We study Gaiotto's conformal limit for the $G^{mathbb{R}}$-Hitchin equations, when $G^{mathbb{R}}$ is a simple real Lie group admitting a $Theta$-positive structure. We identify a family of flat connections coming from certain solutions to the equations for which the conformal limit exists and admits the structure of an oper. We call this new class of opers appearing in the conformal limit $Theta$-positive opers. The two families involved are parameterized by the same base space. This space is a generalization of the base of Hitchin's integrable system in the case when the structure group is a split real group.
{"title":"Conformal limits in Cayley components and $Θ$-positive opers","authors":"Georgios Kydonakis, Mengxue Yang","doi":"arxiv-2408.06198","DOIUrl":"https://doi.org/arxiv-2408.06198","url":null,"abstract":"We study Gaiotto's conformal limit for the $G^{mathbb{R}}$-Hitchin\u0000equations, when $G^{mathbb{R}}$ is a simple real Lie group admitting a\u0000$Theta$-positive structure. We identify a family of flat connections coming\u0000from certain solutions to the equations for which the conformal limit exists\u0000and admits the structure of an oper. We call this new class of opers appearing\u0000in the conformal limit $Theta$-positive opers. The two families involved are\u0000parameterized by the same base space. This space is a generalization of the\u0000base of Hitchin's integrable system in the case when the structure group is a\u0000split real group.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225262","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 tensor functor called $alpha$-induction produces a new unitary fusion category from a Frobenius algebra, or a $Q$-system, in a braided unitary fusion category. A bi-unitary connection, which is a finite family of complex number subject to some axioms, realizes an object in any unitary fusion category. It also gives a characterization of a finite-dimensional nondegenerate commuting square in subfactor theory of Jones and realizes a certain $4$-tensor appearing in recent studies of $2$-dimensional topological order. We study $alpha$-induction for bi-unitary connections, and show that flatness of the resulting $alpha$-induced bi-unitary connections implies commutativity of the original Frobenius algebra. This gives a converse of our previous result and answers a question raised by R. Longo. We furthermore give finer correspondence between the flat parts of the $alpha$-induced bi-unitary connections and the commutative Frobenius subalgebras studied by B"ockenhauer-Evans.
{"title":"Flatness of $α$-induced bi-unitary connections and commutativity of Frobenius algebras","authors":"Yasuyuki Kawahigashi","doi":"arxiv-2408.05501","DOIUrl":"https://doi.org/arxiv-2408.05501","url":null,"abstract":"The tensor functor called $alpha$-induction produces a new unitary fusion\u0000category from a Frobenius algebra, or a $Q$-system, in a braided unitary fusion\u0000category. A bi-unitary connection, which is a finite family of complex number\u0000subject to some axioms, realizes an object in any unitary fusion category. It\u0000also gives a characterization of a finite-dimensional nondegenerate commuting\u0000square in subfactor theory of Jones and realizes a certain $4$-tensor appearing\u0000in recent studies of $2$-dimensional topological order. We study\u0000$alpha$-induction for bi-unitary connections, and show that flatness of the\u0000resulting $alpha$-induced bi-unitary connections implies commutativity of the\u0000original Frobenius algebra. This gives a converse of our previous result and\u0000answers a question raised by R. Longo. We furthermore give finer correspondence\u0000between the flat parts of the $alpha$-induced bi-unitary connections and the\u0000commutative Frobenius subalgebras studied by B\"ockenhauer-Evans.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198786","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 $mathbb{G}$ be a compact quantum group and $Asubseteq B$ an inclusion of $sigma$-finite $mathbb{G}$-dynamical von Neumann algebras. We prove that the $mathbb{G}$-inclusion $Asubseteq B$ is strongly equivariantly amenable if and only if it is equivariantly amenable, using techniques from the theory of non-commutative $L^p$-spaces. In particular, if $(A, alpha)$ is a $mathbb{G}$-dynamical von Neumann algebra with $A$ $sigma$-finite, the action $alpha: A curvearrowleft mathbb{G}$ is strongly (inner) amenable if and only if the action $alpha: A curvearrowleft mathbb{G}$ is (inner) amenable. By duality, we also obtain the same result for $mathbb{G}$ a discrete quantum group, so that, in particular, a discrete quantum group is inner amenable if and only it is strongly inner amenable. This result can be seen as a dynamical generalization of Tomatsu's result on the amenability/co-amenability duality. We provide an example of a co-amenable (non-Kac) compact quantum group that acts non-amenably on a von Neumann algebra. By duality, this gives an explicit example of an amenable discrete quantum group that acts non-amenably on a von Neumann algebra.
