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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'' 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(''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. showed that Exp(''G'') has the structure of a λ-ring. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Exp algebra」の詳細全文を読む スポンサード リンク
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