Tangent Lie Algebra of a Diffeomorphism Group and Application to Holonomy Theory

In this paper we introduce the notion of tangent space T o G of a (not necessary smooth) subgroup G of the diffeomorphism group D i f f ∞ ( M ) of a compact manifold M . We prove that T o G is a Lie subalgebra of the Lie algebra of smooth vector fields on M . The construction can be generalized to subgroups of any (finite- or infinite-dimensional) Lie groups. The tangent Lie algebra T o G introduced this way is a generalization of the classical Lie algebra in the smooth cases. As a working example we discuss in detail the tangent structure of the holonomy group and fibered holonomy group of Finsler manifolds.