让 $mathbb{G}$ 是一个紧凑的量子群，而 $Asubseteq B$ 是$sigma$-finite$mathbb{G}$-dynamical von Neumann algebras 的一个包含。我们利用非交换$L^p$空间理论中的技术证明，$mathbb{G}$包含$A/subseteq B$是强等变可容性的，当且仅当它是等变可容性的。特别是，如果 $(A, alpha)$ 是一个具有 $A$ $sigma$ 有限性的 $mathbb{G}$ 动态 von Neumann 代数，那么作用$alpha：当且仅当动作$alpha：A curvearrowleft mathbb{G}$ 是（内部）可处理的。通过对偶性，我们对离散量子群的 $mathbb{G}$ 也得到了同样的结果，因此，只有当且仅当一个离散量子群是强内可容性的时候，它才是内可容性的。我们举例说明了一个在 von Neumann 代数上非可门地作用的可门（非 Kac）紧凑量子群。根据对偶性，这给出了一个非可门性地作用于 von Neumann 代数的可门性离散量子群的实例。
{"title":"Amenable actions of compact and discrete quantum groups on von Neumann algebras","authors":"K. De Commer, J. De Ro","doi":"arxiv-2408.05571","DOIUrl":"https://doi.org/arxiv-2408.05571","url":null,"abstract":"Let $mathbb{G}$ be a compact quantum group and $Asubseteq B$ an inclusion\u0000of $sigma$-finite $mathbb{G}$-dynamical von Neumann algebras. We prove that\u0000the $mathbb{G}$-inclusion $Asubseteq B$ is strongly equivariantly amenable if\u0000and only if it is equivariantly amenable, using techniques from the theory of\u0000non-commutative $L^p$-spaces. In particular, if $(A, alpha)$ is a\u0000$mathbb{G}$-dynamical von Neumann algebra with $A$ $sigma$-finite, the action\u0000$alpha: A curvearrowleft mathbb{G}$ is strongly (inner) amenable if and only\u0000if the action $alpha: A curvearrowleft mathbb{G}$ is (inner) amenable. By\u0000duality, we also obtain the same result for $mathbb{G}$ a discrete quantum\u0000group, so that, in particular, a discrete quantum group is inner amenable if\u0000and only it is strongly inner amenable. This result can be seen as a dynamical\u0000generalization of Tomatsu's result on the amenability/co-amenability duality.\u0000We provide an example of a co-amenable (non-Kac) compact quantum group that\u0000acts non-amenably on a von Neumann algebra. By duality, this gives an explicit\u0000example of an amenable discrete quantum group that acts non-amenably on a von\u0000Neumann algebra.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225263","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}
$q$-Yangians can be viewed as quantum deformations of the upper triangular loop Lie algebras, and also be viewed as deformation of the Yangian algebra. In this paper, we study the twisted $q$-Yangians as coideal subalgebras of the quantum affine algebra introduced by Molev, Ragoucy and Sorba. We investigate the invariant theory of the quantum symmetric spaces in affine types $AI, AII$ and use the Sklyanin determinants to study the invariant theory and show that they also obey classical type identities similar to the quantum coordinate algebras of finite types.
