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 Z 2 -grading A = A 0 ⊕ A 1 with an associative and commutative A 0 . Simple algebras are described in the following cases: (1) A is four dimensional over an algebraically closed field of characteristic not 2, (2) A 0 is an algebra with the zero product, and (3) A 0 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.