname{Re}(w) = 2 \sum_{h=1}^k \operatorname{Re}(z_0^h \bar z_h)$.

desk verdict The algebraic construction of k-nondegenerate CR algebras is solid, but the central local-model theorem is not proved as written. read the letter →

arxiv 2506.00897 v2 pith:VLZ55X7Y submitted 2025-06-01 math.DG

Stefano Marini , Costantino Medori This is my paper
classification math.DG MSC 32V0532V4032V3532V3053C3053C10
keywords CRgeometryhomogeneousmanifoldsk-nondegeneracyLeviformalgebrassu(2)representationshypersurfacemodelsequivalenceproblem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper addresses the question of whether homogeneous CR hypersurfaces can be $k$-nondegenerate for arbitrarily large $k$. It constructs, for every integer $k \ge 1$, a homogeneous hypersurface of CR codimension one from the Lie algebra $\mathfrak{su}(2)$ extended by its unique irreducible representation of dimension $2k+1$, and proves that this manifold is $k$-nondegenerate. The proof gives the full nested sequence of iterated Levi kernels and exhibits a local model equation, $\operatorname{Re}(w) = 2 \sum_{h=1}^k \operatorname{Re}(z_0^h \bar z_h)$, in $\mathbb{C}^{k+2}$. The result is an explicit family of homogeneous $k$-nondegenerate CR hypersurfaces for every order $k$, together with local normal forms and vector fields generating their symmetry algebras.

What carries the argument

The carrying mechanism is the CR algebra $(\mathfrak{g}_\tau,\mathfrak{f})$: here $\mathfrak{g}_\tau = \mathfrak{su}(2) \ltimes V_\tau$ is the semidirect product of $\mathfrak{su}(2)$ with the unique real irreducible $\mathfrak{su}(2)$-module $V_\tau$ of dimension $2k+1$, and $\mathfrak{f} = \mathfrak{b} \oplus V_+$ is the Borel subalgebra of upper-triangular matrices in $\mathfrak{sl}_2(\mathbb{C})$ together with the positive-weight part of $V_\tau$. The order of nondegeneracy is read off the Freeman sequence $\mathfrak{f}_{r+1} = \{ Z \in \mathfrak{f}_r : [Z, \tau(\mathfrak{f})] \subseteq \mathfrak{f}_r + \tau(\mathfrak{f}) \}$, which collapses to the Cartan subalgebra $\langle H \rangle$ after exactly $k$ steps. On the model hypersurface the same brackets are realized by explicit holomorphic vector fields $Z_h, Z'_h, W, A_{hj}, A'_h, E, J, K, Z_\pm, Z'_\pm$, whose commutation relations reproduce the $\mathfrak{su}(2)$-action and weight gradation of $V_\tau$.

What would settle it

Compute the left kernels $F^r$ of the iterated Levi forms of the model hypersurface $\operatorname{Re}(w) = 2 \sum_{h=1}^k \operatorname{Re}(z_0^h \bar z_h)$ at the origin for a fixed $k \ge 3$. The theorem predicts the strict chain $F^1 \supsetneq F^2 \supsetneq \cdots \supsetneq F^{k-1} \supsetneq F^k = \{0\}$; if the chain collapses earlier, or if $F^k$ is nonzero, the claimed order of nondegeneracy is wrong. A stronger check would compare the full CR automorphism algebra of this hypersurface with the real form of the Lie algebra generated by the vector fields in Propositions 4.1–4.5.

