$δ$-Novikov and $δ$-Novikov--Poisson algebras
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909608545615872 |
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| author | Kaygorodov, Ivan |
| author_facet | Kaygorodov, Ivan |
| contents | This article considers the structure and properties of $δ$-Novikov algebras, a generalization of Novikov algebras characterized by a scalar parameter $δ$. It looks like $δ$-Novikov algebras have a richer structure than Novikov algebras. So, unlike Novikov algebras, they have non-commutative simple finite-dimensional algebras for $δ=-1.$ Additionally, we introduce $δ$-Novikov--Poisson algebras, extending several theorems from the classical Novikov--Poisson algebras. Specifically, we consider the commutator structure $[a, b] = a \circ b - b \circ a$ of $δ$-Novikov algebras, proving that when $δ\neq 1$, these algebras are metabelian Lie-admissible. Moreover, we prove that every metabelian Lie algebra can be embedded into a suitable $δ$-Novikov algebra with respect to the commutator product. We further consider the construction of $δ$-Poisson and transposed $δ$-Poisson algebras through $δ$-derivations on the commutative associative algebras. Finally, we analyze the operad associated with the variety of $δ$-Novikov algebras, proving that it is not Koszul for any value of $δ$. This result extends known results for the Novikov operad $(δ=1)$ and the bicommutative operad $(δ=0)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_08043 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $δ$-Novikov and $δ$-Novikov--Poisson algebras Kaygorodov, Ivan Rings and Algebras This article considers the structure and properties of $δ$-Novikov algebras, a generalization of Novikov algebras characterized by a scalar parameter $δ$. It looks like $δ$-Novikov algebras have a richer structure than Novikov algebras. So, unlike Novikov algebras, they have non-commutative simple finite-dimensional algebras for $δ=-1.$ Additionally, we introduce $δ$-Novikov--Poisson algebras, extending several theorems from the classical Novikov--Poisson algebras. Specifically, we consider the commutator structure $[a, b] = a \circ b - b \circ a$ of $δ$-Novikov algebras, proving that when $δ\neq 1$, these algebras are metabelian Lie-admissible. Moreover, we prove that every metabelian Lie algebra can be embedded into a suitable $δ$-Novikov algebra with respect to the commutator product. We further consider the construction of $δ$-Poisson and transposed $δ$-Poisson algebras through $δ$-derivations on the commutative associative algebras. Finally, we analyze the operad associated with the variety of $δ$-Novikov algebras, proving that it is not Koszul for any value of $δ$. This result extends known results for the Novikov operad $(δ=1)$ and the bicommutative operad $(δ=0)$. |
| title | $δ$-Novikov and $δ$-Novikov--Poisson algebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2505.08043 |