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11.
Shuffle type products are well known in mathematics and physics. They are intimately related to Loday's dendriform algebras and were extensively used to give explicit constructions of free 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, 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. Our construction is related to Loday's unital commutative tridendriform algebra, 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. 相似文献
12.
Thanks to an algebraic duality property of reduced states, the Schmidt best approximation theorems have important corollaries
in the rigorous theory of two-electron moleculae. In turn, the “harmonium model” or “Moshinsky atom” constitutes a non-trivial
laboratory bench for energy functionals proposed over the years (1964–today), purporting to recover the full ground state
of the system from knowledge of the reduced 1-body matrix. That model is usually regarded as solvable; however, some important
aspects of it, in particular the exact energy and full state functionals—unraveling the “phase dilemma” for the system—had
not been calculated heretofore. The solution is given here, made plain by working with Wigner quasiprobabilities on phase
space. It allows in principle for thorough discussions of the merits of several approximate functionals popular in the theoretical
chemical physics literature; in this respect, at the end we focus on Gill’s “Wigner intracule” method for the correlation
energy. 相似文献
13.
Kurusch Ebrahimi-Fard 《Journal of Pure and Applied Algebra》2008,212(2):320-339
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. 相似文献
14.
In this brief Letter, we would like to report on an observation concerning the relation between Rota–Baxter operators and Loday-type algebras, i.e. dendriform di- and tri-algebras. It is shown that associative algebras equipped with a Rota–Baxter operator of arbitrary weight always give such dendriform structures. 相似文献
15.
The quasi-shuffle product and mixable shuffle product are both generalizations of the shuffle product and have both been studied
quite extensively recently. We relate these two generalizations and realize quasi-shuffle product algebras as subalgebras
of mixable shuffle product algebras. As an application, we obtain Hopf algebra structures in free Rota–Baxter algebras. 相似文献
16.
A Magnus- and Fer-Type Formula in Dendriform Algebras 总被引:1,自引:0,他引:1
We provide a refined approach to the classical Magnus (Commun. Pure Appl. Math. 7:649–673, [1954]) and Fer expansion (Bull. Classe Sci. Acad. R. Belg. 44:818–829, [1958]), unveiling a new structure by using the language of dendriform and pre-Lie algebras. The recursive formula for the logarithm
of the solutions of the equations X=1+λ
a
≺
X and Y=1−λ
Y
≻
a in A[[λ]] is provided, where (A,≺,≻) is a dendriform algebra. Then we present the solutions to these equations as an infinite product expansion of exponentials.
Both formulae involve the pre-Lie product naturally associated with the dendriform structure. Several applications are presented.
相似文献
17.
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–Plebański–Strichartz formula for the continuous Baker–Campbell–Hausdorff series. 相似文献
18.
Kurusch Ebrahimi-Fard José M. Gracia-Bondía Frédéric Patras 《Communications in Mathematical Physics》2007,276(2):519-549
Motivated by recent work of Connes and Marcolli, based on the Connes–Kreimer approach to renormalization, we augment the latter
by a combinatorial, Lie algebraic point of view. Our results rely both on the properties of the Dynkin idempotent, one of
the fundamental Lie idempotents in the theory of free Lie algebras, and on properties of Hopf algebras encapsulated in the
notion of associated descent algebras. Besides leading very directly to proofs of the main combinatorial aspects of the renormalization
procedures, the new techniques give rise to an algebraic approach to the Galois theory of renormalization. In particular,
they do not depend on the geometry underlying the case of dimensional regularization and the Riemann–Hilbert correspondence.
This is illustrated with a discussion of the BPHZ renormalization scheme. 相似文献
19.
20.
Moving beyond the classical additive and multiplicative approaches, we present an “exponential” method for perturbative renormalization.
Using Dyson’s identity for Green’s functions as well as the link between the Faà di Bruno Hopf algebra and the Hopf algebras
of Feynman graphs, its relation to the composition of formal power series is analyzed. Eventually, we argue that the new method
has several attractive features and encompasses the BPHZ method. The latter can be seen as a special case of the new procedure
for renormalization scheme maps with the Rota–Baxter property. To our best knowledge, although very natural from group-theoretical
and physical points of view, several ideas introduced in the present paper seem to be new (besides the exponential method,
let us mention the notions of counter-factors and of order n bare coupling constants). 相似文献