{"title":"Cyclic Cohomology of Coalgebras, Coderivations and De Rham Cohomology","authors":"M. Farinati, A. Solotar","doi":"10.1201/9780429187919-6","DOIUrl":"https://doi.org/10.1201/9780429187919-6","url":null,"abstract":"","PeriodicalId":403117,"journal":{"name":"Hopf Algebras and Quantum Groups","volume":"19 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":"122980324","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}
{"title":"Lifting of Nichols Algebras of Type A 2 and Pointed Hopf Algebras of Order p 4","authors":"N. Andruskiewitsch, H. Schneider","doi":"10.1201/9780429187919-1","DOIUrl":"https://doi.org/10.1201/9780429187919-1","url":null,"abstract":"","PeriodicalId":403117,"journal":{"name":"Hopf Algebras and Quantum Groups","volume":"29 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":"124735403","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 : 2019-05-07DOI: 10.1201/9780429187919-14
A. Tyc
{"title":"On Regularity of the Algebra of Covariants for Actions of Pointed Hopf Algebras on Regular Commutative Algebras","authors":"A. Tyc","doi":"10.1201/9780429187919-14","DOIUrl":"https://doi.org/10.1201/9780429187919-14","url":null,"abstract":"","PeriodicalId":403117,"journal":{"name":"Hopf Algebras and Quantum Groups","volume":"17 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":"122433455","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 : 2019-05-07DOI: 10.1201/9780429187919-15
Yinhuo Zhang
In this paper, we generalize Majid’s bicrossproduct construction. We start with a pair (A,B) of two regular multiplier Hopf algebras. We assume that B is a right A-module algebra and that A is a left B-comodule coalgebra. We recall and discuss the two notions in the first sections of the paper. The right action of A on B gives rise to the smash product A#B. The left coaction of B on A gives a possible coproduct ∆# on A#B. We will discuss in detail the necessary compatibility conditions between the action and the coaction for ∆# to be a proper coproduct on A#B. The result is again a regular multiplier Hopf algebra. Majid’s construction is obtained when we have Hopf algebras. We also look at the dual case, constructed from a pair (C,D) of regular multiplier Hopf algebras where now C is a left D-module algebra while D is a right C-comodule coalgebra. We will show that indeed, these two constructions are dual to each other in the sense that a natural pairing of A with C and of B with D will yield a duality between A#B and the smash product C#D. We show that the bicrossproduct of algebraic quantum groups is again an algebraic quantum group (i.e. a regular multiplier Hopf algebra with integrals). The -algebra case will also be considered. Some special cases will be treated and they will be related with other constructions available in the literature. Finally, the basic example, coming from a (not necessarily finite) group G with two subgroups H and K such that G = KH and H ∩K = {e} (where e is the identity of G) will be used throughout the paper for motivation and illustration of the different notions and results. The cases where either H or K is a normal subgroup will get special attention. March 2009 (Version 2.1) 1 Department of Mathematics, Hasselt University, Agoralaan, B-3590 Diepenbeek (Belgium). E-mail: Lydia.Delvaux@uhasselt.be 2 Department of Mathematics, K.U. Leuven, Celestijnenlaan 200B (bus 2400), B-3001 Heverlee (Belgium). E-mail: Alfons.VanDaele@wis.kuleuven.be 3 Department of Mathematics, Southeast University, Nanjing 210096, China. E-mail: shuanhwang@seu.edu.cn
{"title":"A Survey on Multiplier Hopf Algebras","authors":"Yinhuo Zhang","doi":"10.1201/9780429187919-15","DOIUrl":"https://doi.org/10.1201/9780429187919-15","url":null,"abstract":"In this paper, we generalize Majid’s bicrossproduct construction. We start with a pair (A,B) of two regular multiplier Hopf algebras. We assume that B is a right A-module algebra and that A is a left B-comodule coalgebra. We recall and discuss the two notions in the first sections of the paper. The right action of A on B gives rise to the smash product A#B. The left coaction of B on A gives a possible coproduct ∆# on A#B. We will discuss in detail the necessary compatibility conditions between the action and the coaction for ∆# to be a proper coproduct on A#B. The result is again a regular multiplier Hopf algebra. Majid’s construction is obtained when we have Hopf algebras. We also look at the dual case, constructed from a pair (C,D) of regular multiplier Hopf algebras where now C is a left D-module algebra while D is a right C-comodule coalgebra. We will show that indeed, these two constructions are dual to each other in the sense that a natural pairing of A with C and of B with D will yield a duality between A#B and the smash product C#D. We show that the bicrossproduct of algebraic quantum groups is again an algebraic quantum group (i.e. a regular multiplier Hopf algebra with integrals). The -algebra case will also be considered. Some special cases will be treated and they will be related with other constructions available in the literature. Finally, the basic example, coming from a (not necessarily finite) group G with two subgroups H and K such that G = KH and H ∩K = {e} (where e is the identity of G) will be used throughout the paper for motivation and illustration of the different notions and results. The cases where either H or K is a normal subgroup will get special attention. March 2009 (Version 2.1) 1 Department of Mathematics, Hasselt University, Agoralaan, B-3590 Diepenbeek (Belgium). E-mail: Lydia.Delvaux@uhasselt.be 2 Department of Mathematics, K.U. Leuven, Celestijnenlaan 200B (bus 2400), B-3001 Heverlee (Belgium). E-mail: Alfons.VanDaele@wis.kuleuven.be 3 Department of Mathematics, Southeast University, Nanjing 210096, China. E-mail: shuanhwang@seu.edu.cn","PeriodicalId":403117,"journal":{"name":"Hopf Algebras and Quantum Groups","volume":"1 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":"133238568","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}
{"title":"Survey of Cross Product Bialgebras","authors":"Y. Bespalov, B. Drabant","doi":"10.1201/9780429187919-2","DOIUrl":"https://doi.org/10.1201/9780429187919-2","url":null,"abstract":"","PeriodicalId":403117,"journal":{"name":"Hopf Algebras and Quantum Groups","volume":"170 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":"114188881","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}
{"title":"A Generalized Power Map for Hopf Algebras","authors":"Y. Kashina","doi":"10.1201/9780429187919-8","DOIUrl":"https://doi.org/10.1201/9780429187919-8","url":null,"abstract":"","PeriodicalId":403117,"journal":{"name":"Hopf Algebras and Quantum Groups","volume":"30 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":"125966509","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}
{"title":"Separable Functors for the Category of Doi–Hopf Modules II","authors":"S. Caenepeel, B. Ion, G. Militaru, Shenglin Zhu","doi":"10.1201/9780429187919-5","DOIUrl":"https://doi.org/10.1201/9780429187919-5","url":null,"abstract":"","PeriodicalId":403117,"journal":{"name":"Hopf Algebras and Quantum Groups","volume":"36 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":"122776765","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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