Pith. sign in

REVIEW 3 major objections 6 minor 32 references

Kinematical superspaces

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read This paper classifies all $N{=}1$ $d=4$ kinematical and aristotelian Lie superalgebras with spatial isotropy and nonzero supercharge bracket, and all 27 simply-connected homogeneous superspaces they define.

desk verdict The algebraic classification of N=1 d=4 kinematical superalgebras is solid and important; the list of 27 homogeneous superspaces is conditional on an unproved folklore bijection that the authors honestly flag. read the letter →

arxiv 1908.11278 v2 pith:D7PGS2LN submitted 2019-08-29 hep-th math.DGmath.RA

José Figueroa-O'Farrill , Ross Grassie This is my paper · ORCID
classification hep-thmath.DGmath.RA MSC 17B7017B8122E7053C3058A50
keywords kinematicalLiesuperalgebrashomogeneoussuperspacesspatialisotropyquaternionicformalismJacobiidentitiescentralextensionsN=1supersymmetrygeometriclimits
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 answers the question of which supersymmetric versions of four-dimensional kinematical spacetimes exist, completely, for $N{=}1$ with spatial isotropy. It classifies up to isomorphism every Lie superalgebra extending a kinematical or aristotelian (boost-free) Lie algebra by four real supercharges whose bracket is nonzero, obtaining 43 isomorphism classes, some with essential continuous parameters, and it also determines their nontrivial central extensions. It then classifies the simply-connected homogeneous $(4|4)$-dimensional superspaces arising from these algebras, finding 27, all reductive. The result matters because it turns known examples such as Minkowski and anti de Sitter superspace into an exhaustive list, and it shows exactly which spacetimes admit no supersymmetry of this kind.

What carries the argument

The central object is a quaternionic reformulation: the rotation algebra is identified with the imaginary quaternions $\operatorname{Im}\mathbb{H}$, the four supercharges with $\mathbb{H}$, and every bracket is written as quaternion multiplication, making rotational covariance manifest. A supersymmetric extension is then encoded by quaternionic parameters $h,b,p\in\mathbb{H}$, $c_0\in\mathbb{R}$ and $c_1,c_2,c_3\in\operatorname{Im}\mathbb{H}$, and the Jacobi identity reduces to algebraic relations such as $[b,h]=\lambda b+\mu p$ and $c_0 h=\tfrac12 c_1+c_2b+c_3p$. The automorphism group of the underlying kinematical Lie algebra, acting by quaternion conjugation and rescaling, is used to select one representative per orbit. For the superspaces, the carrying object is the super Lie pair $(\mathfrak{s},\mathfrak{h})$, where $\mathfrak{h}$ is spanned by rotations and boosts; classifying such pairs up to automorphism of $\mathfrak{s}$ gives the homogeneous superisation of the corresponding spacetime.

What would settle it

A computer algebra rerun of the Jacobi identities in the full 22-parameter space, for each kinematical Lie algebra $K_1$--$K_{18}$ and aristotelian $A_1$--$A_{3\pm}$, with the automorphism group quotient taken, would settle the algebraic classification: any $[Q,Q]\neq0$ solution not isomorphic to a row of Tables 4 and 6 would refute it. For the superspace list, the decisive check is to prove or disprove the folklore correspondence of Section 4.1 by constructing the homogeneous supermanifold for each pair and verifying that all candidates are covered.

Watch

Extended reading notes

Core claim

The central discovery is that the possible $N{=}1$ supersymmetric extensions are controlled by a 22-dimensional space of rotation-equivariant brackets, and that imposing the super-Jacobi identities leaves precisely the 43 classes in Tables 4 and 6. Some kinematical Lie algebras---euclidean, $\mathfrak{so}(4,1)$ and $\mathfrak{so}(5)$---admit no such supersymmetric extension, because the four-dimensional spinor representation of $\mathfrak{so}(3)$ does not extend to a representation of these algebras. Many spacetimes admit more than one inequivalent superisation, including continuous one-parameter families, and most of the listed superalgebras are not obtained as contractions of the anti de Sitter superalgebra $\mathfrak{osp}(1|4)$. There are also effective super Lie pairs whose underlying Lie pair is not effective, meaning the boost generators act trivially on the spacetime but act as $R$-symmetries on the fermionic directions.

Load-bearing premise

The superspace classification stands on the unproved folklore claim that every effective, geometrically realisable super Lie pair really corresponds to a smooth homogeneous supermanifold; if that correspondence has exceptions, the 27-entry table could be incomplete or contain entries with no genuine geometry. The paper itself states that it knows of no proof of this correspondence.

