374 research outputs found
Quadri-algebras
We introduce the notion of quadri-algebras. These are associative algebras
for which the multiplication can be decomposed as the sum of four operations in
a certain coherent manner. We present several examples of quadri-algebras: the
algebra of permutations, the shuffle algebra, tensor products of dendriform
algebras. We show that a pair of commuting Baxter operators on an associative
algebra gives rise to a canonical quadri-algebra structure on the underlying
space of the algebra. The main example is provided by the algebra End(A) of
linear endomorphisms of an infinitesimal bialgebra A. This algebra carries a
canonical pair of commuting Baxter operators: \beta(T)=T\ast\id and
\gamma(T)=\id\ast T, where denotes the convolution of endomorphisms.
It follows that End(A) is a quadri-algebra, whenever A is an infinitesimal
bialgebra. We also discuss commutative quadri-algebras and state some
conjectures on the free quadri-algebra
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
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
Three osculating walkers
We consider three directed walkers on the square lattice, which move
simultaneously at each tick of a clock and never cross. Their trajectories form
a non-crossing configuration of walks. This configuration is said to be
osculating if the walkers never share an edge, and vicious (or:
non-intersecting) if they never meet. We give a closed form expression for the
generating function of osculating configurations starting from prescribed
points. This generating function turns out to be algebraic. We also relate the
enumeration of osculating configurations with prescribed starting and ending
points to the (better understood) enumeration of non-intersecting
configurations. Our method is based on a step by step decomposition of
osculating configurations, and on the solution of the functional equation
provided by this decomposition
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