Given a Semisimple Lie algebra. We can use category of finite dimensional representations of g and fiber functor to reconstruct this Lie algebra.

Moreover, using the same formalism we can associate a Lie algebra to locally compact Lie group G which is isomorphic to Lie(G) and we can from a semisimple Lie algebra g to construct a Lie group G such that Lie(G)=g

I just did a search of the literature, and I have a positive statement and a negative statement:

First, the positive statement. If we have two groups G and H such that , then their Abelianizations are isomorphic. In particular if G and H are Abelian, then if and only if . This is proved using group homology, see

Guram Donadze and Manuel Ladra, On the groups with isomorphic integral group rings, Int. J. Algebra vol. 3 (2009), no. 11, 525–529.

The original result about Abelian groups and their integral group rings was known for a long time, I think at least by Higman, though I can’t find the precise source.

For the negative answer, I’ll point you to

Martin Hertweck, A counterexample to the isomorphism problem for integral group rings, Annals of Math., 154 (2001), 115–138.

There he constructs a finite solvable group X with order whose integral group ring contains a group of units Y such that but Y and X are nonisomorphic. This paper is also cohomological in nature.

I like your question, it made me do a little bit of hunting. Here is the summary:

Of course, by Wedderburn’s theorem for semisimple algebras we know that . The last statement you made is true though: if we have a monoid algebra , then the group-like elements (those such that ) is a monoid isomorphic to M.

The self-conjugate, tensor-preserving property looks similar to the group-like property for the Hopf algebra, but I don’t see how to conclude Tannaka’s theorem for finite groups from this fact.