We define two coproducts for cycle-free oriented graphs, thus building up two
commutative con- nected graded Hopf algebras, such that one is a
comodule-coalgebra on the other, thus generalizing the result obtained
previously for Hopf algebras of rooted trees.Comment: 7 pages, error on Proposition 1 corrected, one figure adde

Manchon, Dominique

English

arXiv

International audienceWe define two coproducts for cycle-free oriented graphs, thus building up two commutative con- nected graded Hopf algebras, such that one is a comodule-coalgebra on the other, thus generalizing the result obtained previously for Hopf algebras of rooted trees

HAL Clermont Université

On bialgebras and Hopf algebras of oriented graphs

Dominique Manchon

Confluentes Mathematici

Numérisation de Documents Anciens Mathématiques

April 7, 2012 17:36 WSPC/S1793-7442 251-CM 1240003Confluentes Mathematici, Vol. 4, No. 1 (2012) 1240003 (10 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S1793744212400038ON BIALGEBRAS AND HOPF ALGEBRASOF ORIENTED GRAPHSDOMINIQUE MANCHONUniversity of Blaise Pascal,C.N.R.S.-UMR 6620, 63177 Aubière, Francemanchon@math.univ-bpclermont.frReceived 12 January 2011Revised 4 July 2011Published 10 April 2012We define two coproducts for cycle-free oriented graphs, thus building up twocommutative connected graded Hopf algebras, such that one is a comodule-coalgebraon the other, thus generalizing the result obtained in [2] for Hopf algebras of rootedtrees.Keywords: Graphs; Hopf algebras; bialgebras; posets; admissible cuts.AMS Subject Classification: 05C20, 05C38, 81T181. IntroductionHopf algebras of graphs have been introduced by Kreimer [8, 4–6] in order toexplain the combinatorics of renormalization in Quantum Field Theory. Whereasthe product is free commutative, the coproduct is defined by suitable subgraphsand contracted graphs, and depends on the type of graphs considered.We focus on various Hopf algebras of oriented graphs: after giving the basicdefinitions we detail two examples: oriented graphs in general, and then locally one-particle irreducible graphs. We show on concrete computations that the coproductof a locally 1PI oriented graph Γ depends on whether one takes the local 1PI-nessof Γ into account or not.We also explore a third example, the Hopf algebra HCF of oriented cycle-freegraphs. The associated poset structure on the set of vertices yields still anothercoproduct which generalizes the coproduct of rooted trees given by admissiblecuts [8, 7]. We show that the Hopf algebra HCFc thus obtained is a comodule-coalgebra on the Hopf algebra HCF. Modulo discarding the external edges, thisgeneralizes the results of [2] on Hopf algebras of rooted trees.1240003-1April 7, 2012 17:36 WSPC/S1793-7442 251-CM 1240003D. Manchon2. Oriented Feynman Graphs2.1. Basic definitionsAn oriented Feynman graph is an oriented (non-planar) graph with a finite numberof vertices and edges, which can be internal or external. An internal edge is an edgeconnected at both ends to a vertex (which can be the same in case of a self-loop),an external edge is an edge with one open end, the other end being connected toa vertex. An oriented Feynman graph will be called vacuum graph, tadpole graph,self-energy graph, respectively interaction graph if its number of external edges is0, 1, 2, respectively > 2.A cycle in an oriented Feynman graph is a finite collection (e1, . . . , en) of orientedinternal edges such that the target of ek coincides with the source of ek+1 for anyk = 1, . . . , n modulo n. The loop number of a graph Γ is given by:L(Γ) = I(Γ)− V (Γ) + 1, (1)where I(Γ) is the number of internal edges of the graph Γ and where V (Γ) is