Paper detail

On existence of PI-exponents of unital algebras

We construct a family of unital non-associative algebras $\{T_α\vert~ 2<α\in\mathbb R\}$ such that $\underline{exp}(T_α)=2$, whereas $α\le\overline{exp}(T_α)\leα+1$. In particular, it follows that ordinary PI-exponent of codimension growth of algebra $T_α$ does not exist for any $α> 2$. This is the first example of a unital algebra whose PI-exponent does not exist.