Journal of Algebra最新文献

{"title":"Essential semigroups and branching rules","authors":"Andrei Gornitskii","doi":"10.1016/j.jalgebra.2025.04.047","DOIUrl":"10.1016/j.jalgebra.2025.04.047","url":null,"abstract":"<div><div>Let <span><math><mi>g</mi></math></span> be a semisimple complex Lie algebra of finite dimension and <span><math><mi>h</mi></math></span> be a semisimple subalgebra. We present an approach to find the branching rules for the pair <span><math><mi>g</mi><mo>⊃</mo><mi>h</mi></math></span>. According to an idea of Zhelobenko the information on restriction to <span><math><mi>h</mi></math></span> of all irreducible representations of <span><math><mi>g</mi></math></span> is contained in one associative algebra, which we call the <em>branching algebra</em>. We use an <em>essential semigroup</em> Σ, which parametrizes certain bases in every finite-dimensional irreducible representations of <span><math><mi>g</mi></math></span>, and describe the branching rules for <span><math><mi>g</mi><mo>⊃</mo><mi>h</mi></math></span> in terms of a certain subsemigroup <span><math><msup><mrow><mi>Σ</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> of Σ. If <span><math><msup><mrow><mi>Σ</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> is finitely generated, then the semigroup algebra corresponding to <span><math><msup><mrow><mi>Σ</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> is a toric degeneration of the branching algebra. We propose an algorithm to find a description of <span><math><msup><mrow><mi>Σ</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> in this case. We give examples by deriving the branching rules for <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>⊃</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>⊃</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⊃</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>⊃</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, and <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>⊃</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"681 ","pages":"Pages 190-205"},"PeriodicalIF":0.8,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144169748","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Frobenius splitting and cohomology vanishing for the cotangent bundles of the flag varieties of GLn","authors":"Rudolf Tange","doi":"10.1016/j.jalgebra.2025.05.003","DOIUrl":"10.1016/j.jalgebra.2025.05.003","url":null,"abstract":"<div><div>Let <em>k</em> be an algebraically closed field of characteristic <span><math><mi>p</mi><mo>&gt;</mo><mn>0</mn></math></span>, let <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the general linear group over <em>k</em>, let <em>P</em> be a parabolic subgroup of <em>G</em>, and let <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> be the Lie algebra of its unipotent radical. We show that the Kumar-Lauritzen-Thomsen splitting of the cotangent bundle <span><math><mi>G</mi><msup><mrow><mo>×</mo></mrow><mrow><mi>P</mi></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> of <span><math><mi>G</mi><mo>/</mo><mi>P</mi></math></span> has top degree <span><math><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>dim</mi><mo>⁡</mo><mo>(</mo><mi>G</mi><mo>/</mo><mi>P</mi><mo>)</mo></math></span>. The component of that degree is therefore given by the <span><math><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-th power of a function <em>f</em>. We give a formula for <em>f</em> and deduce that it vanishes on the exceptional locus of the resolution <span><math><mi>G</mi><msup><mrow><mo>×</mo></mrow><mrow><mi>P</mi></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mi>P</mi></mrow></msub><mo>→</mo><mover><mrow><mi>O</mi></mrow><mo>‾</mo></mover></math></span> where <span><math><mover><mrow><mi>O</mi></mrow><mo>‾</mo></mover></math></span> is the closure of the Richardson orbit of <em>P</em>. As a consequence we obtain that the higher cohomology groups of a line bundle on <span><math><mi>G</mi><msup><mrow><mo>×</mo></mrow><mrow><mi>P</mi></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> associated to a dominant weight are zero. The splitting of <span><math><mi>G</mi><msup><mrow><mo>×</mo></mrow><mrow><mi>P</mi></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> given by <span><math><msup><mrow><mi>f</mi></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> can be seen as a generalisation of the Mehta-Van der Kallen splitting of <span><math><mi>G</mi><msup><mrow><mo>×</mo></mrow><mrow><mi>B</mi></mrow></msup><mi>u</mi></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"679 ","pages":"Pages 56-64"},"PeriodicalIF":0.8,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144137744","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Amitsur-Small rings","authors":"Adam Chapman ,&nbsp;Elad Paran","doi":"10.1016/j.jalgebra.2025.04.049","DOIUrl":"10.1016/j.jalgebra.2025.04.049","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>D</mi><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span> denote the ring of polynomials in <em>n</em> central variables over a division ring <em>D</em>. We say that <em>D</em> is an <em>Amitsur-Small ring</em> if for any maximal left ideal in <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, <span><math><mi>M</mi><mo>∩</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is a maximal left ideal in <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, for all <span><math><mi>n</mi><mo>∈</mo><mi>N</mi></math></span> and <span><math><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></math></span>. We