A Lie group equipped with a
compatible real algebraic structure is called a locally Nash group. We prove some
general facts about locally Nash groups, then we classify the one-dimensional locally
Nash groups, using a theorem of Weierstrass that characterizes the analytic functions
satisfying an algebraic addition theorem. Besides the standard Nash structure on
the additive group of real numbers, there are locally Nash structures on
the additive reals induced by the exponential function, the sine function,
and by any elliptic function that is real on ℝ. There are no other simply
connected one-dimensional locally Nash groups. Any two quotients of the additive
reals with their standard Nash structure by discrete subgroups are Nash
equivalent. For other locally Nash structures on ℝ, the quotients ℝ∕αZ and
ℝ∕βZ are Nash equivalent if and only if α∕β is rational. The classification of
the one-dimensional Nash groups is equivalent to the classification of the
one-dimensional semialgebraic groups. It is precisely these groups that are definable
over ℝ, so we have also classified the one-dimensional groups definable over
ℝ.
