
Algebra 2013
Two Interacting Coordinate Hopf Algebras of Affine Groups of Formal Series on a CategoryDOI: 10.1155/2013/370618 Abstract: A locally finite category is defined as a category in which every arrow admits only finitely many different ways to be factorized by composable arrows. The large algebra of such categories over some fields may be defined, and with it a group of invertible series (under multiplication). For certain particular locally finite categories, a substitution operation, generalizing the usual substitution of formal power series, may be defined, and with it a group of reversible series (invertible under substitution). Moreover, both groups are actually affine groups. In this contribution, we introduce their coordinate Hopf algebras which are both free as commutative algebras. The semidirect product structure obtained from the action of reversible series on invertible series by antiautomorphisms gives rise to an interaction at the level of their coordinate Hopf algebras under the form of a smash coproduct. 1. Introduction The set of formal power series in one variable , such as, , where , forms a group under the usual multiplication of series (whenever is a commutative ring with a unit). Moreover, the set of series, such as, , , forms a group under another operation, namely, the substitution. For any , and , the substitution of by is defined as the series (the fact that begins with implies that is summable in the usual topology of series). This actually gives rise to a semidirect product of groups (where is the opposite group of ). Actually, this situation may be generalized in the following way. Let be a category in which any arrow admits only finitely many factorizations by composable arrows. Such a category is referred to as a locally finite category. A locally finite category admits a large algebra, that is, the set of all settheoretic maps from the arrows of the category to some base (commutative) ring may be multiplied by a Cauchykind product inherited from the composition of arrows in the category. Now, the set of all series in this large algebra with a coefficient at each identity arrow in the category forms a group under multiplication. Moreover, given a finite semicategory , roughly speaking a category without identities, we may construct the free category over the underlying graph structure of , which is a locally finite category. According to a universal property, we may define in a unique way an evaluation functor that maps formal nonvoid paths in (nonvoid sequences of composable arrows in ) to the result in of their compositions. This gives rise to an operation of substitution on the large algebra of similar to the substitution of formal power
