Example of a gauge transformation which cannot be continuously lifted to the Lie algebra.

Abstract

The habit of writing every element of a Lie group G as exponential of some element of the Lie algebra of G, which is usual in pliysical texts, is of course perfectly legitimate in the usual case (G compact and connected). But it can be somewhat misleading as it induces to think that every continuous gauge transformation (X some subset of space-time, G some compact connected Lie group) can be written in the form with a continuous A Even if, in many circunstances, this inaccuracy has no bad consequences, greater care is necessary when topological considerations are of relevance.