22 research outputs found
From AKNS to derivative NLS hierarchies via deformations of associative products
Using deformations of associative products, derivative nonlinear Schrodinger
(DNLS) hierarchies are recovered as AKNS-type hierarchies. Since the latter can
also be formulated as Gelfand-Dickey-type Lax hierarchies, a recently developed
method to obtain 'functional representations' can be applied. We actually
consider hierarchies with dependent variables in any (possibly noncommutative)
associative algebra, e.g., an algebra of matrices of functions. This also
covers the case of hierarchies of coupled DNLS equations.Comment: 22 pages, 2nd version: title changed and material organized in a
different way, 3rd version: introduction and first part of section 2
rewritten, taking account of previously overlooked references. To appear in
J. Physics A: Math. Ge
Construction of Nijenhuis operators and dendriform trialgebras
Nijenhuis operators are constructed from particular bialgebras called
dendriform- Nijenhuis bialgebras. It turns out that such operators commute with
TD-operators, kind of Baxter-Rota operators, and therefore closely related to
dendriform trialgebras. Examples are given.Comment: 21 page
On Products and Duality of Binary, Quadratic, Regular Operads
Since its introduction by Loday in 1995 with motivation from algebraic
K-theory, dendriform dialgebras have been studied quite extensively with
connections to several areas in mathematics and physics. A few more similar
structures have been found recently, such as the tri-, quadri-, ennea- and
octo-algebras, with increasing complexity in their constructions and
properties. We consider these constructions as operads and their products and
duals, in terms of generators and relations, with the goal to clarify and
simplify the process of obtaining new algebra structures from known structures
and from linear operators.Comment: 22 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
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
Generalized shuffles related to Nijenhuis and TD-algebras
Shuffle and quasi-shuffle products are well-known in the mathematics
literature. They are intimately related to Loday's dendriform algebras, and
were extensively used to give explicit constructions of free commutative
Rota-Baxter algebras. In the literature there exist at least two other
Rota-Baxter type algebras, namely, the Nijenhuis algebra and the so-called
TD-algebra. The explicit construction of the free unital commutative Nijenhuis
algebra uses a modified quasi-shuffle product, called the right-shift shuffle.
We show that another modification of the quasi-shuffle product, the so-called
left-shift shuffle, can be used to give an explicit construction of the free
unital commutative TD-algebra. We explore some basic properties of TD-operators
and show that the free unital commutative Nijenhuis algebra is a TD-algebra. We
relate our construction to Loday's unital commutative dendriform trialgebras,
including the involutive case. The concept of Rota-Baxter, Nijenhuis and
TD-bialgebras is introduced at the end and we show that any commutative
bialgebra provides such objects.Comment: 20 pages, typos corrected, accepted for publication in Communications
in Algebr