Anticommutative Algebras Satisfying Standard Identities of Degree Four
Algebra i logika, Tome 40 (2001) no. 4, pp. 458-483
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We define an anticommutative $\Phi$-algebra $A(D,a)$ whose multiplication generalizes the concept of a Jacobi bracket in the form (4). It is proved that $A(D,a)$ is a $J$-algebra and that it satisfies a standard identity of degree four. A subclass $\mathfrak M$ of algebras $A(D,a)$ over $\Phi$ which is connected with some class of 3-Lie algebras is distinguished. We establish a criterion of being simple for factor algebras of non-Lie algebras in $\mathfrak M$, given a 1-dimensional annihilator, and then use it to construct examples of simple infinite-dimensional (of dimension $p^3-1$) non-Lie $J$-algebras over a field $\Phi$ satisfying standard identities of degree 4, if the characteristic $p$ of $\Phi$ is zero (for $p>2$). Also, the criterion of algebras belonging to $\mathfrak M$ is given.
Keywords:
anticommutative $\Phi$-algebra, Jacobi bracket, simple infinite-dimensional non-Lie $J$-algebra over a field.
@article{AL_2001_40_4_a5,
author = {V. T. Filippov},
title = {Anticommutative {Algebras} {Satisfying} {Standard} {Identities} of {Degree} {Four}},
journal = {Algebra i logika},
pages = {458--483},
publisher = {mathdoc},
volume = {40},
number = {4},
year = {2001},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/AL_2001_40_4_a5/}
}
V. T. Filippov. Anticommutative Algebras Satisfying Standard Identities of Degree Four. Algebra i logika, Tome 40 (2001) no. 4, pp. 458-483. http://geodesic.mathdoc.fr/item/AL_2001_40_4_a5/