Expanding Lie (super)algebras through abelian semigroups

Eduardo Rodr\'iguez; Fernando Izaurieta; Patricio Salgado

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Expanding Lie (super)algebras through abelian semigroups

Fernando Izaurieta, Eduardo Rodríguez, Patricio Salgado

High Energy Physics - Theory · Submitted 21 Jun 2006 · 36 references

Pith's verdict: A clean generalization of Maurer–Cartan expansion whose real payoff is the invariant-tensor construction in section VII; the Z4 'new algebra' claim holds but needs one paragraph of argument the paper doesn't give.

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CitationEduardo Rodr\'iguez, Fernando Izaurieta, Patricio Salgado. "Expanding Lie (super)algebras through abelian semigroups." arXiv:hep-th/0606215 (2006). https://pith.science/paper/SRBOQZEW

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Editor's note

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The real content here isn't the semigroup product itself — Theorem III.1 is a short check that λ_αλ_β[T_A,T_B] closes — but what they build on it: Theorem IV.2 (resonant subalgebras from a subset partition that matches the subspace structure of g) and Theorem VI.1 (reduced algebras from a partition of those subsets). Those two turn 'guess an expansion and check closure' into a mechanical procedure, and they recover the Maurer–Cartan expansions of de Azcárraga et al. as the 0_S-reduction of the S_E case. The worked examples — M algebra, five-brane superalgebra, the D'Auria–Fré-like algebra — are checks against published results and they come out right. That's genuine external validation, not self-consistency dressed up as one. The part I'd actually cite is section VII. The supertrace in the adjoint representation collapses for reduced algebras (their eqs. 157–159 make this explicit), and Theorems VII.1 and VII.2 give a non-trace replacement built from the n-selector. That is a real bottleneck for Chern–Simons constructions and they solve it. Soft spots, in order of size. First, 'new ones arise from different choices' [abstract:s3] is a structural claim, not an isomorphism-class result; the paper never checks whether the Z4 algebra of section V.C is isomorphic to some MC expansion. As it happens the claim survives — the Z4 algebra carries a Z4 grading with {Q′,Q′} ≠ 0 sitting at 3+3 = 2 mod 4, while an S_E expansion is graded additively with truncation, so no grading-preserving map exists — but the paper doesn't make that argument, and a referee should ask for one paragraph spelling it out. Second, minor: in the proof of Theorem VII.2 the line K^j_{i1…(N+1)…in} = 0 is really δ^j_{N+1}; the conclusion is fine because α_{N+1} = 0, but the step as written is imprecise. Third, the D'Auria–Fré-like algebra genuinely differs from the originals — footnote 34 says so and points to Ref. [8] — so the '-like' is honest, not a dodge. Nothing here is load-bearing-broken. The proofs are short enough to check by hand, and the selector identities check out. Who should read it: anyone in the expansion/Chern–Simons supergravity program, and anyone who needs invariant tensors for a non-semisimple algebra. Send it out.

Referee report

Significance

The construction is elementary, generator-level, and checkable by hand, and it strictly generalizes the Maurer–Cartan-form expansion: the recovery of the M algebra (§V.A) and of the five-brane superalgebra (§VI.B) are nontrivial checks against published results by other authors, and the Z_4 example (§V.C) exhibits structure, notably a nonvanishing {Q',Q'} at eq. (118), that the S_E truncation cannot produce. The invariant-tensor theorems of §VII address a genuine bottleneck for Chern–Simons and transgression Lagrangians in 11 dimensions, where the supertrace in the adjoint representation degenerates (eqs. (157)–(159)). The paper is a useful toolbox for the d = 11 supergravity program, and the proofs are short enough that a reader can verify them without auxiliary material.

