Two Interacting Coordinate Hopf Algebras of Affine Groups of Formal Series on a Category

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 (undermultiplication). 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 anti-automorphisms gives rise to an interaction at the level of their coordinate Hopf algebras under the form of a smash coproduct.


Introduction
The set  of formal power series in one variable , such as, 1 + , 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,  +  2 ,  ∈ [[]], forms a group under another operation, namely, the substitution.For any  = ∑ ≥0     ∈ [[]], and  ∈ , the substitution of  by  is defined as the series ∑ ≥0     (the fact that  begins with  implies that (  ) ≥0 is summable in the usual topology of series).This actually gives rise to a semidirect product of groups  ⋊  op (where  op 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 set-theoretic maps from the arrows of the category to some base (commutative) ring may be multiplied by a Cauchy-kind product inherited from the composition of arrows in the category.Now, the set I of all series in this large algebra with a coefficient 1 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 series.The set M 1 of all series in the large algebra which are zero on the identity arrows and 1 on the formal paths of length one (the arrows of ) forms a group under substitution.Parallelizing the usual situation of formal power series, it appears that this group acts by anti-automorphisms on the group I and therefore defines a semidirect product I ⋊ M op 1 .Moreover both groups I and M 1 are actually affine groups and so admit coordinate Hopf algebras which appear to be free as commutative algebras.In this contribution, we present the constructions of both affine groups I and M 1 and introduce their coordinate Hopf algebras which may be thought as generalizing some wellknown Hopf algebras.

Basic Definitions and Notations
In this paper,  denotes a commutative ring with a unit.In general, -algebras are not assumed to possess a unit nor Algebra to be commutative but they are associative.The notation -CAlg stands for the category of commutative -algebras with a unit and unit preserving algebra maps, and the homsets are denoted by -CAlg(, ).In what follows, if  is algebra with a unit, then 1  is its identity.
Let  be any set, and let  ∈   .The support of  is the set { ∈  : () ̸ = 0}.Such a map  is said to be finitely supported or has a finite support when its support is a finite set.The set of all finitely supported maps from  to  is denoted by  () .It is free as a module with basis {  :  ∈ } where   is the map such that   () = 0 if  ̸ =  and   () = 1.In what follows, we identify  ∈  with its image   in  () so that any map  ∈  () may be written in a unique way as a linear combination ∑ ∈ ().The module  () is actually a submodule of the product   .There is a duality bracket between   and  () , namely, the -bilinear form ⟨⋅ | ⋅⟩ :   ×  () →  such that for every  ∈   ,  ∈  () , (the sum has only finitely many nonzero terms because  is finitely supported).It is obviously a two-sided nondegenerate.
In particular, for every  ∈   and every  ∈ , () = ⟨ | ⟩, and for every  ∈  () ,  = ∑ ∈ ⟨ | ⟩.We observe that for each  ∈ ,   may be identified with an element of When  is equipped with the discrete topology, and   has the product topology, then it becomes a (Hausdorff) complete topological module (the addition and   → − are continuous, and scalar multiplication is jointly continuous), and it is the completion of the topological -module  ()  (equipped with the product topology inherited from that of   ).Moreover, for any  ∈   , the family (⟨ | ⟩) ∈ is summable (see [1]) in   with sum  so that we may represent  as an infinite linear combination  = ∑ ∈ ⟨ | ⟩ (more details may be found in [2][3][4]).Moreover, ⟨⋅ | ⋅⟩ is separately continuous.

