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Dirac Cohomology and (Theta )-correspondence for Complex Dual Pairs
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-02-08 DOI: 10.1007/s10468-025-10319-8
S. Afentoulidis-Almpanis, G. Liu, S. Mehdi
{"title":"Dirac Cohomology and (Theta )-correspondence for Complex Dual Pairs","authors":"S. Afentoulidis-Almpanis,&nbsp;G. Liu,&nbsp;S. Mehdi","doi":"10.1007/s10468-025-10319-8","DOIUrl":"10.1007/s10468-025-10319-8","url":null,"abstract":"<div><p>We study the behavior of Dirac cohomology under Howe’s <span>(Theta )</span>-correspondence in the case of complex reductive dual pairs. More precisely, if <span>((G_1,G_2))</span> is a complex reductive dual pair with <span>(G_1)</span> and <span>(G_2)</span> viewed as real groups, we describe those Harish-Chandra modules <span>(pi _1)</span> of <span>(G_1)</span> with nonzero Dirac cohomology whose <span>(Theta )</span>-liftings <span>(Theta (pi _1))</span> still have nonzero Dirac cohomology. In this case, we compute explicitly the Dirac cohomology of <span>(Theta (pi _1))</span>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 1","pages":"337 - 352"},"PeriodicalIF":0.5,"publicationDate":"2025-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-025-10319-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143553874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Endomorphism Algebras of Equivariant Exceptional Collections
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-02-07 DOI: 10.1007/s10468-025-10313-0
Andreas Krug, Erik Nikolov
{"title":"Endomorphism Algebras of Equivariant Exceptional Collections","authors":"Andreas Krug,&nbsp;Erik Nikolov","doi":"10.1007/s10468-025-10313-0","DOIUrl":"10.1007/s10468-025-10313-0","url":null,"abstract":"<div><p>Given an action of a finite group on a triangulated category with a suitable strong exceptional collection, a construction of Elagin produces an associated strong exceptional collection on the equivariant category. In the untwisted case, we prove that the endomorphism algebra of the induced exceptional collection is the basic reduction of the skew group algebra of the endomorphism algebra of the original exceptional collection.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 1","pages":"193 - 210"},"PeriodicalIF":0.5,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-025-10313-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143553871","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Exhaustion of Supercuspidal Representations of p-adic Spin Groups: Semisimple Characters p-adic Spin Groups 的 Supercuspidal Representations 的穷尽：半简单字符
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-02-04 DOI: 10.1007/s10468-025-10318-9
Ngô Văn Định
{"title":"Exhaustion of Supercuspidal Representations of p-adic Spin Groups: Semisimple Characters","authors":"Ngô Văn Định","doi":"10.1007/s10468-025-10318-9","DOIUrl":"10.1007/s10468-025-10318-9","url":null,"abstract":"<div><p>Let <span>(varvec{widehat{G}})</span> be a spin group over a locally compact non-archimedean local field <span>(varvec{F})</span> of odd residual characteristic. We defined lifted self-dual semisimple characters for <span>(varvec{widehat{G}})</span> and from them, we constructed a large class of supercuspidal representations of <span>(varvec{widehat{G}})</span> when <span>(varvec{F})</span> is of characteristic zero. In this paper, we show that any positive level supercuspidal representation of <span>(varvec{widehat{G}})</span> contains such a character.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 1","pages":"315 - 335"},"PeriodicalIF":0.5,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143553795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Subalgebras of Étale Algebras in Braided Fusion Categories 编织融合类中的Étale代数子代数
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-02-03 DOI: 10.1007/s10468-025-10314-z
Sebastian Burciu
{"title":"Subalgebras of Étale Algebras in Braided Fusion Categories","authors":"Sebastian Burciu","doi":"10.1007/s10468-025-10314-z","DOIUrl":"10.1007/s10468-025-10314-z","url":null,"abstract":"<div><p>In (Davydov et al. Selecta Mathematica (N.S.) <b>19</b>, 237–269 2013, Rem. 3.4) the authors asked the question if any étale subalgebra of an étale algebra in a braided fusion category is also étale. We give a positive answer to this question if the braided fusion category <span>(mathcal {C})</span> is pseudo-unitary and non-degenerate. In the case of a pseudo-unitary fusion category we also give a new description of the lattice correspondence from (Davydov et al. J. für die reine und angewandte Mathematik. <b>677</b>, 135–177 2013, Theorem 4.10). This new description enables us to describe the two binary operations on the lattice of fusion subcategories.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 1","pages":"211 - 238"},"PeriodicalIF":0.5,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143553841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Graded Triangular Bases