Major objections (1)

[§V.C and abstract [abstract:s3]] The abstract's claim that 'new ones arise from different choices' rests entirely on the Z_4 example, and the manuscript supports it only by the observation that Z_4 is not S_E ('Since this example uses a semigroup different from S_E, the algebra obtained does not correspond to a Maurer–Cartan forms power-series expansion'). That is a statement about the procedure, not about the algebra. The claim is in fact defensible on structural grounds that the paper does not state: the Z_4 algebra carries a Z_4 grading with 4·1 = 0, so {Q',Q'} at eq. (118) is nonzero, whereas any S_E expansion with truncation N ≤ 5 would place a level-3 × level-3 bracket at level 6 and truncate it to zero; more generally no grading-preserving map from Z_4 to {0,...,N} exists, since 4φ(1) = φ(0) = 0 forces φ(1) = 0. I would ask the authors to add two or three sentences making this argument explicit (or to soften 'new' to 'not obtainable by the S_E procedure as defined'), so that the headline claim does not rest on a definitional observation alone.

Minor objections (10)

[§VII, paragraph after eq. (152)] The statement that the resonant-subalgebra components 'never vanish (provided that ∀γ, α_γ ≠ 0)' is too strong: the components ⟨T_{a_{p1}} ··· T_{a_{pn}}⟩ of the original g-invariant tensor can vanish independently of the α_γ, as happens for an odd number of osp(32|1) generators. Please qualify the statement by the nonvanishing of the corresponding g-tensor components.
[§VII, eq. (150)] The assertion that the multi-selector ansatz (150) 'is always possible to reduce' to (146) using associativity and closure is stated without an argument. Either give the two-line derivation or drop the claim, since Theorem VII.1 already carries the weight of the section.
[Appendix A] The derivation of condition (A3) analyzes p = 0 to obtain (A10) and then jumps from the p = 1 instance of (A13) to the general solution, without explaining why p = 1 suffices for all p. Since the appendix is offered as a re-derivation of Theorem 3 of Ref. [1], a sentence spelling out the induction (or the monotonicity argument in p) would make it self-contained.
[§V.C, first paragraph] The remark that 'Z_3 seems to have no resonant partition' is an unproved empirical observation. Either give a short check or phrase it explicitly as an observation that the authors have not settled.
[Notation, §V] The symbol Z is used for three different objects: the cyclic group Z_4, the five-index generator Z_{abcde}, and the two-index generator Z^{ab} of the M algebra. A notational distinction (e.g., a different font for the group) would help the reader follow §V.C, where all three appear in the same paragraph.
[§VI.A, text before Fig. 7] Typo: 'the generalized IW contraction doesnot correspond to a resonant subalgebra' should read 'does not'.
[References] Several author names are garbled: 'P. K. Towsend' in Refs. [3] and [4] should be 'Townsend', and 'O. Varel a' in Ref. [21] should be 'Varela'. Ref. [29] also has a broken arXiv identifier ('hep-th/051225 5').
[§V.B, footnote 34] The acknowledgment that the algebra differs from the original D'Auria–Fré superalgebras and belongs instead to the family of Ref. [8] is important for the reader's assessment of the 'like' in the title of the subsection; consider moving this from a footnote into the main text.
[Conclusions] The statement that the toolbox 'has been used to construct a Lagrangian for the M algebra in 11 dimensions [14]' rests on a companion preprint that is not yet published. Since the present manuscript does not exhibit the Lagrangian, please mark Ref. [14] as forthcoming or rephrase so the claim is clearly attributed to work in preparation.
[Figs. 1–3] The captions alternate between 'gray region' and 'shaded area' for what appear to be the same or overlapping regions. Please make the shading conventions uniform and confirm that the three panels remain distinguishable in black-and-white print.

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The paper

The paper claims

Multiplying a Lie algebra by an abelian semigroup produces another Lie algebra, and the semigroup's multiplication table alone controls which subalgebras, reductions, and invariant tensors come out.

The reading

The paper asks whether the known 'expansion' tricks that turn one Lie algebra into a bigger one can be replaced by a single, purely algebraic construction. Its answer is yes: take any finite abelian semigroup $S$ and any Lie (super)algebra $g$, form the direct product, and dress the original structure constants with the semigroup's multiplication table. The authors then show that this one move reproduces the Maurer–Cartan power-series expansions of earlier work as a special case, and that choosing other semigroups gives algebras those methods could not reach. They also give general conditions for carving out subalgebras and quotients, and — the part with the clearest physical payoff — explicit formulas for invariant tensors, the ingredient needed to write Chern–Simons and transgression Lagrangians. For a reader interested in eleven-dimensional supergravity, the concrete draw is that the M algebra and a D'Auria–Fré-like superalgebra both fall out of the orthosymplectic algebra $osp(32|1)$ by this route.

