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Volume 2013 (2013), Article ID 370618, 10 pages
Two Interacting Coordinate Hopf Algebras of Affine Groups of Formal Series on a Category
Université Paris 13, Sorbonne Paris Cité, LIPN, CNRS (UMR 7030), 93430 Villetaneuse, France
Received 19 March 2013; Revised 17 May 2013; Accepted 14 July 2013
Academic Editor: Stefaan Caenepeel
Copyright © 2013 Laurent Poinsot. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 anti-automorphisms gives rise to an interaction at the level of their coordinate Hopf algebras under the form of a smash coproduct.
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 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 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 series. The set of all series in the large algebra which are zero on the identity arrows and 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 and therefore defines a semidirect product . Moreover both groups and 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 and and introduce their coordinate Hopf algebras which may be thought as generalizing some well-known Hopf algebras.
2. 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 to be commutative but they are associative. The notation - stands for the category of commutative -algebras with a unit and unit preserving algebra maps, and the hom-sets are denoted by -. In what follows, if is -algebra with a unit, then is its identity.
Let be any set, and let . The support of is the set . 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 if and . 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 through , namely, for every .
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 ) in with sum so that we may represent as an infinite linear combination (more details may be found in [2–4]). Moreover, is separately continuous.
3. 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 [7–9] for the notions concerning algebraic groups, and to  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 - to the category of groups . When an -group is representable when viewed as a set-valued functor (by composition with the forgetful functor from to the category of sets ), that is, - (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 of , for reasons made clear hereafter. The representable -group is an affine algebraic group when is finitely generated as an -algebra. Since the multiplication , the inversion , and the unit element are natural transformations between representable set-theoretic functors, by Yoneda's lemma they uniquely give rise respectively to a (coassociative) coproduct , an antipode , and a counit that turn the algebra into a commutative Hopf -algebra. It turns that the natural set isomorphisms - become group isomorphisms.
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 , where - 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 . 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 one-point set. By Yoneda's lemma, the multiplication uniquely determines an algebra map -, which is explicitly given by , where and (essentially because is induced by the pair of maps under the isomorphism -, ). Similarly, again by Yoneda's lemma, the unit uniquely determines an algebra map given by (with the identity of a group ). Finally, the inversion gives rise to a unique algebra map explicitly given by , where is the member that induces the identity . This map is of course the antipode of .
In what follows we introduce two affine groups and (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.
Theorem 1. Let be a field. The -group is actually an affine group.
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 [11–13]) and that are generalized hereafter. (1)Let be the -group of invertible formal power series , that is, , if, and only if, , where . Let be an infinite sequence of pairwise distinct variables. Let us compute the coproduct , counit , and antipode of free commutative algebra . According to the previous discussion, is obtained by computing the product in of the series and which is equal to . Then, so that for every , . The counit is given by the set-theoretic map for each . The inverse of is given recursively by , and for all . In particular, for , for each , . It follows that or in other terms for all . (2)Let be the set , that is, , if, and only if, for . This forms a group under the usual composition of formal power series . It is clear that is the coordinate system of . Let us explicitly compute the bialgebra structure of the corresponding coordinate Hopf algebra . Following  (or for instance by the usual Faà di Bruno formula, see [15, 16]), we have and , where is a polynomial in variables. Applying this to and gives the coproduct or equivalently, for each . The counit is given by for each . Let and be the inverse of in . For each , we have . Applying this formula to gives (where by convention we put , ) for each .
4. Categories, Semicategories, and Their Total Algebra
The basic concepts from category theory may be found in  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  for this kind of size issues.
A semicategory, also called a taxonomy, see , (respectively, category) is given by its class of objects , its class of arrows or morphisms , two (non necessarily surjective) maps (and such that , , if is a category), is the domain map, is the codomain map, (and is the identity arrow map, if is a category), and finally a map, called composition, such that and for all , and that satisfies an axiom of associativity: for all such that and , (and two axioms of identity: and for every , if is a category). In general, is denoted by and , and by for each . The set is called the set of composable arrows in . More generally, for every , is the set of -tuple of composable arrows, and by associativity, has an obvious meaning whenever is an -tuple of composable arrows. It is also clear that in this case and . We may also denote by or by the fact that with , . Functors between semicategories are the obvious ones.
