Is there analog of Lie algebra for the case of topological groups which are not necessarily differentiable manifolds, and in particular for finite groups? here by "analog" i mean that it should have similar kind of relations to given group as of a finite dimensional Lie algebra with its corresponding Lie group. for example there should be some analog of "Exponential map"; Every linear representation of group should also be a linear representation of its algebra, and this should be in some sense compatible with the exponential map between the two; etc

thanks,