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 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 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.