Lie group

A finite-dimensional real Lie group is a group $G$, in the sense of Group and abelian group, together with the structure of a finite-dimensional real smooth manifold such that the maps

$m:G\times G⟶G,m(g,h)=gh,$

and

$\mathrm{inv}:G⟶G,\mathrm{inv}(g)=g^{-1},$

are smooth in the sense of Smooth maps between manifolds with boundary. Its identity element is denoted by $e$ (or occasionally $1$).

Unless an item explicitly says otherwise, every Lie group on this page is real, finite-dimensional, and has no manifold boundary. Dimension zero is allowed; for example, any countable discrete group with its discrete zero-dimensional smooth structure is a Lie group. A Lie group cannot be empty because a group contains its identity. The definition applies unchanged in dimension one, uses no metric or nondegeneracy hypothesis, and makes no choice from a family of sets.