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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.29447 |
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| _version_ | 1866915901295558656 |
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| author | Panyushev, Dmitri Yakimova, Oksana |
| author_facet | Panyushev, Dmitri Yakimova, Oksana |
| contents | E.B. Vinberg's theory of quasi-derivations of algebras is extended to a broader framework of near-derivations. This deepens connections between Poisson geometry and Lie theory. Although basic results apply to arbitrary algebras, our substantial applications concern the Poisson algebra $(\mathcal S(\mathfrak q),\{\ ,\,\})$ of a Lie algebra $\mathfrak q$. We develop a method for obtaining quasi-derivations via the use of squares of derivations, which allows us to provide quasi-derivations of the simple Lie algebras. It is shown that (1) a near-derivation $D$ of $(\mathcal S(\mathfrak q),\{\ ,\,\})$ yields a pencil of compatible Poisson brackets on $\mathfrak q^*$ and (2) using $D$ one may naturally construct a Poisson-commutative subalgebra of $\mathcal S(\mathfrak q)$. A special attention is given to near-derivations of $(\mathcal S(\mathfrak q),\{\ ,\,\})$ induced from near-derivations of $\mathfrak q$. This provides some old and new families of compatible Poisson brackets. We also compare properties of near-derivations of $\mathfrak q$ and Nijenhuis operators in $\mathfrak{gl}(\mathfrak q)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29447 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Near-derivations and their applications to Lie algebras Panyushev, Dmitri Yakimova, Oksana Representation Theory E.B. Vinberg's theory of quasi-derivations of algebras is extended to a broader framework of near-derivations. This deepens connections between Poisson geometry and Lie theory. Although basic results apply to arbitrary algebras, our substantial applications concern the Poisson algebra $(\mathcal S(\mathfrak q),\{\ ,\,\})$ of a Lie algebra $\mathfrak q$. We develop a method for obtaining quasi-derivations via the use of squares of derivations, which allows us to provide quasi-derivations of the simple Lie algebras. It is shown that (1) a near-derivation $D$ of $(\mathcal S(\mathfrak q),\{\ ,\,\})$ yields a pencil of compatible Poisson brackets on $\mathfrak q^*$ and (2) using $D$ one may naturally construct a Poisson-commutative subalgebra of $\mathcal S(\mathfrak q)$. A special attention is given to near-derivations of $(\mathcal S(\mathfrak q),\{\ ,\,\})$ induced from near-derivations of $\mathfrak q$. This provides some old and new families of compatible Poisson brackets. We also compare properties of near-derivations of $\mathfrak q$ and Nijenhuis operators in $\mathfrak{gl}(\mathfrak q)$. |
| title | Near-derivations and their applications to Lie algebras |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2603.29447 |