Identities in differential perm algebras
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915971678076928 |
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| author | Mashurov, F. A. Sartayev, B. K. |
| author_facet | Mashurov, F. A. Sartayev, B. K. |
| contents | Let $(P,\cdot,d)$ be a differential perm algebra over a field of characteristic $0$, i.e. an associative algebra satisfying $(ab)c=(ba)c$ equipped with a derivation $d$. We investigate polynomial identities in the algebras obtained from $d$ by the derived operations \[ a\prec b=ab',\quad a\succ b=a'b,\quad a\blacklozenge b=ab'+ba',\quad a\bullet b=a'b+ab',\quad a\Diamond b=ab'-ba',\quad a\circ b=a'b-ab', \] where $a'=d(a)$. Our first result shows that any nontrivial differential polynomial identity (not supported by the right annihilator forced by the perm law) implies a purely differential consequence of the form $a_1'a_2'\cdots a_m'=0$ for some positive integer $m$. We then study the subalgebras of the free differential perm algebra generated by $X$ under $\blacklozenge$ and under $\bullet$, giving explicit generating sets and computing the multilinear dimensions of their homogeneous components. Finally, we construct perm-Witt type Lie and Leibniz algebras arising naturally from differential perm algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_00139 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Identities in differential perm algebras Mashurov, F. A. Sartayev, B. K. Rings and Algebras Let $(P,\cdot,d)$ be a differential perm algebra over a field of characteristic $0$, i.e. an associative algebra satisfying $(ab)c=(ba)c$ equipped with a derivation $d$. We investigate polynomial identities in the algebras obtained from $d$ by the derived operations \[ a\prec b=ab',\quad a\succ b=a'b,\quad a\blacklozenge b=ab'+ba',\quad a\bullet b=a'b+ab',\quad a\Diamond b=ab'-ba',\quad a\circ b=a'b-ab', \] where $a'=d(a)$. Our first result shows that any nontrivial differential polynomial identity (not supported by the right annihilator forced by the perm law) implies a purely differential consequence of the form $a_1'a_2'\cdots a_m'=0$ for some positive integer $m$. We then study the subalgebras of the free differential perm algebra generated by $X$ under $\blacklozenge$ and under $\bullet$, giving explicit generating sets and computing the multilinear dimensions of their homogeneous components. Finally, we construct perm-Witt type Lie and Leibniz algebras arising naturally from differential perm algebras. |
| title | Identities in differential perm algebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2605.00139 |