Identities in differential perm algebras
Abstract
Let (P,·,d) be a differential perm algebra over a field of characteristic zero, i.e. an associative algebra satisfying (ab)c=(ba)c and equipped with a derivation d. We study polynomial identities in the algebras obtained by the derived operations a≺ b=ab', a≻ b=a'b, ablacklozenge b=ab'+ba', a• b=a'b+ab', aDiamond b=ab'−ba', a∘ b=a'b−ab', where a'=d(a). We first prove that any nontrivial differential polynomial identity that does not belong to the right annihilator of the free differential perm algebra implies a differential identity of the form a1'a2'⋯ am'=0 for some positive integer m. We then obtain explicit generating sets and determine the dimensions of the multilinear homogeneous components of the subalgebras of the free differential perm algebra generated by X with respect to the products blacklozenge and •. Finally, we construct perm-Witt type Lie and Leibniz algebras arising naturally from differential perm algebras.
Related
- Differential perm bialgebras, differential perm Yang-Baxter equation, and their applications in related bialgebra structures
- Commutator identities, Lie product identities, and Multiplicative Lie algebras
- On the trace identities of 2-dimensional tr-prime algebras over a finite field
- Graded identities for matrix algebras of order two over a finite field
- Identities and isomorphisms of finite graded matrix algebras
BibTeX
@misc{mashurov2026identities,
title = {{Identities in differential perm algebras}},
author = {F. A. Mashurov and B. K. Sartayev},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.00139},
eprint = {2605.00139},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.00139},
}
RIS
TY - UNPB
TI - Identities in differential perm algebras
AU - F. A. Mashurov
AU - B. K. Sartayev
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2605.00139
UR - https://arxiv.org/abs/2605.00139
AB - Let (P,·,d) be a differential perm algebra over a field of characteristic zero, i.e. an associative algebra satisfying (ab)c=(ba)c and equipped with a derivation d. We study polynomial identities in the algebras obtained by the derived operations a≺ b=ab', a≻ b=a'b, ablacklozenge b=ab'+ba', a• b=a'b+ab', aDiamond b=ab'−ba', a∘ b=a'b−ab', where a'=d(a). We first prove that any nontrivial differential polynomial identity that does not belong to the right annihilator of the free differential perm algebra implies a differential identity of the form a_1'a_2'⋯ a_m'=0 for some positive integer m. We then obtain explicit generating sets and determine the dimensions of the multilinear homogeneous components of the subalgebras of the free differential perm algebra generated by X with respect to the products blacklozenge and •. Finally, we construct perm-Witt type Lie and Leibniz algebras arising naturally from differential perm algebras.
ER -