{"title":"Twisted q-Yangians and Sklyanin determinants","authors":"Naihuan Jing, Jian Zhang","doi":"arxiv-2408.04340","DOIUrl":"https://doi.org/arxiv-2408.04340","url":null,"abstract":"$q$-Yangians can be viewed as quantum deformations of the upper triangular\u0000loop Lie algebras, and also be viewed as deformation of the Yangian algebra. In\u0000this paper, we study the twisted $q$-Yangians as coideal subalgebras of the\u0000quantum affine algebra introduced by Molev, Ragoucy and Sorba. We investigate\u0000the invariant theory of the quantum symmetric spaces in affine types $AI, AII$\u0000and use the Sklyanin determinants to study the invariant theory and show that\u0000they also obey classical type identities similar to the quantum coordinate\u0000algebras of finite types.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949151","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}
Jorge A. Guccione, Juan J. Guccione, Christian Valqui
In this paper we determine all the Hopf q-brace structures on rank one pointed Hopf algebras and compute the socle of each one of them. We also identify which among them are Hopf skew-braces. Then we determine when two Hopf q-brace structures on rank one pointed Hopf algebras are isomorphic, and, finally, we compute all the weak braiding operators on these Hopf algebras.
{"title":"Hopf q-braces structures on rank one pointed Hopf algebras","authors":"Jorge A. Guccione, Juan J. Guccione, Christian Valqui","doi":"arxiv-2408.03863","DOIUrl":"https://doi.org/arxiv-2408.03863","url":null,"abstract":"In this paper we determine all the Hopf q-brace structures on rank one\u0000pointed Hopf algebras and compute the socle of each one of them. We also\u0000identify which among them are Hopf skew-braces. Then we determine when two Hopf\u0000q-brace structures on rank one pointed Hopf algebras are isomorphic, and,\u0000finally, we compute all the weak braiding operators on these Hopf algebras.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949152","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}
Using the completed inductive, projective and injective tensor products of Grothendieck for locally convex topological vector spaces, we develop a systematic theory of locally convex Hopf algebras with an emphasis on Pontryagin-type dualities. We describe how classical Hopf algebras, real and complex Lie groups, as well as compact and discrete quantum groups, can all give rise to natural examples of this theory in a variety of different ways. We also show that the space of all continuous functions on a topological group $ G $ whose topological structures are compactly generated has an $ varepsilon $-Hopf algebra structure, and we can recover $ G $ fully as a topological group from this locally convex Hopf algebra. The latter is done via a generalization of Gelfand duality, which is of its own interest. Certain projective and inductive limits are also considered in this framework, and it is shown that how this can lead to examples seemingly outside of the framework of locally compact quantum groups in the sense of Kustermans-Vaes. As an illustration, we propose a version of the infinite quantum permutation group $ S^{+}_{infty} $, the free orthogonal group $ O^{+}_{infty} $, and the free unitary group $ U^{+}_{infty} $ as certain strict inductive limits, all of which still retain a nice duality. Combined with our duality theory, this may be seen as an alternative tentative approach to the Kac program of developing a Pontryagin-type duality to a wider class, while at the same time, we include many more interesting examples of classical and quantum groups.
