  The role of coalgebras as well as algebraic groups in non-commutative
probability has long been advocated by the school of von Waldenfels and
Sch\"urmann. Another algebraic approach was introduced more recently, based on
shuffle and pre-Lie calculus, and results in another construction of groups of
characters encoding the behaviour of states. Comparing the two, the first
approach, recast recently in a general categorical language by Manzel and
Sch\"urmann, can be seen as largely driven by the theory of universal products,
whereas the second construction builds on Hopf algebras and a suitable
algebraization of the combinatorics of noncrossing set partitions. Although
both address the same phenomena, moving between the two viewpoints is not
obvious. We present here an attempt to unify the two approaches by making
explicit the Hopf algebraic connections between them. Our presentation,
although relying largely on classical ideas as well as results closely related
to Manzel and Sch\"urmann's aforementioned work, is nevertheless original on
several points and fills a gap in the non-commutative probability literature.
In particular, we systematically use the language and techniques of algebraic
groups together with shuffle group techniques to prove that two notions of
algebraic groups naturally associated with free, respectively Boolean and
monotone, probability theories identify. We also obtain explicit formulas for
various Hopf algebraic structures and detail arguments that had been left
implicit in the literature.
