On quadratic Novikov algebras
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866917982907662336 |
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| author | Dong, Xiaofeng Hong, Yanyong |
| author_facet | Dong, Xiaofeng Hong, Yanyong |
| contents | A quadratic Novikov algebra is a Novikov algebra $(A, \circ)$ with a symmetric and nondegenerate bilinear form $B(\cdot,\cdot)$ satisfying $B(a\circ b, c)=-B(b, a\circ c+c\circ a)$ for all $a$, $b$, $c\in A$. This notion appeared in the theory of Novikov bialgebras. In this paper, we first investigate some properties of quadratic Novikov algebras and give a decomposition theorem of quadratic Novikov algebras. Then we present a classification of quadratic Novikov algebras of dimensions $2$ and $3$ over $\mathbb{C}$ up to isomorphism. Finally, a construction of quadratic Novikov algebras called double extension is presented and we show that any quadratic Novikov algebra containing a nonzero isotropic ideal can be obtained by double extensions. Based on double extension, an example of quadratic Novikov algebras of dimension 4 is given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_09044 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On quadratic Novikov algebras Dong, Xiaofeng Hong, Yanyong Rings and Algebras Mathematical Physics A quadratic Novikov algebra is a Novikov algebra $(A, \circ)$ with a symmetric and nondegenerate bilinear form $B(\cdot,\cdot)$ satisfying $B(a\circ b, c)=-B(b, a\circ c+c\circ a)$ for all $a$, $b$, $c\in A$. This notion appeared in the theory of Novikov bialgebras. In this paper, we first investigate some properties of quadratic Novikov algebras and give a decomposition theorem of quadratic Novikov algebras. Then we present a classification of quadratic Novikov algebras of dimensions $2$ and $3$ over $\mathbb{C}$ up to isomorphism. Finally, a construction of quadratic Novikov algebras called double extension is presented and we show that any quadratic Novikov algebra containing a nonzero isotropic ideal can be obtained by double extensions. Based on double extension, an example of quadratic Novikov algebras of dimension 4 is given. |
| title | On quadratic Novikov algebras |
| topic | Rings and Algebras Mathematical Physics |
| url | https://arxiv.org/abs/2504.09044 |