Simple finite-dimensional double algebras

Goncharov, M. E.; Kolesnikov, Pavel

doi:10.1016/j.jalgebra.2017.04.020

Cited by 38 publications

(52 citation statements)

References 9 publications

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“…Rota-Baxter operators lead to the splitting of operads [3,45], and are closely related to noncommutative symmetric functions and Hopf algebras [16,27,56]. Recently the relationship between Rota-Baxter operators and double Poisson algebras were studied in [23]. In the Lie algebra context, a Rota-Baxter operator was introduced independently in the 1980s as the operator form of the classical Yang-Baxter equation that plays important roles in many subfields of mathematics and mathematical physics such as integrable systems and quantum groups [10,47].…”

Section: Rota-baxter Operatorsmentioning

confidence: 99%

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Deformations and Homotopy Theory of Relative Rota–Baxter Lie Algebras

Lazarev

Sheng

Tang

2020

Commun. Math. Phys.

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We determine the $$L_\infty $$ L ∞ -algebra that controls deformations of a relative Rota–Baxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying $$\mathsf {Lie}\mathsf {Rep}$$ Lie Rep pair by the dg Lie algebra controlling deformations of the relative Rota–Baxter operator. Consequently, we define the cohomology of relative Rota–Baxter Lie algebras and relate it to their infinitesimal deformations. A large class of relative Rota–Baxter Lie algebras is obtained from triangular Lie bialgebras and we construct a map between the corresponding deformation complexes. Next, the notion of a homotopy relative Rota–Baxter Lie algebra is introduced. We show that a class of homotopy relative Rota–Baxter Lie algebras is intimately related to pre-Lie$$_\infty $$ ∞ -algebras.

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Section: Rota-baxter Operatorsmentioning

confidence: 99%

“…By (23), we have the short exact sequence of chain complexes which induces a long exact sequence of cohomology groups. Also by(23), c n is given by c n([α]) = [h T α].…”

mentioning

confidence: 99%

Deformations and Homotopy Theory of Relative Rota–Baxter Lie Algebras

Lazarev

Sheng

Tang

2020

Commun. Math. Phys.

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“…There is a close connection between Rota-Baxter operators and Yang-Baxter equation [1]. In last twenty years, Rota-Baxter operators have found important applications in renormalizations of quantum field theory [9], pre-algebras [3], infinitesimal bialgebras [1] and double algebras [14]. See [16] for more on Rota-Baxter operators.…”

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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“…See the well-written introduction in [40] for more details. More recently, people have begun to study averaging operators in double algebras, classical Yang-Baxter equation, conformal algebras, and the procedure of replication in the operad theory [1,15,21,39]. It is not yet clear to us how these aspects of embedding tensors and Lie-Leibniz triples are related.…”

Section: Introductionmentioning

confidence: 99%

The Controlling $$L_\infty $$-Algebra, Cohomology and Homotopy of Embedding Tensors and Lie–Leibniz Triples

Sheng

Tang

Zhu

2021

Commun. Math. Phys.

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In this paper, we first construct the controlling algebras of embedding tensors and Lie–Leibniz triples, which turn out to be a graded Lie algebra and an $$L_\infty $$ L ∞ -algebra respectively. Then we introduce representations and cohomologies of embedding tensors and Lie–Leibniz triples, and show that there is a long exact sequence connecting various cohomologies. As applications, we classify infinitesimal deformations and central extensions using the second cohomology groups. Finally, we introduce the notion of a homotopy embedding tensor which will induce a Leibniz$$_\infty $$ ∞ -algebra. We realize Kotov and Strobl’s construction of an $$L_\infty $$ L ∞ -algebra from an embedding tensor, as a functor from the category of homotopy embedding tensors to that of Leibniz$$_\infty $$ ∞ -algebras, and a functor further to that of $$L_\infty $$ L ∞ -algebras.

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Simple finite-dimensional double algebras

Cited by 38 publications

References 9 publications

Deformations and Homotopy Theory of Relative Rota–Baxter Lie Algebras

Deformations and Homotopy Theory of Relative Rota–Baxter Lie Algebras

Deformations of associative Rota-Baxter operators

The Controlling $$L_\infty $$-Algebra, Cohomology and Homotopy of Embedding Tensors and Lie–Leibniz Triples

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