A General Approach on Coordinate Hopf Algebras of a Group of Series
In this section, we present in a general way the notion of coordinate Hopf algebra on a group-valued functor of valued functions defined on some set , for varying algebras .The result presented here will be used in the sequel to define two coordinate Hopf algebras of two groups of formal series that define a semidirect product.The reader should refer to [5,6] for basic definitions about Hopf algebras, to [7][8][9] for the notions concerning algebraic groups, and to [10] for category-theoretic concepts.
Let  be a commutative -algebra with a unit, where  is a commutative ring with unit.An -group is a functor from -CAlg to the category of groups Grp.When an group  is representable when viewed as a set-valued functor (by composition with the forgetful functor from Grp to the category of sets Set), that is, () ≅ -CAlg(, ) (isomorphic as sets, natural in ) for some commutative -algebra , then it is called an affine group (or proaffine algebraic group when the base ring  is a field), and  (determined up to a unique isomorphism) is referred to as the coordinate Hopf algebra O() of , for reasons made clear hereafter.The representable -group  is an affine algebraic group when O() is finitely generated as an -algebra.Since the multiplication   : () × () → (), the inversion   : () → (), and the unit element   : ⋆ → () are natural transformations between representable settheoretic functors, by Yoneda's lemma they uniquely give rise respectively to a (coassociative) coproduct Δ : In what follows we will be interested in the following situation.Let  be an -group, and let  be a set.We assume that there exists a subset  of  such that, as a set-valued functor,  is isomorphic to Set(, (⋅)), where  : -CAlg → Set is the forgetful functor, that is, for every algebra ,   : () ≅   ⊆   (bijection natural in ), and for every algebra map  :  → ,   (()()) =  ∘   () for each  ∈ ().This is equivalent to say that  is represented by the free commutative -algebra  [𝑋].In this situation, we say that  is the coordinate system of  (obviously when it exists,  is uniquely determined up to bijections).We may view any  ∈   as a formal series ∑ ∈ ⟨ | ⟩.The functor  ×  is represented by the algebra [] ⊗  [], while  represents the constant functor * equal to a onepoint set.By Yoneda's lemma, the multiplication  :  ×  →  uniquely determines an algebra map Δ ∈ is induced by the pair of maps (, ) under the isomorphism In what follows we introduce two affine groups I and M 1 (see Section 6), whose -rational points are groups of formal series on some locally finite categories, and whose coordinate rings are both free commutative algebras (as in the aforementional discussion).As -groups, they interact under the form of an -group semidirect product (a semidirect product natural in its algebra variable).The main result of this paper is the following.Example 2. Let  be a variable,  a commutative ring with unit, and  a commutative -algebra with a unit.We present two examples of coordinate Hopf algebras which are well known (see [11][12][13]) and that are generalized hereafter.

