Birkhoff Type Decompositions and the Baker–Campbell–Hausdorff Recursion

Ebrahimi‐Fard, Kurusch; Guo, Li; Manchon, Dominique

doi:10.1007/s00220-006-0080-7

Cited by 68 publications

(117 citation statements)

References 33 publications

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“…Given the recent progresses on Rota-Baxter algebra in both theoretical and applied aspects [1,2,10,[15][16][17][18][19][24][25][26][27][28][29][30][31], it is desirable to obtain convenient constructions of free Rota-Baxter algebras. To this end, in Section 5, we put together bijections and inclusions among bracketed words, Motzkin path, vertex decorated forests and angularly decorated forests, as well as their various subsets (Theorem 5.1).…”

Section: Outline Of the Papermentioning

confidence: 99%

Operated semigroups, Motzkin paths and rooted trees

2008

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Combinatorial objects such as rooted trees that carry a recursive structure have found important applications recently in both mathematics and physics. We put such structures in an algebraic framework of operated semigroups. This framework provides the concept of operated semigroups with intuitive and convenient combinatorial descriptions, and at the same time endows the familiar combinatorial objects with a precise algebraic interpretation. As an application, we obtain constructions of free Rota-Baxter algebras in terms of Motzkin paths and rooted trees.

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Section: Outline Of the Papermentioning

confidence: 99%

Operated semigroups, Motzkin paths and rooted trees

2008

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“…For more details we refer the reader to the classical papers [2], [3], [6], [45], [46], as well as for instance to the references [22], [19].…”

Section: Rota-baxter Algebras and Bogoliubov's Recursionmentioning

confidence: 99%

“…However, as we mentioned, in renormalization the very RB structure one has to deal with is noncommutative, so that these results do not apply. In [17] it was shown that one can prove a first noncommutative Spitzer (also known as Pollaczeck-Spitzer) identity, and that this identity was related to a so-called BakerCampbell-Hausdorff (BCH) recursion, which is another way, besides Bogoliubov's, to perform recursively the renormalization process [18], [19].…”

Section: Introductionmentioning

confidence: 99%

A noncommutative Bohnenblust–Spitzer identity for Rota–Baxter algebras solves Bogoliubov’s recursion

Ebrahimi‐Fard¹,

Manchon²,

Patras³

2009

J. Noncommut. Geom.

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103

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Abstract. The Bogoliubov recursion is a particular procedure appearing in the process of renormalization in perturbative quantum field theory. It provides convergent expressions for otherwise divergent integrals. We develop here a theory of functional identities for noncommutative Rota-Baxter algebras which is shown to encode, among others, this process in the context of Connes-Kreimer's Hopf algebra of renormalization. Our results generalize the seminal Cartier-Rota theory of classical Spitzer-type identities for commutative Rota-Baxter algebras. In the classical, commutative, case these identities can be understood as deriving from the theory of symmetric functions. Here we show that an analogous property holds for noncommutative Rota-Baxter algebras. That is, we show that functional identities in the noncommutative setting can be derived from the theory of noncommutative symmetric functions. Lie idempotents, and particularly the Dynkin idempotent, play a crucial role in the process. Their action on the pro-unipotent groups such as those of perturbative renormalization is described in detail along the way.

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“…The other operators are also important. For example, the Rota-Baxter operator, which originated from probability study [9], is closely related to the classical Yang-Baxter equation, as well as to operads, to combinatorics and, through the Hopf algebra framework of Connes and Kreimer, to the renormalization of quantum field theory [2,3,8,5,17,19,20,18,24,25,28].…”

Section: P(x)p(y) = P(xp(y) + P(x)y − P(x)p(y))mentioning

confidence: 99%

Rota-Baxter type operators, rewriting systems and Gröbner-Shirshov bases

Guo

Sit

Gao

et al. 2015

ACM Commun. Comput. Algebra

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Abstract. In this paper we apply the methods of rewriting systems and Gröbner-Shirshov bases to give a unified approach to a class of linear operators on associative algebras. These operators resemble the classic Rota-Baxter operator, and they are called Rota-Baxter type operators. We characterize a Rota-Baxter type operator by the convergency of a rewriting system associated to the operator. By associating such an operator to a Gröbner-Shirshov basis, we obtain a canonical basis for the free algebras in the category of associative algebras with that operator. This construction include as special cases several previous ones for free objects in similar categories, such as those of Rota-Baxter algebras and Nijenhuis algebras.

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Birkhoff Type Decompositions and the Baker–Campbell–Hausdorff Recursion

Cited by 68 publications

References 33 publications

Operated semigroups, Motzkin paths and rooted trees

Operated semigroups, Motzkin paths and rooted trees

A noncommutative Bohnenblust–Spitzer identity for Rota–Baxter algebras solves Bogoliubov’s recursion

Rota-Baxter type operators, rewriting systems and Gröbner-Shirshov bases

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