We study Beurling-Fourier algebras of $ q $-deformations of compact semisimple Lie groups. In particular, we show that the space of irreducible representations of the function algebras of their Drinfeld doubles is exhausted by the irreducible representations of weighted Fourier algebras associated to a certain family of central weights.
{"title":"Beurling-Fourier algebras of $ q $-deformations of compact semisimple Lie groups and complexification","authors":"Heon Lee, Christian Voigt","doi":"arxiv-2407.02132","DOIUrl":"https://doi.org/arxiv-2407.02132","url":null,"abstract":"We study Beurling-Fourier algebras of $ q $-deformations of compact\u0000semisimple Lie groups. In particular, we show that the space of irreducible\u0000representations of the function algebras of their Drinfeld doubles is exhausted\u0000by the irreducible representations of weighted Fourier algebras associated to a\u0000certain family of central weights.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141520013","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}
A tensor category $mathcal{C}$ over a field $mathbb{K}$ is said to be invertible if there's a tensor category $mathcal{D}$ such that $mathcal{C}boxtimesmathcal{D}$ is Morita equivalent to $mathrm{Vec}_{mathbb{K}}$. When $mathbb{K}$ is algebraically closed, it is well-known that the only invertible fusion category is $mathrm{Vec}_{mathbb{K}}$, and any invertible multi-fusion category is Morita equivalent to $mathrm{Vec}_{mathbb{K}}$. By contrast, we show that for general $mathbb{K}$ the invertible multi-fusion categories over a field $mathbb{K}$ are classified (up to Morita equivalence) by $H^3(mathbb{K};mathbb{G}_m)$, the third Galois cohomology of the absolute Galois group of $mathbb{K}$. We explicitly construct a representative of each class that is fusion (but not split fusion) in the sense that the unit object is simple (but not split simple). One consequence of our results is that fusion categories with braided equivalent Drinfeld centers need not be Morita equivalent when this cohomology group is nontrivial.
{"title":"Invertible Fusion Categories","authors":"Sean Sanford, Noah Snyder","doi":"arxiv-2407.02597","DOIUrl":"https://doi.org/arxiv-2407.02597","url":null,"abstract":"A tensor category $mathcal{C}$ over a field $mathbb{K}$ is said to be\u0000invertible if there's a tensor category $mathcal{D}$ such that\u0000$mathcal{C}boxtimesmathcal{D}$ is Morita equivalent to\u0000$mathrm{Vec}_{mathbb{K}}$. When $mathbb{K}$ is algebraically closed, it is\u0000well-known that the only invertible fusion category is\u0000$mathrm{Vec}_{mathbb{K}}$, and any invertible multi-fusion category is Morita\u0000equivalent to $mathrm{Vec}_{mathbb{K}}$. By contrast, we show that for\u0000general $mathbb{K}$ the invertible multi-fusion categories over a field\u0000$mathbb{K}$ are classified (up to Morita equivalence) by\u0000$H^3(mathbb{K};mathbb{G}_m)$, the third Galois cohomology of the absolute\u0000Galois group of $mathbb{K}$. We explicitly construct a representative of each\u0000class that is fusion (but not split fusion) in the sense that the unit object\u0000is simple (but not split simple). One consequence of our results is that fusion\u0000categories with braided equivalent Drinfeld centers need not be Morita\u0000equivalent when this cohomology group is nontrivial.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141549790","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}
Braided algebras are algebraic structures consisting of an algebra endowed with a Yang-Baxter operator, satisfying some compatibility conditions. Yang-Baxter Hochschild cohomology was introduced by the authors to classify infinitesimal deformations of braided algebras, and determine obstructions to quadratic deformations. Several examples of braided algebras satisfy a weaker version of commutativity, which is called braided commutativity and involves the Yang-Baxter operator of the algebra. We extend the theory of Yang-Baxter Hochschild cohomology to study braided commutative deformations of braided algebras. The resulting cohomology theory classifies infinitesimal deformations of braided algebras that are braided commutative, and provides obstructions for braided commutative quadratic deformations. We consider braided commutativity for Hopf algebras in detail, and obtain some classes of nontrivial examples.
