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Pathlike Co/Bialgebras and Their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics

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Pathlike Co/Bialgebras and Their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics PATHLIKE CO/BIALGEBRAS AND THEIR ANTIPODES WITH APPLICATIONS TO BI- AND HOPF ALGEBRAS APPEARING IN TOPOLOGY, NUMBER THEORY AND PHYSICS. RALPH M. KAUFMANN AND YANG MO Dedicated to Professor Dirk Kreimer on the occasion of his 60th birthday Abstract. We develop an algebraic theory of flavored and pathlike co-, bi- and Hopf algebras. This is the right framework in which to discuss antipodes for bialgebras naturally appearing in topology, number theory and physics. In particular, we can precisely give conditions the invert- ibility of characters needed for renormalization in the formulation of Connes and Kreimer and in the canonical examples these conditions are met. To construct the antipodes directly, we discuss formal localization constructions and quantum deformations. These allow to construct and naturally explain the appearance Brown style coactions. Using previous results, we can tie all the relevant coalgebras to categorical constructions and the bialgebra structures to Feynman categories. Introduction Although the use of Hopf algebras has a long history, the seminal pa- per [CK98] led to a turbocharged development for their use which has pene- trated deeply into mathematical physics, number theory and also topology, their original realm|see [AFS08] for the early beginnings. The important realization in [CK98, CK00, CK01] was that the renormalization procedure for quantum field theory can be based on a character theory for Hopf al- gebras via a so-called Birkhoff factorization and convolution inverses. The relevant Hopf algebras are those of trees with a coproduct given by a sum over so{called admissible cuts |with the factors of the coproduct being the arXiv:2104.08895v1 [math.QA] 18 Apr 2021 left over stump and the collection of cut off branches| and Hopf algebras of graphs in which the factors of the summands of the coproduct are given by a subgraph and the quotient graph. A planar version of the tree for- malism was pioneered in [Foi02]. The appearance in number theory of this type of coproduct goes back to [Gon05] and was developed further and ap- plied with great success in [Bro15]. In all these cases, the product structure is free. A precursor to these considerations can be identified in [Bau81]. 2010 Mathematics Subject Classification. 18A30, 18A40. Key words and phrases. Feynman Category, bialgebra, Hopf algebra. 1 2 RALPH M. KAUFMANN AND YANG MO In [GCKT20a, GCKT20b], we gave the details to prove the realization an- nounced in [KW17] that all these structures stem from a categorical frame- work where the coproduct corresponds to deconcatenation of morphisms. Such coproduct structures can be traced back to [Ler75] and appear in com- binatorics [JR79]. In [KW17], we developed a theory of so{called Feynman categories, which are essentially a special type of monoidal category and could connect the product structure to the monoidal stucture on the level of morphisms. Special cases are related to operads, the simplicial category and categories whose morphisms are classified by graphs. In [GCKT20b], we could show that the monoidal product in these categories is compatible with the deconcatenation co-product thus yielding bialgebras. This general- izes the Connes{Kreimer tree examples and those of Baues and Goncharov, which have operadic origin [GCKT20a] and comprises the Connes{Kreimer graph examples. The co- and bialgebras which have a categorical interpre- tation also include path coalgebras and incidence coalgebras, see §2. There are two versions of the story, one is symmetric and yields cocommutative bialgebras and the second is non{symmetric and in general yields to non{ cocommutative bialgebras, such as those from planar structures. In this paper, we address the question of constructing antipodes in order to make the bialgebras into Hopf algebras. In [GCKT20a, GCKT20b] this was done by taking a quotient which made the bialgebras connected. This quotient is taken implicitly in the examples discussed above, see [GCKT20a, §1]. The questions that we can now answer is: What structures precisely exist already on the bialgebra level, and what is their role? To this end, we give an algebraic characterization of the bi-algberas in terms of what we call flavored and pathlike coalgebras, which capture and generalize the essential features of categorical coalgberas, especially path algebras for quivers. This allows us effective criteria for the existence of antipodes and convolution invertibility of characters. We also work over an arbitrary ground ring. The origins for the theory of flavored coalgebras can be traced back to [TW74] and with hindsight, several of the quotient construction are foreshadowed in [Sch87] for the special case of incidence bialgebras.1 The existence of antipodes is usually proven via some sort of connectivity of a filtration [Tak71,Qui67]. The