Transposed Poisson structures on Lie incidence algebras
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909153311588352 |
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| author | Kaygorodov, Ivan Khrypchenko, Mykola |
| author_facet | Kaygorodov, Ivan Khrypchenko, Mykola |
| contents | Let $X$ be a finite connected poset, $K$ a field of characteristic zero and $I(X,K)$ the incidence algebra of $X$ over $K$ seen as a Lie algebra under the commutator product. In the first part of the paper we show that any $\frac{1}{2}$-derivation of $I(X,K)$ decomposes into the sum of a central-valued $\frac 12$-derivation, an inner $\frac{1}{2}$-derivation and a $\frac{1}{2}$-derivation associated with a map $σ:X^2_<\to K$ that is constant on chains and cycles in $X$. In the second part of the paper we use this result to prove that any transposed Poisson structure on $I(X,K)$ is the sum of a structure of Poisson type, a mutational structure and a structure determined by $λ:X^2_e\to K$, where $X^2_e$ is the set of $(x,y)\in X^2$ such that $x<y$ is a maximal chain not contained in a cycle. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_00332 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Transposed Poisson structures on Lie incidence algebras Kaygorodov, Ivan Khrypchenko, Mykola Rings and Algebras Let $X$ be a finite connected poset, $K$ a field of characteristic zero and $I(X,K)$ the incidence algebra of $X$ over $K$ seen as a Lie algebra under the commutator product. In the first part of the paper we show that any $\frac{1}{2}$-derivation of $I(X,K)$ decomposes into the sum of a central-valued $\frac 12$-derivation, an inner $\frac{1}{2}$-derivation and a $\frac{1}{2}$-derivation associated with a map $σ:X^2_<\to K$ that is constant on chains and cycles in $X$. In the second part of the paper we use this result to prove that any transposed Poisson structure on $I(X,K)$ is the sum of a structure of Poisson type, a mutational structure and a structure determined by $λ:X^2_e\to K$, where $X^2_e$ is the set of $(x,y)\in X^2$ such that $x<y$ is a maximal chain not contained in a cycle. |
| title | Transposed Poisson structures on Lie incidence algebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2309.00332 |