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Keywords = Manin triple

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16 pages, 309 KB  
Article
Manin Triples and Bialgebras of Left-Alia Algebras Associated with Invariant Theory
by Chuangchuang Kang, Guilai Liu, Zhuo Wang and Shizhuo Yu
Mathematics 2024, 12(3), 408; https://doi.org/10.3390/math12030408 - 26 Jan 2024
Cited by 1 | Viewed by 1267
Abstract
A left-Alia algebra is a vector space together with a bilinear map satisfying the symmetric Jacobi identity. Motivated by invariant theory, we first construct a class of left-Alia algebras induced by twisted derivations. Then, we introduce the notions of Manin triples and bialgebras [...] Read more.
A left-Alia algebra is a vector space together with a bilinear map satisfying the symmetric Jacobi identity. Motivated by invariant theory, we first construct a class of left-Alia algebras induced by twisted derivations. Then, we introduce the notions of Manin triples and bialgebras of left-Alia algebras. Via specific matched pairs of left-Alia algebras, we figure out the equivalence between Manin triples and bialgebras of left-Alia algebras. Full article
17 pages, 299 KB  
Article
3-Hom–Lie Yang–Baxter Equation and 3-Hom–Lie Bialgebras
by Shuangjian Guo, Shengxiang Wang and Xiaohui Zhang
Mathematics 2022, 10(14), 2485; https://doi.org/10.3390/math10142485 - 17 Jul 2022
Cited by 2 | Viewed by 1761
Abstract
In this paper, we first introduce the notion of a 3-Hom–Lie bialgebra and give an equivalent description of the 3-Hom–Lie bialgebras, the matched pairs and the Manin triples of 3-Hom–Lie algebras. In addition, we define O-operators of 3-Hom–Lie algebras and construct solutions [...] Read more.
In this paper, we first introduce the notion of a 3-Hom–Lie bialgebra and give an equivalent description of the 3-Hom–Lie bialgebras, the matched pairs and the Manin triples of 3-Hom–Lie algebras. In addition, we define O-operators of 3-Hom–Lie algebras and construct solutions of the 3-Hom–Lie Yang–Baxter equation in terms of O-operators and 3-Hom–pre-Lie algebras. Finally, we show that a 3-Hom–Lie algebra has a phase space if and only if it is sub-adjacent to a 3-Hom–pre-Lie algebra. Full article
(This article belongs to the Special Issue Hopf-Type Algebras, Lie Algebras, Quantum Groups and Related Topics)
15 pages, 263 KB  
Article
The Classical Hom–Leibniz Yang–Baxter Equation and Hom–Leibniz Bialgebras
by Shuangjian Guo, Shengxiang Wang and Xiaohui Zhang
Mathematics 2022, 10(11), 1920; https://doi.org/10.3390/math10111920 - 3 Jun 2022
Cited by 2 | Viewed by 1870
Abstract
In this paper, we first introduce the notion of Hom–Leibniz bialgebras, which is equivalent to matched pairs of Hom–Leibniz algebras and Manin triples of Hom–Leibniz algebras. Additionally, we extend the notion of relative Rota–Baxter operators to Hom–Leibniz algebras and prove that there is [...] Read more.
In this paper, we first introduce the notion of Hom–Leibniz bialgebras, which is equivalent to matched pairs of Hom–Leibniz algebras and Manin triples of Hom–Leibniz algebras. Additionally, we extend the notion of relative Rota–Baxter operators to Hom–Leibniz algebras and prove that there is a Hom–pre-Leibniz algebra structure on Hom–Leibniz algebras that have a relative Rota–Baxter operator. Finally, we study the classical Hom–Leibniz Yang–Baxter equation on Hom–Leibniz algebras and present its connection with the relative Rota–Baxter operator. Full article
(This article belongs to the Special Issue Hopf-Type Algebras, Lie Algebras, Quantum Groups and Related Topics)
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