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A complete classification is given for the corresponding cases arising from nonderogatory endomorphisms, and from maximal Abelian nilpotent subalgebras (MANS) of class 2, linked to Kravchuk signature. The work also classifies all such algebras up to dimension 8, corrects MASA results for sl(4, R), and derives consequences for Jordan forms and Cartan subalgebras of sl(n, R).",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/on-the-classification-of-2-solvable-frobenius-lie-algebras-research-paper/153424/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/on-the-classification-of-2-solvable-frobenius-lie-algebras-research-paper/153424.png","ImageObject",300,407,{"name":92,"@type":93},"Fahsai","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-29","2026-08-28",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",13,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What decomposition is proved for 2-solvable Frobenius Lie algebras?","Question",{"text":113,"@type":114},"Every 2-solvable Frobenius Lie algebra splits as a semidirect sum of an n-dimensional vector space V and an n-dimensional maximal Abelian subalgebra (MASA) of End(V).","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How are the paper’s main classifications organized?",{"text":118,"@type":114},"The classification covers cases corresponding to nonderogatory endomorphisms, and cases arising from maximal Abelian nilpotent subalgebras (MANS) of class 2, with additional complete classification up to dimension 8.",{"name":120,"@type":111,"acceptedAnswer":121},"What additional algebraic consequences does the paper provide?",{"text":122,"@type":114},"It supplies a simple proof that every nonderogatory real matrix admits a Jordan form and gives a new characterization of Cartan subalgebras of sl(n, R), while also correcting the MASA classification list of sl(4, R).","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},153424,1787876163,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":130,"read_time":145,"preview_page_count":4},549768702563,"https://ap-avatar.wpscdn.com/avatar/8000c4aa63b76e948b?x-image-process=image/resize,m_fixed,w_180,h_180&k=1786536092046926083","Journal of Lie Theory Volume 33 (2023) 799–830 © 2023 Heldermann Verlag  \nOn the Classification of 2-Solvable Frobenius Lie Algebras  \nAndré Diatta, Bakary Manga, Ameth Mbaye  \nCommunicated by R. Avdeev  \nAbstract. We prove that every 2-solvable Frobenius Lie algebra splits as a semidirect sum of an n-dimensional vector space V and an n-dimensional maximal Abelian subalgebra (MASA) of the full space of endomorphisms of V . We supply a complete classification of 2-solvable Frobenius Lie algebras corresponding to nonderogatory endomorphisms, as well as those given by maximal Abelian nilpotent subalgebras (MANS) of class 2, hence of Kravchuk signature (n 􀀀1, 0 , 1) . In low dimensions, we classify all 2-solvable Frobenius Lie algebras in general up to dimension 8 . We correct and complete the classification list of MASAs of sl(4, R) by Winternitz and Zassenhaus. Asa biproduct, we give a simple proof that every nonderogatory endormorphism of a real vector space admits a Jordan form and also provide a new characterization of Cartan subalgebras of sl (n, R) .  \nMathematics Subject Classification: 17B05, 17B08, 15A27, 53A15, 53D15, 22E60, 17B60, 70G45, 16W25, 13B25 .  \nKey Words: Frobenius Lie algebra, 2-step solvable exact symplectic Lie algebra, symplectic Lie group, maximal Abelian subalgebra, nonderogatory endomorphism, cyclic matrix, companion matrix, Kravchuk signature, Cartan subalgebra, Jordan form.  \n1. Introduction  \nThroughout this work, V stands for a real vector space of dimension n. We are mainly interested in the classification of 2-solvable Frobenius Lie algebras. It turns out that such a classification is closely related to that of n-dimensional maximal Abelian subalgebras (MASAs) of the space gl (V ) of linear operators (endomorphisms) on V. The sudy of MASAs traces back to Frobenius and Schur, it has been a vibrant and vivid subject these last decades, in relation with several subjects in mathematics and physics such as the classification of Lie algebras and the study of dynamical systems ([5, 10, 12, 13, 14, 20, 23, 24, 25, 26]) . Our interest in 2-solvable Frobenius Lie algebras is mainly related to their applications in symplectic ([3, 7, 9, 15]) and information geometry ([16]), as well as to the investigation of Lie algebras in general. Frobenius Lie algebras have gained popularity mainly since the work of Ooms ([19, 18]) . Among other results, a characterization of 2-solvable Frobenius Lie algebras is given in [19](Theorem 4.1, Section 4, p. 43) . On the other hand, the family of 2-solvable Frobenius Lie algebras with an Abelian nilradical is studied in [1] . The main results of the present paper are the following.  \n• We show that, if B 􀀚 gl (V ) is Abelian and the semidirect sum B n V is a Frobenius Lie algebra, then B is a MASA of gl (V ) (Theorem 3.1) . Moreover,  \nISSN 0949–5932 / $2.50 © Heldermann Verlag  \n800 Diatta, Manga, Mbaye  \nevery 2-solvable Frobenius Lie algebra is of that form. Two 2-solvable Frobenius Lie algebras B 1 n V and B2 n V are isomorphic if and only if there exists some ϕ in the group GL (V ) of invertible linear operators of V , such that B2 = ϕB1 ϕ−1 (Proposition 3.3) .  \n• In Theorem 3 .6, we classify 2-solvable Frobenius Lie algebras B n V , where B \\sl (V ) is a maximal Abelian nilpotent subalgebra (MANS) of class 2 and hence of Kravchuk signature (n 􀀀 1 , 0 , 1) . Here sl (V ) 􀀚 gl (V ) stands for the subspace of traceless linear operators.  \n• We fully classify, in Theorem 4 .3, the family of 2-solvable Frobenius Lie algebras B n V , where B is the algebra K [ϕ] of the polynomials in some nonderogatory ϕ 2 gl (V ) . We will simply write Gϕ = K[ϕ] n V instead of B n V .  \n• We supply a full classification list of all 2-solvable Frobenius Lie algebras in general, up to and including dimension 8 (Theorem 5.1) .  \n• Theorems 3.11, 4.19 give a new characterization of Cartan subalgebras of sl(V ) .  \n• We correct the classification list of MASAs of sl(4 , R) of [26] and further com","cbCaihQASb0Sn4tS","https://ap.wps.com/l/cbCaihQASb0Sn4tS","pdf",266552,32,"English","# Introduction\n## Classification goals and main results\n# Preliminaries\n## Notation, duality, representations\n## Key definitions and setup","[{\"question\":\"What decomposition is proved for 2-solvable Frobenius Lie algebras?\",\"answer\":\"Every 2-solvable Frobenius Lie algebra splits as a semidirect sum of an n-dimensional vector space V and an n-dimensional maximal Abelian subalgebra (MASA) of End(V).\"},{\"question\":\"How are the paper’s main classifications organized?\",\"answer\":\"The classification covers cases corresponding to nonderogatory endomorphisms, and cases arising from maximal Abelian nilpotent subalgebras (MANS) of class 2, with additional complete classification up to dimension 8.\"},{\"question\":\"What additional algebraic consequences does the paper provide?\",\"answer\":\"It supplies a simple proof that every nonderogatory real matrix admits a Jordan form and gives a new characterization of Cartan subalgebras of sl(n, R), while also correcting the MASA classification list of sl(4, R).\"}]","On the Classification of 2-Solvable Frobenius Lie Algebras - Research paper | PDF",81]