The asymptotic approach to the description of the center of a relatively free Lie-nilpotent algebra

In this work, we consider asymptotic properties of dimensional functions related to relatively free algebras. A notion of the $\mathrm{T}$-space inclusion measure into a relatively free algebra is introduced. We calculate this measure for the center of the relatively free Lie-nilpotent algebra of index $5$ and for the $\mathrm{T}$-space of this algebra generated by the long commutator $[x_1, x_2, x_3, x_4]$. Both of these measures coincide being equal to $1/2$. This fact allows us to obtain an asymptotic description of the center. Also, a probability-theoretical view of the inclusion measure is proposed.