{"title":"Novikov $ ��_{2} $ -Graded Algebras with an Associative 0-Component","authors":"A. S. Panasenko, V. N. Zhelyabin","doi":"10.1134/s0037446624020150","DOIUrl":null,"url":null,"abstract":"<p>In 1974 Kharchenko proved that if a <span>\\( 0 \\)</span>-component of an <span>\\( n \\)</span>-graded associative algebra is PI then this algebra is PI.\nIn the Novikov algebras of characteristic 0 the existence of a polynomial identity is equivalent to the solvability of the commutator ideal.\nWe study a <span>\\( _{2} \\)</span>-graded Novikov algebra <span>\\( N=A+M \\)</span> and prove that if the characteristic of the basic field is not 2 or 3\nand its 0-component <span>\\( A \\)</span> is associative or Lie-nilpotent of index 3 then\nthe commutator ideal <span>\\( [N,N] \\)</span> is solvable.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0037446624020150","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In 1974 Kharchenko proved that if a \( 0 \)-component of an \( n \)-graded associative algebra is PI then this algebra is PI.
In the Novikov algebras of characteristic 0 the existence of a polynomial identity is equivalent to the solvability of the commutator ideal.
We study a \( _{2} \)-graded Novikov algebra \( N=A+M \) and prove that if the characteristic of the basic field is not 2 or 3
and its 0-component \( A \) is associative or Lie-nilpotent of index 3 then
the commutator ideal \( [N,N] \) is solvable.