What carries the argument

The $n$-selector. For semigroup elements $a_1, ..., a_n$ it equals 1 when their product is a given element and 0 otherwise; the two-selector version is just the multiplication table written as a matrix. This single object does three jobs at once: it multiplies structure constants in the expanded bracket, it decides which products of subsets stay inside which other subsets (the 'resonance condition' $S_p S_q$ contained in the intersection of the $S_r$ required by $[V_p, V_q]$), and it slots into the invariant-tensor formula $<T_{(A_1,a_1)} ... T_{(A_n,a_n)}> = c_c K^{c}_{a_1 ... a_n} <T_{A_1} ... T_{A_n}>$. The last formula is the paper's practical contribution: it manufactures nonvanishing invariant tensors that the ordinary supertrace cannot supply, because in the $S_E$ case the supertrace collapses to a single component.

Extended reading notes

Core claim

The central claim is Theorem III.1: if $S$ is an abelian semigroup with two-selector $K^{~}_{}$, i.e. the array that records which product of two semigroup elements equals which third, and $g$ is a Lie (super)algebra with structure constants $C^{C}_{AB}$, then the bracket $[T_{(A,a)}, T_{(B,b)}] = K^{c}_{ab} C^{C}_{AB} T_{(C,c)}$ closes and satisfies Jacobi. Jacobi holds because the selectors are associative and symmetric, so the semigroup never disturbs the algebra's own identity. Everything else in the paper is built on that: a 'resonant' partition of $S$ into subsets whose products land inside the subsets dictated by the grading of $g$ gives a subalgebra of the expanded algebra (Theorem IV.2); a splitting of those subsets into a kept part and a discarded part gives a reduced algebra (Theorem VI.1); and the $n$-selector, the obvious generalization of the two-selector, gives invariant tensors for both the expanded algebra and its reduction (Theorems VII.1 and VII.2). The authors then show that the Maurer–Cartan expansion of de Azcárraga and collaborators is exactly the case where $S$ is the truncated power semigroup $S_E$ followed by setting the zero element's generators to zero, so the older method is a special case rather than a rival.

Load-bearing premise

The claim that different semigroup choices give genuinely new algebras rests on the observation that $Z_4$ is not the truncated power semigroup $S_E$; the paper does not check isomorphism classes, so 'new' here means 'not produced by the earlier procedure as defined' rather than 'provably non-isomorphic to every expanded algebra'.

Editorial extensions

If this is right

The Maurer–Cartan power-series expansion of de Azcárraga et al. is recovered in full, including the truncation conditions, as a zero-element reduction of an $S_E$-expansion, so the older results are subsumed rather than replaced.
The M algebra, with its 583 bosonic generators, is obtained from $osp(32|1)$ by an $S_E$-expansion with truncation order 2 followed by a reduction; the D'Auria–Fré-like superalgebra with an extra fermionic generator follows from order 3.
Choosing the cyclic group $Z_4$ instead of $S_E$ gives a superalgebra with two fermionic charges obeying the same anticommutator and two sets of 'AdS-boost' generators sharing one Lorentz algebra, which the Maurer–Cartan procedure as defined does not produce.
The five-brane superalgebra is recovered as a reduced algebra of a resonant subalgebra, and the generalized Iñönü–Wigner contraction is identified as a particular reduction rather than a subalgebra.
Because invariant tensors are available for resonant subalgebras and their zero-element reductions, Chern–Simons and transgression Lagrangians can be written for these algebras in odd dimensions, which is the stated route to an eleven-dimensional supergravity Lagrangian for the M algebra.