demonstrate the existence of non Amitsur-Small division rings, providing a negative answer to a question of Amitsur and Small from 1978. We show that Hamilton's real quaternion algebra <span><math><mi>H</mi><mo>=</mo><msub><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn><mo>,</mo><mi>R</mi></mrow></msub></math></span> is an Amitsur-Small ring, division rings of degree 3 over their center <em>F</em> are never Amitsur-Small, and division rings of degree 2 are not Amitsur-Small if they are not quaternion algebras <span><math><msub><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn><mo>,</mo><mi>F</mi></mrow></msub></math></span> over a Pythagorean field <em>F</em>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"679 ","pages":"Pages 86-95"},"PeriodicalIF":0.8,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144166641","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The charge statistic in type A via the affine Grassmannian","authors":"Leonardo Patimo","doi":"10.1016/j.jalgebra.2025.04.036","DOIUrl":"10.1016/j.jalgebra.2025.04.036","url":null,"abstract":"<div><div>We give a new construction of Lascoux–Schützenberger's charge statistic in type A which is motivated by the geometric Satake equivalence. We obtain a new formula for the charge statistic in terms of modified crystal operators and an independent proof of this formula which does not rely on tableaux combinatorics.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"679 ","pages":"Pages 169-211"},"PeriodicalIF":0.8,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144166621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A generalization of the Arad–Ward theorem on Hall subgroups","authors":"N. Yang ,&nbsp;A.A. Buturlakin","doi":"10.1016/j.jalgebra.2025.05.001","DOIUrl":"10.1016/j.jalgebra.2025.05.001","url":null,"abstract":"<div><div>For a set of primes <em>π</em>, denote by <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>π</mi></mrow></msub></math></span> the class of finite groups containing a Hall <em>π</em>-subgroup. We establish that <span><math><msub><mrow><mi>E</mi></mrow><mrow><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>∩</mo><msub><mrow><mi>E</mi></mrow><mrow><msub><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math></span> is contained in <span><math><msub><mrow><mi>E</mi></mrow><mrow><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∩</mo><msub><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math></span>. As a corollary, we prove that if <em>π</em> is a set of primes, <em>l</em> is an integer such that <span><math><mn>2</mn><mo>⩽</mo><mi>l</mi><mo>&lt;</mo><mo>|</mo><mi>π</mi><mo>|</mo></math></span> and <em>G</em> is a finite group that contains a Hall <em>ρ</em>-subgroup for every subset <em>ρ</em> of <em>π</em> of size <em>l</em>, then <em>G</em> contains a solvable Hall <em>π</em>-subgroup.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"679 ","pages":"Pages 28-36"},"PeriodicalIF":0.8,"publicationDate":"2025-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144084155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hopf–Galois structures on parallel extensions","authors":"Andrew Darlington","doi":"10.1016/j.jalgebra.2025.04.025","DOIUrl":"10.1016/j.jalgebra.2025.04.025","url":null,"abstract":"<div><div>Let <span><math><mi>L</mi><mo>/</mo><mi>K</mi></math></span> be a finite separable extension of fields of degree <em>n</em>, and let <span><math><mi>E</mi><mo>/</mo><mi>K</mi></math></span> be its Galois closure. Greither and Pareigis showed how to find all Hopf–Galois structures on <span><math><mi>L</mi><mo>/</mo><mi>K</mi></math></span>. We will call a subextension <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>/</mo><mi>K</mi></math></span> of <span><math><mi>E</mi><mo>/</mo><mi>K</mi></math></span> <em>parallel</em> to <span><math><mi>L</mi><mo>/</mo><mi>K</mi></math></span> if <span><math><mo>[</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>:</mo><mi>K</mi><mo>]</mo><mo>=</mo><mi>n</mi></math></span>.</div><div>In this paper, we investigate the relationship between the Hopf–Galois structures on an extension <span><math><mi>L</mi><mo>/</mo><mi>K</mi></math></span> and those on the related parallel extensions. We give an example of a transitive subgroup corresponding to an extension admitting a Hopf–Galois structure but that has a parallel extension admitting no Hopf–Galois structures. We show that once one has such a situation, it can be extended into an infinite family of transitive subgroups admitting this phenomenon. We also investigate this fully in the case of extensions of degree <em>pq</em> with <span><math><mi>p</mi><mo>,</mo><mi>q</mi></math></span> distinct odd primes, and show that there is no example of such an extension admitting the phenomenon.