Conformal classical Yang–Baxter equation, S-equation and $${\mathcal {O}}$$-operators

Hong, Yanyong; Bai, Chengming

doi:10.1007/s11005-019-01243-2

Cited by 14 publications

(11 citation statements)

References 16 publications

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“…Definition 5.2. (see [30]) Let A be an associative conformal algebra and let T : A → A be a C[∂ ]-module homomorphism. For q ∈ C, if there holds…”

Section: Definition 51 Let a Be An Associative Conformal Algebra A C[...mentioning

confidence: 99%

“…It was shown in[30, Corollary 4.6] that an O-operator T : M → L on a Lie conformal algebra L associated with a representation (M, ρ) also induces a left-symmetric conformal algebra structure on M bym • λ n = ρ(T (m)) λ n, ∀ m, n ∈ M.(3.29)If L = A L is the commutator Lie conformal algebra of an associative conformal algebra A and the representation of L on M is induced from the conformal A -bimodule structure on M (see(3.24)), then the two left-symmetric conformal algebra structures above are exactly the same.…”

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confidence: 94%

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$\mathcal{O}$-operators and Nijenhius operators of associative conformal algebras

Yuan¹

2022

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We study O-operators of associative conformal algebras with respect to conformal bimodules. As natural generalizations of O-operators and dendriform conformal algebras, we introduce the notions of twisted Rota-Baxter operators and conformal NS-algebras. We show that twisted Rota-Baxter operators give rise to conformal NS-algebras, the same as O-operators induce dendriform conformal algebras. And we introduce a conformal analog of associative Nijenhius operators and enumerate main properties. By using derived bracket construction of Kosmann-Schwarzbach and a method of Uchino, we obtain a graded Lie algebra whose Maurer-Cartan elements are given by O-operators. This allows us to construct cohomology of O-operators. This cohomology can be seen as the Hochschild cohomology of an associative conformal algebra with coefficients in a suitable conformal bimodule.

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“…Definition 5.2. (see [30]) Let A be an associative conformal algebra and let T : A → A be a C[∂ ]-module homomorphism. For q ∈ C, if there holds…”

Section: Definition 51 Let a Be An Associative Conformal Algebra A C[...mentioning

confidence: 99%

mentioning

confidence: 94%

$\mathcal{O}$-operators and Nijenhius operators of associative conformal algebras

Yuan¹

2022

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“…In particular, the skew-symmetric solutions of conformal classical Yang-Baxter equation give rise to Lie conformal bialgebras. Recently, the authors in [16] showed that r is a skew-symmetric solution of the conformal classical Yang-Baxter equation in a finite Lie conformal algebra A if and only if…”

Section: Introductionmentioning

confidence: 99%

“…and ad * is the coadjoint module over the A. More generally, the authors in [16] introduced the notion of O-operator on a Lie conformal algebra. Thus r is a skew-symmetric solution of the conformal classical Yang-Baxter equation in a finite Lie conformal algebra A if and only if r ♯ 0 = r ♯ λ | λ=0 is an O-operator on the LCMod pair (A; ad * ).…”

Section: Introductionmentioning

confidence: 99%

Conformal $r$-matrix-Nijenhuis structures, symplectic-Nijenhuis structures and $\mathcal{O} N$-structures

Liu¹,

Zhou²,

Yuan³

2022

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In this paper, we first study infinitesimal deformations of a Lie conformal algebra and a Lie conformal algebra with a module (called an LCMod pair), which lead to the notions of Nijenhuis operator on the Lie conformal algebra and Nijenhuis structure on the LCMod pair, respectively. Then by adding compatibility conditions between Nijenhuis structures and O-operators, we introduce the notion of an ON-structure on an LCMod pair and show that an ON-structure gives rise to a hierarchy of pairwise compatible O-operators. In particular, we show that compatible O-operators on a Lie conformal algebra can be characterized by Nijenhuis operators on Lie conformal algebras. Finally, we introduce the notions of conformal r-matrix-Nijenhuis structure and symplectic-Nijenhuis structure on the Lie conformal algebra and study their relations. Contents 1. Introduction 1 2. Preliminaries on Lie conformal algebra 3 3. Nijenhuis operators and Nijenhuis structures on Lie conformal algebras 6 3.1. Nijenhuis operators on Lie conformal algebras 6 3.2. Nijenhuis structures on LCMod pairs 11 4. ON-structures and compatible O-operators on LCMod pairs 13 5. Conformal r-matrix-Nijenhuis structures and symplectic-Nijenhuis structures on Lie conformal algebras 19 5.1. Conformal r-matrix-Nijenhuis structures on Lie conformal algebras 20 5.2. Symplectic-Nijenhuis structures on Lie conformal algebras 21 References 25 2010 Mathematics Subject Classification. 17A30, 17B69, 17B62.

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“…Similarly, a theory of left-symmetric conformal bialgebras was developed in [14], which are equivalent to a class of special Lie conformal algebras named parakähler Lie conformal algebras and the notion of conformal S-equation was introduced in the coboundary case. Moreover, the operator forms of the conformal classical Yang-Baxter equation and the conformal S-equation were studied in [12], which shows that the antisymmetric solutions of the conformal classical Yang-Baxter equation and the symmetric solutions of the conformal S-equation can be interpreted in terms of a kind of operators called O-operators in the conformal sense.…”

Section: Introductionmentioning

confidence: 99%

On antisymmetric infinitesimal conformal bialgebras

Hong

Bai

2021

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In this paper, we construct a bialgebra theory for associative conformal algebras, namely antisymmetric infinitesimal conformal bialgebras. On the one hand, it is an attempt to give conformal structures for antisymmetric infinitesimal bialgebras. On the other hand, under certain conditions, such structures are equivalent to double constructions of Frobenius conformal algebras, which are associative conformal algebras that are decomposed into the direct sum of another associative conformal algebra and its conformal dual as C[∂]-modules such that both of them are subalgebras and the natural conformal bilinear form is invariant. The coboundary case leads to the introduction of associative conformal Yang-Baxter equation whose antisymmetric solutions give antisymmetric infinitesimal conformal bialgebras. Moreover, the construction of antisymmetric solutions of associative conformal Yang-Baxter equation is given from O-operators of associative conformal algebras as well as dendriform conformal algebras.

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Conformal classical Yang–Baxter equation, S-equation and $${\mathcal {O}}$$-operators

Cited by 14 publications

References 16 publications

$\mathcal{O}$-operators and Nijenhius operators of associative conformal algebras

$\mathcal{O}$-operators and Nijenhius operators of associative conformal algebras

Conformal $r$-matrix-Nijenhuis structures, symplectic-Nijenhuis structures and $\mathcal{O} N$-structures

On antisymmetric infinitesimal conformal bialgebras

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