Reading between the lines

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

The construction suggests a general classification programme the authors only gesture at: enumerate finite abelian semigroups up to some small order, compute their resonant partitions against a fixed grading of $osp(32|1)$, and you get a finite catalogue of eleven-dimensional superalgebras with known invariant tensors. That is a concrete, checkable enumeration rather than an open-ended search.
The reduction theorems look like a clean algebraic home for contraction and deformation. If every generalized Iñönü–Wigner contraction is a reduction of a resonant subalgebra, then the space of algebras reachable from a fixed $g$ has a partial order by semigroup refinement, and questions about which contractions are deformable back become questions about that order.
The invariant-tensor formula depends only on the $n$-selector, so it applies verbatim to any semigroup, including non-$S_E$ ones. A natural test is whether the $Z_4$ superalgebra's selector-built tensor gives a Chern–Simons action that is not equivalent to any $S_E$-derived one; if it is equivalent, the 'new algebra' claim weakens considerably.
The paper's own remark that abelianity is essential while finiteness and discreteness might be dropped points at a continuum version: replace the semigroup by a commutative monoid of functions or a commutative algebra, and the selector becomes a kernel. Whether the Jacobi identity survives that limit is the obvious next question.

Original abstract

We propose an outgrowth of the expansion method introduced by de Azcarraga et al. [Nucl. Phys. B 662 (2003) 185]. The basic idea consists in considering the direct product between an abelian semigroup S and a Lie algebra g. General conditions under which relevant subalgebras can systematically be extracted from S \times g are given. We show how, for a particular choice of semigroup S, the known cases of expanded algebras can be reobtained, while new ones arise from different choices. Concrete examples, including the M algebra and a D'Auria-Fre-like Superalgebra, are considered. Finally, we find explicit, non-trace invariant tensors for these S-expanded algebras, which are essential ingredients in, e.g., the formulation of Supergravity theories in arbitrary space-time dimensions.

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We propose an outgrowth of the expansion method introduced by de Azcarraga et al. [Nucl. Phys. B 662 (2003) 185]. The basic idea consists in considering the direct product between an abelian semigroup S and a Lie algebra g. General conditions under which relevant subalgebras can systematically be extracted from S \times g are given. We show how, for a particular choice of semigroup S, the known cases of expanded algebras can be reobtained, while new ones arise from different choices. Concrete examples, including the M algebra and a D'Auria-Fre-like Superalgebra, are considered. Finally, we find explicit, non-trace invariant tensors for these S-expanded algebras, which are essential ingredients in, e.g., the formulation of Supergravity theories in arbitrary space-time dimensions.

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s0	We propose an outgrowth of the expansion method introduced by de Azcarraga et al. [Nucl. Phys. B 662 (2003) 185].	background 0.63	0.71	0.04	asserted 0.95
s1	The basic idea consists in considering the direct product between an abelian semigroup S and a Lie algebra g.	method 0.90	0.31	0.06	asserted 1.00
s2	General conditions under which relevant subalgebras can systematically be extracted from S \times g are given.	contribution 0.65	0.84	0.16	asserted 1.00
s3	We show how, for a particular choice of semigroup S, the known cases of expanded algebras can be reobtained, while new ones arise from different choices.	result 0.77	0.93	0.29	asserted 1.00
s4	Concrete examples, including the M algebra and a D'Auria-Fre-like Superalgebra, are considered.	contribution 0.81	0.81	0.07	asserted 0.96
s5	Finally, we find explicit, non-trace invariant tensors for these S-expanded algebras, which are essential ingredients in, e.g., the formulation of Supergravity theories in arbitrary space-time dimensions.	result 0.92	0.95	0.41	asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.94 · evidence from proof or derivation 1.00.

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Figures

**Figure 3.** (a) shows the resonant subalgebra for the case of superalgebras, and

**Figure 4.** FIG. 4: The M algebra as an

**Figure 5.** FIG. 5: A D’Auria–Fr´e-like Superalgebra regarded here as a

Showing 4 of the paper's 8 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-th/0606215 by the authors.

