Description of local multipliers on finite-dimensional associative algebras

In 2020 F. Arzikulov and N. Umrzaqov introduced the concept of a (linear) local multiplier. They proved that every local left (right) multiplier on the matrix ring over a division ring is a left (right, respectively) multiplier. This paper is devoted to (linear) local weak left (right) multipliers on $5$-dimensional naturally graded $2$-filiform non-split associative algebras. An algorithm for obtaining a common form of the matrices of the weak left (right) multipliers on the $5$-dimensional naturally graded $2$-filiform non-split associative algebras $\lambda^{5}_{1}$ and $\lambda^{5}_{2}$, constructed by I. Karimjanov and M. Ladra, is developed. An algorithm for obtaining a general form of the matrices of the local weak left (right) multipliers on the algebras $\lambda^{5}_{1}$ and $\lambda^{5}_{2}$ is also developed. It turns out that the associative algebras $\lambda^{5}_{1}$ and $\lambda^{5}_{2}$ have a local weak left (right) multiplier that is not a weak left (right, respectively) multiplier.