{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:11:45Z","timestamp":1759335105534,"version":"build-2065373602"},"reference-count":94,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T00:00:00Z","timestamp":1759104000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Mecesup PhD grant"},{"name":"T\u00e9rmino de tesis grant from CONICYT"},{"DOI":"10.13039\/501100002850","name":"Fondecyt","doi-asserted-by":"publisher","award":["1050512","3130445"],"award-info":[{"award-number":["1050512","3130445"]}],"id":[{"id":"10.13039\/501100002850","id-type":"DOI","asserted-by":"publisher"}]},{"name":"DIUBB","award":["102609","GI 153209\/C"],"award-info":[{"award-number":["102609","GI 153209\/C"]}]},{"name":"Becas-Chile postdoctoral grant"},{"name":"Agencia Nacional de Investigaci\u00f3n y Desarrollo (ANID), Chile, v\u00eda Fondecyt Regular","award":["1231133"],"award-info":[{"award-number":["1231133"]}]},{"DOI":"10.13039\/501100004895","name":"European Social Fund","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100004895","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["www.mdpi.com"],"crossmark-restriction":true},"short-container-title":["Axioms"],"abstract":"<jats:p>The contraction method is a procedure that allows to establish non-trivial relations between Lie algebras and has had successful applications in both mathematics and theoretical physics. This work deals with generalizations of the contraction procedure, with a main focus on the so-called S-expansion method, as it includes most of the other generalized contractions. Basically, the S-expansion combines a Lie algebra G with a finite abelian semigroup S in order to define new S-expanded algebras. After giving a description of the main ingredients used in this paper, we present a Java library that automates the S-expansion procedure. With this computational tool, we are able to represent Lie algebras and semigroups, so we can perform S-expansions of Lie algebras using arbitrary semigroups. We explain how the library methods have been constructed and how they work; then, we give a set of example programs aimed to solve different problems. They are presented so that any user can easily modify them to perform their own calculations, without necessarily being an expert in Java. Finally, some comments about further developments and possible new applications are made.<\/jats:p>","DOI":"10.3390\/axioms14100735","type":"journal-article","created":{"date-parts":[[2025,9,30]],"date-time":"2025-09-30T09:12:23Z","timestamp":1759223543000},"page":"735","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Java Library to Perform S-Expansions of Lie Algebras"],"prefix":"10.3390","volume":"14","author":[{"given":"Carlos","family":"Inostroza","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica, Universidad de Concepci\u00f3n, Casilla 160-C, Concepci\u00f3n 4070386, Chile"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9489-3134","authenticated-orcid":false,"given":"Igor","family":"Kondrashuk","sequence":"additional","affiliation":[{"name":"Grupo de Matem\u00e1tica Aplicada & Centro de Ciencias Exactas & Departamento de Ciencias B\u00e1sicas, Univerdidad del B\u00edo-B\u00edo, Campus Fernando May, Av. Andres Bello 720, Casilla 447, Chill\u00e1n 3780227, Chile"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4548-7767","authenticated-orcid":false,"given":"Nelson","family":"Merino","sequence":"additional","affiliation":[{"name":"Instituto de Ciencias Exactas y Naturales, Universidad Arturo Prat, Avenida Playa Brava 3256, Iquique 1111346, Chile"},{"name":"Facultad de Ciencias, Universidad Arturo Prat, Avenida Arturo Prat Chac\u00f3n 2120, Iquique 1110939, Chile"}]},{"given":"Felip","family":"Nadal","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica Corpuscular (IFIC), Edificio Institutos de Investigaci\u00f3n, c\/ Catedr\u00e1tico Jos\u00e9 Beltr\u00e1n, 2., E-46980 Paterna, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1215\/S0012-7094-51-01817-0","article-title":"A class of operator algebras which are determined by groups","volume":"18","author":"Segal","year":"1951","journal-title":"Duke Math. J."