Restricted Universal Enveloping Algebras

Question

Is there example of restricted universal enveloping algebra $uL$ of the $p$-Lie algebra $L$ over field $k$ of characteristic $p > 0$ such that $L$ hasn't nonzero $p$-algebraic elements and global dimension of $uL$ is infinite?

Answer 1

Let $L$ be restricted Lie algebra over $k=\mathbb{G}\mathbb{F}_2$ such that $L$ has generators $x,y,z$ and relations $[x,y]=x^2=y^2=z$. Restricted universal enveloping $uL$ for $L$ has a norm $N$ with values into $k[z]$ and so has no zero divisors. The localization of $uL$ at $z$ is a skew field of dimension 4 over $k(z)$. Hence there are no 2-algebraic elements into $L$. But $L_1 = L\otimes_k \mathbb{G}\mathbb{F}_4$ has 2-algebraic elements of degree 4 and so trivial module $\mathbb{G}\mathbb{F}_4$ over $uL_1$ has no finite projective resolvents. So $uL$ itself has infinite global dimension. Details on norm are omitted for reader's pleasure (if any).