The generalized knots-quivers correspondence extends the original knots-quivers correspondence, by allowing higher level generators of quiver generating series. In this paper we explore the underlined combinatorics of such generating series, relationship with the BPS numbers of a corresponding knot, and new combinatorial interpretations of the coefficients of generating series.
{"title":"Lattice paths and quiver generating series with higher level generators","authors":"Dušan Đorđević, Marko Stošić","doi":"arxiv-2408.01832","DOIUrl":"https://doi.org/arxiv-2408.01832","url":null,"abstract":"The generalized knots-quivers correspondence extends the original\u0000knots-quivers correspondence, by allowing higher level generators of quiver\u0000generating series. In this paper we explore the underlined combinatorics of\u0000such generating series, relationship with the BPS numbers of a corresponding\u0000knot, and new combinatorial interpretations of the coefficients of generating\u0000series.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949159","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}
It is shown that every $2$-shifted Poisson structure on a finitely generated semi-free commutative differential graded algebra $A$ defines a very explicit infinitesimal $2$-braiding on the homotopy $2$-category of the symmetric monoidal dg-category of finitely generated semi-free $A$-dg-modules. This provides a concrete realization, to first order in the deformation parameter $hbar$, of the abstract deformation quantization results in derived algebraic geometry due to Calaque, Pantev, To"en, Vaqui'e and Vezzosi. Of particular interest is the case when $A$ is the Chevalley-Eilenberg algebra of a higher Lie algebra, where the braided monoidal deformations developed in this paper may be interpreted as candidates for representation categories of `higher quantum groups'.
{"title":"Infinitesimal 2-braidings from 2-shifted Poisson structures","authors":"Cameron Kemp, Robert Laugwitz, Alexander Schenkel","doi":"arxiv-2408.00391","DOIUrl":"https://doi.org/arxiv-2408.00391","url":null,"abstract":"It is shown that every $2$-shifted Poisson structure on a finitely generated\u0000semi-free commutative differential graded algebra $A$ defines a very explicit\u0000infinitesimal $2$-braiding on the homotopy $2$-category of the symmetric\u0000monoidal dg-category of finitely generated semi-free $A$-dg-modules. This\u0000provides a concrete realization, to first order in the deformation parameter\u0000$hbar$, of the abstract deformation quantization results in derived algebraic\u0000geometry due to Calaque, Pantev, To\"en, Vaqui'e and Vezzosi. Of particular\u0000interest is the case when $A$ is the Chevalley-Eilenberg algebra of a higher\u0000Lie algebra, where the braided monoidal deformations developed in this paper\u0000may be interpreted as candidates for representation categories of `higher\u0000quantum groups'.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141881191","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 functors between the category of vertex algebras and that of Costello-Gwilliam factorization algebras on the complex plane $mathbb{C}$, without analytic structures such as differentiable vector spaces, nuclear spaces, and bornological vector spaces. We prove that this pair of functors is an adjoint pair and that the functor from vertex algebras to factorization algebras is fully faithful. Also, we identify the class of factorization algebras that are categorically equivalent to vertex algebras. To illustrate, we check the compatibility with the commutative structures and the factorization algebras constructed as factorization envelopes, including the Kac-Moody factorization algebra, the quantum observables of the $betagamma$ system, and the Virasoro factorization algebra.
{"title":"An algebraic construction of functors between vertex algebras and Costello-Gwilliam factorization algebras","authors":"Yusuke Nishinaka","doi":"arxiv-2408.00412","DOIUrl":"https://doi.org/arxiv-2408.00412","url":null,"abstract":"We construct functors between the category of vertex algebras and that of\u0000Costello-Gwilliam factorization algebras on the complex plane $mathbb{C}$,\u0000without analytic structures such as differentiable vector spaces, nuclear\u0000spaces, and bornological vector spaces. We prove that this pair of functors is\u0000an adjoint pair and that the functor from vertex algebras to factorization\u0000algebras is fully faithful. Also, we identify the class of factorization\u0000algebras that are categorically equivalent to vertex algebras. To illustrate,\u0000we check the compatibility with the commutative structures and the\u0000factorization algebras constructed as factorization envelopes, including the\u0000Kac-Moody factorization algebra, the quantum observables of the $betagamma$\u0000system, and the Virasoro factorization algebra.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"56 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141881190","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}
Erik Brodsky, Eva Engel, Connor Panish, Lillian Stolberg
The Type D asymmetric simple exclusion process (ASEP) is a particle system involving two classes of particles that can be viewed from both a probabilistic and an algebraic perspective (arXiv:2011.13473). From a probabilistic perspective, we perform stochastic fusion on the Type D ASEP and analyze the outcome on generator matrices, limits of drift speed, stationary distributions, and Markov self-duality. From an algebraic perspective, we construct a fused Type D ASEP system from a Casimir element of $U_q(so_6)$, using crystal bases to analyze and manipulate various representations of $U_q(so_6)$. We conclude that both approaches produce different processes and therefore the previous method of arXiv:1908.02359, which analyzed the usual ASEP, does not generalize to all finite-dimensional simple Lie algebras.
