4,929 research outputs found
Right-handed Hopf algebras and the preLie forest formula
Three equivalent methods allow to compute the antipode of the Hopf algebras
of Feynman diagrams in perturbative quantum field theory (QFT): the Dyson-Salam
formula, the Bogoliubov formula, and the Zimmermann forest formula. Whereas the
first two hold generally for arbitrary connected graded Hopf algebras, the
third one requires extra structure properties of the underlying Hopf algebra
but has the nice property to reduce drastically the number of terms in the
expression of the antipode (it is optimal in that sense).The present article is
concerned with the forest formula: we show that it generalizes to arbitrary
right-handed polynomial Hopf algebras. These Hopf algebras are dual to the
enveloping algebras of preLie algebras -a structure common to many
combinatorial Hopf algebras which is carried in particular by the Hopf algebras
of Feynman diagrams
EF+EX Forest Algebras
We examine languages of unranked forests definable using the temporal
operators EF and EX. We characterize the languages definable in this logic, and
various fragments thereof, using the syntactic forest algebras introduced by
Bojanczyk and Walukiewicz. Our algebraic characterizations yield efficient
algorithms for deciding when a given language of forests is definable in this
logic. The proofs are based on understanding the wreath product closures of a
few small algebras, for which we introduce a general ideal theory for forest
algebras. This combines ideas from the work of Bojanczyk and Walukiewicz for
the analogous logics on binary trees and from early work of Stiffler on wreath
product of finite semigroups
Preordered forests, packed words and contraction algebras
We introduce the notions of preordered and heap-preordered forests,
generalizing the construction of ordered and heap-ordered forests. We prove
that the algebras of preordered and heap-preordered forests are Hopf for the
cut coproduct, and we construct a Hopf morphism to the Hopf algebra of packed
words. Moreover, we define another coproduct on the preordered forests given by
the contraction of edges. Finally, we give a combinatorial description of
morphims defined on Hopf algebras of forests with values in the Hopf algebras
of shuffes or quasi-shuffles.Comment: 42 pages. arXiv admin note: text overlap with arXiv:1007.1547,
arXiv:1004.5208 by other author
Wreath Products of Forest Algebras, with Applications to Tree Logics
We use the recently developed theory of forest algebras to find algebraic
characterizations of the languages of unranked trees and forests definable in
various logics. These include the temporal logics CTL and EF, and first-order
logic over the ancestor relation. While the characterizations are in general
non-effective, we are able to use them to formulate necessary conditions for
definability and provide new proofs that a number of languages are not
definable in these logics
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
(Non)Commutative Hopf algebras of trees and (quasi)symmetric functions
The Connes-Kreimer Hopf algebra of rooted trees, its dual, and the Foissy
Hopf algebra of of planar rooted trees are related to each other and to the
well-known Hopf algebras of symmetric and quasi-symmetric functions via a pair
of commutative diagrams. We show how this point of view can simplify
computations in the Connes-Kreimer Hopf algebra and its dual, particularly for
combinatorial Dyson-Schwinger equations.Comment: For March 2006 CIRM conference "Renormalization and Galois theories
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