Let $V$ be a finite-dimensional real vector space with the standard topology. We can equip $V$ with the "standard smooth structure," which includes as charts every linear isomorphism $\phi:V\to\mathbb R^n$. With this smooth structure, $V$ becomes a Lie group because $(x, y)\mapsto x+y$ and $x\mapsto -x$ are smooth. Do any nonstandard/exotic smooth structures on $V$ also make these operations smooth?
I'm wondering if perhaps the solution to Hilbert's fifth problem (or more basic principles) rule this out, but I'm too much of a novice in this area to understand what those results entail.
There are also some other posts regarding the relationship between exotic smooth structures and Lie groups [1] [2] [3], but again I'm too much of a novice to see whether those questions are directly relevant to this one.