Random walks on finite quantum groups

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dc.check.embargoformatNot applicableen
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dc.contributor.advisorWills, Stephenen
dc.contributor.authorMcCarthy, J.P.
dc.date.accessioned2017-05-30T11:40:27Z
dc.date.available2017-05-30T11:40:27Z
dc.date.issued2017
dc.date.submitted2017
dc.description.abstractOf central interest in the study of random walks on finite groups are ergodic random walks. Ergodic random walks converge to random in the sense that as the number of transitions grows to infinity, the state-distribution converges to the uniform distribution on G. The study of random walks on finite groups is generalised to the study of random walks on quantum groups. Quantum groups are neither groups nor sets and rather what are studied are finite dimensional algebras that have the same properties as the algebra of functions on an actual group — except for commutativity. The concept of a random walk converging to random — and a metric for measuring the distance to random after k transitions — is generalised from the classical case to the case of random walks on quantum groups. A central tool in the study of ergodic random walks on finite groups is the Upper Bound Lemma of Diaconis and Shahshahani. The Upper Bound Lemma uses the representation theory of the group to generate upper bounds for the distance to random and thus can be used to determine convergence rates for ergodic walks. The representation theory of quantum groups is very well understood and is remarkably similar to the representation theory of classical groups. This allows for a generalisation of the Upper Bound Lemma to an Upper Bound Lemma for quantum groups. The Quantum Diaconis–Shahshahani Upper Bound Lemma is used to study the convergence of ergodic random walks on classical groups Zn, Z n 2 , the dual group Scn as well as the ‘truly’ quantum groups of Kac and Paljutkin and Sekine. Note that for all of these generalisations, restricting to commutative subalgebras gives the same definitions and results as the classical theory.en
dc.description.statusNot peer revieweden
dc.description.versionAccepted Version
dc.format.mimetypeapplication/pdfen
dc.identifier.citationMcCarthy, J. P. 2017. Random walks on finite quantum groups. PhD Thesis, University College Cork.en
dc.identifier.endpage140en
dc.identifier.urihttps://hdl.handle.net/10468/4033
dc.language.isoenen
dc.publisherUniversity College Corken
dc.rights© 2017, J.P. McCarthy.en
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/3.0/en
dc.subjectMathematicsen
dc.subjectQuantum groupsen
dc.subjectQuantum probabilityen
dc.subjectRandom walksen
dc.subjectRepresentation theoryen
dc.thesis.opt-outfalse
dc.titleRandom walks on finite quantum groupsen
dc.typeDoctoral thesisen
dc.type.qualificationlevelDoctoralen
dc.type.qualificationnamePhD (Science)en
ucc.workflow.supervisors.wills@ucc.ie
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