D型非对称简单排斥过程（ASEP）是一个涉及两类粒子的粒子系统，可以从概率和代数的角度来看待它（arXiv:2011.13473）。从概率论的角度，我们对 D 型 ASEP 进行了随机融合，并分析了生成矩阵、漂移速度极限、静态分布和马尔可夫自偶性的结果。从代数的角度看，我们从$U_q(so_6)$的卡西米尔元构建了一个融合的D型ASEP系统，并利用晶体基础分析和处理了$U_q(so_6)$的各种表示。我们得出结论：这两种方法会产生不同的过程，因此 arXiv:1908.02359 以前分析通常 ASEP 的方法并不能推广到所有有限维简单李代数。
{"title":"Comparative Analyses of the Type D ASEP: Stochastic Fusion and Crystal Bases","authors":"Erik Brodsky, Eva Engel, Connor Panish, Lillian Stolberg","doi":"arxiv-2407.21015","DOIUrl":"https://doi.org/arxiv-2407.21015","url":null,"abstract":"The Type D asymmetric simple exclusion process (ASEP) is a particle system\u0000involving two classes of particles that can be viewed from both a probabilistic\u0000and an algebraic perspective (arXiv:2011.13473). From a probabilistic\u0000perspective, we perform stochastic fusion on the Type D ASEP and analyze the\u0000outcome on generator matrices, limits of drift speed, stationary distributions,\u0000and Markov self-duality. From an algebraic perspective, we construct a fused\u0000Type D ASEP system from a Casimir element of $U_q(so_6)$, using crystal bases\u0000to analyze and manipulate various representations of $U_q(so_6)$. We conclude\u0000that both approaches produce different processes and therefore the previous\u0000method of arXiv:1908.02359, which analyzed the usual ASEP, does not generalize\u0000to all finite-dimensional simple Lie algebras.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869726","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}
For a root of unity $zeta$ of odd prime order, we restrict coefficients of non-semisimple quantum representations of mapping class groups associated with the small quantum group $mathfrak{u}_zeta mathfrak{sl}_2$ from $mathbb{Q}(zeta)$ to $mathbb{Z}[zeta]$. We do this by exhibiting explicit bases of states spaces that span $mathbb{Z}[zeta]$-lattices that are invariant under projective actions of mapping class groups.
{"title":"On Integrality of Non-Semisimple Quantum Representations of Mapping Class Groups","authors":"Marco De Renzi, Jules Martel","doi":"arxiv-2407.20644","DOIUrl":"https://doi.org/arxiv-2407.20644","url":null,"abstract":"For a root of unity $zeta$ of odd prime order, we restrict coefficients of\u0000non-semisimple quantum representations of mapping class groups associated with\u0000the small quantum group $mathfrak{u}_zeta mathfrak{sl}_2$ from\u0000$mathbb{Q}(zeta)$ to $mathbb{Z}[zeta]$. We do this by exhibiting explicit\u0000bases of states spaces that span $mathbb{Z}[zeta]$-lattices that are\u0000invariant under projective actions of mapping class groups.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"161 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869727","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 the Factorization Paradox from the bottom up by adapting methods from perturbative renormalization. Just as quantum field theories are plagued with loop divergences that need to be cancelled systematically by introducing counterterms, gravitational path integrals are plagued by wormhole contributions that spoil the factorization of the holographic dual. These wormholes must be cancelled by some stringy effects in a UV complete, holographic theory of quantum gravity. In a simple model of two-dimensional topological gravity, we outline a gravitational analog of the recursive BPHZ procedure in order to systematically introduce ``counter-wormholes" which parametrize the unknown stringy effects that lead to factorization. Underlying this procedure is a Hopf algebra of symmetries which is analogous to the Connes--Kreimer Hopf algebra underlying perturbative renormalization. The group dual to this Hopf algebra acts to reorganize contributions from spacetimes with distinct topology, and can be seen as a gauge group relating various equivalent ways of constructing a factorizing gravitational path integral.
{"title":"Wormhole Renormalization: The gravitational path integral, holography, and a gauge group for topology change","authors":"Elliott Gesteau, Matilde Marcolli, Jacob McNamara","doi":"arxiv-2407.20324","DOIUrl":"https://doi.org/arxiv-2407.20324","url":null,"abstract":"We study the Factorization Paradox from the bottom up by adapting methods\u0000from perturbative renormalization. Just as quantum field theories are plagued\u0000with loop divergences that need to be cancelled systematically by introducing\u0000counterterms, gravitational path integrals are plagued by wormhole\u0000contributions that spoil the factorization of the holographic dual. These\u0000wormholes must be cancelled by some stringy effects in a UV complete,\u0000holographic theory of quantum gravity. In a simple model of two-dimensional\u0000topological gravity, we outline a gravitational analog of the recursive BPHZ\u0000procedure in order to systematically introduce ``counter-wormholes\" which\u0000parametrize the unknown stringy effects that lead to factorization. Underlying\u0000this procedure is a Hopf algebra of symmetries which is analogous to the\u0000Connes--Kreimer Hopf algebra underlying perturbative renormalization. The group\u0000dual to this Hopf algebra acts to reorganize contributions from spacetimes with\u0000distinct topology, and can be seen as a gauge group relating various equivalent\u0000ways of constructing a factorizing gravitational path integral.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"295 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869728","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 introduce quantum super-spherical pairs as coideal subalgebras in general linear and orthosymplectic quantum supergroups. These subalgebras play a role of isotropy subgroups for matrices solving $mathbb{Z}_2$-graded reflection equation. They generalize quantum (pseudo)-symmetric pairs of Letzter-Kolb-Regelskis-Vlaar.
