Hunt's Formula for $SU_q(N)$ and $U_q(N)$

TL;DR

This paper extends Hunt's formula to quantum groups $SU_q(N)$ and $U_q(N)$, decomposing Lévy process generators into Gaussian and jump parts, with a focus on their structure and uniqueness.

Contribution

It introduces a Hunt type formula for quantum groups, detailing the decomposition of generators into Gaussian and jump components, and analyzes their structure on subgroups.

Findings

Decomposition of generators into Gaussian and jump parts.

Uniqueness of the decomposition given a projection.

Extension of Hunt's formula to quantum groups.

Abstract

We provide a Hunt type formula for the infinitesimal generators of L\'evy process on the quantum groups $SU_q(N)$ and $U_q(N)$. In particular, we obtain a decomposition of such generators into a gaussian part and a `jump type' part determined by a linear functional that resembles the functional induced by the L\'evy measure. The jump part on $SU_q(N)$ decomposes further into parts that live on the quantum subgroups $SU_q(n)$, $n\le N$. Like in the classical Hunt formula for locally compact Lie groups, the ingredients become unique once a certain projection is chosen. There are analogous result for $U_q(N)$.