31 research outputs found
Dendriform Equations
We investigate solutions for a particular class of linear equations in
dendriform algebras. Motivations as well as several applications are provided.
The latter follow naturally from the intimate link between dendriform algebras
and Rota-Baxter operators, e.g. the Riemann integral or Jackson's q-integral.Comment: improved versio
Twisted dendriform algebras and the pre-Lie Magnus expansion
In this paper an application of the recently introduced pre-Lie Magnus
expansion to Jackson's q-integral and q-exponentials is presented. Twisted
dendriform algebras, which are the natural algebraic framework for Jackson's
q-analogues, are introduced for that purpose. It is shown how the pre-Lie
Magnus expansion is used to solve linear q-differential equations. We also
briefly outline the theory of linear equations in twisted dendriform algebras.Comment: improved version; accepted for publication in the Journal of Pure &
Applied Algebr
The Magnus expansion, trees and Knuth's rotation correspondence
W. Magnus introduced a particular differential equation characterizing the
logarithm of the solution of linear initial value problems for linear
operators. The recursive solution of this differential equation leads to a
peculiar Lie series, which is known as Magnus expansion, and involves Bernoulli
numbers, iterated Lie brackets and integrals. This paper aims at obtaining
further insights into the fine structure of the Magnus expansion. By using
basic combinatorics on planar rooted trees we prove a closed formula for the
Magnus expansion in the context of free dendriform algebra. From this, by using
a well-known dendriform algebra structure on the vector space generated by the
disjoint union of the symmetric groups, we derive the
Mielnik-Pleba\'nski-Strichartz formula for the continuous
Baker-Campbell-Hausdorff series
The combinatorics of Bogoliubov's recursion in renormalization
We describe various combinatorial aspects of the Birkhoff-Connes-Kreimer
factorization in perturbative renormalisation. The analog of Bogoliubov's
preparation map on the Lie algebra of Feynman graphs is identified with the
pre-Lie Magnus expansion. Our results apply to any connected filtered Hopf
algebra, based on the pro-nilpotency of the Lie algebra of infinitesimal
characters.Comment: improved version, 20 pages, CIRM 2006 workshop "Renormalization and
Galois Theory", Org. F. Fauvet, J.-P. Rami
The pre-Lie structure of the time-ordered exponential
The usual time-ordering operation and the corresponding time-ordered
exponential play a fundamental role in physics and applied mathematics. In this
work we study a new approach to the understanding of time-ordering relying on
recent progress made in the context of enveloping algebras of pre-Lie algebras.
Various general formulas for pre-Lie and Rota-Baxter algebras are obtained in
the process. Among others, we recover the noncommutative analog of the
classical Bohnenblust-Spitzer formula, and get explicit formulae for operator
products of time-ordered exponentials
Dendriform-Tree Setting for Fully Non-commutative Fliess Operators
This paper provides a dendriform-tree setting for Fliess operators with
matrix-valued inputs. This class of analytic nonlinear input-output systems is
convenient, for example, in quantum control. In particular, a description of
such Fliess operators is provided using planar binary trees. Sufficient
conditions for convergence of the defining series are also given
An operational calculus for the Mould operad
The operad of moulds is realized in terms of an operational calculus of
formal integrals (continuous formal power series). This leads to many
simplifications and to the discovery of various suboperads. In particular, we
prove a conjecture of the first author about the inverse image of non-crossing
trees in the dendriform operad. Finally, we explain a connection with the
formalism of noncommutative symmetric functions.Comment: 16 pages, one reference added and minor changes in v
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