Watch

Extended reading notes

Core claim

The central claim is Theorem 4.7: the homogeneous $k$-nondegenerate CR hypersurface associated with the CR algebra $(\mathfrak{g}_\tau = \mathfrak{su}(2) \ltimes V_\tau,\ \mathfrak{f} = \mathfrak{b} \oplus V_+)$ is locally CR-equivalent to the hypersurface $\operatorname{Re}(w) = 2 \sum_{h=1}^{k} \operatorname{Re}(z_0^h \bar z_h)$ in $\mathbb{C}^{k+2}$. The proof writes out explicit holomorphic vector fields on this model that satisfy the bracket relations of the CR algebra, with the Cartan element grading the weight spaces of $V_\tau$ exactly as the algebra requires. The iterated Levi forms are computed through the Freeman sequence $\mathfrak{f}_0 = \mathfrak{f} \supsetneq \mathfrak{f}_1 \supsetneq \cdots \supsetneq \mathfrak{f}_{k-1} \supsetneq \mathfrak{f}_k = \mathfrak{f} \cap \tau(\mathfrak{f})$, where each $\mathfrak{f}_h$ is spanned by the Cartan element together with the weight spaces $v_{h+1}, \dots, v_k$; this chain forces the order of nondegeneracy to be exactly $k$. The paper also shows that the standard Levi form has constant rank two and mixed signature, and that the higher-order Levi kernels reproduce the same nested structure.

Load-bearing premise

The proof assumes that a real hypersurface whose CR symmetry algebra contains a copy of the constructed Lie algebra, with the same stabilizer and the same higher Levi kernels, is automatically locally CR-equivalent to the homogeneous model; that equivalence is asserted rather than demonstrated.

Editorial extensions

If this is right

  • For every $k \ge 1$ there is a homogeneous CR hypersurface of hypersurface type that is $k$-nondegenerate, so homogeneous examples with arbitrarily high Levi nondegeneracy order exist.
  • Every member of the family has an explicit local normal form in $\mathbb{C}^{k+2}$ and explicit vector fields generating its infinitesimal CR automorphism algebra, so the family can serve as a concrete testbed for $k$-nondegenerate equivalence problems.
  • The iterated Levi kernels are computed explicitly as a nested sequence of weight-space subbundles, giving a concrete realization of the Freeman sequence for arbitrary $k$.
  • The $k=1$ and $k=2$ cases are compared with known classifications of homogeneous Levi nondegenerate and $2$-nondegenerate hypersurfaces, connecting the new family to existing low-order results.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same semidirect-product recipe, replacing $\mathfrak{su}(2)$ by another compact semisimple Lie algebra and $V_\tau$ by one of its irreducible modules, is a natural place to look for homogeneous $k$-nondegenerate CR manifolds of higher CR codimension; the paper does not pursue this generalization.
  • The explicit weighted-homogeneous equation invites a direct application of equivalence-problem machinery, such as absolute parallelisms or Cartan connections, to the whole family; that step is not carried out in the paper.
  • A direct computation of the full CR automorphism algebra of the model for a fixed $k$ could reveal whether the listed vector fields generate the whole symmetry algebra or only a subalgebra; if it is larger, the model would carry more symmetry than the homogeneous space $G_\tau/T_\tau$ exhibits.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 7 minor

Summary. The paper constructs, for every integer k ≥ 1, a homogeneous CR hypersurface associated with the CR algebra (gτ = su(2) ⋉ Vτ, f = b ⊕ V+), where V is the (2k+1)-dimensional irreducible representation of sl2(C). Lemma 3.3 computes the Freeman sequence explicitly and shows that the CR algebra is k-nondegenerate of CR codimension 1. Theorem 3.6 realizes the corresponding homogeneous CR manifold as Gτ/Tτ. Section 4 proposes the explicit local model Re(w) = 2 Σ_{h=1}^k Re(z_0^h \bar z_h) in C^{k+2}, and Theorem 4.7 claims that this model is locally CR-equivalent to the homogeneous manifold. The proof of Theorem 4.7 proceeds by exhibiting holomorphic vector fields on the model that generate a copy of gτ, but the final step asserting that the CR distribution and structure can be rebuilt from this data is not a proof of local equivalence. The algebraic core of the paper is sound, but the central equivalence theorem is not fully established.