Editorial extensions

If this is right

  • Every $N{=}1$, $d=4$, spatially isotropic homogeneous superspace is one of the 27 entries in Table 14, so a candidate model can be checked against a complete list.
  • Several spacetimes admit no superisation at all: the euclidean, spherical and hyperbolic riemannian spaces, de Sitter spacetime, carrollian de Sitter and the carrollian light cone.
  • Many galilean and aristotelian superspaces come in continuous families with essential parameters, so inequivalent supersymmetry algebras can share the same underlying spacetime.
  • The boost generators can become pure $R$-symmetries in the superspace, producing aristotelian superspaces with $R$-symmetry that have no classical boost action on the body.
  • All 27 superspaces are reductive, so each carries a canonical invariant connection and the associated Killing-superalgebra description applies.

Reading between the lines

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

  • If the folklore correspondence between effective super Lie pairs and homogeneous superisations is eventually proved, Table 14 becomes a theorem rather than a classification resting on an unproved premise; if it fails, the table would need revision. This is the main gap a reader should watch.
  • The quaternionic parametrisation of the extension data is likely reusable for $N{=}2$ in four dimensions, where the spinor module has quaternionic dimension two, so an analogous but larger parameter space could support an exhaustive classification.
  • The existence of continuous families of superspaces over a fixed spacetime suggests that non-relativistic or ultra-relativistic limits of supersymmetric field theories may have inequivalent quantum theories differing only by how supercharges transform under boosts, a distinction the classical spacetime alone cannot see.
  • The observed sharing of the Poincaré supergroup by Minkowski and carrollian anti de Sitter superspaces points toward an ultra-relativistic supersymmetry duality that the paper flags for future work; testing it on correlation functions would be a natural next step.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper classifies N=1, d=4 kinematical and aristotelian Lie superalgebras with spatial isotropy, dropping parity and time-reversal assumptions and requiring [Q,Q] ≠ 0. The authors introduce a quaternionic formalism in which the r-equivariant brackets live in a 22-dimensional parameter space; Lemmas 1 and 2 reduce the Jacobi constraints, and each of the 18 kinematical Lie algebras from Table 2 is then analysed case by case. The result is 43 isomorphism classes (some with essential continuous parameters) listed in Tables 4 and 6, together with their automorphisms, non-trivial central extensions, and compatible gradings. The paper then classifies super Lie pairs (s,h), selects the effective and geometrically realisable ones, and obtains a list of 27 homogeneous superspaces in Table 14, all claimed to be reductive. It also computes low-rank invariant tensors and studies contractions and non-contracting limits between the superspaces.

Significance. The algebraic classification is a substantial and carefully executed contribution. The quaternionic method is elegant, makes rotational equivariance transparent, and recovers the Poincaré and osp(1|4) superalgebras as unique classes while producing many new families, including one-parameter families whose parameter is argued to be essential. The automorphism tables and central-extension tables are valuable reference material, and the limit diagram in Section 5 provides a useful global picture. The geometric classification of 27 homogeneous superspaces is the paper's headline claim, but it depends on a folklore correspondence stated without proof in Section 4.1; if that correspondence is supplied or the claims are appropriately qualified, the paper would be a standard reference for kinematical superspaces. No machine-checked code is provided, but the derivations are explicit and checkable.

