Extensions of solvable Lie algebras with naturally graded filiform nilradical

In this work, we consider extensions of solvable Lie algebras with naturally graded filiform nilradicals. Note that there exist two naturally graded filiform Lie algebras nn,1 and Q(2n). We find all one-dimensional extensions of solvable Lie algebras with nilradical n(n,1). We prove that there exists a unique non-split central extension of solvable Lie algebras with nilradical n(n,1) of maximal codimension. Moreover, all one-dimensional extensions of solvable Lie algebras with nilradical n(n,1) whose codimension is equal to one are found and we compared these solvable algebras with the solvable algebras with nilradicals that are one-dimensional central extension of algebra n(n,1).