Significance. If Theorem 4.7 were rigorously proved, the paper would supply the first systematic family of homogeneous k-nondegenerate CR hypersurfaces for arbitrarily large k, with an explicit local normal form and explicit infinitesimal automorphisms. The construction via CR algebras is natural, and the Freeman-sequence computation in Lemma 3.3 is explicit and checkable. The paper also correctly relates the model to known results for k = 1 (Doubrov–Medvedev–The) and k = 2 (Sykes), and to Labovskii's example. The vector-field computations in Propositions 4.1–4.4 give a concrete description of the model's symmetry algebra. However, the main theorem's proof currently lacks the load-bearing equivalence argument, so the significance is conditional on filling that gap.

major comments (3)
  1. [Section 4, proof of Theorem 4.7, final paragraph] The claimed local CR equivalence between Gτ/Tτ and the hypersurface (32) is not established. The proof ends with the assertion 'The CR distribution and structure can be rebuild from this data', but this is not a proof: one must show that the constructed vector fields define a local transitive action with isotropy exactly t, that the induced CR structure coincides with the standard one on (32), and that the resulting CR algebra is isomorphic to (gτ, f). Alternatively, one could cite a rigidity theorem (e.g., a Cartan–Tanaka type statement) that makes coincident CR algebras imply local equivalence. No such argument or reference is supplied. Because Theorem 4.7 is the central result of the paper, this gap is load-bearing.
  2. [Section 4, proof of Theorem 4.7, final paragraph, with Lemma 4.4] The proposed generator set for f is internally inconsistent. Lemma 4.4 establishes the dictionary Z_+ ↦ X↑ and −Z_- ↦ X↓, while f = b ⊕ V+ contains X↑ and V+, not X↓. The proof nonetheless states that 'the vector fields H, Z_-, and the set {ad^j(Z_-)(A'_k)}, 0 ≤ j ≤ k−1, generate a complex Lie subalgebra isomorphic to f'. Under the paper's own dictionary this is false; for k = 1, the bracket [Z_-, A'_1] = i ∂/∂z_1 produces an element outside the stated span, so H, Z_-, A'_1 do not generate the three-dimensional algebra f. Moreover, if these are understood as holomorphic vector fields on the model, their values at the origin do not span the required (k+1)-dimensional CR tangent space: only Z_- is nonzero at the origin, while H, Z_+, and all ad^j(Z_-)(A'_k) vanish there. The proof must specify whether f is represented by holomorphic vector fields or by their conjugates, and must verify the pointwise identification with H^{0,1}_0.
  3. [Section 4, paragraph before Proposition 4.1] The observation that the Levi form and higher-order Levi forms of the model (32) coincide with those of the CR algebra (gτ, f) is asserted without proof. This coincidence is used to connect the explicit model to the homogeneous manifold, but it is not itself sufficient for local equivalence without a rigidity theorem. Either a direct computation of the Freeman sequence for (32) or a proof via the constructed transitive symmetry algebra is needed. As written, the paragraph is an unsupported assertion at a key point of the argument.
minor comments (7)
  1. [Abstract] 'This paper present' should be 'This paper presents'.
  2. [Equations (33)–(36) and throughout Section 4] The notation z_h^0 is confusing: it should be written as z_0^h. As typeset, A'_h = i z_h^0 ∂/∂z_h can easily be misread as i z_h ∂/∂z_h, which changes the weight computations in Lemma 4.4.
  3. [Lemma 3.3, proof] 'Fremann sequence' should be 'Freeman sequence'.
  4. [Section 2] 'Similary' should be 'Similarly'.
  5. [Definition 1.1] The notation F^{(0)} is used in the definition of k-nondegeneracy but is not defined; it should be specified as F^{(0)} = H^{0,1}M.
  6. [Theorem 3.6] The phrase 'Gτ = SU(2) × Vτ as a topological direct product' is misleading because Gτ is a semidirect product; either use ⋉ or explicitly say 'as a topological space'.
  7. [Proposition 4.2(5)] 'The element K + E act as −(k+1)Id on the abelian algebra W' should specify that the adjoint action of K + E acts as −(k+1) times the identity on W.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the k-nondegeneracy and local model are derived by direct computation and checked against external benchmarks; self-citations are background only.

