Random walks on finite quantum groups

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McCarthy, J.P.
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University College Cork
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Of 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.
Mathematics , Quantum groups , Quantum probability , Random walks , Representation theory
McCarthy, J. P. 2017. Random walks on finite quantum groups. PhD Thesis, University College Cork.