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Martin boundaries of the duals of free unitary quantum groups

Part of: Selfadjoint operator algebras Linear algebraic groups and related topics Markov processes

Published online by Cambridge University Press:  31 May 2019

Sara Malacarne  and
Sergey Neshveyev
Sara Malacarne
Affiliation:
Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, NO-0316 Oslo, Norway email saramal@math.uio.no
Sergey Neshveyev
Affiliation:
Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, NO-0316 Oslo, Norway email sergeyn@math.uio.no
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Abstract

Given a free unitary quantum group $G=A_{u}(F)$, with $F$ not a unitary $2\times 2$ matrix, we show that the Martin boundary of the dual of $G$ with respect to any $G$-${\hat{G}}$-invariant, irreducible, finite-range quantum random walk coincides with the topological boundary defined by Vaes and Vander Vennet. This can be thought of as a quantum analogue of the fact that the Martin boundary of a free group coincides with the space of ends of its Cayley tree.

Keywords

quantum groupMartin boundaryGromov boundaryrandom walks on trees

MSC classification

Primary: 20G42: Quantum groups (quantized function algebras) and their representations 46L65: Quantizations, deformations 60J50: Boundary theory
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 6 , June 2019 , pp. 1171 - 1193
Copyright
© The Authors 2019 

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Footnotes

The research leading to these results received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement no. 307663.

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