Dražen Adamović, Chunrui Ai, Xingjun Lin, Jinwei Yang
In this paper, we show Kazhdan-Lusztig categories, that is, the categories of lower bounded generalized weight modules for certain affine vertex operator superalgebras that are locally finite modules of the underlying finite dimensional Lie superalgebra, are semisimple. Those are all representation categories of affine vertex operator superalgebras at conformal but non admissible levels. As a consequence, the categories of finite length generalized modules for these affine vertex operator superalgebras have braided tensor category structures.
{"title":"Semisimplicity of module categories of certain affine vertex operator superalgebras","authors":"Dražen Adamović, Chunrui Ai, Xingjun Lin, Jinwei Yang","doi":"arxiv-2409.11797","DOIUrl":"https://doi.org/arxiv-2409.11797","url":null,"abstract":"In this paper, we show Kazhdan-Lusztig categories, that is, the categories of\u0000lower bounded generalized weight modules for certain affine vertex operator\u0000superalgebras that are locally finite modules of the underlying finite\u0000dimensional Lie superalgebra, are semisimple. Those are all representation\u0000categories of affine vertex operator superalgebras at conformal but non\u0000admissible levels. As a consequence, the categories of finite length\u0000generalized modules for these affine vertex operator superalgebras have braided\u0000tensor category structures.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253461","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 derive the explicit form of the basic monodromy operator for the quantum loop superalgebra $mathrm{U}_q(mathcal{L}(mathfrak{sl}_{2|1}))$. Two significant additional results emerge from this derivation: simple expressions for the generating functions of the the images of the root vectors of $mathrm{U}_q(mathcal{L}(mathfrak{sl}_{2|1}))$ under the Jimbo homomorphism and explicit expressions for certain central elements of the quantum superalgebra $mathrm{U}_q(mathfrak{gl}_{2|1})$. Furthermore, we establish the relationship between these central elements and those obtained by using the Drinfeld partial trace method.
{"title":"Basic monodromy operator for quantum superalgebra","authors":"A. V. Razumov","doi":"arxiv-2409.11097","DOIUrl":"https://doi.org/arxiv-2409.11097","url":null,"abstract":"We derive the explicit form of the basic monodromy operator for the quantum\u0000loop superalgebra $mathrm{U}_q(mathcal{L}(mathfrak{sl}_{2|1}))$. Two\u0000significant additional results emerge from this derivation: simple expressions\u0000for the generating functions of the the images of the root vectors of\u0000$mathrm{U}_q(mathcal{L}(mathfrak{sl}_{2|1}))$ under the Jimbo homomorphism\u0000and explicit expressions for certain central elements of the quantum\u0000superalgebra $mathrm{U}_q(mathfrak{gl}_{2|1})$. Furthermore, we establish the\u0000relationship between these central elements and those obtained by using the\u0000Drinfeld partial trace method.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253462","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 construct a 2-functor from the Kac-Moody 2-category for the extended quantum affine sl(3) to the homotopy 2-category of bounded chain complexes with values in the Kac-Moody 2-category for quantum gl(3), categorifying the evaluation map between the corresponding quantum Kac-Moody algebras.
