I have trouble in switching between Lie algebras and algebraic groups. Let me ask a more precise question as an example.

Let $k$ be algebraically closed of characteristic zero. Let $G$ be a linear algebraic group over $k$ and let $\mathfrak{g}$ be its Lie Algebra. Consider a toral subalgebra (i.e. made of semisimple elements) $\mathfrak{t}$ of $\mathfrak{g}$. Is this the toral algebra of some torus $T<G$?

More generally, how do we cope with proofs of results that in the $\mathbb{C}$ case require use of the exponential map?

  • $\begingroup$ No: check with $G$ the 2-torus $\endgroup$ – YCor Feb 19 '17 at 5:30

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