Higher Homotopy Hopf Algebras Found: A Ten Year Retrospective
aa r X i v : . [ m a t h . A T ] D ec HIGHER HOMOTPY HOPF ALGEBRA FOUND: A TEN YEARRETROSPECTIVE
RONALD UMBLE Abstract.
The search for higher homotopy Hopf algebras (known today as A ∞ -bialgebras) began in 1996 during a conference at Vassar College honoringJim Stasheff in the year of his 60th birthday. In a talk entitled ”In Search ofHigher Homotopy Hopf Algebras”, I indicated that a DG Hopf algebra could bethought of as some (unknown) higher homotopy structure with trivial higherorder structure and deformed using a graded version of Gerstenhaber andSchack’s bialgebra deformation theory. In retrospect, the bi(co)module struc-ture encoded in Gerstenhaber and Schack’s differential defining deformationcohomology detects some (but not all) of the A ∞ -bialgebra structure relations.Nevertheless, this motivated the discovery of A ∞ -bialgebras by S. Saneblidzeand myself in 2005. To Murray Gerstenhaber and Jim Stasheff Introduction
In a preprint dated June 14, 2004, Samson Saneblidze and I announced the def-inition of A ∞ -bialgebras [SU05], marking approximately six years of collaborationthat continues to this day. Unknown to us at the time, A ∞ -bialgebras are ubiqui-tous and fundamentally important. Indeed, over a field F , the bialgebra structureon the singular chains of a loop space Ω X pulls back along a quasi-isomorphism f : H ∗ (Ω X ; F ) → C ∗ (Ω X ) to an A ∞ -bialgebra structure on homology in a canon-ical way [SU08b].Many have tried unsuccessfully to define A ∞ -bialgebras. The illusive ingredientin the definition turned out to be an explicit diagonal ∆ P on the permutahedra P = ⊔ n ≥ P n , the first construction of which was given by S. Saneblidze and myselfin [SU04]. This paper is an account of the historical events leading up to thediscovery of A ∞ -bialgebras and the truly remarkable role played by ∆ P in thisregard. Although the ideas and examples presented here are quite simple, theyrepresent and motivate general theory in [SU04], [SU05], [SU08a], and [SU08b].Through their work in the theory of PROPs and the related area of infinity Liebialgebras, many authors have contributed indirectly to this work, most notablyM. Chas and D. Sullivan [CS04], V. Godin [God08], J-L. Loday [Lod06], M. Markl[Mar06], T. Pirashvili [Pir02], B. Shoikhet [Sho03], and B. Vallette [Val04]; forextensive bibliographies see [Sul07] and [Mar06]. Date : January 16, 2009.1991
Mathematics Subject Classification.
Primary 55P35, 55P99; Secondary 52B05.
Key words and phrases.
Hopf algebra, A ∞ -bialgebra, operad, matrad, associahedron,permutahedron. This research funded in part by a Millersville University faculty research grant. Several new results spin off of this discussion and are included here: Example1 in Section 3 introduces the first example of a bialgebra H endowed with an A ∞ -algebra structure that is compatible with the comultiplication. Example 2 inSection 4, introduces the first example of a “non-operadic” A ∞ -bialgebra with anon-trivial operation ω , : H ⊗ → H ⊗ . And in Section 5 we prove Theorem1: Given a DG bialgebra ( H, d, µ, ∆) and a Gerstenhaber-Schack 2 -cocycle µ n ∈ Hom − n ( H ⊗ n , H ) , n ≥ , let H = ( H [[ t ]] , d, µ, ∆) . Then ( H [[ t ]] , d, µ, ∆ , tµ n ) isa linear deformation of H as a simple Hopf A ( n )- algebra. The Historical Context
