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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 1., 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 onedimensional groups definable over ℝ. © 1992 by Pacific Journal of Mathematics.

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Pacific Journal of Mathematics

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