Throughout the present work, we extend the study of Zinbiel algebras to Zinbiel superalgebras. In particular, we show that all Zinbiel superalgebras over an arbitrary field are nilpotent in the same way as occurs for Zinbiel algebras. In addition, and since the most important cases of nilpotent algebras or superalgebras are those with maximal nilpotency index, we study the complex null-filiform Zinbiel superalgebra proving that it is unique up to isomorphism. After that, we characterize the naturally graded filiform ones and obtain low-dimensional classifications.

The first part of this work is supported by Ministerio de Economia y Competitividad (Spain), grant PID2020-115155GB-I00 (European FEDER support included, EU); FCT UIDB/MAT/00212/2020, UIDP/MAT/00212/2020, and by the Spanish Government through the Ministry of Universities grant 'Margarita Salas', funded by the European Union - NextGenerationEU. The second part of this work is supported by the Russian Science Foundation under grant 22-71-10001.