{"title":"Deformation Cohomology for Braided Commutativity","authors":"Masahico Saito, Emanuele Zappala","doi":"arxiv-2407.02663","DOIUrl":"https://doi.org/arxiv-2407.02663","url":null,"abstract":"Braided algebras are algebraic structures consisting of an algebra endowed\u0000with a Yang-Baxter operator, satisfying some compatibility conditions.\u0000Yang-Baxter Hochschild cohomology was introduced by the authors to classify\u0000infinitesimal deformations of braided algebras, and determine obstructions to\u0000quadratic deformations. Several examples of braided algebras satisfy a weaker\u0000version of commutativity, which is called braided commutativity and involves\u0000the Yang-Baxter operator of the algebra. We extend the theory of Yang-Baxter\u0000Hochschild cohomology to study braided commutative deformations of braided\u0000algebras. The resulting cohomology theory classifies infinitesimal deformations\u0000of braided algebras that are braided commutative, and provides obstructions for\u0000braided commutative quadratic deformations. We consider braided commutativity\u0000for Hopf algebras in detail, and obtain some classes of nontrivial examples.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141549789","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 a unital associative algebra ${mathcal{SV}ir!}_{q,k}$, having $q$ and $k$ as complex parameters, generated by the elements $K^pm_m$ ($pm mgeq 0$), $T_m$ ($min mathbb{Z}$), and $G^pm_m$ ($min mathbb{Z}+{1over 2}$ in the Neveu-Schwarz sector, $min mathbb{Z}$ in the Ramond sector), satisfying relations which are at most quartic. Calculations of some low-lying Kac determinants are made, providing us with a conjecture for the factorization property of the Kac determinants. The analysis of the screening operators gives a supporting evidence for our conjecture. It is shown that by taking the limit $qrightarrow 1$ of ${mathcal{SV}ir!}_{q,k}$ we recover the ordinary ${mathcal N}=2$ superconformal algebra. We also give a nontrivial Heisenberg representation of the algebra ${mathcal{SV}ir!}_{q,k}$, making a twist of the $U(1)$ boson in the Wakimoto representation of the quantum affine algebra $U_q(widehat{mathfrak{sl}}_2)$, which naturally follows from the construction of ${mathcal{SV}ir!}_{q,k}$ by gluing the deformed $Y$-algebras of Gaiotto and Rap$check{mathrm{c}}$'{a}k.
{"title":"A quantum deformation of the ${mathcal N}=2$ superconformal algebra","authors":"H. Awata, K. Harada, H. Kanno, J. Shiraishi","doi":"arxiv-2407.00901","DOIUrl":"https://doi.org/arxiv-2407.00901","url":null,"abstract":"We introduce a unital associative algebra ${mathcal{SV}ir!}_{q,k}$, having\u0000$q$ and $k$ as complex parameters, generated by the elements $K^pm_m$ ($pm\u0000mgeq 0$), $T_m$ ($min mathbb{Z}$), and $G^pm_m$ ($min mathbb{Z}+{1over\u00002}$ in the Neveu-Schwarz sector, $min mathbb{Z}$ in the Ramond sector),\u0000satisfying relations which are at most quartic. Calculations of some low-lying\u0000Kac determinants are made, providing us with a conjecture for the factorization\u0000property of the Kac determinants. The analysis of the screening operators gives\u0000a supporting evidence for our conjecture. It is shown that by taking the limit\u0000$qrightarrow 1$ of ${mathcal{SV}ir!}_{q,k}$ we recover the ordinary\u0000${mathcal N}=2$ superconformal algebra. We also give a nontrivial Heisenberg\u0000representation of the algebra ${mathcal{SV}ir!}_{q,k}$, making a twist of the\u0000$U(1)$ boson in the Wakimoto representation of the quantum affine algebra\u0000$U_q(widehat{mathfrak{sl}}_2)$, which naturally follows from the construction\u0000of ${mathcal{SV}ir!}_{q,k}$ by gluing the deformed $Y$-algebras of Gaiotto\u0000and Rap$check{mathrm{c}}$'{a}k.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"111 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529604","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}
Christopher Beem, Anirudh Deb, Mario Martone, Carlo Meneghelli, Leonardo Rastelli
In this paper, we construct the associated vertex operator algebras for all $mathcal{N}=2$ superconformal field theories of rank one. We give a uniform presentation through free-field realizations, which turns out to be a particularly suitable framework for this task. The elementary building blocks of the construction are dictated by the low energy degrees of freedom on the Higgs branch, which are well understood for rank-one theories. We further analyze the interplay between Higgs and Coulomb data on the moduli space of vacua, which tightly constrain the overall structure of the free field realizations. Our results suggest a plausible bottom-up classification scheme for low-rank SCFTs incorporating vertex algebra techniques.
