Hao Liwith an appendix by Myungbo Shim, Shoma Sugimotowith an appendix by Myungbo Shim
We introduce a new concept named shift system. This is a purely Lie algebraic setting to develop the geometric representation theory of Feigin-Tipunin construction. After reformulating the discussion in past works of the second author under this new setting, as an application, we extend almost all the main results of these works to the (multiplet) principal W-algebra at positive integer level associated with a simple Lie algebra $mathfrak{g}$ and Lie superalgebra $mathfrak{osp}(1|2n)$, respectively. This paper also contains an appendix by Myungbo Shim on the relationship between Feigin-Tipunin construction and recent quantum field theories.
{"title":"Shift system and its applications","authors":"Hao Liwith an appendix by Myungbo Shim, Shoma Sugimotowith an appendix by Myungbo Shim","doi":"arxiv-2409.07381","DOIUrl":"https://doi.org/arxiv-2409.07381","url":null,"abstract":"We introduce a new concept named shift system. This is a purely Lie algebraic\u0000setting to develop the geometric representation theory of Feigin-Tipunin\u0000construction. After reformulating the discussion in past works of the second\u0000author under this new setting, as an application, we extend almost all the main\u0000results of these works to the (multiplet) principal W-algebra at positive\u0000integer level associated with a simple Lie algebra $mathfrak{g}$ and Lie\u0000superalgebra $mathfrak{osp}(1|2n)$, respectively. This paper also contains an\u0000appendix by Myungbo Shim on the relationship between Feigin-Tipunin\u0000construction and recent quantum field theories.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187292","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given the maximal compact subalgebra $mathfrak{k}(A)$ of a split-real Kac-Moody algebra $mathfrak{g}(A)$ of type $A$, we study certain finite-dimensional representations of $mathfrak{k}(A)$, that do not lift to the maximal compact subgroup $K(A)$ of the minimal Kac-Moody group $G(A)$ associated to $mathfrak{g}(A)$ but only to its spin cover $Spin(A)$. Currently, four elementary of these so-called spin representations are known. We study their (ir-)reducibility, semi-simplicity, and lift to the group level. The interaction of these representations with the spin-extended Weyl-group is used to derive a partial parametrization result of the representation matrices by the real roots of $mathfrak{g}(A)$.
{"title":"Higher spin representations of maximal compact subalgebras of simply-laced Kac-Moody-algebras","authors":"Robin Lautenbacher, Ralf Köhl","doi":"arxiv-2409.07247","DOIUrl":"https://doi.org/arxiv-2409.07247","url":null,"abstract":"Given the maximal compact subalgebra $mathfrak{k}(A)$ of a split-real\u0000Kac-Moody algebra $mathfrak{g}(A)$ of type $A$, we study certain\u0000finite-dimensional representations of $mathfrak{k}(A)$, that do not lift to\u0000the maximal compact subgroup $K(A)$ of the minimal Kac-Moody group $G(A)$\u0000associated to $mathfrak{g}(A)$ but only to its spin cover $Spin(A)$.\u0000Currently, four elementary of these so-called spin representations are known.\u0000We study their (ir-)reducibility, semi-simplicity, and lift to the group level.\u0000The interaction of these representations with the spin-extended Weyl-group is\u0000used to derive a partial parametrization result of the representation matrices\u0000by the real roots of $mathfrak{g}(A)$.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note, after recalling a proof of the Macdonald identities for untwisted affine root systems, we derive the Macdonald identities for twisted affine root systems.
{"title":"Macdonald Identities: revisited","authors":"K. Iohara, Y. Saito","doi":"arxiv-2409.07317","DOIUrl":"https://doi.org/arxiv-2409.07317","url":null,"abstract":"In this note, after recalling a proof of the Macdonald identities for\u0000untwisted affine root systems, we derive the Macdonald identities for twisted\u0000affine root systems.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we prove that if (A, B, C) is a recollement of abelian categories, then wakamatsu tilting (resp. weak support tau-tilting) subcategories in A and C can induce wakamatsu tilting (resp. weak support tau-tilting) subcategories in B, and the converses hold under natural assumptions. As an application, we mainly consider the relationship of tau-cotorsion torsion triples in (A, B, C).
在本文中，我们证明了如果（A，B，C）是abeliancategories的重列，那么A和C中的若松倾斜（respect. weak support tau-tilting）子类可以诱导B中的若松倾斜（respect. weak supporttau-tilting）子类，并且在自然假设下对话成立。作为应用，我们主要考虑（A, B, C）中的tau-cotorsion扭转三元组的关系。
{"title":"Wakamatsu tilting subcategories and weak support tau-tilting subcategories in recollement","authors":"Yongduo Wang, Hongyang Luo, Jian He, Dejun Wu","doi":"arxiv-2409.07026","DOIUrl":"https://doi.org/arxiv-2409.07026","url":null,"abstract":"In this article, we prove that if (A, B, C) is a recollement of abelian\u0000categories, then wakamatsu tilting (resp. weak support tau-tilting)\u0000subcategories in A and C can induce wakamatsu tilting (resp. weak support\u0000tau-tilting) subcategories in B, and the converses hold under natural\u0000assumptions. As an application, we mainly consider the relationship of\u0000tau-cotorsion torsion triples in (A, B, C).","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $Gamma$ be the free group $F_n$ of $n$ generators, resp. the fundamental group $pi_1(Sigma_g)$ of a closed, connnected, orientatble surface of genus $g geq 2$. We show that the charater variety of irreducible, resp. Zariski dense, Anosov representations of $Gamma$ into $SL(n, C)$ is a complex manifold of (complex) dimension $(n-1)(n^2-1)$, resp. $(2g-2) (n^2-1)$. For $Gamma=pi_1(Sigma_g)$, we also show that these character varieties are holomorphic symplectic manifolds.
