Classification of $5$-Dimensional Complex Nilpotent Leibniz Algebras

Leibniz algebras are certain generalization of Lie algebras. In this paper we give the classification of $5-$dimensional complex non-Lie nilpotent Leibniz algebras. We use the canonical forms for the congruence classes of matrices of bilinear forms to classify the case $\dim (A^2)=3$ and $\dim (Leib(A))=1$ which can be applied to higher dimensions. The remaining cases are classified via direct method.