full rationale

The central construction is not circular. Lemma 3.3 computes the Freeman sequence (24) directly from the bracket relations and the weight decomposition (21), so the k-nondegeneracy of the CR algebra is established in the paper itself rather than imported from a prior result. Theorem 3.6 assembles the homogeneous manifold from this CR algebra using standard facts (Mostow [37]). The local model (32) is taken from Labovskii [23, Example 1] and is compared with the external classifications for k=1,2 ([11], [39]); the vector-field computations in Propositions 4.1-4.3 and Lemma 4.4 are direct calculations on (32). The proof of Theorem 4.7 does contain a genuine gap: the final sentence 'The CR distribution and structure can be rebuilt from this data' asserts local equivalence without exhibiting a diffeomorphism or invoking a rigidity theorem, and there are internal algebraic inconsistencies in the proposed dictionary (Lemma 4.4 states [H,A'_k]=2kA'_k while direct computation gives 0; Theorem 4.7 says H, Z_-, and ad^j(Z_-)(A'_k) generate f, while the paper's own isomorphism maps Z_- to X_down in tau(f), not f). These are correctness/completeness problems, not circularity: no fitted parameter is renamed as a prediction, and no theorem from the authors' prior work is used to force the conclusion. The self-citations ([24]-[27]) supply background on graded Lie algebras, higher-order Levi forms, and flag orbits; they are not load-bearing for Theorem 4.7. Hence the score is 1 for minor non-load-bearing self-citations, not for circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No numerical constants are fitted to data, so the free-parameter ledger is empty; k is a free integer parameter of the construction (the desired nondegeneracy depth), not an ad hoc constant. No new physical entities are introduced. The CR algebra (gτ = su(2)⋉Vτ, f = b⊕V+) is a mathematical construction, essentially from the authors' earlier [25], and its k-nondegeneracy is verified by direct computation of the Freeman sequence, which gives an independent checkable handle. The main axiomatic weight is the unproven rigidity step in the proof of Theorem 4.7.

assumptions (4)
  • standard math Finite-dimensional irreducible complex representations of sl2(C) are classified by dimension, are unique up to isomorphism, and have each weight occurring with multiplicity one.
    Used in Lemma 3.3, Theorem 3.6, and the proof of Theorem 4.7 to conclude that the degree-2k polynomial module V is irreducible and that weight spaces are one-dimensional; standard textbook content (cf. [46]).
  • domain assumption A real-analytic CR hypersurface is locally induced as a real hypersurface of some C^N, and when Tanaka regular it admits local coordinates of the form Re(w) = F(z, z-bar).
    Invoked at the start of Section 4 to place the model in C^{k+2}; this is Tanaka's embedding theorem [41], standard for real-analytic hypersurface-type CR manifolds.
  • domain assumption Homogeneous CR manifolds are in correspondence with CR algebras (gτ, f), so k-nondegeneracy can be read off the Freeman sequence of the algebra.
    Section 2 framework from [30, 33]; the paper cites this as established theory and uses it to translate geometric k-nondegeneracy into algebra.
  • ad hoc to paper If a real hypersurface admits a local Lie algebra of CR vector fields isomorphic to gτ with isotropy t, and its iterated Levi forms coincide with those of Gτ/Tτ, then the hypersurface is locally CR-equivalent to Gτ/Tτ.
    This rigidity and transitivity step is the load-bearing premise of the proof of Theorem 4.7; the paper asserts it ('The CR distribution and structure can be rebuilt from this data') but does not prove it or cite a theorem establishing it. If it fails, the local model theorem is not established.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On Homogeneous CR Manifolds of Arbitrary Order of Levi Nondegeneracy." pith.science (2026). https://pith.science/paper/VLZ55X7Y

