Operator Forms of the Classical Yang–baxter Equation in Lie Superalgebras

Wang, Yan; Hou, Dongping; Bai, Chengming

doi:10.1142/s0219887810004452

Cited by 5 publications

(4 citation statements)

References 27 publications

(33 reference statements)

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“…The triple (A, [ , ], R) refers to a Rota-Baxter Lie superalgebra, see [54]. We introduce the notion of super O-operators of pre-Lie superalgebras and study some properties over Lie superalgebras and pre-Lie superalgebras.…”

Section: R(x)r(y) = R R(x)y + X R(y)mentioning

confidence: 99%

Rota–Baxter Operators on Pre-Lie Superalgebras

Abdaoui

Mabrouk

Makhlouf

2017

Bull. Malays. Math. Sci. Soc.

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In this paper, we study Rota-Baxter operators and super O-operator of associative superalgebras, Lie superalgebras, pre-Lie superalgebras and L-dendriform superalgebras. Then we give some properties of pre-Lie superalgebras constructed from associative superalgebras, Lie superalgebras and L-dendriform superalgebras. Moreover, we provide all Rota-Baxter operators of weight zero on complex pre-Lie superalgebras of dimensions 2 and 3.

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Section: R(x)r(y) = R R(x)y + X R(y)mentioning

confidence: 99%

Rota–Baxter Operators on Pre-Lie Superalgebras

Abdaoui

Mabrouk

Makhlouf

2017

Bull. Malays. Math. Sci. Soc.

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“…As a natural generalization of Poisson algebras, Poisson superalgebras contains classes of Lie superalgebras and associative superalgebras. Some similar results on Classical Yang-Baxter equation (CYBE) on Lie superalgebras, Associative Yang-Baxter equation (AYBE) on associative superalgebras, and PYBE on Poisson algebras prompt us to study PYBE on Poisson superalgebras; see [Agu00], [WHB10], and [Xu94]. Our primary objective is to give a systematic study on skew-symmetric solutions of PYBE in terms of O-operators of Poisson superalgebras.…”

Section: Introductionmentioning

confidence: 92%

Poisson Yang-Baxter equations and $\mathcal{O}$-operators of Poisson superalgebras

Shan¹,

Zhang²

2021

Preprint

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We investigate connections between O-operators of Poisson superalgebras and skewsymmetric solutions of the Poisson Yang-Baxter equation (PYBE). We prove that a skew-symmetric solution of the PYBE on a Poisson superalgebra can be interpreted as an O-operator associated to the co-regular representation. We show that this connection can be enhanced with symplectic forms when considering non-degenerate skew-symmetric solutions. We also show that O-operators associated to a general representation could give skew-symmetric solutions of the PYBE in certain semi-direct product of Poisson superalgebras.

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“…In 2008, Kong and Bai classified all compatible LSSA on the super-Virasoro algebras satisfying some restricted conditions ( [10]). A close relation between LSSAs and the graded classical Yang-Baxter equation in Lie superalgebras was obtained in [16]. As a subclass of LSSAs, Novikov superalgebras were studied extensively in [17,18,11,1,19,20,21].…”

Section: Introductionmentioning

confidence: 95%

Left-symmetric Structures on Complex Simple Lie Superalgebras

Zhang

2013

Preprint

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A well-known fact is that there does not exist any compatible left-symmetric structures on a finite-dimensional complex semisimple Lie algebra (see [5]). This result is not valid in semisimple Lie superalgebra case. In this paper, we study the compatible Left-symmetric superalgebra (LSSA for short) structures on complex simple Lie superalgebras. We prove that there is not any compatible LSSA structure on a finite-dimensional complex simple Lie superalgebra except for the classical simple Lie superalgebra A(m, n)(m n) and Cartan simple Lie superalgebra W(n)(n ≥ 3). We also classify all compatible LSSAs with a right-identity on A(0, 1).

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Operator Forms of the Classical Yang–baxter Equation in Lie Superalgebras

Cited by 5 publications

References 27 publications

Rota–Baxter Operators on Pre-Lie Superalgebras

Rota–Baxter Operators on Pre-Lie Superalgebras

Poisson Yang-Baxter equations and $\mathcal{O}$-operators of Poisson superalgebras

Left-symmetric Structures on Complex Simple Lie Superalgebras

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