{"title":"Evaluation 2-Functors for Kac-Moody 2-Categories of Type A2","authors":"Marco Mackaay, James Macpherson, Pedro Vaz","doi":"arxiv-2409.10434","DOIUrl":"https://doi.org/arxiv-2409.10434","url":null,"abstract":"We construct a 2-functor from the Kac-Moody 2-category for the extended\u0000quantum affine sl(3) to the homotopy 2-category of bounded chain complexes with\u0000values in the Kac-Moody 2-category for quantum gl(3), categorifying the\u0000evaluation map between the corresponding quantum Kac-Moody algebras.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253463","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 $V$ be a simple, non-negatively-graded, rational, $C_2$-cofinite, and self dual vertex operator algebra, $g_1, g_2, g_3$ be three commuting finitely ordered automorphisms of $V$ such that $g_1g_2=g_3$ and $g_i^T=1$ for $i=1, 2, 3$ and $Tin N$. Suppose $M^1$ is a $g_1$-twisted module. For any $n, min frac{1}{T}N$, we construct an $A_{g_3, n}(V)$-$A_{g_2, m}(V)$-bimodule $mathcal{A}_{g_3, g_2, n, m}(M^1)$ associated to the quadruple $(M^1, g_1, g_2, g_3)$. Given an $A_{g_2, m}(V)$-module $U$, an admissible $g_3$-twisted module $mathcal{M}(M^1, U)$ is constructed. For the quadruple $(V, 1, g, g)$ for some $gin text{Aut}(V)$, $mathcal{A}_{g, g, n, m}(V)$ coincides with the $A_{g, n}(V)$-$A_{g, m}(V)$-bimodules $A_{g, n, m}(V)$ constructed by Dong-Jiang, and $mathcal{M}(V, U)$ is the generalized Verma type admissible $g$-twisted module generated by $U$. For an irreducible $g_1$-twisted module $M^1$ and an irreducible $g_2$-twisted module $M^2$, we give a construction of tensor product of $M^1$ and $M^2$ using the bimodule theory developed in this paper. As an application, a twisted version of the fusion rules theorem is established.
{"title":"Bimodules over twisted Zhu algebras and a construction of tensor product of twisted modules for vertex operator algebras","authors":"Yiyi Zhu","doi":"arxiv-2409.08995","DOIUrl":"https://doi.org/arxiv-2409.08995","url":null,"abstract":"Let $V$ be a simple, non-negatively-graded, rational, $C_2$-cofinite, and\u0000self dual vertex operator algebra, $g_1, g_2, g_3$ be three commuting finitely\u0000ordered automorphisms of $V$ such that $g_1g_2=g_3$ and $g_i^T=1$ for $i=1, 2,\u00003$ and $Tin N$. Suppose $M^1$ is a $g_1$-twisted module. For any $n, min\u0000frac{1}{T}N$, we construct an $A_{g_3, n}(V)$-$A_{g_2, m}(V)$-bimodule\u0000$mathcal{A}_{g_3, g_2, n, m}(M^1)$ associated to the quadruple $(M^1, g_1,\u0000g_2, g_3)$. Given an $A_{g_2, m}(V)$-module $U$, an admissible $g_3$-twisted\u0000module $mathcal{M}(M^1, U)$ is constructed. For the quadruple $(V, 1, g, g)$\u0000for some $gin text{Aut}(V)$, $mathcal{A}_{g, g, n, m}(V)$ coincides with the\u0000$A_{g, n}(V)$-$A_{g, m}(V)$-bimodules $A_{g, n, m}(V)$ constructed by\u0000Dong-Jiang, and $mathcal{M}(V, U)$ is the generalized Verma type admissible\u0000$g$-twisted module generated by $U$. For an irreducible $g_1$-twisted module\u0000$M^1$ and an irreducible $g_2$-twisted module $M^2$, we give a construction of\u0000tensor product of $M^1$ and $M^2$ using the bimodule theory developed in this\u0000paper. As an application, a twisted version of the fusion rules theorem is\u0000established.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253465","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 derive explicit closed formulas for the Kirillov-Kostant-Souriau (KKS) coaction maps of open path regularized holonomies of the Knizhnik-Zamolodchikov (KZ) equation, and the corresponding Poisson brackets for the Lie algebra ${rm gl}(N, mathbb{C})$. Our main technical tool is a certain projection of the generalized pentagon equation of cite{AFR2024}.
{"title":"Poisson brackets and coaction maps of regularized holonomies of the KZ equation","authors":"Anton Alekseev, Florian Naef, Muze Ren","doi":"arxiv-2409.08894","DOIUrl":"https://doi.org/arxiv-2409.08894","url":null,"abstract":"We derive explicit closed formulas for the Kirillov-Kostant-Souriau (KKS)\u0000coaction maps of open path regularized holonomies of the Knizhnik-Zamolodchikov\u0000(KZ) equation, and the corresponding Poisson brackets for the Lie algebra ${rm\u0000gl}(N, mathbb{C})$. Our main technical tool is a certain projection of the\u0000generalized pentagon equation of cite{AFR2024}.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253466","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}
Sebastiano Carpi, Christopher Raymond, Yoh Tanimoto, James E. Tener
We give an equivalence of categories between: (i) M"obius vertex algebras which are equipped with a choice of generating family of quasiprimary vectors, and (ii) (not-necessarily-unitary) M"obius-covariant Wightman conformal field theories on the unit circle. We do not impose any technical restrictions on the theories considered (such as finite-dimensional conformal weight spaces or simplicity), yielding the most general equivalence between these two axiomatizations of two-dimensional chiral conformal field theory. This provides new opportunities to study non-unitary vertex algebras using the lens of algebraic conformal field theory and operator algebras, which we demonstrate by establishing a non-unitary version of the Reeh-Schlieder theorem.
