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Cofree Compositions of Coalgebras Stefan Forcey, Aaron Lauve, Frank Sottile

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Cofree Compositions of Coalgebras Stefan Forcey, Aaron Lauve, Frank Sottile Cofree compositions of coalgebras Stefan Forcey, Aaron Lauve, Frank Sottile To cite this version: Stefan Forcey, Aaron Lauve, Frank Sottile. Cofree compositions of coalgebras. 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.363-374. hal-01215100 HAL Id: hal-01215100 https://hal.inria.fr/hal-01215100 Submitted on 13 Oct 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. FPSAC 2011, Reykjav´ık, Iceland DMTCS proc. AO, 2011, 363–374 Cofree compositions of coalgebras Stefan Forcey1, Aaron Lauve2y and Frank Sottile3z 1Department of Theoretical and Applied Mathematics, The University of Akron, Akron, OH, USA 2Department of Mathematics, Loyola University of Chicago, Chicago, IL, USA 3Department of Mathematics, Texas A&M University, College Station, TX, USA Abstract. We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and the theory of species. We prove that the composition of two cofree coalgebras is cofree and give conditions which imply that the composition is a one-sided Hopf algebra. These conditions hold when one coalgebra is a graded Hopf operad D and the other is a connected graded coalgebra with coalgebra map to D. We conclude with examples of these structures, where the factor coalgebras have bases indexed by the vertices of multiplihedra, composihedra, and hypercubes. Resum´ e.´ Nous developpons´ la notion de composition de coalgebres,` qui apparaˆıt naturellement dans la theorie´ des categories´ d’ordre superieur´ et dans la theorie´ des especes.` Nous montrons que la composee´ de deux coalgebres` colibres est colibre et nous donnons des conditions qui impliquent que la composee´ est une algebre` de Hopf uni- laterale.´ Ces conditions sont valables quand l’une des coalgebres` est une operade´ de Hopf graduee´ D et l’autre est une coalgebre` graduee´ connexe avec un morphisme vers D. Nous concluons avec des exemples de ces structures, ou` les coalgebres` composees´ ont des bases indexees´ par les sommets de multipliedres,` de composiedres,` et d’hypercubes. Keywords: multiplihedron, cofree coalgebra, Hopf algebra, operad, species 1 Introduction The Hopf algebras of ordered trees (Malvenuto and Reutenauer (1995)) and of planar binary trees (Loday and Ronco (1998)) are cofree coalgebras that are connected by cellular maps from permutahedra to as- sociahedra. Related polytopes include the multiplihedra (Stasheff (1970)) and the composihedra (Forcey (2008b)), and it is natural to study what Hopf structures may be placed on these objects. The map from permutahedra to associahedra factors through the multiplihedra, and in (Forcey et al. (2010)) we used this factorization to place Hopf structures on bi-leveled trees, which correspond to vertices of multiplihedra. Multiplihedra form an operad module over associahedra. This leads to painted trees, which also corre- spond to the vertices of the multiplihedra. In terms of painted trees, the Hopf structures of (Forcey et al. (2010)) are related to the operad module structure. We generalize this in Section 3, defining the functorial construction of a graded coalgebra D ◦ C from graded coalgebras C and D. We show that this composition of coalgebras preserves cofreeness. In Section 4 we give sufficient conditions, when D is a Hopf algebra, ySupported in part by NSA grant H98230-10-1-0362. zSupported in part by NSF grants DMS-0701050 and DMS-1001615. 1365–8050 c 2011 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France 364 Stefan Forcey, Aaron Lauve and Frank Sottile for the composition of coalgebras D ◦ C (and C ◦ D) to be a one-sided Hopf algebra. These conditions also guarantee that a composition is a Hopf module and a comodule algebra over D. This composition (also known as substitution) is familiar from the theories of operads and species. If a species is a monoid with respect to ◦ then it is also an operad (Aguiar and Mahajan, 2010, App. B). In Section 4 we show that an operad D of connected graded coalgebras is automatically a Hopf algebra. We discuss three examples related to well-known objects from category theory and algebraic topology and show that the Hopf algebra of simplices of (Forcey and Springfield (2010)) is cofree as a coalgebra. 