@misc{pith2026250600897,
  author       = {Pith},
  title        = {Pith review of: On Homogeneous CR Manifolds of Arbitrary Order of Levi Nondegeneracy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VLZ55X7Y}},
  note         = {Machine review of arXiv:2506.00897}
}
abstract

This paper present homogeneous CR hypersurfaces satisfying the $CR$-invariant property of being $k$-nondegenerate for an arbitrary integer $k\geq 1$. The construction of such homogeneous manifolds are based on $CR$ algebras defined by irreducible representations of $\mathfrak{su}(2)$. An explicit study of the iterated Levi forms with their respective kernels, along with the local model equation, is given.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages

  1. [25]

    , Higher order Levi forms on homogeneous CR manifolds, Math. Z. 299 (2021), 563–589

    work page 2021

  2. [23]

    On dimensions of the groups of biholomorphic automorphisms of real-analytic hypersurfaces, Mat

    Labovskii, A.S. On dimensions of the groups of biholomorphic automorphisms of real-analytic hypersurfaces, Mat. Zametki 61 (1997), no. 3, 349–358; translation in Math. Notes 61 (1997), no. 3-4, 287–294

    work page 1997

  3. [11]

    and The, D

    Doubrov, B., Medvedev, A. and The, D. Homogeneous Levi non-degenerate hypersurfaces in C3, Math. Z. 297 (2021), 669–709

    work page 2021

  4. [39]

    https://www.iumj.indiana.edu

    Sykes, D., Homogeneous 2-nondegenerate CR manifolds of hypersurface type in low dimensions, Indiana Univer- sity Mathematics Journal, to appear. https://www.iumj.indiana.edu

    work page

  5. [1]

    The CR structure of minimal orbits in complex flag manifolds, J

    Altomani, A., Medori, C., and Nacinovich, M. The CR structure of minimal orbits in complex flag manifolds, J. Lie Theory 16 (2006), 483–530

    work page 2006

  6. [2]

    , On the topology of minimal orbits in complex flag manifolds, Tohoku Math. J. (2) 60 (2008), 403–422

    work page 2008

  7. [3]

    , Orbits of real forms in complex flag manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)9 (2010), 69–109

    work page 2010

  8. [4]

    Abelian extensions of semisimple graded CR algebras , Adv

    Altomani, A., Nacinovich, M. Abelian extensions of semisimple graded CR algebras , Adv. Geom. 4 (2004), 433–457

    work page 2004

Show all 47 references
  1. [5]

    Real submanifolds in complex space and their mappings , Princeton Math

    Baouendi, M.S., Ebenfelt, M.S., and Rothschild, L.P. Real submanifolds in complex space and their mappings , Princeton Math. Ser., 47 Princeton University Press, Princeton, NJ, 1999. xii+404 pp. ISBN:0-691-00498-6

    1999

  2. [6]

    K., Modification of Poincar´e’s construction and its application in CR-geometry of hypersurfaces in C4, Izvestiya: Mathematics 86 (2022), Issue 5, 852–875, DOI: 10.1070/IM9249

    Beloshapka, V . K., Modification of Poincar´e’s construction and its application in CR-geometry of hypersurfaces in C4, Izvestiya: Mathematics 86 (2022), Issue 5, 852–875, DOI: 10.1070/IM9249

    2022 doi

  3. [7]

    Beloshapka, V .K.,Model Surfaces: Weighted Approach, Russ. J. Math. Phys. 30 (2023), 25–45

    2023

  4. [8]

    Chapters 1–3 , Elements of Mathematics (Berlin), Springer-Verlag, Berlin, (1989), Translated from the French, Reprint of the 1975 edition

    Bourbaki, N., Lie groups and Lie algebras. Chapters 1–3 , Elements of Mathematics (Berlin), Springer-Verlag, Berlin, (1989), Translated from the French, Reprint of the 1975 edition

    1989

  5. [9]

    Chapters 7–9 , Elements of Mathematics (Berlin), Springer-Verlag, Berlin, (2005), Translated from the 1975 and 1982 French originals by Andrew Pressley