major comments (3)
  1. [4.1] The one-to-one correspondence between effective, geometrically realisable super Lie pairs and homogeneous superisations of homogeneous manifolds is asserted as mathematical folklore, and the authors explicitly state that they know no proof or reference for it. Since Section 4.1 also declares all supermanifolds in the paper to be split, and the cited reference [18] is phrased for spin manifolds, the converse direction is not established for non-split or non-spin situations. Consequently Table 14 currently classifies split superisations of the Table 1 spacetimes under an assumed bijection, and the abstract's unqualified statement '27 homogeneous superspaces' is stronger than what is proved. The authors should either prove the correspondence, give a precise reference, or explicitly state the classification as conditional on this folklore result.
  2. [3.5 and 4.2] The restriction to r-fixing automorphisms is introduced with 'Without loss of generality' at the start of Section 3.5, but no justification is given. The automorphisms of k generated by B and P are in particular inner automorphisms of the Lie superalgebra s and generically do not fix the rotational subalgebra r. If such automorphisms can relate two admissible subalgebras h that both contain the fixed r, then the orbit classification in Section 4.2 using only the r-fixing automorphism groups of Tables 8 and 9 could overcount the super Lie pairs and hence the homogeneous superspaces in Table 14. The authors should supply the missing conjugacy argument that every Aut(s)-orbit on admissible subalgebras is represented by an Aut_r(s)-orbit, or alternatively compute the action of the full automorphism group on admissible subalgebras.
  3. [3.1.6] The claim that parameters such as λ are 'essential' is load-bearing for the classification because several families in Tables 4 and 6 are presented as genuinely one-parameter families. The argument given is that the difference of brackets is a cocycle and 'one can check' it is not a coboundary. This check is not shown. Since the entire classification of isomorphism classes depends on this point, at least one explicit non-coboundary computation should be included, or a reference should be given where the same method is carried out.
minor comments (6)
  1. [Abstract and 4.1] The abstract and several introductory statements refer to 'homogeneous superspaces' without qualification. Given the splitness assumption and the folklore caveat in Section 4.1, the wording should say 'split homogeneous superspaces' or otherwise make the standing assumptions explicit.
  2. [2.5] The sentence dismissing automorphisms of k that transform r ('their description... will not play a rôle in our discussion') is too terse; it should be expanded or connected to the argument in Section 3.5.
  3. [3.1.6-3.1.15] The repeated phrase 'This same argument shows that the parameters appearing in other Lie superalgebras are essential as well' would benefit from a single explicit example worked out in detail, so that the reader can verify the cocycle/coboundary test.
  4. [Table 4] The table lists 35 rows, but some rows represent one-parameter families and some represent single isomorphism classes; the caption should clarify the counting that yields the total of 43 isomorphism classes.
  5. [4.2] The color coding in Table 12 (blue, green, grey) is informative but may be illegible in monochrome prints; adding a symbol such as a star, dagger, or footnote marker would improve readability.
  6. [2.4] In the proof of Lemma 2, the notation 'Q(sc1ss)' is ambiguous; adding parentheses, such as Q(s c1 s s), would make the quaternion multiplication easier to follow.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the superalgebra classification is derived by solving Jacobi identities against external even Lie algebra inputs, and the geometric caveat in Section 4.1 is a missing proof, not a circular reduction.

full rationale

The algebraic classification in Tables 4 and 6 is self-contained: the paper fixes a 22-dimensional space of r-equivariant brackets (Eqs. (2.7)-(2.15)), imposes the four Jacobi components (Lemmas 1, 2 and the case-by-case analysis of Section 3), and quotients by the automorphism group G described in Eq. (2.40). The known superalgebras (Poincare S14 and osp(1|4) S15) are outputs of this procedure, not inputs. The input even Lie algebra list (Table 2) and the homogeneous spacetime list (Table 1) are taken from [7,16] and [4], but these concern only s-bar_0 and are external to the claim being derived; [7] is independent and [4] is a separate published classification, so citing them is not circular. The super Lie pair classification in Section 4 uses the admissible subalgebras and automorphisms from [4] and Section 3.5, and does not presuppose Table 14. The only notable caveat is a non-circular gap: Section 4.1 states that "there is a one-to-one correspondence between (isomorphism classes of) effective, geometrically realisable super Lie pairs and (isomorphism classes of) homogeneous superisations of homogeneous manifolds. To the best of our knowledge, this result is part of the mathematical folklore and we are not aware of any reference where this result is proved or even stated as such." The paper also restricts to split supermanifolds. Thus Table 14 is a classification of split superisations, and its unrestricted reading as a classification of all homogeneous (4|4)-dimensional superspaces is conditional on that unproved bijection. This is a completeness/correctness risk, not a reduction of the output to the input. No fitted parameter is renamed as a prediction, and no load-bearing uniqueness theorem is imported from the authors' own prior work.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The core algebraic classification is derived from first principles, namely the Jacobi identities, in the paper. The listed continuous parameters (lambda, gamma, chi) are outputs of the classification, not fitted inputs, so they are not counted as free parameters. The main external inputs are the prior classifications of kinematical Lie algebras and admissible subalgebras, plus an unproved folklore correspondence for superisations. No new physical entities are introduced.