{"title":"On Character Variety of Anosov Representations","authors":"Krishnendu Gongopadhyay, Tathagata Nayak","doi":"arxiv-2409.07316","DOIUrl":"https://doi.org/arxiv-2409.07316","url":null,"abstract":"Let $Gamma$ be the free group $F_n$ of $n$ generators, resp. the fundamental\u0000group $pi_1(Sigma_g)$ of a closed, connnected, orientatble surface of genus\u0000$g geq 2$. We show that the charater variety of irreducible, resp. Zariski\u0000dense, Anosov representations of $Gamma$ into $SL(n, C)$ is a complex\u0000manifold of (complex) dimension $(n-1)(n^2-1)$, resp. $(2g-2) (n^2-1)$. For\u0000$Gamma=pi_1(Sigma_g)$, we also show that these character varieties are\u0000holomorphic symplectic manifolds.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study linear versions of Reedy categories in relation with finite dimensional algebras and abelian model structures. We prove that, for a linear Reedy category $mathcal{C}$ over a field, the category of left $mathcal{C}$--modules admits a highest weight structure, which in case $mathcal{C}$ is finite corresponds to a quasi-hereditary algebra with an exact Borel subalgebra. We also lift complete cotorsion pairs and abelian model structures to certain categories of additive functors indexed by linear Reedy categories, generalizing analogous results from the hereditary case.
{"title":"Linear Reedy categories, quasi-hereditary algebras and model structures","authors":"Georgios Dalezios, Jan Stovicek","doi":"arxiv-2409.06823","DOIUrl":"https://doi.org/arxiv-2409.06823","url":null,"abstract":"We study linear versions of Reedy categories in relation with finite\u0000dimensional algebras and abelian model structures. We prove that, for a linear\u0000Reedy category $mathcal{C}$ over a field, the category of left\u0000$mathcal{C}$--modules admits a highest weight structure, which in case\u0000$mathcal{C}$ is finite corresponds to a quasi-hereditary algebra with an exact\u0000Borel subalgebra. We also lift complete cotorsion pairs and abelian model\u0000structures to certain categories of additive functors indexed by linear Reedy\u0000categories, generalizing analogous results from the hereditary case.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"67 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187296","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We classify $n$-representation infinite algebras $Lambda$ of type ~A. This type is defined by requiring that $Lambda$ has higher preprojective algebra $Pi_{n+1}(Lambda) simeq k[x_1, ldots, x_{n+1}] ast G$, where $G leq operatorname{SL}_{n+1}(k)$ is finite abelian. For the classification, we group these algebras according to a more refined type, and give a combinatorial characterisation of these types. This is based on so-called height functions, which generalise the height function of a perfect matching in a Dimer model. In terms of toric geometry and McKay correspondence, the types form a lattice simplex of junior elements of $G$. We show that all algebras of the same type are related by iterated $n$-APR tilting, and hence are derived equivalent. By disallowing certain tilts, we turn this set into a finite distributive lattice, and we construct its maximal and minimal elements.
{"title":"A classification of $n$-representation infinite algebras of type Ã","authors":"Darius Dramburg, Oleksandra Gasanova","doi":"arxiv-2409.06553","DOIUrl":"https://doi.org/arxiv-2409.06553","url":null,"abstract":"We classify $n$-representation infinite algebras $Lambda$ of type ~A. This\u0000type is defined by requiring that $Lambda$ has higher preprojective algebra\u0000$Pi_{n+1}(Lambda) simeq k[x_1, ldots, x_{n+1}] ast G$, where $G leq\u0000operatorname{SL}_{n+1}(k)$ is finite abelian. For the classification, we group\u0000these algebras according to a more refined type, and give a combinatorial\u0000characterisation of these types. This is based on so-called height functions,\u0000which generalise the height function of a perfect matching in a Dimer model. In\u0000terms of toric geometry and McKay correspondence, the types form a lattice\u0000simplex of junior elements of $G$. We show that all algebras of the same type\u0000are related by iterated $n$-APR tilting, and hence are derived equivalent. By\u0000disallowing certain tilts, we turn this set into a finite distributive lattice,\u0000and we construct its maximal and minimal elements.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The S-dual $(mathbf G^veecurvearrowrightmathbf M^vee)$ of the pair $(mathbf Gcurvearrowrightmathbf M)$ of a smooth affine algebraic symplectic manifold $mathbf M$ with hamiltonian action of a complex reductive group $mathbf G$ was introduced implicitly in [arXiv:1706.02112] and explicitly in [arXiv:1807.09038] under the cotangent type assumption. The definition was a modification of the definition of Coulomb branches of gauge theories in [arXiv:1601.03586]. It was motivated by the S-duality of boundary conditions of 4-dimensional $mathcal N=4$ super Yang-Mills theory, studied by Gaiotto and Witten [arXiv:0807.3720]. It is also relevant to the relative Langlands duality proposed by Ben-Zvi, Sakellaridis and Venkatesh. In this article, we review the definition and properties of S-dual.
