Non-existence of genuine (compact) quantum symmetries of compact, connected smooth manifolds

2018/05/14 by Goswami, Debashish · 1 citation
#FOS: Mathematics #Operator Algebras (math.OA) #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.1805.05765

Abstract

Suppose that a compact quantum group \mathcal Q acts faithfully on a smooth, compact, connected manifold M, i.e. has a C^∗ (co)-action α on C(M), such that α(C^∞(M)) ⊆ C^∞(M, \mathcal Q) and the linear span of α(C^∞(M))(1 ⊗ \mathcal Q) is dense in C^∞(M, \mathcal Q) with respect to the Frechet topology. It was conjectured by the author quite a few years ago that \mathcal Q must be commutative as a C^∗ algebra i.e. \mathcal Q ≅ C(G) for some compact group G acting smoothly on M. The goal of this paper is to prove the truth of this conjecture. A remarkable aspect of the proof is the use of probabilistic techniques involving Brownian stopping time.

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