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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Online-Zugang: | https://arxiv.org/abs/2305.07371 |
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| _version_ | 1866914693264703488 |
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| author | Kolesnikov, Pavel Mashurov, Farukh Sartayev, Bauyrzhan |
| author_facet | Kolesnikov, Pavel Mashurov, Farukh Sartayev, Bauyrzhan |
| contents | For a non-associative algebra $A$ with a derivation $d$, its derived algebra $A^{(d)}$ is the same space equipped with new operations $a\succ b = d(a)b$, $a\prec b = ad(b)$, $a,b\in A$. Given a variety ${\rm Var}$ of algebras, its derived variety is generated by all derived algebras $A^{(d)}$ for all $A$ in ${\rm Var}$ and for all derivations $d$ of $A$. The same terminology is applied to binary operads governing varieties of non-associative algebras. For example, the operad of Novikov algebras is the derived one for the operad of (associative) commutative algebras. We state a sufficient condition for every algebra from a derived variety to be embeddable into an appropriate differential algebra of the corresponding variety. We also find that for ${\rm Var} = {\rm Zinb}$, the variety of Zinbiel algebras, there exist algebras from the derived variety (which coincides with the class of pre-Novikov algebras) that cannot be embedded into a Zinbiel algebra with a derivation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_07371 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Pre-Novikov Algebras and Derived Zinbiel Variety Kolesnikov, Pavel Mashurov, Farukh Sartayev, Bauyrzhan Rings and Algebras 17A36, 17A30, 18M60 For a non-associative algebra $A$ with a derivation $d$, its derived algebra $A^{(d)}$ is the same space equipped with new operations $a\succ b = d(a)b$, $a\prec b = ad(b)$, $a,b\in A$. Given a variety ${\rm Var}$ of algebras, its derived variety is generated by all derived algebras $A^{(d)}$ for all $A$ in ${\rm Var}$ and for all derivations $d$ of $A$. The same terminology is applied to binary operads governing varieties of non-associative algebras. For example, the operad of Novikov algebras is the derived one for the operad of (associative) commutative algebras. We state a sufficient condition for every algebra from a derived variety to be embeddable into an appropriate differential algebra of the corresponding variety. We also find that for ${\rm Var} = {\rm Zinb}$, the variety of Zinbiel algebras, there exist algebras from the derived variety (which coincides with the class of pre-Novikov algebras) that cannot be embedded into a Zinbiel algebra with a derivation. |
| title | On Pre-Novikov Algebras and Derived Zinbiel Variety |
| topic | Rings and Algebras 17A36, 17A30, 18M60 |
| url | https://arxiv.org/abs/2305.07371 |