{"title":"A theory of locally convex Hopf algebras","authors":"Hua Wang","doi":"arxiv-2408.03805","DOIUrl":"https://doi.org/arxiv-2408.03805","url":null,"abstract":"Using the completed inductive, projective and injective tensor products of\u0000Grothendieck for locally convex topological vector spaces, we develop a\u0000systematic theory of locally convex Hopf algebras with an emphasis on\u0000Pontryagin-type dualities. We describe how classical Hopf algebras, real and\u0000complex Lie groups, as well as compact and discrete quantum groups, can all\u0000give rise to natural examples of this theory in a variety of different ways. We\u0000also show that the space of all continuous functions on a topological group $ G\u0000$ whose topological structures are compactly generated has an $ varepsilon\u0000$-Hopf algebra structure, and we can recover $ G $ fully as a topological group\u0000from this locally convex Hopf algebra. The latter is done via a generalization\u0000of Gelfand duality, which is of its own interest. Certain projective and\u0000inductive limits are also considered in this framework, and it is shown that\u0000how this can lead to examples seemingly outside of the framework of locally\u0000compact quantum groups in the sense of Kustermans-Vaes. As an illustration, we\u0000propose a version of the infinite quantum permutation group $ S^{+}_{infty} $,\u0000the free orthogonal group $ O^{+}_{infty} $, and the free unitary group $\u0000U^{+}_{infty} $ as certain strict inductive limits, all of which still retain\u0000a nice duality. Combined with our duality theory, this may be seen as an\u0000alternative tentative approach to the Kac program of developing a\u0000Pontryagin-type duality to a wider class, while at the same time, we include\u0000many more interesting examples of classical and quantum groups.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"76 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949154","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 discuss the rationality of Lorentzian lattice conformal field theory (LLCFT) recently constructed in arXiv:2312.16296 and obtain equivalent characterizations of rationality generalising Wendland's rational Narain CFT characterization. We then describe the construction of a modular tensor category (MTC) associated to rational LLCFTs. We explicitly construct the modular data and braiding and fusing matrices for the MTC. As a concrete example, we show that the LLCFT based on a certain even, self-dual Lorentzian lattice of signature $(m,n)$ with $m$ even realises the $D(mbmod 8)$ level 1 Kac-Moody MTC.
{"title":"Rationality of Lorentzian Lattice CFTs And The Associated Modular Tensor Category","authors":"Ranveer Kumar Singh, Madhav Sinha, Runkai Tao","doi":"arxiv-2408.02744","DOIUrl":"https://doi.org/arxiv-2408.02744","url":null,"abstract":"We discuss the rationality of Lorentzian lattice conformal field theory\u0000(LLCFT) recently constructed in arXiv:2312.16296 and obtain equivalent\u0000characterizations of rationality generalising Wendland's rational Narain CFT\u0000characterization. We then describe the construction of a modular tensor\u0000category (MTC) associated to rational LLCFTs. We explicitly construct the\u0000modular data and braiding and fusing matrices for the MTC. As a concrete\u0000example, we show that the LLCFT based on a certain even, self-dual Lorentzian\u0000lattice of signature $(m,n)$ with $m$ even realises the $D(mbmod 8)$ level 1\u0000Kac-Moody MTC.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949156","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 a premodular category $mathcal{C}$, we show that its $R$-symbol can be recovered from its $T$-matrice, fusion coefficients and some 2nd generalized Frobenius-Schur indicators. In particular, if $mathcal{C}$ is modular, its $R$-symbols for a certain gauge choice are completely determined by its modular data.
{"title":"Recovering R-symbols from modular data","authors":"Siu-Hung Ng, Eric C Rowell, Xiao-Gang Wen","doi":"arxiv-2408.02748","DOIUrl":"https://doi.org/arxiv-2408.02748","url":null,"abstract":"Given a premodular category $mathcal{C}$, we show that its $R$-symbol can be\u0000recovered from its $T$-matrice, fusion coefficients and some 2nd generalized\u0000Frobenius-Schur indicators. In particular, if $mathcal{C}$ is modular, its\u0000$R$-symbols for a certain gauge choice are completely determined by its modular\u0000data.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949155","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}
Monoidal categories with additional structure such as a braiding or some form of duality abound in quantum topology. They often appear in tandem with Frobenius algebras inside them. Motivations for this range from the theory of module categories to the construction of correlators in conformal field theory. We generalize the Baez-Dolan microcosm principle to consistently describe all these types of algebras by extending it to cyclic and modular algebras in the sense of Getzler-Kapranov. Our main result links the microcosm principle for cyclic algebras to the one for modular algebras via Costello's modular envelope. The result can be understood as a local-to-global construction for various flavors of Frobenius algebras that substantially generalizes and unifies the available, and often intrinsically semisimple methods using for example triangulations, state-sum constructions or skein theory. Several applications of the main result in conformal field theory are presented: We classify consistent systems of correlators for open conformal field theories and show that the genus zero correlators for logarithmic conformal field theories constructed by Fuchs-Schweigert can be uniquely extended to handlebodies. This establishes a very general correspondence between full genus zero conformal field theory in dimension two and skein theory in dimension three.