thenumber of vertices. We shall mainly focus on cycle-free oriented graphs, for whichthere exists a poset structure on the set of vertices: namely, v < w if and only ifthere exists a path from v to w, i.e. a collection (e1, . . . , en) of edges such that thetarget of ek coincides with the source of ek+1 for k = 1, . . . , n− 1, and such that v(respectively w) is the source (respectively the target) of e1 (respectively en).The edges (internal or external) will be of different types labeled by a positiveinteger (1, 2, 3, . . .), each type being represented by the way the corresponding edgeis drawn (full, dashed, wavy, various colours, etc.). Let τ(e) ∈ N∗ be the type ofthe edge e. For any vertex v let st(v) be the star of v, i.e. the set of all half-edgesattached to v (hence a self-loop yields two half-edges). Hence the valence of thevertex is given by the cardinal of st(v). Finally to each vertex v we associate itstype T (v), defined as the sequence (n1, . . . , nr) of positive integers where nj standsfor the number of edges of type j in st(v). The orientation does not enter into thedefinition of the type of a vertex.A one-particle irreducible graph (in short, 1PI graph) is a connected graph whichremains connected when we cut any internal edge. A disconnected graph is said tobe locally 1PI if any of its connected components is 1PI.2.2. Connected subgraphs, covering subgraphsand contracted graphsLet Γ be an oriented Feynman graph, let V(Γ) be the set of its vertices, and let Pbe a nonempty subset of V(Γ). The subgraph Γ(P ) associated to P is defined asfollows: the internal edges of Γ(P ) are the internal edges of Γ with source and targetin P , and the external edges are the external edges of Γ with source or target in P ,as well as the internal edges of Γ with one end in P and the other end outside P .The orientations of the edges of γP are obviously derived from their orientation inΓ. The subgraph Γ(P ) is connected if and only if for any v, w ∈ P one can go from1240003-2April 7, 2012 17:36 WSPC/S1793-7442 251-CM 1240003Hopf Algebras of Oriented GraphsFig. 1. A graph Γ together with a covering subgraph γ and the contracted graph Γ/γ.v to w by following internal edges of Γ with both ends in P , forwards or backwards.We set by convention Γ(∅) = 1, where 1 is the empty graph. For any Q ⊆ P ⊆ V(Γ)we obviously have:Γ(P )(Q) = Γ(Q). (2)A covering subgraph of Γ is an oriented Feynman graph γ (in general disconnected),given by a collection {Γ(P1), . . . ,Γ(Pn)} of connected subgraphs such that Pj ∩Pk = ∅ for j = k, and such that any vertex of Γ belongs to Pj for some (unique)j ∈ {1, . . . , n}. Covering subgraphs of Γ are in one-to-one correspondence withpartitions of V(Γ) into connected subsets, which refine the partition into connectedcomponents. For any covering subgraph γ, the contracted graph Γ/γ is defined byshrinking all connected components of γ inside Γ onto a point.The following proposition is straightforward:Proposition 1. Let Γ be an oriented cycle-free Feynman graph. Let γ be a coveringsubgraph and let V = P1  · · · Pn be the associated partition of V . If Γ/γ is cycle-free, then Pj is a convex subset of the poset V(Γ) for any j ∈ {1, . . . , n}.Note that the converse is not true, as shown by the following counterexample:1240003-3April 7, 2012 17:36 WSPC/S1793-7442 251-CM 1240003D. ManchonThe residue of the graph Γ is the contracted graph Γ/Γ, where the coveringsubgraph is the graph Γ itself. The associated partition of V(Γ) is the coarsestpossible, i.e. it is given by its connected components. The residue is the only graphwith no internal edge and the same external edges than those of Γ. At the otherextreme, the