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.
For each element g of G introduce a countable set of variables gi for i>0. Define exp(gt) to be the formal power series in t
\exp(gt)=1+g1t+g
| 3+ … . | |
| 3t |
The exp ring of G is the commutative ring generated by all the elements gi with the relations
\exp((g+h)t)=\exp(gt)\exp(ht)
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 gi to 0.
showed that Exp(G) has the structure of a λ-ring.