{"title":"Free field realizations for rank-one SCFTs","authors":"Christopher Beem, Anirudh Deb, Mario Martone, Carlo Meneghelli, Leonardo Rastelli","doi":"arxiv-2407.01674","DOIUrl":"https://doi.org/arxiv-2407.01674","url":null,"abstract":"In this paper, we construct the associated vertex operator algebras for all\u0000$mathcal{N}=2$ superconformal field theories of rank one. We give a uniform\u0000presentation through free-field realizations, which turns out to be a\u0000particularly suitable framework for this task. The elementary building blocks\u0000of the construction are dictated by the low energy degrees of freedom on the\u0000Higgs branch, which are well understood for rank-one theories. We further\u0000analyze the interplay between Higgs and Coulomb data on the moduli space of\u0000vacua, which tightly constrain the overall structure of the free field\u0000realizations. Our results suggest a plausible bottom-up classification scheme\u0000for low-rank SCFTs incorporating vertex algebra techniques.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"74 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141519938","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 tridiagonal algebra is defined by two generators and two relations, called the tridiagonal relations. Special cases of the tridiagonal algebra include the $q$-Onsager algebra, the positive part of the $q$-deformed enveloping algebra $U_q({widehat{mathfrak{sl}}}_2)$, and the enveloping algebra of the Onsager Lie algebra. In this paper, we introduce the $S_3$-symmetric tridiagonal algebra. This algebra has six generators. The generators can be identified with the vertices of a regular hexagon, such that nonadjacent generators commute and adjacent generators satisfy a pair of tridiagonal relations. For a $Q$-polynomial distance-regular graph $Gamma$ we turn the tensor power $V^{otimes 3}$ of the standard module $V$ into a module for an $S_3$-symmetric tridiagonal algebra. We investigate in detail the case in which $Gamma$ is a Hamming graph. We give some conjectures and open problems.
{"title":"The $S_3$-symmetric tridiagonal algebra","authors":"Paul Terwilliger","doi":"arxiv-2407.00551","DOIUrl":"https://doi.org/arxiv-2407.00551","url":null,"abstract":"The tridiagonal algebra is defined by two generators and two relations,\u0000called the tridiagonal relations. Special cases of the tridiagonal algebra\u0000include the $q$-Onsager algebra, the positive part of the $q$-deformed\u0000enveloping algebra $U_q({widehat{mathfrak{sl}}}_2)$, and the enveloping\u0000algebra of the Onsager Lie algebra. In this paper, we introduce the $S_3$-symmetric tridiagonal algebra. This\u0000algebra has six generators. The generators can be identified with the vertices\u0000of a regular hexagon, such that nonadjacent generators commute and adjacent\u0000generators satisfy a pair of tridiagonal relations. For a $Q$-polynomial\u0000distance-regular graph $Gamma$ we turn the tensor power $V^{otimes 3}$ of the\u0000standard module $V$ into a module for an $S_3$-symmetric tridiagonal algebra. We investigate in detail the case in which $Gamma$ is a Hamming graph. We\u0000give some conjectures and open problems.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"237 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141520015","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 every prime number p and integer $n>1$, a simple, involutive, non-degenerate set-theoretic solution $(X,r$) of the Yang-Baxter equation of cardinality $|X| = p^n$ is constructed. Furthermore, for every non-(square-free) positive integer m which is not the square of a prime number, a non-simple, indecomposable, irretractable, involutive, non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation of cardinality $|X| = m$ is constructed. A recent question of Castelli on the existence of singular solutions of certain type is also answered affirmatively.
{"title":"Simple solutions of the Yang-Baxter equation of cardinality $p^n$","authors":"Ferran Cedo, Jan Okninski","doi":"arxiv-2407.07907","DOIUrl":"https://doi.org/arxiv-2407.07907","url":null,"abstract":"For every prime number p and integer $n>1$, a simple, involutive,\u0000non-degenerate set-theoretic solution $(X,r$) of the Yang-Baxter equation of\u0000cardinality $|X| = p^n$ is constructed. Furthermore, for every\u0000non-(square-free) positive integer m which is not the square of a prime number,\u0000a non-simple, indecomposable, irretractable, involutive, non-degenerate\u0000set-theoretic solution $(X,r)$ of the Yang-Baxter equation of cardinality $|X|\u0000= m$ is constructed. A recent question of Castelli on the existence of singular\u0000solutions of certain type is also answered affirmatively.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"70 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141612960","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 consider the Etingof-Kazhdan quantum vertex algebra $mathcal{V}^c(mathfrak{gl}_N)$ associated with the trigonometric $R$-matrix of type $A$. By combining Li's theory of $phi$-coordinated modules and the ideas from our previous paper, we introduce the notion of deformed $phi$-coordinated quantum vertex algebra module. We show that the orthogonal twisted $h$-Yangians and restricted modules for the generalized orthogonal twisted $h$-Yangians can be equipped with the structure of (truncated) deformed $phi$-coordinated $mathcal{V}^c(mathfrak{gl}_N)$-module and demonstrate its applications.
