On simple left-symmetric algebras Full article
| Journal |
Journal of Algebra
ISSN: 0021-8693 , E-ISSN: 1090-266X |
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| Output data | Year: 2023, Volume: 621, Pages: 58-86 Pages count : 29 DOI: 10.1016/j.jalgebra.2023.01.009 | ||||||||
| Tags | Left-symmetric algebra; Lie-solvable algebra; Nilpotent algebra; Novikov algebra;Pre-Lie algebraSimple algebra; | ||||||||
| Authors |
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| Affiliations |
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Funding (1)
| 1 | Russian Science Foundation | 21-11-00286 |
Abstract:
We prove that the multiplication algebra M(A) of any simple finite-dimensional left-symmetric nonassociative algebra A over a field of characteristic zero coincides with the right multiplication algebra R(A). In particular, A does not contain any proper right ideal. These results immediately give a description of simple finite-dimensional Novikov algebras over an algebraically closed field of characteristic zero [29]. The structure of finite-dimensional simple left-symmetric nonassociative algebras from a very narrow class A of algebras with the identities [[x,y],[z,t]]=[x,y]([z,t]u)=0 is studied in detail. We prove that every such algebra A admits a Z2-grading A=A0⊕A1 with an associative and commutative A0. Simple algebras are described in the following cases: (1) A is four dimensional over an algebraically closed field of characteristic not 2, (2) A0 is an algebra with the zero product, and (3) A0 is simple; in the last two cases, the description is given in terms of root systems. A necessary and sufficient condition for A to be complete is given. © 2023 Elsevier Inc.
Cite:
Pozhidaev A.
, Umirbaev U.
, Zhelyabin V.
On simple left-symmetric algebras
Journal of Algebra. 2023. V.621. P.58-86. DOI: 10.1016/j.jalgebra.2023.01.009 WOS Scopus РИНЦ OpenAlex
On simple left-symmetric algebras
Journal of Algebra. 2023. V.621. P.58-86. DOI: 10.1016/j.jalgebra.2023.01.009 WOS Scopus РИНЦ OpenAlex
Dates:
| Submitted: | Aug 30, 2022 |
| Published online: | Jan 24, 2023 |
| Published print: | Feb 9, 2023 |
Identifiers:
| Web of science: | WOS:000944733300001 |
| Scopus: | 2-s2.0-85147577840 |
| Elibrary: | 60461665 |
| OpenAlex: | W4317929715 |