2 Preliminaries L We work over a fixed field K of characteristic zero. For a graded vector space V = n Vn, we write jvj = n and say v has degree n if v 2 Vn. 2.1 Hopf algebras and cofree coalgebras A coalgebra C is a vector space C equipped with a coassociative coproduct ∆: C ! C ⊗ C and counit P 0 00 ": C! K. For c 2 C, write ∆(c) as (c) c ⊗ c . Coassociativity means that X X X (c0)0 ⊗ (c0)00 ⊗ c00 = c0 ⊗ (c00)0 ⊗ (c00)00 = c0 ⊗ c00 ⊗ c000 ; (c);(c0) (c);(c00) (c) P 0 00 P 0 00 and the counital condition reads (c) "(c )c = (c) c "(c ) = c. A Hopf algebra is a unital associative algebra H that is also a coalgebra whose structure maps (coproduct ∆ and counit ") are algebra homomor- phisms, with the additional condition of having an antipode. See (Montgomery (1993)) for more details. L Takeuchi (1971) showed that a graded bialgebra H = ( n≥0 Hn; ·; ∆;") that is connected (H0 = K) is a Hopf algebra. A one-sided Hopf algebra H = (H; u; m; ∆; "; S) is allowed to have only a one-sided unit u and to satisfy only one of m(S ⊗1)∆ = u" and m(1⊗S)∆ = u". This relaxes the standard notion appearing in the literature (Green et al. (1980)), where only the antipode is allowed to be one-sided. L ⊗n The graded cofree coalgebra on a vector space V is C(V ) := n≥0 V with counit the projection ": C(V ) ! K = V ⊗0 and the deconcatenation coproduct: writing “n” for the tensor product in V ⊗n, we have n X ∆(c1n · · · ncn) = (c1n · · · nci) ⊗ (ci+1n · · · ncn) : i=0 Observe that V is the set of primitive elements of C(V ). A graded coalgebra C is cofree if C' C(PC), where PC is the space of primitive elements of C. Many coalgebras arising in combinatorics are cofree. 2.2 Cofree Hopf algebras on trees We describe three cofree Hopf algebras built on rooted planar binary trees: ordered trees Sn, binary trees Y , and (left) combs C on n internal nodes. Set S := S S and define Y and C similarly. n n · n≥0 n · · 2.2.1 Constructions on trees The nodes of a tree t 2 Yn form a poset. An ordered tree w = w(t) is a linear extension of this node poset of t. This linear extension is indicated by placing a permutation in the gaps between its leaves, which gives a bijection between ordered trees and permutations. The map τ : Sn !Yn sends an ordered Cofree compositions of coalgebras 365 tree w(t) to its underlying tree t. The map κ: Yn ! Cn shifts all nodes of a tree to the right branch from the root. Set S0 = Y0 = C0 = . Note that jCnj = 1 for all n ≥ 0. 3 4 1 2 2 4 1 3 1 4 2 3 - - τ κ ordered trees S· - binary trees Y· - left combs C· Splitting an ordered tree w along the path from a leaf to the root yields an ordered forest (where the nodes in the forest are totally ordered) or a pair of ordered trees, ? ? 2 5 1 4 3 2 5 1 4 3 2 5 1 2 3 1 4 3 2 1 g or ; −−! ; ; : g Write w ! (w0; w1) when the ordered forest (w0; w1) (or pair of ordered trees) is obtained by splitting w. (Context will determine how to interpret the result.) We may graft an ordered forest ~w = (w0; : : : ; wn) onto an ordered tree v 2 Sn, obtaining the tree ~w=v as follows. First increase each label of v so that its nodes are greater than the nodes of ~w, and then graft th tree wi onto the i leaf of v. For example, 1 4 3 2 7 5 1 ! 3 2 6 4 if (~w; v) = ; ; ; ; ; ; 7 5 1 3 2 6 4 3 2 8 11 7 5 1 10 6 9 4 8 11 10 9 then ~w=v = = . Splitting and grafting make sense for trees in Y·. They also work for C· if, after grafting a forest of combs onto the leaves of a comb, one applies κ to the resulting planar binary tree to get a new comb. 2.2.2 Three cofree Hopf algebras L th Let SSym := n≥0 SSymn be the graded vector space whose n graded piece has basis fFw j w 2 Sng. Define YSym and CSym similarly. The set maps τ and κ induce vector space maps τ and κ, τ (Fw) = Fτ(w) and κ(Ft) = Fκ(t). Fix X 2 fS; Y; Cg. For w 2 X· and v 2 Xn, set 366 Stefan Forcey, Aaron Lauve and Frank Sottile X Fw · Fv := F(w0;:::;wn)=v ; g w!(w0;:::;wn) the sum over all (ordered) forests obtained by splitting w at a multiset of n leaves. For w 2 X·, set X ∆(Fw) := Fw0 ⊗ Fw1 ; g w!(w0;w1) the sum over all splittings of w at one leaf.
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