partition of V(Γ) into singletons (i.e. the finest possible) gives rise tothe unique covering subgraph γ0 without internal edges. The contracted graph Γ/γ0is then equal to Γ. Given two covering subgraphs γ and δ, say that γ contains δ ifthe partition associated with δ refines the partition associated with γ. In this caseδ can also be seen as a covering subgraph of γ.3. Some Bialgebras and Hopf Algebras of Graphs3.1. The full bialgebra of oriented Feynman graphsLet H̃ be the vector space spanned by (connected or not) oriented Feynman graphs.The product is given by concatenation, hence H̃ = S(V ), where V is the vectorspace spanned by connected oriented Feynman graphs. The unit 1 is identified withthe empty graph, and the coproduct is given by:∆(Γ) =∑γ covering subgraph of Γγ ⊗ Γ/γ. (3)This is obviously an algebra morphism, and we have:(∆⊗ I)∆(Γ) =∑δ covering subgraph of γ, γ covering subgraph of Γδ ⊗ γ/δ ⊗ Γ/γ,(I ⊗∆)∆(Γ) =∑δ covering subgraph of Γ, eγ covering subgraph of Γ/δδ ⊗ γ̃ ⊗ (Γ/δ)/γ̃.There is an obvious bijection γ → γ̃ = γ/δ from covering subgraphs of Γ containingδ onto covering subgraphs of Γ/δ, given by shrinking δ. As we have the obvious“transitive shrinking property”:Γ/γ = (Γ/δ)/(γ/δ), (4)the two expressions coincide, hence ∆ is coassociative. The co-unit is given byε(1) = 1 and ε(Γ) = 0 for any nonempty graph Γ. The bialgebra H̃ is graded by thenumber of internal edges (and even multi-graded by the numbers of internal edgesof various given types). The elements of degree zero are the residues, i.e. the graphswithout internal edges. Any residue graph R is grouplike, i.e. ∆(R) = R⊗R. Asan example of coproduct computation (with only one type of edges), we have:(5)1240003-4April 7, 2012 17:36 WSPC/S1793-7442 251-CM 1240003Hopf Algebras of Oriented Graphs3.2. The Hopf algebra of oriented Feynman graphsThe Hopf algebra H is obtained from H̃ by identifying all degree zero elements withthe unit 1, namely:H = H̃/J , (6)where J is the (bi-)ideal generated by the elements Γ − 1 where Γ is any graphwithout internal edges. The bialgebra H is obviously connected graded, hence itis a Hopf algebra, which can be identified as a commutative algebra with S(W ),where W is the vector space spanned by connected oriented Feynman graphs withat least one internal edge. The coproduct computation (5) yields:(7)3.3. Locally 1PI graphsA similar construction holds for locally 1PI graphs: the bialgebra H̃1PI is given byS(V1PI), where V1PI is the vector space spanned by connected oriented 1PI Feynmangraphs. The coproduct is given by:∆(Γ) =∑γ locally 1PI covering subgraph of Γγ ⊗ Γ/γ, (8)and is coassociative due to the fact that the transitive shrinking property of Sec. 3.1still makes sense for locally 1PI covering subgraphs. The (multi-) grading given bythe number of internal edges is still relevant, but an alternative grading is givenby the loop number. The associated Hopf algebra H1PI is built up similarly to Hin Sec. 3.2, by identifying the elements of degree zero with the unit 1. Note that,for both gradings, the elements of degree zero are the residues: it comes from thefact that a graph Γ with loop number L(Γ) = 0 which is locally 1PI cannot haveany internal edge. Here is an example of coproduct computation, in H̃1PI and H1PIrespectively:(9)(10)3.4. Cycle-free graphsLet Γ be a cycle-free oriented Feynman graph. In view of Proposition 1, we saythat a covering subgraph γ of Γ is poset-compatible if the contracted graph Γ/γ iscycle-free. It implies that all elements of the associated partition are convex subsets1240003-5April 7, 2012 17:36 WSPC/S1793-7442 251-CM 1240003D. Manchonof the poset V(Γ). The bialgebra H̃CF is given