Weighted infinitesimal unitary bialgebras on rooted forests and weighted cocycles

Zhang, Yi; Chen, Dan; Gao, Xing; Luo, Yanfeng

doi:10.2140/pjm.2019.302.741

Cited by 16 publications

(19 citation statements)

References 36 publications

(73 reference statements)

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“…Weighted (relative) Rota-Baxter operators are generalization of (relative) Rota-Baxter operators. They are related to post-algebras [5], weighted infinitesimal bialgebras [27], weighted associative Yang-Baxter equations [27], combinatorics of planar rooted forests [26], and play an important role in mathematical physics [5]. Some classification result of weighted Rota-Baxter operators on matrix algebras are given in [15].…”

Section: Introductionmentioning

confidence: 99%

Deformations of associative Rota-Baxter operators

Das

2020

Journal of Algebra

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Weighted Rota-Baxter operators on associative algebras are closely related to modified Yang-Baxter equations, splitting of algebras, weighted infinitesimal bialgebras, and play an important role in mathematical physics. For any λ ∈ k, we construct a differential graded Lie algebra whose Maurer-Cartan elements are given by λ-weighted relative Rota-Baxter operators. Using such characterization, we define the cohomology of a λ-weighted relative Rota-Baxter operator T , and interpret this as the Hochschild cohomology of a suitable algebra with coefficients in an appropriate bimodule. We study linear, formal and finite order deformations of T from cohomological points of view. Among others, we introduce Nijenhuis elements that generate trivial linear deformations and define a second cohomology class to any finite order deformation which is the obstruction to extend the deformation. In the end, we also consider the cohomology of λ-weighted relative Rota-Baxter operators in the Lie case and find a connection with the case of associative algebras.

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Section: Introductionmentioning

confidence: 99%

Deformations of associative Rota-Baxter operators

Das

2020

Journal of Algebra

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“…Rota-Baxter operators with arbitrary weight (also called weighted Rota-Baxter operators) was considered in [4,5]. They are related with tridendriform algebras [12], post-Lie algebras and modified Yang-Baxter equations [4], weighted infinitesimal bialgebras and weighted Yang-Baxter equations [32], combinatorics of rooted forests [31], among others. Recently, the authors in [18] defined the cohomology of Rota-Baxter operators of weight 1 on Lie algebras and Lie groups.…”

Section: Introductionmentioning

confidence: 99%

Cohomology of weighted Rota-Baxter Lie algebras and Rota-Baxter paired operators

Das

2021

Preprint

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In this paper, we define representations and cohomology of weighted Rota-Baxter Lie algebras. As applications of cohomology, we study abelian extensions and formal 1-parameter deformations weighted Rota-Baxter Lie algebras. Finally, we consider weighted Rota-Baxter paired operators that induces a weighted Rota-Baxter Lie algebra and a representation of it. We define suitable cohomology for such paired operators that govern deformation.

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“…The concept of infinitesimal bialgebras was introduced by S. Joni and G.-C. Rota in [17] and further studied by M. Aguiar in [7,8,9]. There have been several interesting developments of infinitesimal bialgebras in combinatorics and in other areas of mathematics, see [13,15,16,32,33]. On the other hand, the notion of infinitesimal BiHom-bialgebras is introduced and studied in [18,19,23].…”

Section: Introductionmentioning

confidence: 99%