{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,5,11]],"date-time":"2022-05-11T15:46:57Z","timestamp":1652284017656},"reference-count":22,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","issue":"2","license":[{"start":{"date-parts":[[2021,6,1]],"date-time":"2021-06-01T00:00:00Z","timestamp":1622505600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,6,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We connect the theorems of Rentschler [18] and Dixmier [10] on locally nilpotent derivations and automorphisms of the polynomial ring <jats:italic>A<\/jats:italic>\n                  <jats:sub>0<\/jats:sub> and of the Weyl algebra <jats:italic>A<\/jats:italic>\n                  <jats:sub>1<\/jats:sub>, both over a field of characteristic zero, by establishing the same type of results for the family of algebras \n<jats:disp-formula id=\"j_cm-2021-0024_eq_001\">\n                     <jats:alternatives>\n                        <jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cm-2021-0024_eq_001.png\" \/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\">\n                           <m:mrow>\n                              <m:msub>\n                                 <m:mi>A<\/m:mi>\n                                 <m:mi>h<\/m:mi>\n                              <\/m:msub>\n                              <m:mo>=<\/m:mo>\n                              <m:mrow>\n                                 <m:mo>\u3008<\/m:mo>\n                                 <m:mrow>\n                                    <m:mi>x<\/m:mi>\n                                    <m:mo>,<\/m:mo>\n                                    <m:mi>y<\/m:mi>\n                                    <m:mo>|<\/m:mo>\n                                    <m:mi>y<\/m:mi>\n                                    <m:mi>x<\/m:mi>\n                                    <m:mo>\u2212<\/m:mo>\n                                    <m:mi>x<\/m:mi>\n                                    <m:mi>y<\/m:mi>\n                                    <m:mo>=<\/m:mo>\n                                    <m:mi>h<\/m:mi>\n                                    <m:mrow>\n                                       <m:mo>(<\/m:mo>\n                                       <m:mi>x<\/m:mi>\n                                       <m:mo>)<\/m:mo>\n                                    <\/m:mrow>\n                                 <\/m:mrow>\n                                 <m:mo>\u3009<\/m:mo>\n                              <\/m:mrow>\n                              <m:mo>,<\/m:mo>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>{A_h} = \\left\\langle {x,y|yx - xy = h\\left( x \\right)} \\right\\rangle ,<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:disp-formula>\n, where <jats:italic>h<\/jats:italic> is an arbitrary polynomial in <jats:italic>x<\/jats:italic>. In the second part of the paper we consider a field \ud835\udd3d of prime characteristic and study \ud835\udd3d[<jats:italic>t<\/jats:italic>]-comodule algebra structures on <jats:italic>A<jats:sub>h<\/jats:sub>\n                  <\/jats:italic>. We also compute the Makar-Limanov invariant of absolute constants of <jats:italic>A<jats:sub>h<\/jats:sub>\n                  <\/jats:italic> over a field of arbitrary characteristic and show how this subalgebra determines the automorphism group of <jats:italic>A<jats:sub>h<\/jats:sub>\n                  <\/jats:italic>.