Remark 3. If is a semicategory, then there is the possibility that some objects do not correspond to domain or codomain of arrows (this is a kind of isolated point in a graph), because the map is not assumed to be onto, while it is for a category.
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 together with the following constants of structure (see ) 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 , that is, for all , . A homomorphism between semigroups (respectively, monoid) with zero is a usual homomorphism of semigroups (monoids) that preserves the zeroes. The contracted algebra (see ) 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. Now, the arrows of any semicategory (respectively, category) together with a new element added form a semigroup with zero with multiplication given for every by if, and only if, , and otherwise. Moreover, any functor between two semi-categories (respectively, categories) defines a homomorphism of semigroups with zero by extending with . This provides a functor from the category of all semi-categories (respectively, categories) to the category of semigroups with zero. Moreover, it is clear that (as - and -algebras). We observe that is a monoid with zero if, and only if, is a category with only one object, that is, is a usual monoid (under composition). In this case, is isomorphic to the usual (unital) algebra of the monoid . If is finite, then is a unital algebra with identity .
A finite decomposition semigroup (respectively, monoid) with zero is a semigroup (respectively, monoid) with zero such that for all , the set is finite (it is a generalization of the property (D) from ). This makes it possible to consider a topological completion for the contracted algebra . As module is isomorphic to the module of all finitely supported maps from to . Under this isomorphism, we may equip with the product topology, with (and ) discrete, inherited from the product -module . Its topological completion is with its product topology. Any element of may be represented in a unique way as a formal series as in Section 2. Moreover, because is a finite decomposition semigroup with zero, the multiplication of may be uniquely extended by uniform continuity to an associative and jointly continuous multiplication on given by The module with this product is called the total contracted algebra of , and it is denoted by . 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 be a continuous homomorphism of algebras (if is a monoid and is unital, then we also assume that ), then there exists a unique continuous homomorphism of algebras such that for every (in addition, if is a monoid, and is unital, then respects the units). All the details of this construction may be found in .
In a similar way, let us call a finite decomposition semicategory (respectively, category) a semicategory (respectively, category) such that for every , the set is finite (it is again a generalization of the property (D) from  that may be found in [21–23]). Once more this allows us to define a topological completion for given as the -algebra (which is as an -module) of all formal series , with product .
Remark 4. Let . We have
It is not difficult to see that is a finite decomposition semicategory (respectively, category) if, and only if, is a finite decomposition semigroup, and in one of these equivalent cases, (as topological - and -algebras). We observe that if is a category, then is the two-sided identity of . The -algebra satisfies the following universal property: let be a complete topological -algebra, and let be a continuous homomorphism of algebras from (with the product topology inherited from ) to . Then, there is a unique continuous homomorphism of -algebras such that for every . Let be a semicategory (respectively, a category). Let such that if , then , (and if is a category, for every ). If is a semicategory, then is a functor between and (seen as a semicategory with only one object). If is a category, then this is no more a functor because it should be the case that (the unit of ). In any case, it is a homomorphism of semigroups with zero from to . Therefore, it may be extended uniquely to a homomorphism of algebras by linearity. If it happens that it is continuous, then it may be itself extended as a continuous endomorphism of algebra . It satisfies . If is a category, then we observe that so that respects the unit.
Let be a directed graph or quivers (see ), that is, it is given by a set of vertices, a set of edges, and two maps and (or simply when there is no risk of confusion) . We may define the free semicategory (respectively, category ) generated by as follows: , , , (in particular for , for all , and these -tuples are called paths of length ), (where denotes the empty word, and is a path of length zero), , , , , for every such that , then , , and for every . It is easy to check that and are indeed a semicategory and a category, respectively. Moreover, is obtained from by free adjunctions of units for each object (and the obvious extensions of , , and of the composition). This means the following: let be a semicategory. We denote by the category obtained from by adjunctions of the identities on each object of , the trivial extension of the domain and codomain maps, and the composition. Then, for every category seen as a semicategory in an evident way, and any functor there is a unique functor such that for every . A morphism of directed graph from to is a pair of maps , such that , . We observe that any semicategory (respectively, category) is obviously also a directed graph by forgetting some of its structure. Moreover, any functor between semi-categories (respectively, categories) is also a morphism between the underlying directed graphs.The following universal property is satisfied. Let be a directed graph, and let be a semicategory (respectively, category). Let be a morphism between and seen as a directed graph. Then, there is a unique functor from (respectively, ) to such that for every , , and for every .
