{"title":"The Exponential Map for Hopf Algebras","authors":"Ghaliah Alhamzi, E. Beggs","doi":"10.3842/SIGMA.2022.017","DOIUrl":null,"url":null,"abstract":". We give an analogue of the classical exponential map on Lie groups for Hopf ∗ -algebras with differential calculus. The major difference with the classical case is the interpretation of the value of the exponential map, classically an element of the Lie group. We give interpretations as states on the Hopf algebra, elements of a Hilbert C ∗ -bimodule of 12 densities and elements of the dual Hopf algebra. We give examples for complex valued functions on the groups S 3 and Z , Woronowicz’s matrix quantum group C q [ SU 2 ] and the Sweedler–Taft algebra.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2022-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Symmetry Integrability and Geometry-Methods and Applications","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.3842/SIGMA.2022.017","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
. We give an analogue of the classical exponential map on Lie groups for Hopf ∗ -algebras with differential calculus. The major difference with the classical case is the interpretation of the value of the exponential map, classically an element of the Lie group. We give interpretations as states on the Hopf algebra, elements of a Hilbert C ∗ -bimodule of 12 densities and elements of the dual Hopf algebra. We give examples for complex valued functions on the groups S 3 and Z , Woronowicz’s matrix quantum group C q [ SU 2 ] and the Sweedler–Taft algebra.
期刊介绍:
Scope
Geometrical methods in mathematical physics
Lie theory and differential equations
Classical and quantum integrable systems
Algebraic methods in dynamical systems and chaos
Exactly and quasi-exactly solvable models
Lie groups and algebras, representation theory
Orthogonal polynomials and special functions
Integrable probability and stochastic processes
Quantum algebras, quantum groups and their representations
Symplectic, Poisson and noncommutative geometry
Algebraic geometry and its applications
Quantum field theories and string/gauge theories
Statistical physics and condensed matter physics
Quantum gravity and cosmology.