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"510","DOI":"10.1073\/pnas.39.6.510","article-title":"On the contraction of groups and their representations","volume":"39","author":"Wigner","year":"1953","journal-title":"Proc. Nat. Acad. Sci. USA"},{"unstructured":"G\u00fcrseyed, F. (1964). Contractions of Lie groups and their representations. Group Theoretical Concepts in Elementary Particle Physics, Gordon and Breach.","key":"ref_3"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1063\/1.1724208","article-title":"Contractions of Lie groups","volume":"2","author":"Saletan","year":"1961","journal-title":"J. Math. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"59","DOI":"10.2307\/1970484","article-title":"On the deformations of rings and algebras","volume":"79","author":"Gerstenhaber","year":"1964","journal-title":"Ann. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/S0002-9904-1966-11401-5","article-title":"Cohomology and deformations in graded Lie algebras","volume":"72","author":"Nijenhuis","year":"1966","journal-title":"Bull. A Math. Soc."},{"key":"ref_7","first-page":"89","article-title":"Deformations of Lie algebra structures","volume":"171","author":"Nijenhuis","year":"1967","journal-title":"J. Math. Mech."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"339","DOI":"10.2140\/pjm.1967.22.339","article-title":"On the rigidity of semi-direct products of Lie algebras","volume":"22","author":"Richardson","year":"1967","journal-title":"Pac. J. Math."},{"doi-asserted-by":"crossref","unstructured":"Barut, A.O., and Ratzka, R. (1986). Theory of Group Representations and Applications, World Scientific.","key":"ref_9","DOI":"10.1142\/0352"},{"doi-asserted-by":"crossref","unstructured":"Gilmore, R. (1974). Lie Groups, Lie Algebras, and Some of their Applications, Wiley-Interscience.","key":"ref_10","DOI":"10.1063\/1.3128987"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4519","DOI":"10.1063\/1.530905","article-title":"Contractions of Lie algebras: Generalized Inonu-Wigner contractions versus graded contractions","volume":"36","year":"1995","journal-title":"J. Math. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2028","DOI":"10.1063\/1.529222","article-title":"Contraction of Lie algebra representations","volume":"32","year":"1991","journal-title":"J. Math. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1505","DOI":"10.1142\/S0129055X00000605","article-title":"Contractions, generalized In\u00f6n\u00fc and Wigner contractions and deformations of finite-dimensional Lie algebras","volume":"12","year":"2000","journal-title":"Rev. Math. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"853","DOI":"10.1143\/PTP.109.853","article-title":"Wess-Zumino term for the AdS superstring and generalized Inonu-Wigner contraction","volume":"109","author":"Hatsuda","year":"2003","journal-title":"Prog. Theor. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/S0550-3213(03)00342-0","article-title":"Generating Lie and gauge free differential (super)algebras by expanding Maurer-Cartan forms and Chern-Simons supergravity","volume":"662","author":"Izquierdo","year":"2003","journal-title":"Nucl. Phys. B"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1375","DOI":"10.1088\/0264-9381\/21\/10\/010","article-title":"Extensions, expansions, Lie algebra cohomology and enlarged superspaces","volume":"21","author":"Izquierdo","year":"2004","journal-title":"Class. Quant. Grav."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2738","DOI":"10.1007\/s10773-007-9385-3","article-title":"Expansions of algebras and superalgebras and some applications","volume":"46","author":"Izquierdo","year":"2007","journal-title":"Int. J. Theor. Phys."},{"doi-asserted-by":"crossref","unstructured":"Nakahara, M. (1990). Geometry, Topology and Physics, Institute of Physics Publishing.","key":"ref_18","DOI":"10.1887\/0750306068"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"123512","DOI":"10.1063\/1.2390659","article-title":"Expanding Lie (super)algebras through Abelian semigroups","volume":"47","author":"Izaurieta","year":"2006","journal-title":"J. Math. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"255207","DOI":"10.1088\/1751-8113\/45\/25\/255207","article-title":"A generalized action for (2 + 1)-dimensional Chern-Simons gravity","volume":"45","author":"Fierro","year":"2012","journal-title":"J. Phys. A"},{"unstructured":"Zaitsev, G.A. Group-Invariant Study of the Sets of Limiting Geometrie and Special Lie Subalgebras. In Talk thesises of All-USSR Geometric Conference; Kiev, USSR, 1961. pp. 1\u201348. (In Russian).","key":"ref_21"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1007\/BF02874051","article-title":"Contraction of group representations II","volume":"65","author":"Celeghini","year":"1981","journal-title":"Nuovo Cimento B"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"123515","DOI":"10.1063\/1.2400834","article-title":"Contractions of low-dimensional Lie algebras","volume":"47","author":"Nesterenko","year":"2006","journal-title":"J. Math. Phys."},{"unstructured":"Vaneeva, O.O. (2012, January 17\u201321). S-Expansions of Three-Dimensional Lie Algebras. Proceedings of the 6th International Workshop on Group Analysis of Differential Equations and Integrable Systems, Protaras, Cyprus.","key":"ref_24"},{"unstructured":"Zanelli, J. (2025, August 23). Lecture Notes on Chern-Simons (Super-)Gravities, Available online: https:\/\/arxiv.org\/abs\/hep-th\/0502193.","key":"ref_25"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"675","DOI":"10.1140\/epjc\/s10052-008-0540-7","article-title":"Eleven-dimensional gauge theory for the M algebra as an Abelian semigroup expansion of osp(32|1)","volume":"54","author":"Izaurieta","year":"2008","journal-title":"Eur. Phys. J. C"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"073511","DOI":"10.1063\/1.3171923","article-title":"Dual Formulation of the Lie Algebra S-expansion Procedure","volume":"50","author":"Izaurieta","year":"2009","journal-title":"J. Math. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"278","DOI":"10.1016\/j.physletb.2006.07.058","article-title":"Lie-algebra expansions, Chern-Simons theories and the Einstein-Hilbert Lagrangian","volume":"640","author":"Edelstein","year":"2006","journal-title":"Phys. Lett. B"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/j.physletb.2009.06.017","article-title":"Standard General Relativity from Chern-Simons Gravity","volume":"678","author":"Izaurieta","year":"2009","journal-title":"Phys. Lett. B"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1016\/j.physletb.2013.07.019","article-title":"Even-dimensional General Relativity from Born-Infeld gravity","volume":"725","author":"Concha","year":"2013","journal-title":"Phys. Lett. B"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2741","DOI":"10.1140\/epjc\/s10052-014-2741-6","article-title":"Chern-Simons and Born-Infeld gravity theories and Maxwell algebras type","volume":"74","author":"Concha","year":"2014","journal-title":"Eur. Phys. J. C"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1016\/j.physletb.2015.01.038","article-title":"Generalized Poincar\u00e9 algebras and Lovelock-Cartan gravity theory","volume":"742","author":"Concha","year":"2015","journal-title":"Phys. Lett. B"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"L35","DOI":"10.1088\/0264-9381\/15\/5\/001","article-title":"Born-Infeld-Einstein Actions?","volume":"15","author":"Deser","year":"1998","journal-title":"Class. Quant. Grav."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"124026","DOI":"10.1103\/PhysRevD.85.124026","article-title":"Black hole for the Einstein-Chern-Simons gravity","volume":"85","author":"Quinzacara","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2479","DOI":"10.1140\/epjc\/s10052-013-2479-6","article-title":"Stellar equilibrium in Einstein-Chern-Simons gravity","volume":"73","author":"Quinzacara","year":"2013","journal-title":"Eur. Phys. J. C"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3087","DOI":"10.1140\/epjc\/s10052-014-3087-9","article-title":"Accelerated FRW Solutions in Chern-Simons Gravity","volume":"74","author":"Salgado","year":"2014","journal-title":"Eur. Phys. J. C"},{"key":"ref_37","first-page":"49","article-title":"Static solutions in Einstein-Chern-Simons gravity","volume":"1606","author":"Quinzacara","year":"2016","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1016\/j.physletb.2016.02.037","article-title":"Generalized Galilean algebras and Newtonian gravity","volume":"755","author":"Rubio","year":"2016","journal-title":"Phys. Lett. B"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"013503","DOI":"10.1063\/1.3036177","article-title":"S-Expansion of Higher-Order Lie Algebras","volume":"50","author":"Caroca","year":"2009","journal-title":"J. Math. Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"123527","DOI":"10.1063\/1.3272997","article-title":"Generating Higher-Order Lie Algebras by Expanding Maurer Cartan Forms","volume":"50","author":"Caroca","year":"2009","journal-title":"J. Math. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"043519","DOI":"10.1063\/1.3579990","article-title":"Generating infinite-dimensional algebras from loop algebras by expanding Maurer-Cartan forms","volume":"52","author":"Caroca","year":"2011","journal-title":"J. Math. Phys."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"225201","DOI":"10.1088\/1751-8113\/46\/22\/225201","article-title":"Bianchi spaces and their three-dimensional isometries as S-expansions of two-dimensional isometries","volume":"46","author":"Caroca","year":"2013","journal-title":"J. Phys. A"},{"unstructured":"Bianchi, L. (1898). Sugli Spazi a tre Dimensioni che Ammettono un Gruppo Continuo di Movimenti, Accademia Nazionale delle Scienze. Memorie di Matematica e di Fisica della Societa Italiana delle Scienze, Serie Terza Tomo XI.","key":"ref_43"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"2171","DOI":"10.1023\/A:1015357132699","article-title":"On the Three-Dimensional Spaces Which Admit a Continuous Group of Motions","volume":"33","author":"Bianchi","year":"2001","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"234147","DOI":"10.1155\/2009\/234147","article-title":"Semi-simple extension of the (super)Poincare algebra","volume":"2009","author":"Soroka","year":"2009","journal-title":"Adv. High Energy Phys."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"2689","DOI":"10.1142\/S0217732311037078","article-title":"Gauged AdS-Maxwell algebra and gravity","volume":"26","author":"Durka","year":"2011","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1250023","DOI":"10.1142\/S021773231250023X","article-title":"AdS-Maxwell superalgebra and supergravity","volume":"27","author":"Durka","year":"2012","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1016\/j.physletb.2013.11.009","article-title":"so(D-1,1)\u2297so(D-1,2) algebras and gravity","volume":"728","author":"Salgado","year":"2014","journal-title":"Phys. Lett. B"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1016\/j.physletb.2016.06.016","article-title":"New family of Maxwell like algebras","volume":"759","author":"Concha","year":"2016","journal-title":"Phys. Lett. B"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"084077","DOI":"10.1103\/PhysRevD.89.084077","article-title":"Topological gravity and transgression holography","volume":"89","author":"Salgado","year":"2014","journal-title":"Phys. Rev. D"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1016\/j.physletb.2018.10.066","article-title":"Minimal AdS-Lorentz supergravity in three-dimensions","volume":"788","author":"Fierro","year":"2019","journal-title":"Phys. Lett. B"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1128","DOI":"10.1016\/j.nuclphysb.2014.07.022","article-title":"Maxwell Superalgebras and Abelian Semigroup Expansion","volume":"886","author":"Concha","year":"2014","journal-title":"Nucl. Phys. B"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"090","DOI":"10.1007\/JHEP09(2014)090","article-title":"N = 1 Supergravity and Maxwell superalgebras","volume":"1409","author":"Concha","year":"2014","journal-title":"J. High Energy Phys."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"009","DOI":"10.1007\/JHEP08(2015)009","article-title":"Generalized supersymmetric cosmological term in N=1 Supergravity","volume":"1508","author":"Concha","year":"2015","journal-title":"J. High Energy Phys."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/j.physletb.2015.09.005","article-title":"Chern-Simons supergravity in D=3 and Maxwell superalgebra","volume":"750","author":"Concha","year":"2015","journal-title":"Phys. Lett. B"},{"key":"ref_56","first-page":"007","article-title":"On the Supersymmetric Extension of Gauss-Bonnet like Gravity","volume":"1609","author":"Ipinza","year":"2016","journal-title":"J. High Energy Phys."