{"title":"Associating deformed $φ$-coordinated modules for the quantum affine vertex algebra with orthogonal twisted $h$-Yangians","authors":"Lucia Bagnoli, Slaven Kožić","doi":"arxiv-2407.00515","DOIUrl":"https://doi.org/arxiv-2407.00515","url":null,"abstract":"We consider the Etingof-Kazhdan quantum vertex algebra\u0000$mathcal{V}^c(mathfrak{gl}_N)$ associated with the trigonometric $R$-matrix\u0000of type $A$. By combining Li's theory of $phi$-coordinated modules and the\u0000ideas from our previous paper, we introduce the notion of deformed\u0000$phi$-coordinated quantum vertex algebra module. We show that the orthogonal\u0000twisted $h$-Yangians and restricted modules for the generalized orthogonal\u0000twisted $h$-Yangians can be equipped with the structure of (truncated) deformed\u0000$phi$-coordinated $mathcal{V}^c(mathfrak{gl}_N)$-module and demonstrate its\u0000applications.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"354 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141520014","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 the realization of the Reshetikhin-Semenov-Tian-Shansky (RS) superalgebra for two parameter quantum affine superalgebra $U_{p, q}(widehat{mathfrak{gl}(m|n))}$. We find a simple coproduct for the Drinfeld generators and obtain a Hopf superalgebra structure for this quantum affine superalgebra.
{"title":"$RLL$-Realization and Its Hopf Superalgebra Structure of $U_{p, q}(widehat{mathfrak{gl}(m|n))}$","authors":"Naihong Hu, Naihuan Jing, Xin Zhong","doi":"arxiv-2407.00406","DOIUrl":"https://doi.org/arxiv-2407.00406","url":null,"abstract":"We establish the realization of the Reshetikhin-Semenov-Tian-Shansky (RS)\u0000superalgebra for two parameter quantum affine superalgebra $U_{p,\u0000q}(widehat{mathfrak{gl}(m|n))}$. We find a simple coproduct for the Drinfeld\u0000generators and obtain a Hopf superalgebra structure for this quantum affine\u0000superalgebra.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141520016","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 a new deformation of multiple zeta value (MZV). It has one parameter $omega$ satisfying $0
我们引入了多重zeta值（MZV）的一种新变形。它有一个参数$omega$，满足$0＜omega＜2$，并在$omega to +0$的极限中恢复 MZV。通过使用多重积分，它被定义在与多重zeta值（$q$MZV）类似的$q$代数框架中。我们证明了我们的变形多重zeta值满足$q$MZV所满足的双重洗牌关系。我们还证明了广濑、佐藤和关对 ($q$)MZV 所证明的扩展双奥氏体关系，方法是使用多元积分，其积分项包含由 Ruijsenaars 提出的双曲伽马函数。
{"title":"A new deformation of multiple zeta value","authors":"Yoshihiro Takeyama","doi":"arxiv-2406.19641","DOIUrl":"https://doi.org/arxiv-2406.19641","url":null,"abstract":"We introduce a new deformation of multiple zeta value (MZV). It has one\u0000parameter $omega$ satisfying $0<omega<2$ and recovers MZV in the limit as\u0000$omega to +0$. It is defined in the same algebraic framework as a\u0000$q$-analogue of multiple zeta value ($q$MZV) by using a multiple integral. We\u0000prove that our deformed multiple zeta value satisfies the double shuffle\u0000relations which are satisfied by $q$MZVs. We also prove the extended double\u0000Ohno relations, which are proved for ($q$)MZVs by Hirose, Sato and Seki, by\u0000using a multiple integral whose integrand contains the hyperbolic gamma\u0000function due to Ruijsenaars.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141520018","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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