L\'evy processes on the Lorentz-Lie algebra

TL;DR

This paper investigates Levy processes on the Lorentz-Lie algebra, constructing a Schürmann triple for a unique non-trivial cocycle and analyzing its properties and decompositions on subalgebras.

Contribution

It constructs a Schürmann triple for the unique non-trivial cocycle on the Lorentz group Lie algebra and studies the associated Levy process and its subalgebra decompositions.

Findings

Identification of the unique irreducible unitary representation with a non-trivial cocycle.

Construction of a Schürmann triple for this cocycle.

Analysis of Levy process properties and subalgebra decompositions.

Abstract

L\'evy processes in the sense of Sch\"urmann on the Lie algebra of the Lorentz grouop are studied. It is known that only one of the irreducible unitary representations of the Lorentz group admits a non-trivial one-cocycle. A Sch\"urmann triple is constructed for this cocycle and the properties of the associated L\'evy process are investigated. The decommpositions of the restrictions of this triple to the Lie subalgebras $so(3)$ and $so(2,1)$ are described.