<\/jats:p>","DOI":"10.2478\/cm-2021-0024","type":"journal-article","created":{"date-parts":[[2021,7,15]],"date-time":"2021-07-15T23:54:36Z","timestamp":1626393276000},"page":"269-279","source":"Crossref","is-referenced-by-count":0,"title":["Actions of the additive group <i>G<sub>a<\/sub>\n                  <\/i> on certain noncommutative deformations of the plane"],"prefix":"10.46298","volume":"29","author":[{"given":"Ivan","family":"Kaygorodov","sequence":"first","affiliation":[{"name":"CMCC, Universidade Federal do ABC , Santo Andr\u00e9 , Brazil"}]},{"given":"Samuel A.","family":"Lopes","sequence":"additional","affiliation":[{"name":"CMUP, Departamento de Matem\u00e1tica, Faculdade de Ci\u00eancias , Universidade do Porto , Rua do Campo Alegre s\/n, 4169\u2013007 Porto , Portugal"}]},{"given":"Farukh","family":"Mashurov","sequence":"additional","affiliation":[{"name":"Suleyman Demirel University , Kaskelen , Kazakhstan ; Kazakh-British Technical University , Almaty , Kazakhstan"}]}],"member":"25203","published-online":{"date-parts":[[2021,7,15]]},"reference":[{"key":"2022040122095110821_j_cm-2021-0024_ref_001","doi-asserted-by":"crossref","unstructured":"[1] J. Alev, F. Dumas: Invariants du corps de Weyl sous l\u2019action de groupes finis. Communications in Algebra 25 (5) (1997) 1655\u20131672.","DOI":"10.1080\/00927879708825943"},{"key":"2022040122095110821_j_cm-2021-0024_ref_002","doi-asserted-by":"crossref","unstructured":"[2] V. Bavula, D. Jordan: Isomorphism problems and groups of automorphisms for generalized Weyl algebras. Transactions of the American Mathematical Society 353 (2) (2001) 769\u2013794.","DOI":"10.1090\/S0002-9947-00-02678-7"},{"key":"2022040122095110821_j_cm-2021-0024_ref_003","doi-asserted-by":"crossref","unstructured":"[3] A. Belov-Kanel, M. Kontsevich: The Jacobian conjecture is stably equivalent to the Dixmier conjecture. Moscow Mathematical Journal 7 (2) (2007) 209\u2013218.","DOI":"10.17323\/1609-4514-2007-7-2-209-218"},{"key":"2022040122095110821_j_cm-2021-0024_ref_004","doi-asserted-by":"crossref","unstructured":"[4] G. Benkart, S.A. Lopes, M. Ondrus: A parametric family of subalgebras of the Weyl algebra II. Irreducible modules. In: Recent developments in algebraic and combinatorial aspects of representation theory, vol. 602 of Contemporary Mathematics. American Mathematical Society (2013) 73\u201398.","DOI":"10.1090\/conm\/602\/12027"},{"key":"2022040122095110821_j_cm-2021-0024_ref_005","doi-asserted-by":"crossref","unstructured":"[5] G. Benkart, S.A. Lopes, M. Ondrus: Derivations of a parametric family of subalgebras of the Weyl algebra. Journal of Algebra 424 (2015) 46\u201397.","DOI":"10.1016\/j.jalgebra.2014.11.007"},{"key":"2022040122095110821_j_cm-2021-0024_ref_006","doi-asserted-by":"crossref","unstructured":"[6] G. Benkart, S.A. Lopes, M. Ondrus: A parametric family of subalgebras of the Weyl algebra I. Structure and automorphisms. Transactions of the American Mathematical Society 367 (3) (2015) 1993\u20132021.","DOI":"10.1090\/S0002-9947-2014-06144-8"},{"key":"2022040122095110821_j_cm-2021-0024_ref_007","doi-asserted-by":"crossref","unstructured":"[7] A. Crachiola, L. Makar-Limanov: On the rigidity of small domains. Journal of Algebra 284 (1) (2005) 1\u201312.","DOI":"10.1016\/j.jalgebra.2004.09.015"},{"key":"2022040122095110821_j_cm-2021-0024_ref_008","doi-asserted-by":"crossref","unstructured":"[8] A.J. Crachiola: The hypersurface x + x2y + z2 + t3 = 0 over a field of arbitrary characteristic. Proceedings of the American Mathematical Society 134 (5) (2006) 1289\u20131298.","DOI":"10.1090\/S0002-9939-05-08171-2"},{"key":"2022040122095110821_j_cm-2021-0024_ref_009","doi-asserted-by":"crossref","unstructured":"[9] S.D. Crode, I.P. Shestakov: Locally nilpotent derivations and automorphisms of free associative algebra with two generators. Communications in Algebra 48 (7) (2020) 3091\u20133098.","DOI":"10.1080\/00927872.2020.1729363"},{"key":"2022040122095110821_j_cm-2021-0024_ref_010","doi-asserted-by":"crossref","unstructured":"[10] J. Dixmier: Sur les alg\u00e8bres de Weyl. Bulletin de la Soci\u00e9t\u00e9 math\u00e9matique de France 96 (1968) 209\u2013242.10.24033\/bsmf.1667","DOI":"10.24033\/bsmf.1667"},{"key":"2022040122095110821_j_cm-2021-0024_ref_011","doi-asserted-by":"crossref","unstructured":"[11] J. Dixmier: Enveloping algebras. American Mathematical Society (1996). Vol. 11 of Graduate Studies in Mathematics. Revised reprint of the 1977 translation.10.1090\/gsm\/011","DOI":"10.1090\/gsm\/011"},{"key":"2022040122095110821_j_cm-2021-0024_ref_012","doi-asserted-by":"crossref","unstructured":"[12] V. Drensky, L. Makar-Limanov: Locally nilpotent derivations of free algebra of rank two. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 15 (2019) Paper No. 091.","DOI":"10.3842\/SIGMA.2019.091"},{"key":"2022040122095110821_j_cm-2021-0024_ref_013","doi-asserted-by":"crossref","unstructured":"[13] H.W.E. Jung: \u00dcber ganze birationale Transformationen der Ebene. Journal f\u00fcr die Reine und Angewandte Mathematik 1942 (184) (1942) 161\u2013174.","DOI":"10.1515\/crll.1942.184.161"},{"key":"2022040122095110821_j_cm-2021-0024_ref_014","doi-asserted-by":"crossref","unstructured":"[14] I. Kaygorodov, I. Shestakov, U. Umirbaev: Free generic Poisson fields and algebras. Communications in Algebra 46 (4) (2018) 1799\u20131812.","DOI":"10.1080\/00927872.2017.1358269"},{"key":"2022040122095110821_j_cm-2021-0024_ref_015","doi-asserted-by":"crossref","unstructured":"[15] L. Makar-Limanov: On the hypersurface x + x2y + z2 + t3 = 0 in \u21024 or a \u21023-like threefold which is not C3. Israel Journal of Mathematics 96 (2) (1996) 419\u2013429.","DOI":"10.1007\/BF02937314"},{"key":"2022040122095110821_j_cm-2021-0024_ref_016","doi-asserted-by":"crossref","unstructured":"[16] L. Makar-Limanov, U. Turusbekova, U. Umirbaev: Automorphisms and derivations of free Poisson algebras in two variables. Journal of Algebra 322 (9) (2009) 3318\u20133330.","DOI":"10.1016\/j.jalgebra.2008.01.005"},{"key":"2022040122095110821_j_cm-2021-0024_ref_017","doi-asserted-by":"crossref","unstructured":"[17] M. Miyanishi: Ga-action of the a ne plane. Nagoya Mathematical Journal 41 (1971) 97\u2013100.10.1017\/S0027763000014094","DOI":"10.1017\/S0027763000014094"},{"key":"2022040122095110821_j_cm-2021-0024_ref_018","unstructured":"[18] R. Rentschler: Op\u00e9rations du groupe additif sur le plan a ne. Comptes rendus de l\u2019Acad\u00e9mie des Sciences, S\u00e9r. A-B 267 (1968) 384\u2013387."},{"key":"2022040122095110821_j_cm-2021-0024_ref_019","doi-asserted-by":"crossref","unstructured":"[19] G. Restuccia, H.J. Schneider: On actions of infinitesimal group schemes. Journal of Algebra 261 (2) (2003) 229\u2013244.","DOI":"10.1016\/S0021-8693(02)00683-X"},{"key":"2022040122095110821_j_cm-2021-0024_ref_020","unstructured":"[20] G. Restuccia, H.J. Schneider: On actions of the additive group on the Weyl algebra. Atti della Accademia Peloritana dei Pericolanti-Classe di Scienze Fisiche, Matematiche e Naturali 83 (1) (2005) 9pp."},{"key":"2022040122095110821_j_cm-2021-0024_ref_021","unstructured":"[21] Y. Tsuchimoto: Endomorphisms of Weyl algebra and p-curvatures. Osaka Journal of Mathematics 42 (2) (2005) 435\u2013452."},{"key":"2022040122095110821_j_cm-2021-0024_ref_022","doi-asserted-by":"crossref","unstructured":"[22] A. Van den Essen: Polynomial automorphisms and the Jacobian conjecture. Birkh\u00e4user (2000). Progress in Mathematics, vol. 190.10.1007\/978-3-0348-8440-2","DOI":"10.1007\/978-3-0348-8440-2_10"}],"container-title":["Communications in Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.sciendo.com\/pdf\/10.2478\/cm-2021-0024","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,5,11]],"date-time":"2022-05-11T15:24:30Z","timestamp":1652282670000},"score":1,"resource":{"primary":{"URL":"https:\/\/cm.episciences.org\/9543"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,1]]},"references-count":22,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,7,15]]},"published-print":{"date-parts":[[2021,6,1]]}},"alternative-id":["10.2478\/cm-2021-0024"],"URL":"https:\/\/doi.org\/10.2478\/cm-2021-0024","relation":{},"ISSN":["2336-1298"],"issn-type":[{"value":"2336-1298","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,6,1]]}}}