assumptions (5)
  • domain assumption The classification of ten-dimensional kinematical Lie algebras with space isotropy (Table 2) and their automorphisms is correct, taken from [4], [7] and [16].
    Used throughout Section 3 as the even part s_0 of each superalgebra; if a kinematical algebra were missing, the superalgebra list would be incomplete.
  • domain assumption The classification of admissible subalgebras h = r + V of kinematical Lie algebras in [4, Sections 3.1-3.2] is correct; the paper determines super Lie pairs by inspecting that classification.
    Section 4.2 states that vectorial subspaces V = alpha B + beta P define admissible subalgebras and that the results of [4] are used by inspection.
  • domain assumption Effective, geometrically realisable super Lie pairs (s,h) are in bijection with homogeneous superisations of homogeneous manifolds.
    Section 4.1 states this is part of the mathematical folklore with no reference; it underlies the classification of homogeneous superspaces in Table 14.
  • standard math Standard quaternionic representation theory of sp(1): the spinor module is H, and r-equivariant maps V x S to S and S⊙S to k are classified by quaternionic parameters h,b,p,c_i,c0.
    Sections 2.3-2.4 derive the 22-dimensional parameter space using Schur-type arguments and the decomposition of the symmetric square of the spinor.
  • standard math Hochschild-Serre spectral sequence gives H^2(s) isomorphic to H^2(s,r) for central extensions relative to the rotational subalgebra.
    Section 3.4 invokes reference [17] to reduce central extension computations to r-invariant 2-cochains.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Kinematical superspaces." pith.science (2026). https://pith.science/paper/D7PGS2LN

@misc{pith2026190811278,
  author       = {Pith},
  title        = {Pith review of: Kinematical superspaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D7PGS2LN}},
  note         = {Machine review of arXiv:1908.11278}
}
abstract

We classify $N{=}1$ $d=4$ kinematical and aristotelian Lie superalgebras with spatial isotropy, but not necessarily parity nor time-reversal invariance. Employing a quaternionic formalism which makes rotational covariance manifest and simplifies many of the calculations, we find a list of $43$ isomorphism classes of Lie superalgebras, some with parameters, whose (nontrivial) central extensions are also determined. We then classify their corresponding simply-connected homogeneous $(4|4)$-dimensional superspaces, resulting in a list of $27$ homogeneous superspaces, some with parameters, all of which are reductive. We determine the invariants of low rank and explore how these superspaces are related via geometric limits.

Figures

Figures reproduced from arXiv: 1908.11278 by the authors.

Figure 1
Figure 1. Homogeneous superspaces and their limits. (Numbers are hyperlinked to the corresponding superspaces in [PITH_FULL_IMAGE:figures/full_fig_p047_1.png] view at source ↗
Figure 2
Figure 2. Limits between superisable spacetimes limχ→∞ SM11χ = SM90, all other non-contracting limits between superspaces induce limits between the underlying spacetimes which arise from contractions of the kinematical Lie algebras: the limits |λ| → ∞ of SM5λ and SM6λ induce the contraction dSG → G, whereas the limits |λ| → ∞ of SM7γ,λ, SM8γ,λ and SM9λ induce the contractions dSGγ → G, where γ = 1 for SM9λ. 6. Conclusions In … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

32 extracted references · 27 canonical work pages

  1. [18]

    Possible Supersymmetric Kinematics

    C.-G. Huang and L. Li, “Possible Supersymmetric Kinema tics,” Chin. Phys. C39 (2015), no. 9, 093103, arXiv:1409.5498 [hep-th]

    work page Pith review arXiv 2015

  2. [1]

    Hence all the superspaces in Table 14 are reductive

    In [ 4] it is shown that all the homogeneous spacetimes in Table 1 are reductive with the exception of the carrollian light-co ne KINEMATICAL SUPERSP ACES 41 LC, which in any case does not admit any (4|4)-dimensional superisation. Hence all the superspaces in Table 14 are reductive. Let (s, h) be the super Lie pair associated with one of the homogeneous s...

    work page

  3. [2]

    We say that a super Lie pair (s, h) is effective if h does not contain an ideal of s. We observe that the condition of being geometrically realis able has nothing to do with supersym- metry , whereas the condition of being effective does take int o account the whole superalgebra. It is thus possible, and indeed we will see examples below , that a g eometric...

    work page

  4. [3]

    Killing superalgebras for Lorentzian four-manifolds

    P . de Medeiros, J. Figueroa-O’Farrill, and A. Santi, “Ki lling superalgebras for Lorentzian four-manifolds,” J. High Energy Physics 2016 (2016), no. 6, 1–50, arXiv:1605.00881 [hep-th]

    work page Pith review arXiv 2016

  5. [4]

    We start by classifying the supe r Lie pairs associated with the kinematical Lie superalgebras

