Lie 3-algebras and deformations of relative Rota-Baxter operators on 3-Lie algebras

Tang, Rong; Hou, Shuai; Sheng, Yunhe

doi:10.1016/j.jalgebra.2020.09.017

Cited by 30 publications

(29 citation statements)

References 83 publications

(91 reference statements)

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“…Remark 2.10. Note that Lie n-algebras and n-Lie algebras are two different n-generalizations of Lie algebras [20] and n-Lie algebras are special Lie n-algebras [1]. Correspondingly, our pre Lie n-algebras (generalized pre-Lie algebras of order n in [12]) are different from n-pre Lie algebras in [12] and n-pre Lie algebras are special pre Lie n-algebras.…”

Section: Preliminariesmentioning

confidence: 99%

Relation among $n$-ary algebras and homotopy algebras

Wang¹,

Wu²

2021

Preprint

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We present P L∞-algebras in the form of composition of maps and show that a P L∞-algebra V can be described by a nilpotent coderivation on coalgebra P * V of degree −1. Using coalgebra maps among T * V , ∧ * V , P * V , we show that every A∞-algebra carries a P L∞algebra structure and every P L∞-algebra carries an L∞-algebra structure. In particular, we obtain a pre Lie n-algebra structure on an arbitrary partially associative n-algebra and deduce pre Lie n-algebras are n-Lie admissible.

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Section: Preliminariesmentioning

confidence: 99%

Relation among $n$-ary algebras and homotopy algebras

Wang¹,

Wu²

2021

Preprint

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“…Later Das and Misha also determined the L ∞ -structures underlying the cohomology theory for Rota-Baxter associative algebras of weight zero [17]. There are some other related work [62,63,11,12,13,15,16]. These work all concern Rota-Baxter operators of weight zero.…”

Section: Introductionmentioning

confidence: 99%

Deformations and homotopy theory of Rota-Baxter algebras of any weight

Wang¹,

Zhou²

2021

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This paper studies formal deformations and homotopy theory of Rota-Baxter algebras of any weight. We define an L ∞ -algebra, which controls simultaneous deformations of associative products and Rota-Baxter operators. As a consequence, we develop a cohomology theory of Rota-Baxter algebras of any weight and justify it by interpreting lower degree cohomology groups as formal deformations and abelian extensions. The notion of homotopy Rota-Baxter algebras is introduced and it is shown that the operad governing homotopy Rota-Baxter algebras is a minimal model of the operad of Rota-Baxter algebras.

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“…Rota-Baxter operators on super-type algebras were studied in [1], which build relationships between associative superalgebras, Lie superalgebras, L-dendriform superalgebras and pre-Lie superalgebras. The notion of a Rota-Baxter operator on a 3-Lie algebra was given in [6], and the cohomology theory of Rota-Baxter operators on 3-Lie algebras was developed in [27]. Generally, Rota-Baxter operators can be defined on operads, which give rise to splittings of operads [5,23].…”

Section: Introductionmentioning

confidence: 99%

Relative Rota-Baxter operators and symplectic structures on Lie-Yamaguti algebras

Sheng¹,

Zhao²

2021

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In this paper, first we introduce the notion of a quadratic Lie-Yamaguti algebra and show that the invariant bilinear form in a quadratic Lie-Yamaguti algebra induces an isomorphism between the adjoint representation and the coadjoint representation. Then we introduce the notions of relative Rota-Baxter operators on Lie-Yamaguti algebras and pre-Lie-Yamaguti algebras. We prove that a pre-Lie-Yamaguti algebra gives rise to a Lie-Yamaguti algebra naturally and a relative Rota-Baxter operator induces a pre-Lie-Yamaguti algebra. Finally we study symplectic structures on Lie-Yamaguti algebra, which give rise to relative Rota-Baxter operators as well as pre-Lie-Yamaguti algebras. As applications, we study phase spaces of Lie-Yamaguti algebras, and show that there is a one-to-one correspondence between phase spaces of Lie-Yamaguti algebras and Manin triples of pre-Lie-Yamaguti algebras.

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Lie 3-algebras and deformations of relative Rota-Baxter operators on 3-Lie algebras

Cited by 30 publications

References 83 publications

Relation among $n$-ary algebras and homotopy algebras

Relation among $n$-ary algebras and homotopy algebras

Deformations and homotopy theory of Rota-Baxter algebras of any weight

Relative Rota-Baxter operators and symplectic structures on Lie-Yamaguti algebras

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