Categories, Semicategories, and Their Total Algebra
The basic concepts from category theory may be found in [10] but are recalled hereafter.When viewed as algebraic objects, the categories are always considered as small categories in the sense that their classes of objects and arrows form usual sets (in some given universe).The reader should refer to [10] for this kind of size issues.
Let  be any commutative ring with a unit, and let  be a commutative -algebra with a unit.We may define the algebra of , denoted by ⟨⟩, as the free -module with basis Arr() together with the following constants of structure (see [18]) that define an associative product: It becomes an -algebra in an evident way.There is another way, that will be useful, to define this algebra.A semigroup (respectively, monoid) with zero is a usual semigroup (respectively, monoid), say , together with a two-sided absorbing element 0  , that is, for all  ∈ , 0  = 0  = 0  .A homomorphism between semigroups (respectively, monoid) with zero is a usual homomorphism of semigroups (monoids) Algebra that preserves the zeroes.The contracted algebra  0 [] (see [19]) is then defined as the free -module with basis  together with the associative product given for every ,  ∈  by We observe that it is a unital algebra when  is a monoid with zero.A finite decomposition semigroup (respectively, monoid) with zero  is a semigroup (respectively, monoid) with zero such that for all  ∈  \ {0  }, the set {(, ) ∈  2 :  = } is finite (it is a generalization of the property (D) from [18]).This makes it possible to consider a topological completion for the contracted algebra  0 [].As module  0 [] is isomorphic to the module  (\{0  }) of all finitely supported maps from \{0  } to .Under this isomorphism, we may equip  0 [] with the product topology, with  (and ) discrete, inherited from the product -module  \{0  } .Its topological completion is  \{0  } with its product topology.Any element  of  \{0  } may be represented in a unique way as a formal series ∑ ∈\{0  } ⟨ | ⟩ as in Section 2.Moreover, because  is a finite decomposition semigroup with zero, the multiplication of  0 [] may be uniquely extended by uniform continuity to an associative and jointly continuous multiplication on  \{0  } given by The module  \{0  } with this product is called the total contracted algebra of , and it is denoted by  0 [[]].It satisfies the following universal problem.Let  be a finite decomposition semigroup (respectively, monoid) with zero, let  be a complete topological -algebra (respectively, unital -algebra), and let  :  0 [] →  be a continuous homomorphism of algebras (if  is a monoid and  is unital, then we also assume that (1  ) = 1  ), then there exists a unique continuous homomorphism of algebras φ :  0 [[]] →  such that φ() = () for every  ∈  0 [] (in addition, if  is a monoid, and  is unital, then φ respects the units).All the details of this construction may be found in [20].
Equivalently, a semigroup (respectively, monoid) with zero  is said to be locally finite [20] if for every  ̸ = 0  , for all , the set of all -tuples ( 1 , . . .,   ), (  ̸ = 1  ,  = 1, . . ., , when  is assumed to be a monoid), such that  1 ⋅ ⋅ ⋅   =  is finite.It is clear that a semicategory (respectively, category) is locally finite, if, and only if, the semigroup (or monoid if  has only one object since in this case Arr() is actually a monoid) with zero Arr() ⊔ {0} is locally finite.
For a locally finite semicategory (respectively, category), we define the length ℓ() of an arrow  as the supremum in N of the lengths of decompositions of .In particular, if  is a locally finite category, ℓ() = 0 if, and only if,  is an identity.For each free semicategory  + (respectively, category  * ) over a directed graph , the length of a path ( 1 , . . .,   ) ∈ Arr( + ) is , while the length of an identity (, ) ∈ Arr( * ) is zero.
It is clear that the algebra ⟨⟨⟩⟩ of a locally finite semicategory (respectively, category)  is filtered by this order function.Another way to define this filtration is the following.Let us define M ≥ () = { ∈ ⟨⟨⟩⟩ | () ≥ } for each  ≥ 0. Therefore, M ≥0 () = ⟨⟨⟩⟩, and It is clear that ⋂ ≥0 M ≥ () = (0) so that the filtration is separated and decreasing.With this filtration, ⟨⟨⟩⟩ becomes a (Hausdorff) complete topological -algebra (with  and  discrete) when (M ≥ ())  is considered as a basis of neighborhoods of zero (see [20,26]).Observe that if there is some ℓ such that Arr() ℓ is infinite, then this topology is strictly finer than the product topology (in any case, namely, Arr() ℓ is either finite or infinite for each ℓ, then it is finer since the projections   → ⟨ | ⟩ are continuous in the topology induced by the filtration for all ).For instance, let (  ) ≥0 be an infinite sequence of arrows such that   ̸ =   for each  ̸ = , and ℓ(  ) = ℓ for each .Let   = ∑  =0   .Then, (  )  converges to  = ∑ ≥0   in ⟨⟨⟩⟩ with the product topology, but does not converge for the topology induced by the filtration (since ( −   ) = ℓ for all ).Nevertheless when for each ℓ, Arr() ℓ is finite, then both topologies coincide.