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1140\/epjc\/s10052-017-4615-1","article-title":"In\u00f6n\u00fc-Wigner Contraction and D=2+1 Supergravity","volume":"77","author":"Concha","year":"2017","journal-title":"Eur. Phys. J. C"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"145202","DOI":"10.1088\/1751-8121\/aa5c0b","article-title":"Resonant algebras and gravity","volume":"50","author":"Durka","year":"2017","journal-title":"J. Phys. A: Math. Theor."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"081701","DOI":"10.1063\/1.4991378","article-title":"Infinite S-Expansion with Ideal Subtraction and Some Applications","volume":"58","author":"Ravera","year":"2017","journal-title":"J. Math. Phys."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"365204","DOI":"10.1088\/1751-8113\/46\/36\/365204","article-title":"General properties of the expansion methods of Lie algebras","volume":"46","author":"Andrianopoli","year":"2013","journal-title":"J. Phys. A"},{"unstructured":"(2025, August 23). Available online: https:\/\/github.com\/SemigroupExp\/Sexpansion\/releases\/tag\/v1.0.0.","key":"ref_61"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1145\/1241706.1241710","article-title":"A survey of computer applications to semigroups and related structures","volume":"12","author":"Plemmons","year":"1969","journal-title":"ACM Sigsam"},{"unstructured":"Izaurieta, F. (2006). Semigroup Expansion and M-Supergravity in Eleven Dimensions. arXiv.","key":"ref_63"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1016\/j.geomphys.2014.08.013","article-title":"Einstein-Hilbert action with cosmological term from Chern-Simons gravity","volume":"86","author":"Salgado","year":"2014","journal-title":"J. Geom. Phys."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1090\/S0002-9939-1955-0069814-7","article-title":"SWAC computes 126 distinct semigroups of order 4","volume":"6","author":"Forsythe","year":"1955","journal-title":"Proc. Amer. Math. Soc."},{"key":"ref_66","first-page":"13","article-title":"Semigroups of order five","volume":"62","author":"Motzkin","year":"1956","journal-title":"Bull. Amer. Math. Soc."},{"unstructured":"Plemmons, R. (1967). Construction and analysis of non-equivalent finite semigroups. Computational Problems in Abstract Algebra, Pergamon.","key":"ref_67"},{"key":"ref_68","first-page":"2","article-title":"There are 15973 semigroups of order 6","volume":"2","author":"Plemmons","year":"1967","journal-title":"Math Algorithms"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/BF02194655","article-title":"Die Halbgruppen der Ordnungen \u2264 7","volume":"14","author":"Wick","year":"1977","journal-title":"Semigroup Forum"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1007\/BF02573467","article-title":"Semigroups of order 8","volume":"49","author":"Satoh","year":"1994","journal-title":"Semigroup Forum"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/s10472-009-9140-y","article-title":"The monoids of orders eight, nine & ten","volume":"56","author":"Distler","year":"2009","journal-title":"Ann. Math. Artif. Intell."},{"key":"ref_72","first-page":"61","article-title":"The monoids of order eight and nine","volume":"Volume 5144","author":"Autexier","year":"2008","journal-title":"Lecture Notes in Computer Science Proceedings of the Artificial Intelligence and Symbolic Computation, 8th International Conference, AISC 2008, Birmingham, UK, 31 July\u20132 August 2008"},{"unstructured":"Distler, A., and Mitchell, J.D. (2025, August 23). Smallsemi\u2014A library of Small Semigroups. Available online: https:\/\/gap-packages.github.io\/smallsemi\/.","key":"ref_73"},{"unstructured":"Distler, A., Kelsey, T., and Mitchell, J.D. (2025, August 23). Available online: https:\/\/circa.st-andrews.ac.uk\/research-software\/.","key":"ref_74"},{"key":"ref_75","first-page":"883","article-title":"The Semigroups of Order 10","volume":"Volume 7514","author":"Distler","year":"2012","journal-title":"Principles and Practice of Constraint Programming 18th International Conference, CP 2012, Qu\u00efbec City, QC, Canada, 8\u201312 October 2012"},{"unstructured":"Hildebrant, J. (2001). Handbook of Finite Semigroup Programs (LSU Mathematics Electronic Preprint Series), preprint 2001-24.","key":"ref_76"},{"unstructured":"(2025, August 23). Available online: http:\/\/math.nist.gov\/javanumerics\/jama\/.","key":"ref_77"},{"unstructured":"(2025, August 23). Available online: https:\/\/github.com\/SemigroupExp\/Sexpansion\/wiki.","key":"ref_78"},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"023516","DOI":"10.1063\/1.4941135","article-title":"Geometrical aspects of the Lie algebra S-expansion procedure","volume":"57","author":"Artebani","year":"2016","journal-title":"J. Math. Phys."