${\mathbb{Z}}_n$--graded Independence

Abstract

We generalize results of Mingo and Nica on graded independence from the context of ${\mathbb{Z}}_2$--graded (Fermionic) noncommutative probability spaces to that of ${\mathbb{Z}}_n$--graded noncommutative probability spaces. We show that for $q$ a primitive $n$-th root of unity, the $q$-cumulants defined by Nica linearize the addition of homogeneous ${\mathbb{Z}}_n$--graded independent random variables.

16 pages, Latex. Minor revision