Lie Groups and Algebras

Lie Groups are formalised as locales, building on the AFP theory of Smooth Manifolds. We formalise the diffeomorphism group of a manifold, and the action of a Lie group on a manifold. The general linear group is shown to be a Lie group by proving properties of the determinant, and matrix inverses. We also develop a theory of smooth vector fields on a $C^{\mathrm{\infty }}$ manifold $M$, defined as smooth maps from the manifold to its tangent bundle $TM$. We employ a shortcut that avoids difficulties in defining the tangent bundle as a manifold, but which still leads to vector fields with the properties one would expect. Notably, they are derivations $C^{\mathrm{\infty }}(M)\to C^{\mathrm{\infty }}(M)$. We then construct the Lie algebra of a Lie group as an algebra of left-invariant smooth vector fields.