Substitution of Series without Constant Terms
Let  be a commutative ring with a unit, and let  be a commutative -algebra with a unit.Let  be any semicategory.
From now on let us assume that  is finite, that is, the sets Arr() = E() and Ob() = V() are finite (if  is a category, then it is sufficient to assume that Arr() is finite, because   is onto,  = 0, 1, but when  is a semicategory that lacks identities, then we need in addition to assume that Ob() is finite, because   ,  = 0, 1, is not assumed to be onto anymore).Then, for each ℓ ∈ N, there are only finitely many elements in Arr( * ) ℓ (in particular, Arr( * ) 0 = {id  :  ∈ Ob()}), so that the product topology and the filtration coincide.According to Lemma 5,   is continuous for the product topologies, and we may extend it by continuity to the completion ⟨⟨⟩⟩ of ⟨⟩ for the product topologies.We obtain a continuous algebra endomorphism, again denoted by   , of ⟨⟨⟩⟩ such that In particular,   (∑ ∈Ob() id  ) = ∑ ∈Ob()   (id  ) = ∑ ∈Ob() id  , so that both   : ⟨⟩ → ⟨⟨⟩⟩ and its continuous extension   : ⟨⟨⟩⟩ → ⟨⟨⟩⟩ respect the units (∑ ∈Ob() id  ∈ ⟨⟩ because Ob() is finite).Lemma 6.The series   () is given by the following.
(4) If  is a field say K, then M() is a group under ∘.
Remark 14.As in Remark 13, if  is the semicategory with a single object and a single arrow, then the coordinate Hopf algebra of M 1 is the usual Faà di Bruno Hopf algebra (see [11,15]).
Let K be a field.Let  and  be two Hopf algebras over K.According to [29], if  is commutative and  is an comodule bialgebra, then the tensor algebra  ⊗ K  admits a structure of Hopf algebra, called the smash coproduct, and is denoted by #.An affine group on a field K will be referred to as a proaffine algebraic group.
Example 15 (see [29,30] Example 15 may be applied on the two affine groups of series over a category previously introduced.Let  be a commutative ring with a unit and let  be a finite semicategory.We consider the group-valued functors of series I and M 1 over the locally finite category  * so that the results of Sections 6.1 and 6.2 hold.Let  be any commutative algebra with a unit.For every  ∈ I() and  ∈ M 1 (),  ∘  ∈ I() (by Lemma 6).Moreover,   : I → I() is invertible with inverse   ∘−1 .Actually, it defines a group antihomomorphism  : M 1 () → Aut(I()) by () =   .Therefore, we obtain a structure of semidirect product I() ⋊ M op 1 () on I() × M 1 () by ( 1 ,  1 ) ⋊ ( 2 ,  2 ) = ( 1 ( 2 ∘  1 ),  2 ∘  1 ) and with (1 K⟨⟨ * ⟩⟩ , ) as identity (where  op denotes the opposite group of a group ).The inverse of (, ) is given by ( −1 ∘  ∘−1 ,  ∘−1 ).This construction is easily seen to be natural in  so that we have a semidirect -group functor I ⋊ M op 1 .
Remark 16.This situation generalizes the case of formal power series in one-variable  where we have a semidirect product  ⋊  (using the notations from Example 2) given by exactly the same formula ( 1 ,  1 ) ⋊ ( 2 ,  2 ) = ( 1 ( 2 ∘  1 ),  2 ∘  1 ) and with (1, ) as unit element.See for instance [31].Now, let us assume that  is a field K (in order to apply the results from [29,30]).According to Example 15, we have a structure of Hopf algebra given by the smash coproduct O(I)#O(M  Remark 17.In the paper [32], the authors described a similar interaction between two Hopf algebras of functions on the so-called B-series [33,34]. an antipode  : O() → O(), and a counit O() →  that turn the algebra O() into a commutative Hopf -algebra.It turns that the natural set isomorphisms () ≅ -CAlg(O(), ) become group isomorphisms.

Algebra 3 Theorem 1 .
Let K be a field.The K-group I ⋊ M op 1 is actually an affine group.
).Let  and  be two proaffine algebraic groups, represented by the coordinate Hopf algebras O() and O().Let us assume that we have a semidirect product  ⋊   of proaffine algebraic groups, that is,  :  ×  →  is a natural transformation that induces a group action from  on  by proaffine algebraic group automorphisms.Then, according to Yoneda's lemma,  induces an algebra mapO() → O() ⊗ K O() that turns O() into a O()-comodule bialgebra.Thus, O() # O()is a Hopf algebra that can be shown to be isomorphic to O( ⋊  ).