Relations between infinitesimal non-commutative cumulants

Boolean, free and monotone cumulants as well relations among them, have proven to be important in the study of non-commutative probability theory. Quite notably, Boolean were successfully used infinite divisibility via Bercovici--Pata bijection. On other hand, recent years concept infinitesimal has been developed, together with notion which can useful context combinatorial questions. In this paper, we show that known free, still hold framework. Our approach is based on use Grassmann algebra. Formulas involving obtained by applying a formal derivation formulas. The between various types turn out captured shuffle algebra moment-cumulant In formulation, (free, monotone) are represented elements Lie characters over particular Hopf The latter consists graded connected double tensor defined space neither commutative nor cocommutative. note it shown how naturally extends space. basic step replacing base field target linear maps field. We also consider analog map.