    H/o.sc/m.sc/o.sc/g.sc/e.sc/n.sc/e.sc/o.sc/u.sc/s.sc /s.sc/u.sc/p.sc/e.sc/r.sc/s.sc/p.sc/a.sc/c.sc/e.sc/s.sc In this section, we classify the simply-connected (4|4)-dimensional homogeneous kinematical and aristotelian superspaces. We start by classifying the supe r Lie pairs associated with the kinematical Lie superalgebras. After determining the super Lie...

    work page

  6. [5]

    These are all the rotationally invariant tensors of rank 1

    Dually , there is a rotationally invariant line in m∗, which is the span of η. These are all the rotationally invariant tensors of rank 1. Let us now consider rank 2. As a representation of Sp(1), m ⊗ m has the following invariants. First of all, we haveH2, which is the only invariant featuring H. Another invariant is P2 := ∑ iPi ⊗Pi, which corresponds to...

    work page

  7. [6]

    internal

    L/i.sc/m.sc/i.sc/t.sc/s.sc /b.sc/e.sc/t.sc/w.sc/e.sc/e.sc/n.sc /s.sc/u.sc/p.sc/e.sc/r.sc/s.sc/p.sc/a.sc/c.sc/e.sc/s.sc In this section, we exhibit some limits between the superspa ces in Table 14 and interpret them in terms of contractions of the underlying Lie superalgebras. As we will show , a limit between two superspaces induces a lim it of the underl...

    work page

  8. [7]

    moduli space

    C/o.sc/n.sc/c.sc/l.sc/u.sc/s.sc/i.sc/o.sc/n.sc/s.sc In this paper, we have answered the question: What are the possible super-kinematics? by classifying (N=1d=4) kinematical Lie superalgebras and their corresponding su perspaces. The Lie superalgebras were classified by solving the Jacobi identities in a quaternionic reformulation, which made the computati...

    work page

Show all 32 references
  1. [8]

    mostly minus

    This map is surjective and, moreover, so(V)-equivariant becauseη, ⟨− , − ⟩ are so(V)- invariant and Clifford action is so(V)-equivariant. The Jacobi identity [[s,s],s] = 0 is trivially satisfied because [s,s] ∈ V andV acts trivially on S. This defines the Poincaré superalgebra s....

  2. [9]

    Extension of the Algeb ra of Poincare Group Generators and Violation of P Invarianc e,

    Yu. A. Golfand and E. P . Likhtman, “Extension of the Algeb ra of Poincare Group Generators and Violation of P Invarianc e,” JETP Lett. 13 (1971) 323–326. [Pisma Zh. Eksp. Teor. Fiz.13,452(1971)]

    1971

  3. [10]

    Nonlinear Realization of Supersymmetry in d e Sitter Space,

    B. Zumino, “Nonlinear Realization of Supersymmetry in d e Sitter Space,” Nucl. Phys. B127 (1977) 189–201

    1977

  4. [11]

    Spatially isot ropic homogeneous spacetimes,

    J. Figueroa-O’Farrill and S. Prohazka, “Spatially isot ropic homogeneous spacetimes,” JHEP 01 (2019) 229, arXiv:1809.01224 [hep-th]

    2019 arXiv

  5. [12]

    Ge ometry and BMS Lie algebras of spatially isotropic homogene ous spacetimes,

    J. Figueroa-O’Farrill, R. Grassie, and S. Prohazka, “Ge ometry and BMS Lie algebras of spatially isotropic homogene ous spacetimes,” arXiv:1905.00034 [hep-th]

    1905 arXiv

  6. [13]

    Possible kinematics,

    H. Bacry and J.-M. Lévy-Leblond, “Possible kinematics, ” J. Math. Phys. 9 (1968) 1605–1614

    1968

  7. [14]

    Classification of ten-dimensiona l kinematical groups with space isotropy ,

    H. Bacry and J. Nuyts, “Classification of ten-dimensiona l kinematical groups with space isotropy ,” J. Math. Phys. 27 (1986), no. 10, 2455–2457

    1986

  8. [15]

    Possible superkinematics,

    J. Rembieliński and W . T ybor, “Possible superkinematics,” Acta Phys. Polon. B 15 (1984), no. 7, 611–615

    1984

  9. [16]

    Kinematical supe ralgebras,

    V . Hussin, J. Negro, and M. A. del Olmo, “Kinematical supe ralgebras,” J. Phys. A 32 (1999), no. 27, 5097–5121

    1999

  10. [17]

