Pub Date : 2019-05-07DOI: 10.1201/9780429187919-12
S. Raianu
{"title":"An Easy Proof for the Uniqueness of Integrals","authors":"S. Raianu","doi":"10.1201/9780429187919-12","DOIUrl":"https://doi.org/10.1201/9780429187919-12","url":null,"abstract":"","PeriodicalId":403117,"journal":{"name":"Hopf Algebras and Quantum Groups","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133145755","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}
To a vector space V equipped with a non-quasiclassical involutary solution of the quantum Yang-Baxter equation and a partition $lambda$, we associate a vector space $Vl$ and compute its dimension. The functor $Vmapsto Vl$ is an analogue of the well-known Schur functor. The category generated by the objects $Vl$ is called the Schur-Weyl category. We suggest a way to construct some related twisted varieties looking like orbits of semisimple elements in sl(n)^*. We consider in detail a particular case of such "twisted orbits", namely the twisted non-quasiclassical hyperboloid and we define the twisted Casimir operator on it. In this case, we obtain a formula looking like the Weyl formula, and describing the asymptotic behavior of the function $N(la)={sharp la_ileqla}$, where $la_i$ are the eigenvalues of this operator.
{"title":"Schur–Weyl Categones and Non‐Quasiclassical Weyl Type Formula","authors":"D. Gurevich, Z. Mriss","doi":"10.1201/9780429187919-7","DOIUrl":"https://doi.org/10.1201/9780429187919-7","url":null,"abstract":"To a vector space V equipped with a non-quasiclassical involutary solution of the quantum Yang-Baxter equation and a partition $lambda$, we associate a vector space $Vl$ and compute its dimension. The functor $Vmapsto Vl$ is an analogue of the well-known Schur functor. The category generated by the objects $Vl$ is called the Schur-Weyl category. We suggest a way to construct some related twisted varieties looking like orbits of semisimple elements in sl(n)^*. We consider in detail a particular case of such \"twisted orbits\", namely the twisted non-quasiclassical hyperboloid and we define the twisted Casimir operator on it. In this case, we obtain a formula looking like the Weyl formula, and describing the asymptotic behavior of the function $N(la)={sharp la_ileqla}$, where $la_i$ are the eigenvalues of this operator.","PeriodicalId":403117,"journal":{"name":"Hopf Algebras and Quantum Groups","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1999-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132562862","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}
Coalgebra-Galois extensions generalise Hopf-Galois extensions, which can be viewed as non-commutative torsors. In this paper it is analysed when a coalgebra-Galois extension is a separable, split, or strongly separable extension.
{"title":"Coalgebra‐Galois Extensions from the Extension Theory Point of View","authors":"T. Brzeziński","doi":"10.1201/9780429187919-4","DOIUrl":"https://doi.org/10.1201/9780429187919-4","url":null,"abstract":"Coalgebra-Galois extensions generalise Hopf-Galois extensions, which can be viewed as non-commutative torsors. In this paper it is analysed when a coalgebra-Galois extension is a separable, split, or strongly separable extension.","PeriodicalId":403117,"journal":{"name":"Hopf Algebras and Quantum Groups","volume":"116 2-3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1999-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132462645","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}
Pub Date : 1998-08-12DOI: 10.1201/9780429187919-10
D. Nikshych, L. Vainerman
We establish the equivalence of three versions of a finite dimensional quantum groupoid: a generalized Kac algebra introduced by T. Yamanouchi, a weak $C^*$-Hopf algebra introduced by G. Bohm, F. Nill and K. Szlachanyi (with an involutive antipode), and a Kac bimodule -- an algebraic version of a Hopf bimodule, the notion introduced by J.-M. Vallin. We also study the structure and construct examples of finite dimensional quantum groupoids.
我们建立了有限维量子群样的三个版本的等价性:由T. Yamanouchi引入的广义Kac代数，由G. Bohm, F. Nill和K. Szlachanyi引入的弱$C^*$-Hopf代数(具有对合对极)，以及由J.-M引入的Hopf双模的代数版本的Kac双模。Vallin。我们还研究了有限维量子群的结构和构造实例。
{"title":"Algebraic Versions of a Finite‐Dimensional Quantum Groupoid","authors":"D. Nikshych, L. Vainerman","doi":"10.1201/9780429187919-10","DOIUrl":"https://doi.org/10.1201/9780429187919-10","url":null,"abstract":"We establish the equivalence of three versions of a finite dimensional quantum groupoid: a generalized Kac algebra introduced by T. Yamanouchi, a weak $C^*$-Hopf algebra introduced by G. Bohm, F. Nill and K. Szlachanyi (with an involutive antipode), and a Kac bimodule -- an algebraic version of a Hopf bimodule, the notion introduced by J.-M. Vallin. We also study the structure and construct examples of finite dimensional quantum groupoids.","PeriodicalId":403117,"journal":{"name":"Hopf Algebras and Quantum Groups","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133012797","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}
Pub Date : 1900-01-01DOI: 10.1201/9780429187919-11
F. Panaite, F. Oystaeyen
{"title":"Quasi‐Hopf Algebras and the Centre of a Tensor Category","authors":"F. Panaite, F. Oystaeyen","doi":"10.1201/9780429187919-11","DOIUrl":"https://doi.org/10.1201/9780429187919-11","url":null,"abstract":"","PeriodicalId":403117,"journal":{"name":"Hopf Algebras and Quantum Groups","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125043812","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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