References

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The asymptotic symmetry algebra of three-dimensional Maxwell Chern-Simons gravity with spin-3 fields is a new nonlinear algebra, hs3max-bms3, which is also obtained as the flat limit of three copies of the W3 algebra.
Newton-Hooke/Carrollian expansions of (A)dS and Chern-Simons gravity
hep-th 2019-12 accept novelty 5.0 of 10

Expanding the (anti-)de Sitter algebra produces infinite families of Newton-Hooke and Carroll symmetry algebras with a cosmological constant, whose three-dimensional Chern-Simons actions reproduce extended Newton-Hook...
Non-Relativistic Gravity Theory based on an Enlargement of the Extended Bargmann Algebra
hep-th 2019-05 conditional novelty 5.0 of 10

A new non-relativistic three-dimensional Chern-Simons gravity theory, built from an enlarged Extended Bargmann algebra, reduces to Maxwellian Exotic Bargmann gravity in the flat limit.
Lie Algebra Expansions and Actions for Non-Relativistic Gravity
hep-th 2019-04 conditional novelty 5.0 of 10

A single algebraic expansion of the Poincaré algebra, truncated at a chosen order, reproduces the algebras and actions of several non-relativistic gravity theories, with a derived rule for which truncations admit an i...
On the Maxwell supergravity and flat limit in 2+1 dimensions
hep-th 2018-06 conditional novelty 5.0 of 10

A minimal three-dimensional Maxwell Chern-Simons supergravity action is constructed from an S-expansion of the super-Lorentz algebra and recovered as the zero-cosmological-constant limit of a minimal AdS-Lorentz supergravity.
Asymptotic symmetries of three-dimensional Chern-Simons gravity for the Maxwell algebra
hep-th 2018-05 conditional novelty 5.0 of 10

The asymptotic symmetry of 3D Chern-Simons gravity with the Maxwell algebra is a deformed bms3 algebra with three central charges, the third proportional to the Maxwell coupling.
Generalized Chern-Simons higher-spin gravity theories in three dimensions
hep-th 2017-12 conditional novelty 5.0 of 10

Semigroup expansion of sl(3,R) produces spin-3 extensions of the Maxwell and AdS-Lorentz algebras, yielding Chern-Simons gravity actions including a flat-space model with vanishing cosmological constant.
Pure Lovelock gravity and Chern-Simons theory
hep-th 2016-03 conditional novelty 5.0 of 10

Pure Lovelock gravity actions in odd dimensions can be recovered as a limit of Chern-Simons actions for an expanded AdS algebra, though the full dynamics require extra assumptions imposed by hand.
Topological gravity and transgression holography
hep-th 2014-01 conditional novelty 5.0 of 10

Even-dimensional Poincaré topological gravity actions of the form curvature-to-the-n times an adjoint scalar arise as the boundary term of a transgression field theory in one higher dimension.
The Extended Cartan Homotopy Formula and a Subspace Separation Method for Chern--Simons Theory
hep-th 2006-03 conditional novelty 5.0 of 10

A triangle identity for transgression forms gives a systematic way to separate Chern-Simons Lagrangians into gauge-algebra subspace pieces and boundary terms, illustrated in 5D supergravity.
Minimal AdS-Lorentz supergravity in three-dimensions
hep-th 2014-01 conditional novelty 4.5 of 10

A three-dimensional Chern-Simons supergravity with N=1 AdS-Lorentz symmetry is constructed from a semigroup expansion of the AdS superalgebra, with invariant tensors, Casimir operators, and BTZ-like Killing spinors wo...
3D Carrollian gravity from 2D Euclidean symmetry
hep-th 2024-12 conditional novelty 4.0 of 10

Post-Carroll-Newtonian Chern-Simons gravities are systematically obtained by semigroup-expanding 2D Euclidean B_k algebras, recovering known Carrollian models as subcases.
Extended $D=3$ Bargmann supergravity from a Lie algebra expansion
hep-th 2019-04 conditional novelty 4.0 of 10

