On simple left-symmetric algebras

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...

Full description

Saved in:
Bibliographic Details
Published in:Journal of algebra 2023-05, Vol.621, p.58-86
Main Authors: Pozhidaev, Alexandr, Umirbaev, Ualbai, Zhelyabin, Viktor
Format: Article
Language:English
Subjects:
Left-symmetric algebra
Lie-solvable algebra
Nilpotent algebra
Novikov algebra
Pre-Lie algebra
Simple algebra
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2023.01.009