369 research outputs found
Rota-Baxter Algebras and Dendriform Algebras
In this paper we study the adjoint functors between the category of
Rota-Baxter algebras and the categories of dendriform dialgebras and
trialgebras. In analogy to the well-known theory of the adjoint functor between
the category of associative algebras and Lie algebras, we first give an
explicit construction of free Rota-Baxter algebras and then apply it to obtain
universal enveloping Rota-Baxter algebras of dendriform dialgebras and
trialgebras. We further show that free dendriform dialgebras and trialgebras,
as represented by binary planar trees and planar trees, are canonical
subalgebras of free Rota-Baxter algebras.Comment: Typos corrected and the last section on analog of
Poincare-Birkhoff-Witt theorem deleted for a gap in the proo
Free Rota-Baxter algebras and rooted trees
A Rota-Baxter algebra, also known as a Baxter algebra, is an algebra with a
linear operator satisfying a relation, called the Rota-Baxter relation, that
generalizes the integration by parts formula. Most of the studies on
Rota-Baxter algebras have been for commutative algebras. Two constructions of
free commutative Rota-Baxter algebras were obtained by Rota and Cartier in the
1970s and a third one by Keigher and one of the authors in the 1990s in terms
of mixable shuffles. Recently, noncommutative Rota-Baxter algebras have
appeared both in physics in connection with the work of Connes and Kreimer on
renormalization in perturbative quantum field theory, and in mathematics
related to the work of Loday and Ronco on dendriform dialgebras and
trialgebras.
This paper uses rooted trees and forests to give explicit constructions of
free noncommutative Rota--Baxter algebras on modules and sets. This highlights
the combinatorial nature of Rota--Baxter algebras and facilitates their further
study. As an application, we obtain the unitarization of Rota-Baxter algebras.Comment: 23 page
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
New identities in dendriform algebras
Dendriform structures arise naturally in algebraic combinatorics (where they
allow, for example, the splitting of the shuffle product into two pieces) and
through Rota-Baxter algebra structures (the latter appear, among others, in
differential systems and in the renormalization process of pQFT). We prove new
combinatorial identities in dendriform dialgebras that appear to be strongly
related to classical phenomena, such as the combinatorics of Lyndon words,
rewriting rules in Lie algebras, or the fine structure of the
Malvenuto-Reutenauer algebra. One of these identities is an abstract
noncommutative, dendriform, generalization of the Bohnenblust-Spitzer identity
and of an identity involving iterated Chen integrals due to C.S. Lam.Comment: 16 pages, LaTeX. Concrete examples and applications adde
On matrix differential equations in the Hopf algebra of renormalization
We establish Sakakibara's differential equations in a matrix setting for the
counter term (respectively renormalized character) in Connes-Kreimer's Birkhoff
decomposition in any connected graded Hopf algebra, thus including Feynman
rules in perturbative renormalization as a key example.Comment: 22 pages, typos correcte
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
Rota-Baxter algebras and new combinatorial identities
The word problem for an arbitrary associative Rota-Baxter algebra is solved.
This leads to a noncommutative generalization of the classical Spitzer
identities. Links to other combinatorial aspects, particularly of interest in
physics, are indicated.Comment: 8 pages, improved versio
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