Expanding D=3 N=2 superPoincaré in a formal parameter reproduces the extended Bargmann superalgebra and its Chern-Simons action, matching a 2016 result by a simpler route.
Chern-Simons Supergravity in D=3 and Maxwell superalgebra
hep-th 2015-07 conditional novelty 4.0 of 10

A D=3 Chern-Simons supergravity action is built from the minimal Maxwell superalgebra, with the invariant tensor and gauge transformations written out explicitly.
N=1 Supergravity and Maxwell superalgebras
hep-th 2014-07 conditional novelty 4.0 of 10

The N=1, D=4 pure supergravity action (Einstein-Hilbert plus Rarita-Schwinger) is recovered, modulo boundary terms, as the alpha4 piece of a MacDowell-Mansouri-like action built from the minimal Maxwell superalgebra s...
Maxwell Superalgebras and Abelian Semigroup Expansion
hep-th 2014-05 conditional novelty 4.0 of 10

The Abelian semigroup expansion of osp(4|N) reproduces known D=4 Maxwell superalgebras and generates new families sM_{m+2} and sM^(N)_{m+2}.
Standard General Relativity from Chern-Simons Gravity
hep-th 2009-05 conditional novelty 4.0 of 10

By S-expanding the anti-de Sitter algebra with a finite abelian semigroup, the authors build a five-dimensional Chern-Simons Lagrangian whose coupling-constant-zero limit is exactly the Einstein-Hilbert action.
Review on Non-Relativistic Gravity
gr-qc 2022-12 accept novelty 3.0 of 10

Expanding general relativity in powers of 1/c yields a 'type II' Newton-Cartan geometry whose action reproduces Newton's Poisson equation off shell and whose torsionful solutions describe strong gravitational fields.
A non-lorentzian primer
hep-th 2022-06 accept novelty 3.0 of 10

A structured primer that assembles the kinematical algebras, homogeneous spacetimes, particle actions, gravity theories and limiting field theories of non-lorentzian physics into a single self-contained account.

How later papers used it

73 later papers cite it

Evidence: substantial. 40 of the citing papers share no author with it, and 17 of those carry a confident label from Pith's citation classifier.

Pith read 37 of the 276 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 2 used its method or tool. 5 more papers by the same authors are listed: 5 used its method or tool.

Uses its method or tool 7 of 35 papers with this label · 5 by the same authors

Combined this paper's S-expansion method with the Chern-Simons formalism to construct three-dimensional non-relativistic supergravity actions.

Non-Lorentzian Supergravity and Kinematical Superalgebras (2024) · 1.00 · paraphrased: the licence does not allow quoting
Since the direct product of $S$ with a Lie algebra is again a Lie algebra [ref] , one can define the expanded Nappi-Witten algebra [ref] S×={ S⊗{ ,_a,} }. whose generators are obtained by considering the tensor product of the semigroup elements and the generators of the $$ algebra, i.e. ^(i)_a=λ_i ⊗_a , ^(i)=λ_i⊗ , ^(i) =λ_i ⊗ , The commutation relations of the expanded Nappi-Witten algebra are [ ^(i),^(j)_a ] = ϵ_a^ b ^(i♢j)_b , [ ^(i)_a,^(j)_b ] = -ϵ_ab^(i♢j) .

Non-relativistic spin-3 symmetries in 2+1 dimensions from expanded/extended Nappi-Witten algebras (2022) · Expansions of higher-spin Nappi-Witten algebras · 0.82

By the same authors

Followed this paper's definitions to show that a partition satisfies relations and that a reduced algebra corresponds to it.

Maxwell Superalgebras and Abelian Semigroup Expansion (2014) (same authors) · 0.81 · paraphrased: the licence does not allow quoting
Applied this paper's S-expansion method to construct a non-relativistic Maxwell-Bloch superalgebra and its non-degenerate invariant tensor.

Non-Relativistic Chern-Simons Supergravity with Torsion (2026) (same authors) · 0.97 · paraphrased: the licence does not allow quoting
Followed this paper's definitions to consider the S-expansion of the Lie algebra so(4,2) using a finite abelian semigroup.