Let be a directed graph. Let be a semigroup (respectively, monoid) with zero. Any morphism of directed graphs between to , seen as a directed graph with only one vertex, is then given by a set-theoretic map without any property to satisfy. Let . This gives rise to a map (respectively, ) by (and in addition, for each ) which is shown to be a functor between semi-categories. It is unique with the property that for every (and in addition, for each object ).
It is quite clear that (respectively, ) is a finite decomposition semicategory (respectively, category), or equivalently, (respectively, ) is a finite decomposition semigroup with zero (and even a finite decomposition monoid with zero if has only one vertex). Moreover (respectively, ) satisfies a stronger property. It is a locally finite semicategory (respectively, category). Let be a semicategory (respectively, a category). Let . Any sequence of composable arrows in (where in addition we assume that is not an identity for each , if is a category) is called a proper decomposition of length of , if, and only if, . A proper decomposition of is then a proper decomposition of length of for some . The set of all proper decompositions of length of is denoted by , and the set of all proper decompositions of is denoted by . We assume that each identity arrow has only one decomposition, which has length zero, the empty decomposition (if is a category). We observe that for each (if is a category), , and for every , while , for every nonidentity arrow of . A semicategory (respectively, category) is said to be locally finite [21–23] if for every arrow , is a finite set. A locally finite semicategory (respectively, category) is also a finite decomposition semicategory (respectively, category) because is assumed to be finite (and this suffices for the case of semi-categories because ), and (if is a category). In particular, in a locally finite category, no nonidentity arrow may be left or right invertible. Indeed, if is right invertible, then there is such that (so that ), and then is a proper decomposition of of arbitrary length.
Equivalently, a semigroup (respectively, monoid) with zero is said to be locally finite  if for every , for all , the set of all -tuples , (, , when is assumed to be a monoid), such that 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 is actually a monoid) with zero is locally finite.
For a locally finite semicategory (respectively, category), we define the length of an arrow as the supremum in of the lengths of decompositions of . In particular, if is a locally finite category, if, and only if, is an identity. For each free semicategory (respectively, category ) over a directed graph , the length of a path is , while the length of an identity is zero.
Let be a locally finite semicategory (respectively, category). For every (respectively, ), let us define . In particular, in a locally finite category, . It is clear that in a locally finite semicategory (respectively, category), (respectively, ) which is a disjoint sum. According to , we have for every . This length is extended to an order function defined by (the infimum being taken in ). In particular, if, and only if, . The following hold. (1) for every object (if is a category). (2). (3).
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 for each . Therefore, , and is a two-sided ideal in (improper if is a semicategory). It is clear that so that the filtration is separated and decreasing. With this filtration, becomes a (Hausdorff) complete topological -algebra (with and discrete) when is considered as a basis of neighborhoods of zero (see [20, 26]). Observe that if there is some such that is infinite, then this topology is strictly finer than the product topology (in any case, namely, is either finite or infinite for each , then it is finer since the projections are continuous in the topology induced by the filtration for all ). For instance, let be an infinite sequence of arrows such that for each , and for each . Let . Then, converges to in with the product topology, but does not converge for the topology induced by the filtration (since for all ). Nevertheless when for each , is finite, then both topologies coincide.
5. 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. Forgetting the composition of arrows, we obtain a directed graph . Let (respectively, ) be the free semicategory (respectively, category) generated by . By universality of , there exists a unique functor such that for every . This functor is obviously onto on arrows (and the identity on objects). The paths in are denoted by , and for each such path there is a unique integer and a unique sequence of composable arrows in such that . The composition of paths will be denoted by juxtaposition (we reserve for the composition in ). Let and be two composable arrows in , and such that , . Then, and are composable in . Indeed, . Because is a functor, we have then . More generally, if are composable in and for each , such that , then is an -tuple of composable paths in and . We also observe that for every , .
