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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 泊松超代数的普适包络代数
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 代数模范畴的根幂零指标的推广
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 关于高级专员代表的一般职级
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}))) 对的弱g恒等式 ((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
Some Examples of Bicrossed Products with the Rapid Decay Property 具有快速衰减特性的双交叉产物的几个例子
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-12-30 DOI: 10.1007/s10468-024-10308-3
Hua Wang
{"title":"Some Examples of Bicrossed Products with the Rapid Decay Property","authors":"Hua Wang","doi":"10.1007/s10468-024-10308-3","DOIUrl":"10.1007/s10468-024-10308-3","url":null,"abstract":"<div><p>We consider bicrossed products obtained by twisting compact semi-direct products by a suitable finite subgroup. We give a practical criterion for the rapid decay property and polynomial growth of the dual of such bicrossed products under a mild restriction. Using this theory, we construct concrete new examples of discrete quantum groups possessing the rapid decay property but not growing polynomially.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 1","pages":"101 - 123"},"PeriodicalIF":0.5,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143554030","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
A Note on the Quantum Berezinian for the Double Yangian of the Lie Superalgebra (mathfrak {gl}_{m|n}) 关于 Lie Superalgebra (mathfrak {gl}_{m|n}) 的双扬琴量子贝雷津尼的注释
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-12-28 DOI: 10.1007/s10468-024-10310-9
Lucia Bagnoli, Slaven Kožić
{"title":"A Note on the Quantum Berezinian for the Double Yangian of the Lie Superalgebra (mathfrak {gl}_{m|n})","authors":"Lucia Bagnoli,&nbsp;Slaven Kožić","doi":"10.1007/s10468-024-10310-9","DOIUrl":"10.1007/s10468-024-10310-9","url":null,"abstract":"<div><p>In this note, we generalize the notion of quantum Berezinian to the double Yangian <span>(textrm{DY}(mathfrak {gl}_{m|n}))</span> of the Lie superalgebra <span>( mathfrak {gl}_{m|n} )</span>. We show that its coefficients form a family of algebraically independent topological generators of the center of <span>(textrm{DY}(mathfrak {gl}_{m|n}))</span>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 1","pages":"143 - 155"},"PeriodicalIF":0.5,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143553920","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
Representations of the Super-Yangian of Type D(n, m) D（n, m）型超杨延的表示
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-12-28 DOI: 10.1007/s10468-024-10304-7
A. I. Molev
{"title":"Representations of the Super-Yangian of Type D(n, m)","authors":"A. I. Molev","doi":"10.1007/s10468-024-10304-7","DOIUrl":"10.1007/s10468-024-10304-7","url":null,"abstract":"<div><p>We consider the classification problem for finite-dimensional irreducible representations of the Yangians associated with the orthosymplectic Lie superalgebras <span>(mathfrak {osp}_{2n|2m})</span> with <span>(ngeqslant 2)</span>. We give necessary conditions for an irreducible highest weight representation to be finite-dimensional. We conjecture that these conditions are also sufficient and prove the conjecture for a class of representations with linear highest weights. The arguments are based on a new type of odd reflections for the Yangian associated with <span>(mathfrak {osp}_{2|2})</span>. In the Appendix, we construct an isomorphism between the Yangians associated with the Lie superalgebras <span>(mathfrak {osp}_{2|2})</span> and <span>(mathfrak {gl}_{1|2})</span>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 1","pages":"25 - 45"},"PeriodicalIF":0.5,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-024-10304-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143553919","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
Extremal Weight Crystals Over Affine Lie Algebras of Infinite Rank 无限秩仿射李代数上的极值权晶体
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-12-21 DOI: 10.1007/s10468-024-10302-9
Taehyeok Heo
{"title":"Extremal Weight Crystals Over Affine Lie Algebras of Infinite Rank","authors":"Taehyeok Heo","doi":"10.1007/s10468-024-10302-9","DOIUrl":"10.1007/s10468-024-10302-9","url":null,"abstract":"<div><p>We explain extremal weight crystals over affine Lie algebras of infinite rank using combinatorial models: a spinor model due to Kwon, and an infinite rank analogue of Kashiwara-Nakashima tableaux due to Lecouvey. In particular, we show that Lecouvey’s tableau model is isomorphic to an extremal weight crystal of level zero. Using these combinatorial models, we explain an algebra structure of the Grothendieck ring for a category consisting of some extremal weight crystals.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 1","pages":"1 - 24"},"PeriodicalIF":0.5,"publicationDate":"2024-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-024-10302-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143554018","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 Quantum Lakshmibai-Seshadri Paths for an Arbitrary Weight 任意权的量子Lakshmibai-Seshadri路径的推广
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-12-11 DOI: 10.1007/s10468-024-10298-2
Takafumi Kouno, Satoshi Naito
{"title":"A Generalization of Quantum Lakshmibai-Seshadri Paths for an Arbitrary Weight","authors":"Takafumi Kouno,&nbsp;Satoshi Naito","doi":"10.1007/s10468-024-10298-2","DOIUrl":"10.1007/s10468-024-10298-2","url":null,"abstract":"<div><p>We construct an injective weight-preserving map (called the forgetful map) from the set of all admissible subsets in the quantum alcove model associated to an arbitrary weight. The image of this forgetful map can be explicitly described by introducing the notion of “interpolated quantum Lakshmibai-Seshadri (QLS for short) paths”, which can be thought of as a generalization of quantum Lakshmibai-Seshadri paths. As an application, we reformulate, in terms of interpolated QLS paths, an identity of Chevalley type for the graded characters of Demazure submodules of a level-zero extremal weight module over a quantum affine algebra, which is a representation-theoretic analog of the Chevalley formula for the torus-equivariant <i>K</i>-group of a semi-infinite flag manifold.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 6","pages":"2321 - 2353"},"PeriodicalIF":0.5,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-024-10298-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142994451","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
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