by S(VCF), where VCF is the vectorspace spanned by connected oriented cycle-free Feynman graphs. The coproduct isgiven by:∆(Γ) =∑γ poset-compatible covering subgraph of Γγ ⊗ Γ/γ, (11)and is coassociative due to the fact that the transitive shrinking property of Sec. 3.1still makes sense for poset-compatible covering subgraphs of a cycle-free graph. The(multi-) grading given by the number of internal edges is still relevant, and theassociated Hopf algebra HCF is built up similarly to H in Sec. 3.2, by identifyingthe elements of degree zero with the unit 1.Note that, contrarily to the previous examples, the orientation of the edgesenters here in an essential way. Our favorite coproduct computation takes the fol-lowing form, in H̃CF and HCF respectively:(12)(13)3.5. Cycle-free locally 1PI graphsWe can combine Secs. 3.3 and 3.4: the bialgebra H̃CF1PI of cycle-free locally 1PIgraphs is given by the intersection H̃CF ∩ H̃1PI. This is the free commutative algebraon the vector space spanned by the space VCF1PI of connected cycle-free locally 1PIgraphs, and the coproduct is given by:∆(Γ) =∑γposet-compatible locally 1PIcovering subgraph of Γγ ⊗ Γ/γ, (14)and the associated Hopf algebra HCF1PI is obtained by identifying the residuegraphs with the empty graph 1. Details are left to the reader.4. A Comodule–Coalgebra on the Bialgebra of OrientedCycle-Free Graphs4.1. Another Hopf algebra structure on oriented cycle-free graphsConsider the bialgebra H̃CF = S(VCF) of Sec. 3.4. We keep the same commutativeproduct, but we define another coproduct as follows. For any cycle-free orientedgraph Γ we set:∆c(Γ) =∑V1V2=V(Γ), V2<V1Γ(V1)⊗ Γ(V2). (15)1240003-6April 7, 2012 17:36 WSPC/S1793-7442 251-CM 1240003Hopf Algebras of Oriented GraphsThe inequality V2 < V1 means that for any comparable v1 ∈ V1 and v2 ∈ V2 wehave v2 < v1 in the poset V(Γ). Such a pair of disjoint subsets will be called anadmissible cut. It matches the usual notion of admissible cut when the graph Γ isa rooted tree [8, 7, 11]. Note, however, that the relation < on the set of subsets ofV(Γ) is not transitive. The coproduct is obviously coassociative, as we have:(I ⊗∆c)∆c(Γ) = (∆c ⊗ I)∆c(Γ)=∑V1V2V3=V(Γ), V3<V2<V1Γ(V1)⊗ Γ(V2)⊗ Γ(V3), (16)where the notation V3 < V2 < V1 means V3 < V2, V2 < V1 and V3 < V1. Thiscoproduct is also an algebra morphism. We denote by HCFc the connected gradedHopf algebra given by this coproduct. It is naturally isomorphic to H̃CF as a com-mutative algebra, but the grading is now given by the number of vertices. As anexample, we have:(17)4.2. The comodule–coalgebra structure on HCFcThe coproduct ∆ on the bialgebra H̃CF can also be seen as a left (respectivelyright) coaction Φ : HCFc → H̃CF ⊗HCFc, respectively Ψ : HCFc → HCFc ⊗ H̃CF.Theorem 2. The left coaction map Φ verifies :(Id eHCF ⊗∆c) ◦ Φ = m1,3 ◦ (Φ⊗ Φ) ◦∆c, (18)i.e. the following diagram commutes:HCFc Φ ∆cH̃CF ⊗HCFcI⊗∆cHCFc ⊗HCFcΦ⊗ΦH̃CF ⊗HCFc ⊗ H̃CF ⊗HCFcm1,3 H̃CF ⊗HCFc ⊗HCFcwherem1,3 : H̃CF ⊗HCFc ⊗ H̃CF ⊗HCFc → H̃CF ⊗HCFc ⊗HCFca⊗ b⊗ c⊗ d → ac⊗ b⊗ d.In other words, HCFc is a H̃CF-comodule coalgebra, i.e. a coalgebra in the categoryof H̃CF-comodules.1240003-7April 7, 2012 17:36 WSPC/S1793-7442 251-CM 1240003D. ManchonProof. This result is a direct generalization of Theorem 8 in [2] and is proved ina similar way: the verification is immediate for the empty graph. We have for anynonempty graph:(Id eHCF ⊗∆c) ◦ Φ(Γ)= (Id eHCF ⊗∆c)∑γ poset-compatible coveringsubgraph of Γγ ⊗ Γ/γ=∑γ poset-compatible coveringsubgraph of Γ∑U1U2=V(Γ/γ), U2<U1γ ⊗ (Γ/γ)(U1)⊗ (Γ/γ)(U2).On the other hand, we compute:m1,3 ◦ (Φ⊗ Φ) ◦∆c(Γ)= m1,3 ◦ (Φ⊗ Φ) ∑V1V2=V(Γ), V2<V1Γ(V1)⊗ Γ(V2)= m1,3∑V1V2=V(Γ), V2<V1∑γ′ poset-compatible