On mixed identities of endomorphs, bimodules, and $\omega$-algebras
A. P. Pozhidaev Sobolev Institute of Mathematics, Novosibirsk, Russia
Abstract:
We describe mixed multilinear identities of degree
$3$ on right endomorphs of arbitrary algebras over a field
$F$ of characteristic not equal to
$2$. As a consequence, we obtain irreducible bimodules over
$M_n(F)$ in the variety defined by the monoassociativity identity and in the variety of
$(1,1)$ālgebras. We construct a broad class of right-symmetric bimodules, including irreducible right-symmetric
$M_n(F)$b̄imodules. We introduce a class of
$\omega$-right-symmetric algebras
${\mathcal A}_\omega$ with an
$\omega$īdentity, which generalizes the class of right-symmetric algebras, where
$\omega:{\mathcal A}\times {\mathcal A} \to F$ is a bilinear skew-symmetric form on
${\mathcal A}$. We also describe the structure of finite-dimensional algebras
${\mathcal A}_\omega$, in particular, simple algebras of this kind. We prove that the commutator algebra
${\mathcal A}^{(-)}$ of an arbitrary
$\omega$-right-symmetric algebra
${\mathcal A}$ is an
$\omega$-Lie algebra and that
${\mathcal A}^{(-)}$ is solvable of degree
$\leq 3$ in the finite-dimensional case.
Keywords:
endomorph, right-symmetric algebra, simple algebra, pre-Lie algebra, mixed identity, bimodule, irreducible bimodule, right-symmetric bimodule, $\omega$-Lie algebra.
UDC:
512.554
MSC: 35R30 Received: 25.02.2025
Revised: 24.03.2025
Accepted: 25.04.2025
DOI:
10.33048/smzh.2025.66.413
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