Standard General Relativity from Chern-Simons Gravity (2009) (same authors) · 0.99 · paraphrased: the licence does not allow quoting
Obtained the three-dimensional AdS-Carroll algebra by means of this paper's S-expansion method.

3D Carrollian gravity from 2D Euclidean symmetry (2025) (same authors) · 0.99 · paraphrased: the licence does not allow quoting
Obtained its new non-relativistic superalgebras from two relativistic N=2 AdS-L superalgebras considering this paper's semigroup expansion procedure.

Three-dimensional non-relativistic extended supergravity with cosmological constant (2020) (same authors) · 0.97 · paraphrased: the licence does not allow quoting

How the literature uses it

60 arXiv papers cite this in 242 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 138 sentences without a confident label

Builds on it 2 papers

[ref] , can be generalized so as to obtain expanded higher-order Lie algebras.

S-Expansion of Higher-Order Lie Algebras · Comments · 0.93
In this work, we present a supersymmetric extension of the three-dimensional exotic Newtonian CS gravity introduced in [ref] by applying the semigroup expansion ($S$-expansion) [ref] to the $𝒩=2$ AdS supergravity theory.

Three-dimensional exotic Newtonian supergravity theory with cosmological constant · Introduction · 0.87

Uses its method or tool 30 papers

In the following sections, we will apply the $S$-expansion method [ref] to the super Nappi-Witten algebra in order to obtain different NR superalgebras.

Non-relativistic three-dimensional supergravity theories and semigroup expansion method · Nappi-Witten superalgebra and Chern-Simons action · 1.00
By applying the S-expansion method [ref] , and inspired by the results obtained in [ref] , we construct the corresponding NR MB superalgebra together with its non-degenerate invariant tensor, which constitutes the essential ingredient for defining a well-defined CS supergravity action.

Non-Relativistic Chern-Simons Supergravity with Torsion · Non-relativistic regime of the Mielke-Baekler supergravity · 0.97
It is the purpose of this Letter to show that the SSEP algebra $𝔰𝔬 ( D-1,1 ) ⊕𝔰𝔬 ( D-1,2 )$ can be obtained from the AdS algebra $𝔰𝔬 ( D-1,2 )$ via the $S$-expansion procedure with an appropriate semigroup $S$ [ref] .

A generalized action for $\left( 2 + 1 \right)$-dimensional Chern--Simons gravity · Introduction · 0.85

Compares with it 3 papers

In this work we have found a relation between two procedures for the expansion of Lie algebras, namely a relation between the so-called `power series expansion method' of Ref. [ref] and the `$S$-expansion Procedure' of Ref. [ref] .

Dual Formulation of the Lie Algebra S-expansion Procedure · Comments and Possible Developments · 0.99
The present work shows the difference between the possibilities of the $S$-expansion procedure [ref] (using semigroups) and the MC forms expansion (using a parameter).

Standard General Relativity from Chern-Simons Gravity · Comments and Possible Developments · 0.91
In a sense, Galilean contraction procedures provide an opposite construction to semigroup expansions which also give rise to Galilean subalgebras [ref] .

Asymmetric Galilean conformal algebras · Discussion · 0.88

Cites it as background 18 papers

In Refs. [ref] , [ref] , [ref] was proposed a natural outgrowth of power series expansion method, which is based on combining the structure constant of the algebra $𝔤$ with the inner law of a semigroup $S$ in order to define the Lie bracket of a new $S$-expanded algebra.

Einstein-Hilbert action with cosmological term from Chern-Simons gravity · Introduction · 0.87
An alternative expansion procedure to the method of power series expansion is the $S$-expansion method [ref] .

Dual Formulation of the Lie Algebra S-expansion Procedure · Introduction · 0.80
Some time ago was shown that the standard, five-dimensional General Relativity can be obtained from Chern-Simons gravity theory for a certain Lie algebra $𝔅$ [ref] , which was obtained from the AdS algebra and a particular semigroup $S$ by means of the S-expansion procedure introduced in Refs. [ref] .

Stellar equilibrium in Einstein-Chern-Simons gravity · Introduction · 0.95

Discussion

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