{"title":"The Cyclic and Modular Microcosm Principle in Quantum Topology","authors":"Lukas Woike","doi":"arxiv-2408.02644","DOIUrl":"https://doi.org/arxiv-2408.02644","url":null,"abstract":"Monoidal categories with additional structure such as a braiding or some form\u0000of duality abound in quantum topology. They often appear in tandem with\u0000Frobenius algebras inside them. Motivations for this range from the theory of\u0000module categories to the construction of correlators in conformal field theory.\u0000We generalize the Baez-Dolan microcosm principle to consistently describe all\u0000these types of algebras by extending it to cyclic and modular algebras in the\u0000sense of Getzler-Kapranov. Our main result links the microcosm principle for\u0000cyclic algebras to the one for modular algebras via Costello's modular\u0000envelope. The result can be understood as a local-to-global construction for\u0000various flavors of Frobenius algebras that substantially generalizes and\u0000unifies the available, and often intrinsically semisimple methods using for\u0000example triangulations, state-sum constructions or skein theory. Several\u0000applications of the main result in conformal field theory are presented: We\u0000classify consistent systems of correlators for open conformal field theories\u0000and show that the genus zero correlators for logarithmic conformal field\u0000theories constructed by Fuchs-Schweigert can be uniquely extended to\u0000handlebodies. This establishes a very general correspondence between full genus\u0000zero conformal field theory in dimension two and skein theory in dimension\u0000three.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949157","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}
This paper is about the positive part $U_q^+$ of the $q$-deformed enveloping algebra $U_q(widehat{mathfrak{sl}}_2)$. The algebra $U_q^+$ admits an embedding, due to Rosso, into a $q$-shuffle algebra $mathbb{V}$. The underlying vector space of $mathbb{V}$ is the free algebra on two generators $x,y$. Therefore, the algebra $mathbb{V}$ has a basis consisting of the words in $x,y$. Let $U$ denote the image of $U_q^+$ under the Rosso embedding. In our first main result, we find all the words in $x,y$ that are contained in $U$. One type of solution is called alternating. The alternating words have been studied by Terwilliger. There is another type of solution, which we call doubly alternating. In our second main result, we display many commutator relations involving the doubly alternating words. In our third main result, we describe how the doubly alternating words are related to the alternating words.
{"title":"Doubly alternating words in the positive part of $U_q(widehat{mathfrak{sl}}_2)$","authors":"Chenwei Ruan","doi":"arxiv-2408.02633","DOIUrl":"https://doi.org/arxiv-2408.02633","url":null,"abstract":"This paper is about the positive part $U_q^+$ of the $q$-deformed enveloping\u0000algebra $U_q(widehat{mathfrak{sl}}_2)$. The algebra $U_q^+$ admits an\u0000embedding, due to Rosso, into a $q$-shuffle algebra $mathbb{V}$. The\u0000underlying vector space of $mathbb{V}$ is the free algebra on two generators\u0000$x,y$. Therefore, the algebra $mathbb{V}$ has a basis consisting of the words\u0000in $x,y$. Let $U$ denote the image of $U_q^+$ under the Rosso embedding. In our\u0000first main result, we find all the words in $x,y$ that are contained in $U$.\u0000One type of solution is called alternating. The alternating words have been\u0000studied by Terwilliger. There is another type of solution, which we call doubly\u0000alternating. In our second main result, we display many commutator relations\u0000involving the doubly alternating words. In our third main result, we describe\u0000how the doubly alternating words are related to the alternating words.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949158","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：