On simple left-symmetric algebras
Pozhidaev A. Umirbaev U. Zhelyabin V.
1 May 2023Academic Press Inc.
Journal of Algebra
2023#62158 - 86 pp.
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.
Left-symmetric algebra , Lie-solvable algebra , Nilpotent algebra , Novikov algebra , Pre-Lie algebra , Simple algebra
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Sobolev Institute of Mathematics of SB RAS, Novosibirsk, Russian Federation
Wayne State University, Detroit, MI, United States
Al-Farabi Kazakh National University, Almaty, Kazakhstan
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
Sobolev Institute of Mathematics of SB RAS
Wayne State University
Al-Farabi Kazakh National University
Institute of Mathematics and Mathematical Modeling
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