Asymptotic Behavior in Lie Nilpotent Relatively Free Algebras and Extended Grassmann Algebras

The present paper is devoted to estimating the codimensions $c_n$ for the variety of associative algebras given by the identity $[x_1,\dots,x_l]=0$ for odd $l$. The characteristic of the ground field is different from $2$ and $3$. The construction of the so-called extended Grassmann algebra is applied very effectively.