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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2506.07061 |
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| _version_ | 1866918050173812736 |
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| author | Ma, Tianshui Zhao, Chan Zheng, Huihui |
| author_facet | Ma, Tianshui Zhao, Chan Zheng, Huihui |
| contents | Special left Alia algebras were introduced by Dzhumadil'daev in [J. Math. Sci. (N.Y.) 161(2009), 11-30] when studying the classification of algebras with skew-symmetric identity of degree 3. A special left Alia algebra (resp. coalgebra) $(A, [,]_{(f,g)})$ (resp. $(A, Δ_{(F,G)})$) is constructed by a commutative associative algebra (resp. cocommutative coassociative coalgebra) $(A, \cdot)$ (resp. $(A, δ)$) together with two linear maps $f, g: A\longrightarrow A$ (resp. $F, G: A\longrightarrow A$). We find that if $((A, \cdot), f)$ (resp. $((A, δ), F)$) is a Nijenhuis associative algebra (resp. coassociative coalgebra) such that $f\circ g=g\circ f$ (resp. $F\circ G=G\circ F$), then $((A, [,]_{(f,g)}), f)$ (resp. $((A, Δ_{(F,G)}), F)$) is a Nijenhuis left Alia algebra (resp. coalgebra). A bialgebraic structure, named Nijenhuis associative D-bialgebra and denoted by $((A, \cdot, δ), f, F)$, for $((A, \cdot), f)$ and $((A, δ), F)$ was presented in [J. Algebra 639(2024), 150-186]. In this paper, we investigate the bialgebraic structure, named Nijenhuis left Alia bialgebra and denoted by $((A, [,], Δ), N, S)$, for a Nijenhuis left Alia algebra $((A, [,]), N)$ and a Nijenhuis left Alia coalgebra $((A, Δ), S)$, such that Nijenhuis special left Alia bialgebra $((A, [,]_{(f,g)}, Δ_{(F,G)}), f, F)$ can be induced by Nijenhuis commutative cocommutative associative D-bialgebra $((A, \cdot, δ), f, F)$. We also provide a method to construct Nijenhuis operators on a left Alia algebra (resp. coalgebra). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_07061 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bialgebras induced by special left Alia algebras Ma, Tianshui Zhao, Chan Zheng, Huihui Rings and Algebras Special left Alia algebras were introduced by Dzhumadil'daev in [J. Math. Sci. (N.Y.) 161(2009), 11-30] when studying the classification of algebras with skew-symmetric identity of degree 3. A special left Alia algebra (resp. coalgebra) $(A, [,]_{(f,g)})$ (resp. $(A, Δ_{(F,G)})$) is constructed by a commutative associative algebra (resp. cocommutative coassociative coalgebra) $(A, \cdot)$ (resp. $(A, δ)$) together with two linear maps $f, g: A\longrightarrow A$ (resp. $F, G: A\longrightarrow A$). We find that if $((A, \cdot), f)$ (resp. $((A, δ), F)$) is a Nijenhuis associative algebra (resp. coassociative coalgebra) such that $f\circ g=g\circ f$ (resp. $F\circ G=G\circ F$), then $((A, [,]_{(f,g)}), f)$ (resp. $((A, Δ_{(F,G)}), F)$) is a Nijenhuis left Alia algebra (resp. coalgebra). A bialgebraic structure, named Nijenhuis associative D-bialgebra and denoted by $((A, \cdot, δ), f, F)$, for $((A, \cdot), f)$ and $((A, δ), F)$ was presented in [J. Algebra 639(2024), 150-186]. In this paper, we investigate the bialgebraic structure, named Nijenhuis left Alia bialgebra and denoted by $((A, [,], Δ), N, S)$, for a Nijenhuis left Alia algebra $((A, [,]), N)$ and a Nijenhuis left Alia coalgebra $((A, Δ), S)$, such that Nijenhuis special left Alia bialgebra $((A, [,]_{(f,g)}, Δ_{(F,G)}), f, F)$ can be induced by Nijenhuis commutative cocommutative associative D-bialgebra $((A, \cdot, δ), f, F)$. We also provide a method to construct Nijenhuis operators on a left Alia algebra (resp. coalgebra). |
| title | Bialgebras induced by special left Alia algebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2506.07061 |