Lie Rota--Baxter operators on the Sweedler algebra $H_4$
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| Format: | Preprint |
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2024
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| _version_ | 1866913350280019968 |
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| author | Bardakov, Valeriy G. Nikonov, Igor M. Zhelaybin, Viktor N. |
| author_facet | Bardakov, Valeriy G. Nikonov, Igor M. Zhelaybin, Viktor N. |
| contents | If $A$ is an associative algebra, then we can define the adjoint Lie algebra $A^{(-)}$ and Jordan algebra $A^{(+)}$. It is easy to see that any associative Rota--Baxter operator on $A$ induces a Lie and Jordan Rota--Baxter operator on $A^{(-)}$ and $A^{(+)}$ respectively. Are there Lie (Jordan) Rota--Baxter operators, which are not associative Rota--Baxter operators?
In the present article we are studying these questions for the Sweedler algebra $H_4$, that is a 4-dimension non-commutative Hopf algebra. More precisely, we describe the Rota--Baxter operators on Lie algebra on the adjoint Lie algebra $H_4^{(-)}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_08291 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lie Rota--Baxter operators on the Sweedler algebra $H_4$ Bardakov, Valeriy G. Nikonov, Igor M. Zhelaybin, Viktor N. Group Theory Rings and Algebras 16T05, 17B38 If $A$ is an associative algebra, then we can define the adjoint Lie algebra $A^{(-)}$ and Jordan algebra $A^{(+)}$. It is easy to see that any associative Rota--Baxter operator on $A$ induces a Lie and Jordan Rota--Baxter operator on $A^{(-)}$ and $A^{(+)}$ respectively. Are there Lie (Jordan) Rota--Baxter operators, which are not associative Rota--Baxter operators? In the present article we are studying these questions for the Sweedler algebra $H_4$, that is a 4-dimension non-commutative Hopf algebra. More precisely, we describe the Rota--Baxter operators on Lie algebra on the adjoint Lie algebra $H_4^{(-)}$. |
| title | Lie Rota--Baxter operators on the Sweedler algebra $H_4$ |
| topic | Group Theory Rings and Algebras 16T05, 17B38 |
| url | https://arxiv.org/abs/2405.08291 |