    , Lie groups and Lie algebras. Chapters 7–9 , Elements of Mathematics (Berlin), Springer-Verlag, Berlin, (2005), Translated from the 1975 and 1982 French originals by Andrew Pressley

    2005

  6. [10]

    Chern S. S. and Moser J. K. Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219–271

    1974

  7. [12]

    Fels, G., Locally homogeneous finitely nondegenerate CR-manifolds Math. Res. Lett. 14 (2007), 893–922

    2007

  8. [13]

    201 (2008), 1–82

    Fels, G., Kaup, W., Classification of Levi degenerate homogeneous CR-manifolds in dimension 5, Acta Math. 201 (2008), 1–82

    2008

  9. [14]

    Freeman, M., Real submanifolds with degenerate Levi form, Several complex variables (Proc. Sympos. Pure Math. 30, Williams Coll., Williamstown, Mass., (1975)), Part 1, pp. 141–147. Amer. Math. Soc., Providence, R.I., (1977)

    1975

  10. [15]

    Freeman, M., Local Biholomorphic Straightening of Real Submanifolds, Ann. of Math. (2) 106 (1977), 319–352

    1977

  11. [16]

    A First Course , Grad

    Fulton, W., Harris, J., Representation Theory. A First Course , Grad. Texts in Math. 129, Springer-Verlag, New York, 1991. xvi+551 pp

    1991

  12. [17]

    Gregorovi ˇc, J., On equivalence problem for 2–nondegenerate CR geometries with simple models, Adv. Math. 384, (2021), Paper No. 107718, 35 pp

    2021

  13. [18]

    Texts in Math

    Hilton P.J., and Stammbach, U., A Course in Homological Algebra, Grad. Texts in Math. 4, Springer-Verlag, New York, 1997. xii+364 pp

    1997

  14. [19]

    E., Introduction to Lie Algebras and Representation Theory, Grad

    Humphreys, J. E., Introduction to Lie Algebras and Representation Theory, Grad. Texts in Math.9, Springer-Verlag, New York-Berlin, 1978, xii+171 pp

    1978

  15. [20]

    Reduction of five-dimensional uniformly Levi degenerate CR structures to absolute paral- lelisms, J

    Isaev, A., Zaitsev, D. Reduction of five-dimensional uniformly Levi degenerate CR structures to absolute paral- lelisms, J. Geom. Anal. 23 (2013), 1571–1605

    2013

  16. [21]

    and Sykes, D.,New examples of 2-nondegenerate real hypersurfaces inCN with arbitrary nilpotent symbols, J

    Kol ´aˇr, M., Kossovskiy, I. and Sykes, D.,New examples of 2-nondegenerate real hypersurfaces inCN with arbitrary nilpotent symbols, J. Lond. Math. Soc. (2) 110 (2024), no. 2, Paper No. e12962, 36 pp

    2024

  17. [22]

    and Santi, A., On 3-nondegenerate CR manifolds in dimension 7 (I): The transitive case J

    Kruglikov, B. and Santi, A., On 3-nondegenerate CR manifolds in dimension 7 (I): The transitive case J. Reine Angew. Math. 820 (2025), 167–211

    2025

  18. [24]

    and Nacinovich, M., On some classes of Z-graded Lie algebras

    Marini, S., Medori, C. and Nacinovich, M., On some classes of Z-graded Lie algebras. Abh. Math. Semin. Univ. Hambg. 90 (2020), 45–71

    2020

  19. [26]

    , On finitely nondegenerate closed homogeneous CR manifolds , Ann. Mat. Pura Appl. (4) 202 (2023), 2715–2747

    2023

  20. [27]

    Marini, S., Medori, C., Nacinovich, M., and Spiro, A., On transitive contact and CR algebras , Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 20 (2020), 771–795

    2020

  21. [28]

    21 (2017), pp

    Marini, S., and Nacinovich, M., Orbits of real forms, Matsuki duality and CR-cohomology, Complex and symplec- tic geometry, Springer INdAM Ser. 21 (2017), pp. 149–162

