Classification of types A and A_+ from low dimensional standard and non-standard filiform Lie Algebras

In this paper, we study low-dimensional Filiform Lie algebras. Specifically, three-dimensional standard Filiform Lie algebras and five-dimensional non-standard Filiform Lie algebras. The classification method was given in the following stage. For given a low-dimensional Filiform Lie algebra, we compute its second centre. We showed that three-dimensional Filiform Lie algebra-called Heisenberg Lie algebra-is type 𝐴 and 𝐴+ as well. On the other hand, for 𝑛≥3, the standard Filiform Lie algebras are type 𝐴 but not type 𝐴+. In this case, we give a concrete example of case five-dimensional Heisenberg Lie algebra. Moreover, we proved that five-dimensional non-standard Filiform Lie algebra is type 𝐴 but not type 𝐴+. It is still an open problem to classify types 𝐴 and 𝐴+ for the general case of non-standard Filiform Lie algebra of dimension ≥6.