Let and . We define . Let be defined as the morphism of directed graphs (when is a directed graph with only one vertex). Then, it is extended uniquely to a functor by , and for every object of (again, is seen as a semicategory with only one object). According to the universal property of , is uniquely extended by linearity to a homomorphism of algebras, again denoted , from to (we observe that in general it does not preserve the units since is unital only when is finite; however for each object ). We have for each .
Lemma 5. The homomorphism is continuous (for the filtrations).
Proof. By linearity it is sufficient to prove that is continuous. We have . Now, , . Therefore, for all , and then so that is continuous for the filtrations.
From now on let us assume that is finite, that is, the sets and are finite (if is a category, then it is sufficient to assume that is finite, because is onto, , but when is a semicategory that lacks identities, then we need in addition to assume that is finite, because , , is not assumed to be onto anymore). Then, for each , there are only finitely many elements in (in particular, ), 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, , so that both and its continuous extension respect the units ( because is finite).
Lemma 6. The series is given by the following. (1) for every . (2) for every .
Proof. We have (1)Let . We have . We have . (2)Let (that is, ). Each proper decomposition of of the form , , occurs in the sum (6) with coefficient given by where . Collecting all these coefficients yields the expected result for the coefficient of in .
We observe that maps to itself according to the first point of Lemma 6. Let . For every , , then for each , . Therefore, for every (since we also have . Therefore, is the identity on (since is dense in , and is continuous). Moreover, we have . For every , then , so that . But the family is summable, and by associativity of summable families, . Finally for similar reasons, , so that . This means that acts as a two-sided identity for the operation .
Example 7. Let be the semicategory with only one object and one arrow . Then, , and . Moreover, . Finally, , where denotes the usual substitution of formal power series (see for instance ), and is the unit of .
Let be a finite semigroup. Then, (the free semigroup on ), and (the free monoid on , with identity the empty word ). Therefore, (the large algebra of ). We have .
Let be a finite poset. We define a semicategory , where is the strict partial order associated to . Its set of objects is . We have an arrow between and if, and only if, . The composition is the obvious one. Now, consists of the identities for every , and all the chains . Because is finite, it is clear that is finite. Moreover . We have . In particular, if covers , then . We have .
We now denote by , and call it the substitution. We also define .
Lemma 8. The set is a monoid under .
Let be left (respectively, right) invertible (with respect to ). Then, for every , is left (respectively, right) invertible in , and so it is invertible since is commutative. In particular, .
Let such that for every , is invertible in (in particular, ). Then, is invertible (with respect to ), its inverse , denoted by , is an element of such that for every , is the inverse of . In particular, if for every , then is invertible, and its inverse satisfies for every .
If is a field say , then is a group under .
Proof. It is already known that is a binary operation for , with unit given by . It remains to prove associativity. Actually we can prove the following. Let . For every , . Now, let with . The computation of at involves all proper decompositions into nontrivial paths, and then all the proper decompositions of each factor . It is equivalent to sum first over all the proper decompositions and then to consider all manners of regrouping adjacent factors , which yields to the value of at .
Let be left (respectively, right) invertible. Therefore there exists some such that (respectively, ). In particular, for every , (respectively, ), according to Lemma 6.
Let such that for every , is invertible in . For every , let us define . For every , let us define . Now, suppose that such that . By induction on , we define . We define . It is clear that , and that for every . It is easy to see that (because is built to satisfy this property), so that is left invertible. Since for every , is invertible in , it follows that also is left invertible. Therefore there is such that , then so that is invertible with (two-sided) inverse . The second fact becomes obvious.
It is clear that is a submonoid of (since , and for every such that , whenever ). To prove that is a group under (if is the field ), it suffices to prove that any element of is invertible (with respect to ) which is clear according to the previous point.
6. Coordinate Hopf Algebras for Groups of Invertible Series with respect to Multiplication and Substitution
In this section we define two -groups of series over a category, and we use the result from Section 3 to introduce their coordinate Hopf algebras.
6.1. Group of (Unipotent) Invertible Series
Let be a commutative -algebra with a unit. Let be a finite decomposition category. Let , and let . Let us define
Remark 9. We observe that in general for an arbitrary category , is not closed under . Indeed, let us consider the finite category (actually, groupoid) with two objects and two nonidentity arrows , with . It is obviously a finite decomposition category (since it is a finite category). Let and