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-01-29 DOI: 10.1007/s10468-025-10315-y
Jonathan Brundan
{"title":"Graded Triangular Bases","authors":"Jonathan Brundan","doi":"10.1007/s10468-025-10315-y","DOIUrl":"10.1007/s10468-025-10315-y","url":null,"abstract":"<div><p>This article develops a practical technique for studying representations of <span>(Bbbk )</span>-linear categories arising in the categorification of quantum groups. We work in terms of locally unital algebras which are <span>(mathbb {Z})</span>-graded with graded pieces that are finite-dimensional and bounded below, developing a theory of <i>graded triangular bases</i> for such algebras. The definition is a graded extension of the notion of triangular basis as formulated in Brundan and Stroppel (Mem. Amer. Math. Soc. <b>293</b>(1459), vii+152 2024). However, in the general graded setting, finitely generated projective modules often fail to be Noetherian, so that existing results from the study of highest weight categories are not directly applicable. Nevertheless, we show that there is still a good theory of <i>standard modules</i>. In motivating examples arising from Kac-Moody 2-categories, these modules categorify the PBW bases for the modified forms of quantum groups constructed by Wang.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 1","pages":"239 - 280"},"PeriodicalIF":0.5,"publicationDate":"2025-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143554006","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Weyl Group Twists and Representations of Quantum Affine Borel Algebras 韦尔群扭转与量子仿射玻尔代数的表征
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-01-25 DOI: 10.1007/s10468-025-10316-x
Keyu Wang
{"title":"Weyl Group Twists and Representations of Quantum Affine Borel Algebras","authors":"Keyu Wang","doi":"10.1007/s10468-025-10316-x","DOIUrl":"10.1007/s10468-025-10316-x","url":null,"abstract":"<div><p>We define categories <span>(varvec{mathcal {O}}^{varvec{w}})</span> of representations of Borel subalgebras <span>(varvec{mathcal {U}}_{!varvec{q}}varvec{mathfrak {b}})</span> of quantum affine algebras <span>(varvec{mathcal {U}}_{!varvec{q}}hat{varvec{mathfrak {g}}})</span>, which come from the category <span>(varvec{mathcal {O}})</span> twisted by Weyl group elements <span>(varvec{w})</span>. We construct inductive systems of finite-dimensional <span>(varvec{mathcal {U}}_{varvec{q}}varvec{mathfrak {b}})</span>-modules twisted by <span>(varvec{w})</span>, which provide representations in the category <span>(varvec{mathcal {O}}^{varvec{w}})</span>. We also establish a classification of simple modules in these categories <span>(varvec{mathcal {O}}^{varvec{w}})</span>. We explore convergent phenomenon of <span>(varvec{q})</span>-characters of representations of quantum affine algebras, which conjecturally give the <span>(varvec{q})</span>-characters of representations in <span>(varvec{mathcal {O}}^{varvec{w}})</span>. Furthermore, we propose a conjecture concerning the relationship between the category <span>(varvec{mathcal {O}})</span> and the twisted category <span>(varvec{mathcal {O}}^{varvec{w}})</span>, and we propose a possible connection with shifted quantum affine algebras.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 1","pages":"281 - 313"},"PeriodicalIF":0.5,"publicationDate":"2025-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143553838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Universal Enveloping Algebras of Poisson Superalgebras
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-01-10 DOI: 10.1007/s10468-024-10312-7
Thomas Lamkin
{"title":"Universal Enveloping Algebras of Poisson Superalgebras","authors":"Thomas Lamkin","doi":"10.1007/s10468-024-10312-7","DOIUrl":"10.1007/s10468-024-10312-7","url":null,"abstract":"<div><p>In this paper, we define and study the universal enveloping algebra of a Poisson superalgebra. In particular, a new PBW Theorem for Lie-Rinehart superalgebras is proved leading to a PBW Theorem for Poisson superalgebras, we show the universal enveloping algebra of a Poisson Hopf superalgebra (resp. Poisson-Ore extension) is a Hopf superalgebra (resp. iterated Ore extension), and we study the universal enveloping algebra for interesting classes of Poisson superalgebras such as Poisson symplectic superalgebras.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 1","pages":"157 - 191"},"PeriodicalIF":0.5,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-024-10312-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143553954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Generalization of the Nilpotency Index of the Radical of the Module Category of an Algebra
