In mathematics, an exp algebra is a

Hopf algebra Hopf is a German surname. Notable people with the surname include: * Eberhard Hopf (1902–1983), Austrian mathematician * Hans Hopf (1916–1993), German tenor * Heinz Hopf (1894–1971), German mathematician * Heinz Hopf (actor) (1934–2001), Sw ...

Exp(''G'') constructed from an

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...

''G'', and is the

universal Universal is the adjective for universe. Universal may also refer to: Companies * NBCUniversal, a media and entertainment company ** Universal Animation Studios, an American Animation studio, and a subsidiary of NBCUniversal ** Universal TV, a ...

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

''R'' such that there is an exponential map from ''G'' to the group of the

power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...

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 In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the gi ...

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 polynomial In mathematics, a Laurent polynomial (named after Pierre Alphonse Laurent) in one variable over a field \mathbb is a linear combination of positive and negative powers of the variable with coefficients in \mathbb. Laurent polynomials in ''X'' f ...

s 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'' :

\exp(gt) = 1+g_1t+g_2t^2+g_3t^3+\cdots.

The exp ring of ''G'' is the commutative ring generated by all the elements ''g''_''i'' 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

as follows. The

coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The cop ...

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. showed that Exp(''G'') has the structure of a

λ-ring In algebra, a λ-ring or lambda ring is a commutative ring together with some operations λ''n'' on it that behave like the exterior powers of vector spaces. Many rings considered in K-theory carry a natural λ-ring structure. λ-rings also provide ...

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 In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in ''n'' indeterminates, as ''n'' goes to infinity. This ring serves as universal structure in which ...

(or the Hopf algebra of symmetric functions). * suggest that it might be interesting to extend the theory to non-commutative groups ''G''.

References

* *{{citation, mr=0687747 , last=Hoffman, first= P. , title=Exponential maps and λ-rings , journal=J. Pure Appl. Algebra, volume= 27 , year=1983, issue= 2, pages= 131–162, doi=10.1016/0022-4049(83)90011-7, doi-access= Hopf algebras