Infinitely divisible states on finite quantum groups

TL;DR

This paper characterizes infinitely divisible states on finite quantum groups, showing they must be of Poisson type, thus generalizing classical results to the quantum setting.

Contribution

It proves that on finite quantum groups, all infinitely divisible states are of Poisson type, extending classical theorems to quantum groups.

Findings

Infinitely divisible states on finite quantum groups are of Poisson type.

Two proofs are provided for the main theorem.

Generalizes classical results to the quantum group context.

Abstract

In this paper we study the states of Poisson type and infinitely divisible states on compact quantum groups. Each state of Poisson type is infinitely divisible, i.e., it admits $n$-th root for all $n\geq1$. The main result is that on finite quantum groups infinitely divisible states must be of Poisson type. This generalizes B\"oge's theorem concerning infinitely divisible measures (commutative case) and Parthasarathy's result on infinitely divisible positive definite functions (cocommutative case). Two proofs are given.