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"854","DOI":"10.1002\/prop.201600094","article-title":"An Analytic Method for S-Expansion involving Resonance and Reduction","volume":"64","author":"Ipinza","year":"2016","journal-title":"Fortsch. Phys."},{"unstructured":"Inostroza, C., Kondrashuk, I., Merino, N., and Nadal, F. Algorithm to find S-related Lie algebras, unpublished.","key":"ref_81"},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"1259","DOI":"10.1081\/AGB-200053956","article-title":"Degenerations of 7-dimensional nilpotent Lie algebras","volume":"33","author":"Burde","year":"2005","journal-title":"Comm. Algebra"},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"2742","DOI":"10.1016\/j.jalgebra.2010.08.009","article-title":"Lowest dimensional example on non-universality of generalized In\u00f6n\u00fc-Wigner contractions","volume":"324","author":"Popovych","year":"2010","journal-title":"J. Algebra"},{"unstructured":"Concha, P.K., Inostroza, C., Merino, N., and Rodr\u00edguez, E.K. Chern-Simons gravities related with General Relativity, unpublished.","key":"ref_84"},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/0370-2693(75)90504-3","article-title":"Superfield Densities and Action Principle in Curved Superspace","volume":"56B","author":"Arnowitt","year":"1975","journal-title":"Phys. Lett."},{"key":"ref_86","first-page":"187","article-title":"Gauge Fields on Superspaces with Different Holonomy Groups","volume":"22","author":"Akulov","year":"1975","journal-title":"JETP Lett."},{"doi-asserted-by":"crossref","unstructured":"Castellani, L., D\u2019Auria, R., and Fre, P. (1991). Supergravity and Superstrings: A Geometric Perspective, World Scientific.","key":"ref_87","DOI":"10.1142\/9789814542388_0025"},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"064001","DOI":"10.1103\/PhysRevD.74.064001","article-title":"Black Holes in Pure Lovelock Gravities","volume":"74","author":"Cai","year":"2006","journal-title":"Phys. Rev. D"},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"1131","DOI":"10.1007\/s10714-013-1514-0","article-title":"On the static Lovelock black holes","volume":"45","author":"Dadhich","year":"2013","journal-title":"Gen. Rel. Grav."},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"064009","DOI":"10.1103\/PhysRevD.93.064009","article-title":"Dynamical structure of Pure Lovelock gravity","volume":"93","author":"Dadhich","year":"2016","journal-title":"Phys. Rev. D"},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"024055","DOI":"10.1103\/PhysRevD.94.024055","article-title":"Pure Lovelock gravity and Chern-Simons theory","volume":"94","author":"Concha","year":"2016","journal-title":"Phys. Rev. D"},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/j.physletb.2016.09.008","article-title":"Lovelock gravities from Born\u2013Infeld gravity theory","volume":"765","author":"Concha","year":"2017","journal-title":"Phys. Lett. B"},{"unstructured":"Durka, R., Inostroza, C., and Merino, N. Pure Lovelock gravity from Chern-Simons theory, unpublished.","key":"ref_93"},{"unstructured":"Dubrovin, B.A., Fomenko, A.T., and Novikov, S.P. (2012). Modern Geometry\u2014Methods and Applications. Part I: The Geometry of Surfaces, Transformation Groups, and Fields, Springer. Transl. from the Russian by Robert G. Burns. Graduate Texts in Mathematics, 93.","key":"ref_94"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/10\/735\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T04:28:13Z","timestamp":1759292893000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/10\/735"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,29]]},"references-count":94,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2025,10]]}},"alternative-id":["axioms14100735"],"URL":"https:\/\/doi.org\/10.3390\/axioms14100735","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,9,29]]}}}