S-dual $(mathbf G^veecurvearrowrightmathbf M^vee)$ of the pair$(mathbf Gcurvearrowrightmathbf M)$ of a smooth affine algebraic symplecticmanifold $mathbf M$ with hamiltonian action of a complex reductive group$mathbf G$ 在[arXiv:1706.02112]中隐含地提出，并在[arXiv:1807.09038]中根据余切型假设明确地提出。这个定义是对[arXiv:1601.03586]中规理论库仑分支定义的修正。它是由 Gaiotto 和 Witten [arXiv:0807.3720]研究的 4 维 $mathcal N=4$ 超级杨-米尔斯理论边界条件的 S 对偶性激发的。它也与 Ben-Zvi、Sakellaridis 和 Venkatesh 提出的相对朗兰兹对偶性有关。本文回顾了 S 对偶的定义和性质。
{"title":"S-dual of Hamiltonian $mathbf G$ spaces and relative Langlands duality","authors":"Hiraku Nakajima","doi":"arxiv-2409.06303","DOIUrl":"https://doi.org/arxiv-2409.06303","url":null,"abstract":"The S-dual $(mathbf G^veecurvearrowrightmathbf M^vee)$ of the pair\u0000$(mathbf Gcurvearrowrightmathbf M)$ of a smooth affine algebraic symplectic\u0000manifold $mathbf M$ with hamiltonian action of a complex reductive group\u0000$mathbf G$ was introduced implicitly in [arXiv:1706.02112] and explicitly in\u0000[arXiv:1807.09038] under the cotangent type assumption. The definition was a\u0000modification of the definition of Coulomb branches of gauge theories in\u0000[arXiv:1601.03586]. It was motivated by the S-duality of boundary conditions of\u00004-dimensional $mathcal N=4$ super Yang-Mills theory, studied by Gaiotto and\u0000Witten [arXiv:0807.3720]. It is also relevant to the relative Langlands duality\u0000proposed by Ben-Zvi, Sakellaridis and Venkatesh. In this article, we review the\u0000definition and properties of S-dual.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Dylan Johnston, Diego Martín Duro, Dmitriy Rumynin
In this article, we classify disconnected reductive groups over an algebraically closed field with a few caveats. Internal parts of our result are both a classification of finite groups and a classification of integral representations of a fixed finite group. Modulo these classifications - which are impossible in different senses - our main result explicitly tabulates the groups with an efficient algorithm. Besides this, we obtain new results about the representation theory of disconnected reductive groups in characteristic zero. We give two descriptions of their representation rings and prove that their Knutson Index is finite.
{"title":"Disconnected Reductive Groups: Classification and Representations","authors":"Dylan Johnston, Diego Martín Duro, Dmitriy Rumynin","doi":"arxiv-2409.06375","DOIUrl":"https://doi.org/arxiv-2409.06375","url":null,"abstract":"In this article, we classify disconnected reductive groups over an\u0000algebraically closed field with a few caveats. Internal parts of our result are\u0000both a classification of finite groups and a classification of integral\u0000representations of a fixed finite group. Modulo these classifications - which\u0000are impossible in different senses - our main result explicitly tabulates the\u0000groups with an efficient algorithm. Besides this, we obtain new results about\u0000the representation theory of disconnected reductive groups in characteristic\u0000zero. We give two descriptions of their representation rings and prove that\u0000their Knutson Index is finite.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Drawing inspiration from the works of Beligiannis-Marmaridis and Lin, we refine the axioms for a right $(n+2)$-angulated category and give some examples of such categories. Interestingly, we show that the morphism axiom for a right $(n+2)$-angulated category is actually redundant. Moreover, we prove that the higher octahedral axiom is equivalent to the mapping cone axiom for a right $(n+2)$-angulated category.
{"title":"The axioms for right (n+2)-angulated categories","authors":"Jing He, Jiangsha Li","doi":"arxiv-2409.05561","DOIUrl":"https://doi.org/arxiv-2409.05561","url":null,"abstract":"Drawing inspiration from the works of Beligiannis-Marmaridis and Lin, we\u0000refine the axioms for a right $(n+2)$-angulated category and give some examples\u0000of such categories. Interestingly, we show that the morphism axiom for a right\u0000$(n+2)$-angulated category is actually redundant. Moreover, we prove that the\u0000higher octahedral axiom is equivalent to the mapping cone axiom for a right\u0000$(n+2)$-angulated category.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187123","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：