    2017

  22. [29]

    , Mostow’s fibration for canonical embeddings of compact homogeneous CR manifolds, Rend. Semin. Mat. Univ. Padova140 (2018), 1–43

    2018

  23. [30]

    109 (1997), 195–250

    Medori, C., and Nacinovich, M., Levi-Tanaka algebras and homogeneous CR manifolds , Compositio Math. 109 (1997), 195–250

    1997

  24. [31]

    , Classification of semisimple Levi-Tanaka algebras, Ann. Mat. Pura Appl. (4) 174 (1998), 285–349

    1998

  25. [32]

    , The Levi-Malcev theorem for graded CR Lie algebras, Recent advances in Lie theory (Vigo, 2000), Res. Exp. Math. 25 (2002), pp. 341–346

    2002

  26. [33]

    Algebra 287 (2005), 234–274

    , Algebras of infinitesimal CR automorphisms, J. Algebra 287 (2005), 234–274

    2005

  27. [34]

    Medori, C., and Spiro, A., The Equivalence Problem for Five-dimensional Levi Degenerate CR Manifolds , Int. Math. Res. Not. IMRN (2014), no. 20, 5602–5647. ON HOMOGENEOUS CR MANIFOLDS OF ARBITRARY ORDER OF LEVI NONDEGENERACY 15

    2014

  28. [35]

    , Structure equations of Levi degenerate CR hypersurfaces of uniform type , Rend. Semin. Mat. Univ. Po- litec. Torino 73 (2015), 127–150

    2015

  29. [36]

    Merker, J., Pocchiola, S., Explicit Absolute Parallelism for 2-Nondegenerate Real Hypersurfaces M⊂ C3 of Con- stant Levi Rank 1, J. Geom. Anal. 30 (2020), 3233–3242

    2020

  30. [37]

    D., Covariant Fiberings of Klein Spaces, II, Amer

    Mostow, G. D., Covariant Fiberings of Klein Spaces, II, Amer. J. Math. 84 (1962), 466–474

    1962

  31. [38]

    xii+162 pp

    Samelson, H., Notes on Lie Algebras, Springer-Verlag, New York, 1990. xii+162 pp

    1990

  32. [40]

    Sykes, D., Zelenko, I., On geometry of 2-nondegenerate CR structures of hypersurface type and flag structures on leaf spaces of Levi foliations, Adv. Math. 413 (2023), Paper No. 108850, 65 pp

    2023

  33. [41]

    Tanaka, N., On the pseudo-conformal geometry of hypersurfaces of the space of n complex variables, J. Math. Soc. Japan 14 (1962), 397-429

    1962

  34. [42]

    , On generalized graded Lie algebras and geometric structures. I, J. Math. Soc. Japan 19 (1967), 215–254

    1967

  35. [43]

    , On differential systems, graded Lie algebras and pseudogroups, J. Math. Kyoto Univ. 10 (1970), 1–82

    1970

  36. [44]

    , On nondegenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan. J. Math. (N.S.) 2 (1976), 131–190

    1976

  37. [45]

    , Group theory and quantum mechanics, McGraw-Hill, 1964

    Tinkham, M. , Group theory and quantum mechanics, McGraw-Hill, 1964

    1964

  38. [46]

    Graduate Texts in Mathematics, 102

    Varadarajan, V .S., Lie Groups, Lie Algebras, and Their Representations. Graduate Texts in Mathematics, 102. Springer, New York, NY , 1984, xiii+430 pp., ISBN: 978-1-4612-7016-4

    1984

  39. [47]

    A., The action of a real semisimple group on a complex flag manifold

    Wolf, J. A., The action of a real semisimple group on a complex flag manifold. I. Orbit structure and holomorphic arc components, Bull. Amer. Math. Soc. 75 (1969), 1121–1237. Stefano Marini, D epartment of Mathematical, P hysical, and Computer Sciences, U niversity of Parma, P...

    1969

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.