coveringsubgraph of Γ(V1)∑γ′′ poset-compatible coveringsubgraph of Γ(V2)γ′ ⊗ Γ(V1)/γ′ ⊗ γ′′ ⊗ Γ(V2)/γ′′=∑V1V2=V(Γ), V2<V1∑γ′ poset-compatible coveringsubgraph of Γ(V1)∑γ′′ poset-compatible coveringsubgraph of Γ(V2)γ′γ′′ ⊗ Γ(V1)/γ′ ⊗ Γ(V2)/γ′′=∑V1V2=V(Γ), V2<V1∑γ poset-compatible covering subgraph of Γwithout any internal edge between V1 and V2γ ⊗ Γ(V1)/γ ∩ Γ(V1)⊗ Γ(V2)/γ ∩ Γ(V2)=∑γ poset-compatible coveringsubgraph of Γ∑U1U2=V(Γ/γ), U2<U1γ ⊗ (Γ/γ)(U1)⊗ (Γ/γ)(U2),which proves the theorem.1240003-8April 7, 2012 17:36 WSPC/S1793-7442 251-CM 1240003Hopf Algebras of Oriented Graphs5. Discarding External EdgesThe whole discussion can be carried out from the beginning, dealing with graphswith only internal edges, leading to a bialgebra K̃ and a Hopf algebra K, as wellas to their variants K̃1PI, K̃CF, K̃CF1PI, K1PI, KCF and KCF1PI. The definitionof a subgraph remains the same except that we discard the external edges whichcould appear. As an example we compute on our favorite example the analog of thecoproduct ∆ of Secs. 3.1 and 3.2 respectively in this new framework:,(19).(20)The computations for the cycle-free variants are the same except that we discardthe last term. Locally 1PI variants are straightforward and left to the reader. Theadmissible cut coproduct of Sec. 4.1 reads on the “triangle” graph:(21)The coproduct ∆ of K̃CF coincides on rooted forests with the coproduct of thebialgebra H̃ of [2], and the coproduct ∆c coincides on rooted forests with theConnes–Kreimer coproduct ∆CK. The main difference here is that there are no pre-Lie structures [10] associated with these Hopf algebras of Feynman graphs, whichare not right-sided in the sense of [9].It would be interesting to develop objects similar to B-series such that “compo-sition” and “substitution” of these objects are reflected by this construction, thusgeneralizing [3] and [2].AcknowledgmentsResearch partly supported by CNRS, GDR “Renormalisation”. I particularly thankChristian Brouder and Fabien Vignes-Tourneret for useful remarks.References1. Ch. Brouder, Runge–Kutta methods and renormalization, Euro. Phys. J. C 12 (2000)512–534.2. D. Calaque, K. Ebrahimi-Fard and D. Manchon, Two interacting Hopf algebras oftrees, Adv. Appl. Math. 47 (2011) 282–308.3. Ph. Chartier, E. Hairer and G. Vilmart, Numerical integrators based on modifieddifferential equations, Math. Comp. 76 (2007) 1941–1953.4. A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutativegeometry, Commun. Math. Phys. 199 (1998) 203–242.1240003-9April 7, 2012 17:36 WSPC/S1793-7442 251-CM 1240003D. Manchon5. A. Connes and D. Kreimer, Renormalization in quantum field theory and theRiemann–Hilbert problem I: The Hopf algebra structure of graphs and the maintheorem, Commun. Math. Phys. 210 (2000) 249–273.6. A. Connes and D. Kreimer, Renormalization in quantum field theory and theRiemann–Hilbert problem. II. The β-function, diffeomorphisms and the renormal-ization group, Commun. Math. Phys. 216 (2001) 215–241.7. L. Foissy, Les algèbres de Hopf des arbres enracinés décorés I, II, Bull. Sci. Math. 126(2002) 193–239, 249–288.8. D. Kreimer, On the Hopf algebra structure of perturbative quantum field theories,Adv. Theor. Math. Phys. 2 (1998) 303–334.9. J.-L. Loday and M. Ronco, Combinatorial Hopf algebras, in Quantum of Maths, ClayMath. Proc., Vol. 11 (Amer. Math. Soc., 2010), pp. 347–383.10. D. Manchon and A. Säıdi, Lois pré-Lie en interaction, Comm. Alg. 39 (2011) 3662–3680.11. A. Murua, The Hopf algebra of rooted trees, free Lie algebras, and Lie series, Found.Comput. Math. 6 (2006) 387–426.1240003-10

On bialgebras and Hopf algebras of oriented graphs

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