171 research outputs found
Lie theory for Hopf operads
The present article takes advantage of the properties of algebras in the
category of S-modules (twisted algebras) to investigate further the fine
algebraic structure of Hopf operads. We prove that any Hopf operad P carries
naturally the structure of twisted Hopf P-algebra. Many properties of classical
Hopf algebraic structures are then shown to be encapsulated in the twisted Hopf
algebraic structure of the corresponding Hopf operad. In particular, various
classical theorems of Lie theory relating Lie polynomials to words (i.e.
elements of the tensor algebra) are lifted to arbitrary Hopf operads.Comment: 23 pages. Using xyPi
Monotone, free, and boolean cumulants: a shuffle algebra approach
The theory of cumulants is revisited in the "Rota way", that is, by following
a combinatorial Hopf algebra approach. Monotone, free, and boolean cumulants
are considered as infinitesimal characters over a particular combinatorial Hopf
algebra. The latter is neither commutative nor cocommutative, and has an
underlying unshuffle bialgebra structure which gives rise to a shuffle product
on its graded dual. The moment-cumulant relations are encoded in terms of
shuffle and half-shuffle exponentials. It is then shown how to express
concisely monotone, free, and boolean cumulants in terms of each other using
the pre-Lie Magnus expansion together with shuffle and half-shuffle logarithms.Comment: final versio
The splitting process in free probability theory
Free cumulants were introduced by Speicher as a proper analog of classical
cumulants in Voiculescu's theory of free probability. The relation between free
moments and free cumulants is usually described in terms of Moebius calculus
over the lattice of non-crossing partitions. In this work we explore another
approach to free cumulants and to their combinatorial study using a
combinatorial Hopf algebra structure on the linear span of non-crossing
partitions. The generating series of free moments is seen as a character on
this Hopf algebra. It is characterized by solving a linear fixed point equation
that relates it to the generating series of free cumulants. These phenomena are
explained through a process similar to (though different from) the
arborification process familiar in the theory of dynamical systems, and
originating in Cayley's work
Cumulants, free cumulants and half-shuffles
Free cumulants were introduced as the proper analog of classical cumulants in
the theory of free probability. There is a mix of similarities and differences,
when one considers the two families of cumulants. Whereas the combinatorics of
classical cumulants is well expressed in terms of set partitions, the one of
free cumulants is described, and often introduced in terms of non-crossing set
partitions. The formal series approach to classical and free cumulants also
largely differ. It is the purpose of the present article to put forward a
different approach to these phenomena. Namely, we show that cumulants, whether
classical or free, can be understood in terms of the algebra and combinatorics
underlying commutative as well as non-commutative (half-)shuffles and
(half-)unshuffles. As a corollary, cumulants and free cumulants can be
characterized through linear fixed point equations. We study the exponential
solutions of these linear fixed point equations, which display well the
commutative, respectively non-commutative, character of classical, respectively
free, cumulants.Comment: updated and revised version; accepted for publication in PRS
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