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-01-09 DOI: 10.1007/s10468-024-10307-4
Claudia Chaio, Pamela Suarez
{"title":"A Generalization of the Nilpotency Index of the Radical of the Module Category of an Algebra","authors":"Claudia Chaio,&nbsp;Pamela Suarez","doi":"10.1007/s10468-024-10307-4","DOIUrl":"10.1007/s10468-024-10307-4","url":null,"abstract":"<div><p>Let <i>A</i> be a finite dimensional representation-finite algebra over an algebraically closed field. The aim of this work is to generalize the results proven in [8]. Precisely, we determine which vertices of <span>(Q_A)</span> are sufficient to be considered in order to compute the nilpotency index of the radical of the module category of a monomial algebra and a toupie algebra <i>A</i>, when the Auslander-Reiten quiver is not necessarily a component with length.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 1","pages":"81 - 99"},"PeriodicalIF":0.5,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143553953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the General Ranks of QP Representations
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-01-04 DOI: 10.1007/s10468-024-10306-5
JiaRui Fei
{"title":"On the General Ranks of QP Representations","authors":"JiaRui Fei","doi":"10.1007/s10468-024-10306-5","DOIUrl":"10.1007/s10468-024-10306-5","url":null,"abstract":"<div><p>We propose a mutation formula for the general rank from a principal component <span>({{,textrm{PC},}}(delta ))</span> of representations to another one <span>({{,textrm{PC},}}({epsilon }))</span> for a quiver with potential. We give sufficient conditions for the formula to hold. In particular, the formula holds when any of <span>(delta )</span> and <span>({epsilon })</span> is reachable. We discover several related mutation invariants.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 1","pages":"47 - 79"},"PeriodicalIF":0.5,"publicationDate":"2025-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143553902","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Weak G-identities for the Pair ((M_2( mathbb {C}),sl_2( mathbb {C})))
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-01-02 DOI: 10.1007/s10468-024-10309-2
Ramon Códamo, Plamen Koshlukov
{"title":"Weak G-identities for the Pair ((M_2( mathbb {C}),sl_2( mathbb {C})))","authors":"Ramon Códamo,&nbsp;Plamen Koshlukov","doi":"10.1007/s10468-024-10309-2","DOIUrl":"10.1007/s10468-024-10309-2","url":null,"abstract":"<div><p>In this paper we study algebras acted on by a finite group <i>G</i> and the corresponding <i>G</i>-identities. Let <span>(M_2( mathbb {C}))</span> be the <span>(2times 2)</span> matrix algebra over the field of complex numbers <span>( mathbb {C})</span> and let <span>(sl_2( mathbb {C}))</span> be the Lie algebra of traceless matrices in <span>(M_2( mathbb {C}))</span>. Assume that <i>G</i> is a finite group acting as a group of automorphisms on <span>(M_2( mathbb {C}))</span>. These groups were described in the Nineteenth century, they consist of the finite subgroups of <span>(PGL_2( mathbb {C}))</span>, which are, up to conjugacy, the cyclic groups <span>( mathbb {Z}_n)</span>, the dihedral groups <span>(D_n)</span> (of order 2<i>n</i>), the alternating groups <span>( A_4)</span> and <span>(A_5)</span>, and the symmetric group <span>(S_4)</span>. The <i>G</i>-identities for <span>(M_2( mathbb {C}))</span> were described by Berele. The finite groups acting on <span>(sl_2( mathbb {C}))</span> are the same as those acting on <span>(M_2( mathbb {C}))</span>. The <i>G</i>-identities for the Lie algebra of the traceless <span>(sl_2( mathbb {C}))</span> were obtained by Mortari and by the second author. We study the weak <i>G</i>-identities of the pair <span>((M_2( mathbb {C}), sl_2( mathbb {C})))</span>, when <i>G</i> is a finite group. Since every automorphism of the pair is an automorphism for <span>(M_2( mathbb {C}))</span>, it follows from this that <i>G</i> is one of the groups above. In this paper we obtain bases of the weak <i>G</i>-identities for the pair <span>((M_2( mathbb {C}), sl_2( mathbb {C})))</span> when <i>G</i> is a finite group acting as a group of automorphisms.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 1","pages":"125 - 141"},"PeriodicalIF":0.5,"publicationDate":"2025-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143553706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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