problem is that the bialgebras are usually not connected for the standard filtrations, which is why quotients are taken. Concretely, after reviewing the basic structures the generality we need in §1, and providing an extended list of examples, to illustrate and motivate the constructions, in §2, we introduce the concept of a QT–filtration in §3, which generalizes the commonly used filtrations. This is what is needed to make Takeuchi's argument work and results in: Theorem 3.28 Consider a coalgebra C with counit and an algebra A with 8 1 unit η. Given a QT{sequence tFiui“0 on C, such that ExtRpC{F0C; Aq “ 1We thank D. Kreimer for pointing this out PATHLIKE CO/BIALGEBRAS AND THEIR ANTIPODES 3 0, for any element f of HompC; Aq; f is ‹-invertible if and only if the restriction f|F0C is ‹-invertible in HompF0C; Aq. A special type of QT–filtration, which we call the bivariate Quillen filtra- tion (see §4), arises by considering skew primitive elements. These are the essential actors for so{called flavored coalgebras and are key to understand- ing the structures, before taking quotients. In the applications a slightly stronger condition is satisfied for the level 0 part of the filtration, which leads to the notion of pathlike co/bialgebras. Theorem 4.16 Let C be pathlike coalgebra and A algebra and f P HompC; Aq. Q If ExtpC{F¯0C ;Aq “ 0, then f has an ‹-inverse in the convolution alge- bra HompC; Aq if and only if for every grouplike element g, fpgq has an inverse as ring element in A. In particular every grouplike central character is ‹-invertible. These results are then directly applicable to renormalization via Rota- Baxter algebras and Birkhoff decompositions, which are briefly reviewed in §5. A closer inspection of this framework yields two paths of actions. The first is to realize that the bialgebras need not necessarily be Hopf algebras, but only the characters need to be ‹{invertible. Thus restricting the char- acters by mandating certain properties on them or the target algebra will ensure their invertibility in the pathlike case. The second is to formally invert the grouplike elements. Combining the two approaches yields several other universal constructions, through which the characters with special properties factor. These are natural construction spelled out in §5 which also naturally lead to quantum deformations of the algebras. This gener- alizes and explains similar constructions in [GCKT20a]. it also gives an astonishing connection to Brown's coactions [Bro15] and allows to general- ize them to the full framework, see §4.4.1 and §5.2.2. With hindsight, the special role and treatment of ζmp2q is precisely the difference between the two approaches. Setting it to 0 implements the quotient and keeping is as an invertible central elements defines the q{deformation. We then apply the general framework to the case of pathlike bialgebras arising from Feynman categories in §6, giving the details for the simplicial setup of Goncharov and Baues |including Joyal duality|, the Connes{ Kreimer tree bialgebras (with leaves, planar and non{planar), their gen- eralization to operads and finally graphical Feynman categories that are relateted to Kreimer's core Hopf algebra. To be more self-contained, these are given in a new set-theoretical presentation in an appendix. We close with technical conditions on Feynman categories, which are met in all the exam- ples and guarantee that the bialgebras associated to the Feynman categories are pathlike. Theorem 6.15 The bialgebra BisopFq, is pathlike if F is generated by iso- morphisms and a set of generators which are essentially skew primitive or essentially indecomposable. In case that there are homogenous relations, the coalgebra will be graded. 4 RALPH M. KAUFMANN AND YANG MO For a non{symmetric Feynman category BpFq is pathlike, and F is gen- erated by identities and skew primitives, that is indecomposables, which is equivalent to a set of generators which are skew primitive. In case that there are homogenous relations, the coalgebra will be graded. It is necessary that V is discrete. Here F and V are part of the data for a Feynman category, see §2.6. Acknowledgments. This paper is the outgrowth of a long conversation with Dirk Kreimer whom we would like to congratulate and profoundly thank for his interest and support. Much of the impetus for this work has come through conversations with him and his group. We wish to thank the Humboldt University, the KMPB, and the Humboldt Foundation for making these visits possible. We furthermore would like to thank F. Brown, K. Yeats and C. Berger for helpful discussions and the organizers of the conference Algebraic Structures in Perturbative Quantum Field Theory at the IHES, in November 2020. The wonderful interactions that took place there led to part of the results of this paper. 1. Preliminaries on Hopf algebra We collect the relevant results on Hopf algebras to fix a common notation and nomenclature | see e,g, [Car07, MP12] . Throughout this paper, we work over a commutative unital ground ring R (and k be a field).
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