{"title":"Quantum super-spherical pairs","authors":"D. Algethami, A. Mudrov, V. Stukopin","doi":"arxiv-2407.19477","DOIUrl":"https://doi.org/arxiv-2407.19477","url":null,"abstract":"We introduce quantum super-spherical pairs as coideal subalgebras in general\u0000linear and orthosymplectic quantum supergroups. These subalgebras play a role\u0000of isotropy subgroups for matrices solving $mathbb{Z}_2$-graded reflection\u0000equation. They generalize quantum (pseudo)-symmetric pairs of\u0000Letzter-Kolb-Regelskis-Vlaar.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"74 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869798","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}
Motivated by classical investigation of conjugation invariant positive-definite functions on discrete groups, we study tracial central states on universal C*-algebras associated with compact quantum groups, where centrality is understood in the sense of invariance under the adjoint action. We fully classify such states on $q$-deformations of compact Lie groups, on free orthogonal quantum groups, quantum permutation groups and on quantum hyperoctahedral groups.
{"title":"Tracial central states on compact quantum groups","authors":"Amaury Freslon, Adam Skalski, Simeng Wang","doi":"arxiv-2407.19314","DOIUrl":"https://doi.org/arxiv-2407.19314","url":null,"abstract":"Motivated by classical investigation of conjugation invariant\u0000positive-definite functions on discrete groups, we study tracial central states\u0000on universal C*-algebras associated with compact quantum groups, where\u0000centrality is understood in the sense of invariance under the adjoint action.\u0000We fully classify such states on $q$-deformations of compact Lie groups, on\u0000free orthogonal quantum groups, quantum permutation groups and on quantum\u0000hyperoctahedral groups.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869729","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}
By the introduction of locally constant prefactorization algebras at a fixed scale, we show a mathematical incarnation of the fact that observables at a given scale of a topological field theory propagate to every scale over euclidean spaces. The key is that these prefactorization algebras over $mathbb{R}^n$ are equivalent to algebras over the little $n$-disc operad. For topological field theories with defects, we get analogous results by replacing $mathbb{R}^n$ with the spaces modelling corners $mathbb{R}^ptimesmathbb{R}^{q}_{geq 0}$. As a toy example in $1d$, we quantize, once more, constant Poisson structures.
{"title":"Algebras over not too little discs","authors":"Damien Calaque, Victor Carmona","doi":"arxiv-2407.18192","DOIUrl":"https://doi.org/arxiv-2407.18192","url":null,"abstract":"By the introduction of locally constant prefactorization algebras at a fixed\u0000scale, we show a mathematical incarnation of the fact that observables at a\u0000given scale of a topological field theory propagate to every scale over\u0000euclidean spaces. The key is that these prefactorization algebras over\u0000$mathbb{R}^n$ are equivalent to algebras over the little $n$-disc operad. For\u0000topological field theories with defects, we get analogous results by replacing\u0000$mathbb{R}^n$ with the spaces modelling corners\u0000$mathbb{R}^ptimesmathbb{R}^{q}_{geq 0}$. As a toy example in $1d$, we\u0000quantize, once more, constant Poisson structures.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"62 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141777251","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 paper, we focus on a new lower bound quantum cluster algebra which is generated by the initial quantum cluster variables and the quantum projective cluster variables of an acyclic quantum cluster algebra with principle coefficients. We show that the new lower bound quantum cluster algebra coincides with the corresponding acyclic quantum cluster algebra. Moreover, we establish a class of formulas between these generators, and obtain the dual PBW basis of this algebra.
{"title":"On the acyclic quantum cluster algebras with principle coefficients","authors":"Junyuan Huang, Xueqing Chen, Ming Ding, Fan Xu","doi":"arxiv-2407.17685","DOIUrl":"https://doi.org/arxiv-2407.17685","url":null,"abstract":"In this paper, we focus on a new lower bound quantum cluster algebra which is\u0000generated by the initial quantum cluster variables and the quantum projective\u0000cluster variables of an acyclic quantum cluster algebra with principle\u0000coefficients. We show that the new lower bound quantum cluster algebra\u0000coincides with the corresponding acyclic quantum cluster algebra. Moreover, we\u0000establish a class of formulas between these generators, and obtain the dual PBW\u0000basis of this algebra.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141777252","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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