# Exp algebra

In mathematics, an **exp algebra** is a Hopf algebra Exp(*G*) 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 *R*[[*t*]] 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**[*G*] 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(*gt*) 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(*G*) can be made into a commutative and cocommutative Hopf algebra as follows. The coproduct of Exp(*G*) is defined so that all the elements exp(*gt*) are group-like. The antipode is defined by making exp(–*gt*) the antipode of exp(*gt*). The counit takes all the generators *g*_{i} to 0.

Hoffman (1983) showed that Exp(*G*) 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 (or Hopf algebra) is naturally isomorphic to the ring of symmetric functions (or the Hopf algebra of symmetric functions). - Michiel Hazewinkel, Nadiya Gubareni, and V. V. Kirichenko (2010) suggest that it might be interesting to extend the theory to non-commutative groups
*G*.

## References

- Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. V. (2010),
*Algebras, rings and modules. Lie algebras and Hopf algebras*, Mathematical Surveys and Monographs,**168**, Providence, RI: American Mathematical Society, ISBN 978-0-8218-5262-0, MR 2724822, Zbl 1211.16023 - Hoffman, P. (1983), "Exponential maps and λ-rings",
*J. Pure Appl. Algebra*,**27**(2): 131–162, doi:10.1016/0022-4049(83)90011-7, MR 0687747