Exp algebra

In mathematics, an exp algebra is a Hopf algebra Exp constructed from an abelian group G, and is the universal ring R such that there is an exponential map from G to the group of the power series in formal power series|Rt with constant term 1. In other words the functor Exp from abelian groups to commutative rings is adjoint to the functor from commutative rings to abelian groups taking a ring to the group of formal power series with constant term 1.
The definition of the exp ring of G is similar to that of the group ring Z of G, which is the universal ring such that there is an exponential homomorphism from the group to its units. In particular there is a natural homomorphism from the group ring to a completion of the exp ring. However in general the Exp ring can be much larger than the group ring: for example, the group ring of the integers is the ring of Laurent polynomials in 1 variable, while the exp ring is a polynomial ring in countably many generators.

Construction

For each element g of G introduce a countable set of variables g_i for i>0. Define exp to be the formal power series in t
The exp ring of G is the commutative ring generated by all the elements g_i with the relations
for all g, h in G; in other words the coefficients of any power of t on both sides are identified.
The ring Exp can be made into a commutative and cocommutative Hopf algebra as follows. The coproduct of Exp is defined so that all the elements exp are group-like. The antipode is defined by making exp the antipode of exp. The counit takes all the generators g_i to 0.
showed that Exp has the structure of a λ-ring.

Examples

The exp ring of an infinite cyclic group such as the integers is a polynomial ring in a countable number of generators g_i where g is a generator of the cyclic group. This ring is naturally isomorphic to the ring of symmetric functions.
suggest that it might be interesting to extend the theory to non-commutative groups G.