我们给出了以下范畴之间的等价性：(i) 带有准主向量生成族选择的莫比乌斯顶点代数，以及 (ii) 单位圆上的（不一定是单元的）莫比乌斯共变怀特曼共形场理论。我们没有对所考虑的理论施加任何技术限制（如有限维共形权重空间或简单性），从而在二维手性共形场理论的这两个轴化之间产生了最一般的等价性。这为利用代数共形场论和算子代数的视角研究非单元顶点代数提供了新的机会，我们通过建立非单元版本的里赫-施里德尔定理证明了这一点。
{"title":"Non-unitary Wightman CFTs and non-unitary vertex algebras","authors":"Sebastiano Carpi, Christopher Raymond, Yoh Tanimoto, James E. Tener","doi":"arxiv-2409.08454","DOIUrl":"https://doi.org/arxiv-2409.08454","url":null,"abstract":"We give an equivalence of categories between: (i) M\"obius vertex algebras\u0000which are equipped with a choice of generating family of quasiprimary vectors,\u0000and (ii) (not-necessarily-unitary) M\"obius-covariant Wightman conformal field\u0000theories on the unit circle. We do not impose any technical restrictions on the\u0000theories considered (such as finite-dimensional conformal weight spaces or\u0000simplicity), yielding the most general equivalence between these two\u0000axiomatizations of two-dimensional chiral conformal field theory. This provides\u0000new opportunities to study non-unitary vertex algebras using the lens of\u0000algebraic conformal field theory and operator algebras, which we demonstrate by\u0000establishing a non-unitary version of the Reeh-Schlieder theorem.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253467","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}
Masoud Khalkhali, Nathan Pagliaroli, Andrei Parfeni, Brayden Smith
Given a matrix model, by combining the Schwinger-Dyson equations with positivity constraints on its solutions, in the large $N$ limit one is able to obtain explicit and numerical bounds on its moments. This technique is known as bootstrapping with positivity. In this paper we use this technique to estimate the critical points and exponents of several matrix multi-models. As a proof of concept, we first show it can be used to find the well-studied quartic single matrix model's critical phenomena. We then apply the method to several similar ``unsolved" 2-matrix models with various quartic interactions. We conjecture and present strong evidence for the string susceptibility exponent for some of these models to be $gamma = 1/2$, which heuristically indicates that the continuum limit will likely be the Continuum Random Tree. For the other 2-matrix models, we find estimates of new string susceptibility exponents that may indicate a new continuum limit. We then study an unsolved 3-matrix model that generalizes the 3-colour model with cubic interactions. Additionally, for all of these models, we are able to derive explicitly the first several terms of the free energy in the large $N$ limit as a power series expansion in the coupling constants at zero by exploiting the structure of the Schwinger-Dyson equations.