    Kinematical superalgebras and Lie algebras of order 3,

    R. Campoamor-Stursberg and M. Rausch de Traubenberg, “ Kinematical superalgebras and Lie algebras of order 3,” J. Math. Phys. 49 (2008) 063506, arXiv:0801.2630 [hep-th]

    2008 arXiv

  11. [19]

    Galilean Supersymmetry,

    R. Puzalowski, “Galilean Supersymmetry,” Acta Phys. Austriaca 50 (1978) 45

    1978

  12. [20]

    Nonrelativistic Supersymmetry,

    F. Palumbo, “Nonrelativistic Supersymmetry,” in Proceedings of the International Conference on Recent Prog ress in Many Body Theories, held at International Center for Theoretical Phy sics, T rieste, October 2-7, 1978, p. 582. 1978

    1978

  13. [21]

    Nonrelativistic Supersymme try,

    T. E. Clark and S. T. Love, “Nonrelativistic Supersymme try,” Nucl. Phys. B231 (1984) 91–108

    1984

  14. [22]

    Nonrelativistic li mit of supersymmetric theories,

    J. A. de Azcárraga and D. Ginestar, “Nonrelativistic li mit of supersymmetric theories,” J. Math. Phys. 32 (1991) 3500–3508

    1991

  15. [23]

    Kinematical Lie algebras v ia deformation theory,

    J. M. Figueroa-O’Farrill, “Kinematical Lie algebras v ia deformation theory,” J. Math. Phys. 59 (2018), no. 6, 061701, arXiv:1711.06111 [hep-th]

    2018 arXiv

  16. [24]

    Cohomology of Lie algeb ras,

    G. Hochschild and J.-P . Serre, “Cohomology of Lie algeb ras,” Ann. of Math. (2) 57 (1953) 591–603

    1953

  17. [25]

    Superization of homogeneous spin manifolds and geometry of homogeneous supermanifolds,

    A. Santi, “Superization of homogeneous spin manifolds and geometry of homogeneous supermanifolds,” Abh. Math. Semin. Univ. Hambg. 80 (2010), no. 1, 87–144, arXiv:0905.3832 [math.DG]

    2010 arXiv

  18. [26]

    Graded manifolds, graded Lie theory , and p requantization,

    B. Kostant, “Graded manifolds, graded Lie theory , and p requantization,” in Differential geometrical methods in mathematical physics (Proc. Sympos., Univ. Bonn, Bonn, 1975) , pp. 177–306. Lecture Notes in Math., Vol. 570. Springer, Be rlin, 1977

    1975

  19. [27]

    The structure of supermanifolds,

    M. Batchelor, “The structure of supermanifolds,” T rans. Amer. Math. Soc.253 (1979) 329–338

    1979

  20. [28]

    Graded manifolds and graded Lie algebra s,

    J.-L. Koszul, “Graded manifolds and graded Lie algebra s,” in Proceedings of the international meeting on geometry and ph ysics (Florence, 1982), pp. 71–84. Pitagora, Bologna, 1983

    1982

  21. [29]

    Limits of thre e-dimensional gravity and metric kinematical Lie algebras in any dimension,

    J. Matulich, S. Prohazka, and J. Salzer, “Limits of thre e-dimensional gravity and metric kinematical Lie algebras in any dimension,” JHEP 07 (2019) 118, arXiv:1903.09165 [hep-th]

    2019 arXiv

  22. [30]

    Conformal Lie algebras via deformation theory,

    J. M. Figueroa-O’Farrill, “Conformal Lie algebras via deformation theory,” arXiv:1809.03603 [hep-th]

    arXiv

  23. [31]

    On Schrödinger superalgebras,

    C. Duval and P . A. Horvathy , “On Schrödinger superalgebras,” J. Math. Phys. 35 (1994) 2516–2538, arXiv:hep-th/0508079 [hep-th]

    1994 arXiv

  24. [32]

    K illing superalgebras for Lorentzian six-manifolds,

    P . de Medeiros, J. Figueroa-O’Farrill, and A. Santi, “K illing superalgebras for Lorentzian six-manifolds,” arXiv:1804.00319 [hep-th] . KINEMATICAL SUPERSP ACES 51 M/a.sc/x.sc/w.sc/e.sc/l.sc/l.sc I/n.sc/s.sc/t.sc/i.sc/t.sc/u.sc/t.sc/e.sc /a.sc/n.sc/d.sc S/c.sc/h.sc/o.sc/o.sc/...

    arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.