# Are there torsion-free restricted simple Lie algebras?

It is known that a torsion-free group can be simple (see e.g. Rataggi's paper https://www.degruyter.com/abstract/j/jgth.2007.10.issue-3/jgt.2007.028/jgt.2007.028.xml). I would like to know if the situation in the context of modular Lie algebras is similar. More precisily:

Let $$\mathbb{F}$$ be a field of characteristic $$p>0$$. Does there exist a restricted simple Lie algebra over $$\mathbb{F}$$ with no nonzero $$p$$-algebraic element?

I recall that an element $$x$$ of a restricted Lie algebra $$(L,[p])$$ is said to be $$p$$-algebraic if the restricted subalgebra generated by $$x$$ is finite-dimensional.

• Your link is for torsion-free finitely presented simple groups. For arbitrary groups, it's much easier. For instance it's been known for decades that the group of self-homeomorphisms of $\mathbf{R}$ with compact support is simple. Also torsion-free finitely generated simple groups were constructed in the 50s. – YCor Jun 17 '19 at 19:27
• Thank you for the remark. Are you aware of any similar results for restricted Lie algebras? – Nathan Jun 17 '19 at 20:45
• You already asked this question in the post! – YCor Jun 18 '19 at 7:50