Left-symmetric Structures on Complex Simple Lie Superalgebras

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A well-known fact is that there does not exist any compatible left-symmetric structures on a finite-dimensional complex semisimple Lie algebra (see Chu1974). This result is not valid in semisimple Lie superalgebra case. In this paper, we study the compatible Left-symmetric superalgebra (LSSA for short) structures on complex simple Lie superalgebras. We prove that there is not any compatible LSSA structure on a finite-dimensional complex simple Lie superalgebra except for the classical simple Lie superalgebra A(m,n)(m≠ n) and Cartan simple Lie superalgebra W(n)(n≥ 3). We also classify all compatible LSSAs with a right-identity on A(0,1).