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"679 ","pages":"Pages 1-27"},"PeriodicalIF":0.8,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144069639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Corrigendum to “The Gruenberg-Kegel graph of finite solvable rational groups” [J. Algebra 642 (2024) 470–479]","authors":"Sara C. Debón ,&nbsp;Diego García-Lucas ,&nbsp;Ángel del Río","doi":"10.1016/j.jalgebra.2025.03.051","DOIUrl":"10.1016/j.jalgebra.2025.03.051","url":null,"abstract":"","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"677 ","pages":"Pages 618-629"},"PeriodicalIF":0.8,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143918221","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On N-graded vertex algebras associated with Gorenstein algebras","authors":"Alex Keene,&nbsp;Christian Soltermann,&nbsp;Gaywalee Yamskulna","doi":"10.1016/j.jalgebra.2025.04.032","DOIUrl":"10.1016/j.jalgebra.2025.04.032","url":null,"abstract":"<div><div>This paper investigates the algebraic structure of indecomposable <span><math><mi>N</mi></math></span>-graded vertex algebras <span><math><mi>V</mi><mo>=</mo><msubsup><mrow><mo>⨁</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><msub><mrow><mi>V</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, emphasizing the intricate interactions between the commutative associative algebra <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, the Leibniz algebra <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and how non-degenerate bilinear forms on <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> influence their overall structure. We establish foundational properties for indecomposability and locality in <span><math><mi>N</mi></math></span>-graded vertex algebras, with our main result demonstrating the equivalence of locality, indecomposability, and specific structural conditions on semiconformal-vertex algebras. The study of symmetric invariant bilinear forms of semiconformal-vertex algebra is investigated. We also examine the structural characteristics of <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, demonstrating conditions under which certain <span><math><mi>N</mi></math></span>-graded vertex algebras cannot be quasi vertex operator algebras, semiconformal-vertex algebras, or vertex operator algebras, and explore <span><math><mi>N</mi></math></span>-graded vertex algebras <span><math><mi>V</mi><mo>=</mo><msubsup><mrow><mo>⨁</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><msub><mrow><mi>V</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> associated with Gorenstein algebras. Our analysis includes examining the socle, Poincaré duality properties, and invariant bilinear forms of <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and their influence on <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, providing conditions for embedding rank-one Heisenberg vertex operator algebras within <em>V</em>. Supporting examples and detailed theoretical insights further illustrate these algebraic structures.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"678 ","pages":"Pages 729-768"},"PeriodicalIF":0.8,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143934666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Homotopy equivalences and Grothendieck duality over rings with finite Gorenstein weak global dimension","authors":"Junpeng Wang ,&nbsp;Sergio Estrada","doi":"10.1016/j.jalgebra.2025.04.033","DOIUrl":"10.1016/j.jalgebra.2025.04.033","url":null,"abstract":"<div><div>Let <em>R</em> be a ring with Gwgldim<span><math><mo>(</mo><mi>R</mi><mo>)</mo><mo>&lt;</mo><mo>∞</mo></math></span>. We obtain a triangle-equivalence <span><math><mtext>K</mtext><mo>(</mo><mi>R</mi><mtext>-</mtext><mtext>GProj</mtext><mo>)</mo><mo>≃</mo><mtext>K</mtext><mo>(</mo><mi>R</mi><mtext>-</mtext><mtext>GInj</mtext><mo>)</mo></math></span> which restricts to a triangle-equivalence <span><math><mtext>K</mtext><mo>(</mo><mi>R</mi><mtext>-</mtext><mtext>Proj</mtext><mo>)</mo></math></span> <span><math><mo>≃</mo><mtext>K</mtext><mo>(</mo><mi>R</mi><mtext>-</mtext><mtext>Inj</mtext><mo>)</mo></math></span>. This class of rings includes, among others, (left) Gorenstein rings, Ding–Chen rings and the more general Gorenstein <em>n</em>-coherent rings (<span><math><mi>n</mi><mo>∈</mo><mi>N</mi><mo>∪</mo><mo>{</mo><mo>∞</mo><mo>}</mo><mo>,</mo><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>). As application, we establish some triangle-equivalences of Grothendieck duality over Ding–Chen rings and Gorenstein <em>n</em>-coherent rings.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"678 ","pages":"Pages 769-808"},"PeriodicalIF":0.8,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143935669","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the uniqueness of maximal solvable extensions of nilpotent Leibniz superalgebras","authors":"B.A. Omirov ,&nbsp;G.O. Solijanova","doi":"10.1016/j.jalgebra.2025.04.035","DOIUrl":"10.1016/j.jalgebra.2025.04.035","url":null,"abstract":"<div><div>In the present paper under certain conditions the description of the maximal solvable extension of complex finite-dimensional nilpotent Leibniz superalgebras is obtained. Specifically, we establish that under the condition ensuring the fulfillment of Lie's theorem for a maximal solvable extension of a special kind of nilpotent Leibniz superalgebra (which is consistent and <em>d</em>-locally diagonalizable), it is decomposed into a semidirect sum of a nilpotent Leibniz superalgebra and a maximal torus on it. In other words, under certain conditions the direct sum of the nilpotent superalgebra and its torus (as a vector spaces), admits a solvable Leibniz superalgebra structure. In addition, for the left-side action of a maximal torus on nilpotent Leibniz superalgebra, which does not admit <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> as a direct summand and is diagonalizable, we prove the uniqueness of the maximal extension. Along with the answer to Šnobl's conjecture for Lie algebras this result covers several already known results for Lie (super)algebras and Leibniz algebras.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"677 ","pages":"Pages 798-832"},"PeriodicalIF":0.8,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143935173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}