{"title":"Bootstrapping the critical behavior of multi-matrix models","authors":"Masoud Khalkhali, Nathan Pagliaroli, Andrei Parfeni, Brayden Smith","doi":"arxiv-2409.07565","DOIUrl":"https://doi.org/arxiv-2409.07565","url":null,"abstract":"Given a matrix model, by combining the Schwinger-Dyson equations with\u0000positivity constraints on its solutions, in the large $N$ limit one is able to\u0000obtain explicit and numerical bounds on its moments. This technique is known as\u0000bootstrapping with positivity. In this paper we use this technique to estimate\u0000the critical points and exponents of several matrix multi-models. As a proof of\u0000concept, we first show it can be used to find the well-studied quartic single\u0000matrix model's critical phenomena. We then apply the method to several similar\u0000``unsolved\" 2-matrix models with various quartic interactions. We conjecture\u0000and present strong evidence for the string susceptibility exponent for some of\u0000these models to be $gamma = 1/2$, which heuristically indicates that the\u0000continuum limit will likely be the Continuum Random Tree. For the other\u00002-matrix models, we find estimates of new string susceptibility exponents that\u0000may indicate a new continuum limit. We then study an unsolved 3-matrix model\u0000that generalizes the 3-colour model with cubic interactions. Additionally, for\u0000all of these models, we are able to derive explicitly the first several terms\u0000of the free energy in the large $N$ limit as a power series expansion in the\u0000coupling constants at zero by exploiting the structure of the Schwinger-Dyson\u0000equations.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198757","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 establish a structure theorem, analogous to the classical result of Milnor and Moore, for differential graded Hopf algebras: any differential Hopf algebra $H$ that is free as a coalgebra carries an underlying $B_infty$ algebra structure that restricts to the subspace of primitives, and conversely $H$ may be recovered via a universal enveloping differential-2-associative algebra. This extends the work of Loday and Ronco [12] where the ungraded non-differential case was treated, and only the multibrace part of the $B_infty$ structure was found. We show that the multibrace structure of [12] originates from a twisting of a quasi-trivial structure, extending the work of Markl [14] on the $A_infty$ structure underlying any algebra with a square-zero endomorphism. In this framework it is also clear that the multibrace and $A_infty$ structures are compatible, and provide an appropriate $B_infty$ structure for the structure theorem.
{"title":"On differential Hopf algebras and $B_infty$ algebras","authors":"Imma Gálvez-Carrillo, María Ronco, Andy Tonks","doi":"arxiv-2409.06632","DOIUrl":"https://doi.org/arxiv-2409.06632","url":null,"abstract":"We establish a structure theorem, analogous to the classical result of Milnor\u0000and Moore, for differential graded Hopf algebras: any differential Hopf algebra\u0000$H$ that is free as a coalgebra carries an underlying $B_infty$ algebra\u0000structure that restricts to the subspace of primitives, and conversely $H$ may\u0000be recovered via a universal enveloping differential-2-associative algebra.\u0000This extends the work of Loday and Ronco [12] where the ungraded\u0000non-differential case was treated, and only the multibrace part of the\u0000$B_infty$ structure was found. We show that the multibrace structure of [12]\u0000originates from a twisting of a quasi-trivial structure, extending the work of\u0000Markl [14] on the $A_infty$ structure underlying any algebra with a\u0000square-zero endomorphism. In this framework it is also clear that the\u0000multibrace and $A_infty$ structures are compatible, and provide an appropriate\u0000$B_infty$ structure for the structure theorem.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198758","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 study, we develop an analogue of Brownian motion on free unitary quantum groups and provide its limit profile.
在本研究中，我们开发了自由单元量子群上的布朗运动类似物，并提供了其极限剖面。
{"title":"Brownian motion on the unitary quantum group. Construction and asymptotic study","authors":"Jean Delhaye","doi":"arxiv-2409.06552","DOIUrl":"https://doi.org/arxiv-2409.06552","url":null,"abstract":"In this study, we develop an analogue of Brownian motion on free unitary\u0000quantum groups and provide its limit profile.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225240","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 describe a proof of the following folklore theorem: If $cX = G/K$ is the homogeneous space of a simply connected compact semisimple Lie group with Poisson-Lie stabilizers, then the $q$-deformed algebras of regular functions $CC[cX_q]$ with $0
{"title":"Algebraic isomorphisms of quantized homogeneous spaces","authors":"Robert Yuncken","doi":"arxiv-2409.06139","DOIUrl":"https://doi.org/arxiv-2409.06139","url":null,"abstract":"We describe a proof of the following folklore theorem: If $cX = G/K$ is the\u0000homogeneous space of a simply connected compact semisimple Lie group with\u0000Poisson-Lie stabilizers, then the $q$-deformed algebras of regular functions\u0000$CC[cX_q]$ with $0<qleq1$ are mutually non-isomorphic as $*$-algebras.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","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":"142198759","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}
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