Two papers with far-reaching consequences in algebra and topology appearedin 1963. In [Ger63] Murray Gerstenhaber introduced the deformation theory ofassociative algebras and in [Sta63] Jim Stasheff introduced the notion of an A ( n )-algebra. Although the notion of what we now call a “non-Σ operad” appears inboth papers, this connection went unnoticed until after Jim’s visit to the Universityof Pennsylvania in 1983. Today, Gerstenhaber’s deformation theory and Stasheff’shigher homotopy algebras are fundamental tools in algebra, topology and physics.An extensive bibliography of applications appears in [MSS02].By 1990, techniques from deformation theory and higher homotopy structureshad been applied by many authors, myself included [Umb89], [LU92], to classifyrational homotopy types with a fixed cohomology algebra. And it seemed rea-sonable to expect that rational homotopy types with a fixed Pontryagin algebra H ∗ (Ω X ; Q ) could be classified in a similar way. Presumably, such a theory wouldinvolve deformations of DG bialgebras (DGBs) as some higher homotopy structurewith compatible A ∞ -algebra and A ∞ -coalgebra substructures, but the notion ofcompatibility was not immediately clear and an appropriate line of attack seemedillusive. But one thing was clear: If we apply a graded version of Gerstenhaberand Schack’s (G-S) deformation theory [GS92], [LM91], [LM96], [Umb96] and de-form a DGB H as some (unknown) higher homotopy structure, new operations ω j,i : H ⊗ i → H ⊗ j appear and their interactions with the deformed bialgebra oper-ations are partially detected by the differentials. While this is but one small pieceof a very large puzzle, it gave us a clue.During the conference honoring Jim Stasheff in the year of his 60th birthday,held at Vassar College in June 1996, I discussed this particular clue in a talk entitled“ In Search of Higher Homotopy Hopf Algebras ” ([McC98] p. xii). Although G-Sdeformations of DGBs are less constrained than the A ∞ -bialgebras known today,they motivated the definition announced eight years later.Following the Vassar conference, forward progress halted. Questions of struc-tural compatibility seemed mysterious and inaccessible. Then in 1998, Jim Stasheffran across some related work by S. Saneblidze [San96], of the A. Razmadze Mathe-matical Institute in Tbilisi, and suggested that I get in touch with him. Thus beganour long and fruitful collaboration. Over the months that followed, Saneblidze ap-plied techniques of homological perturbation theory to solve the aforementionedclassification problem [San99], but the higher order structure in the limit is implicitand the structure relations are inaccessible. In retrospect, this is not surprisingas explicit structure relations require explicit combinatorial diagonals ∆ P on thepermutahedra P = ⊔ n ≥ P n and ∆ K on the associahedra K = ⊔ n ≥ K n . But suchdiagonals are difficult to construct and were unknown to us at the time. Indeed,
IGHER HOMOTPY HOPF ALGEBRA FOUND: A TEN YEAR RETROSPECTIVE 3 one defines the tensor product of A ∞ -algebras in terms of ∆ K , and the search fora construction of ∆ K had remained a long-standing problem in the theory of op-erads. We announced our construction of ∆ K in 2000 [SU00]; our construction of∆ P followed a year or two later (see [SU04]).3. Two Important Roles For ∆ P The diagonal ∆ P plays two fundamentally important roles in the theory of A ∞ -bialgebras. First, one builds the structure relations from components of free exten-sions of initial maps as higher (co)derivations with respect to ∆ P , and second, ∆ P specifies exactly which of these components to use.To appreciate the first of these roles, recall the following definition given byStasheff in his seminal work on A ∞ -algebras in 1963 [Sta63]: Let A be a gradedmodule, let { µ i ∈ Hom i − (cid:0) A ⊗ i , A (cid:1) } n ≥ be an arbitrary family of maps, and let d be the cofree extension of Σ µ i as a coderivation of the tensor coalgebra T c A (with a shift in dimension). Then ( A, µ i ) is an A ∞ -algebra if d = 0; when thisoccurs, the universal complex ( T c A, d ) is called the tilde-bar construction and thestructure relations in A are the homogeneous components of d = 0. Similarly,let H be a graded module and let { ω j,i ∈ Hom − i − j (cid:0) H ⊗ i , H ⊗ j (cid:1) } i,j ≥ , be anarbitrary family of maps. When (cid:0) H, ω j,i (cid:1) is an A ∞ -bialgebra, there is an associateduniversal complex ( Bd ( H ) , ω ) , called the biderivative of ω = Σ ω j,i (constructedby Saneblidze and myself in [SU05]) whose differential ω is the sum of various(co)free extensions of various subfamilies of (cid:8) ω j,i (cid:9) as ∆ P -(co)derivations. Andindeed, the structure relations in H are the homogeneous components of ω = 0 . To demonstrate the spirit of this, consider a free graded module H of finite typeand an (arbitrary) map ω = µ + µ +∆ with components µ : H ⊗ → H, µ : H ⊗ → H, and ∆ : H → H ⊗ . Extend ∆ as an algebra map ∆ : T a H → T a (cid:0) H ⊗ (cid:1) , extend µ + µ as a coderivation d : T c H → T c H , and extend µ as a coalgebramap µ : T c (cid:0) H ⊗ (cid:1) → T c H. Finally, note that f = ( µ ⊗ µ and g = (1 ⊗ µ ) µ arecoalgebra maps, and extend µ as an ( f, g )-coderivation µ : T c (cid:0) H ⊗ (cid:1) → T c H. Then ω = d + µ + µ + ∆ ∈ L p,q,r,s ≥ Hom (cid:16) ( H ⊗ p ) ⊗ q , ( H ⊗ r ) ⊗ s (cid:17) . Let σ r,s : ( H ⊗ r ) ⊗ s → ( H ⊗ s ) ⊗ r denote the canonical permutation of tensor factorsand define a composition product ⊚ for homogeneous components A and B of ω = d + µ + µ + ∆ by A ⊚ B = (cid:26) A ◦ σ r,s ◦ B, if defined0 , otherwise;when A ⊚ B is defined, ( H ⊗ r ) ⊗ s is the target of B , and ( H ⊗ s ) ⊗ r is the source of A. Then (cid:0)
H, µ, µ , ∆ (cid:1) is an A ∞ -infinity bialgebra if ω ⊚ ω = 0 . Note that ∆ µ and( µ ⊗ µ ) σ , (∆ ⊗ ∆) are the homogeneous components of ω ⊚ ω in Hom (cid:0) H ⊗ , H ⊗ (cid:1) ;consequently, ω ⊚ ω = 0 implies the Hopf relation∆ µ = ( µ ⊗ µ ) σ , (∆ ⊗ ∆) . Now if (
H, µ, ∆) is a bialgebra, the operations µ t , µ t , and ∆ t in a G-S deformationof H satisfy(3.1) ∆ t µ t = (cid:2) µ t ( µ t ⊗ ⊗ µ t + µ t ⊗ µ t (1 ⊗ µ t ) (cid:3) σ , ∆ ⊗ t RONALD UMBLE and the homogeneous components of ω ⊚ ω = 0 in Hom (cid:0) H ⊗ , H ⊗ (cid:1) are exactlythose in (3.1). So this is encouraging.Recall that the permutahedron P is a point 0 and P is an interval 01. In thesecases ∆ P agrees with the Alexander-Whitney diagonal on the simplex:∆ P (0) = 0 ⊗ and ∆ P (01) = 0 ⊗
01 + 01 ⊗ . If X is an n -dimensional cellular complex, let C ∗ ( X ) denote the cellular chains of X. When X has a single top dimensional cell, we denote it by e n . An A ∞ -algebrastructure { µ n } n ≥ on H is encoded operadically by a family of chain maps { ξ : C ∗ ( P n − ) → Hom ( H ⊗ n , H ) } , which factor through the map θ : C ∗ ( P n − ) → C ∗ ( K n ) induced by cellular projec-tion P n − → K n given by A. Tonks [Ton97] and satisfy ξ (cid:0) e n − (cid:1) = µ n . The factthat ( ξ ⊗ ξ ) ∆ P (cid:0) e (cid:1) = µ ⊗ µ and( ξ ⊗ ξ ) ∆ P (cid:0) e (cid:1) = µ ( µ ⊗ ⊗ µ + µ ⊗ µ (1 ⊗ µ )are components of µ and µ suggests that we extend a given µ n as a higher coderiva-tion µ n : T c ( H ⊗ n ) → T c H with respect to ∆ P . Indeed, an A ∞ -bialgebra of theform ( H, ∆ , µ n ) n ≥ is defined in terms of the usual A ∞ -algebra relations togetherwith the relations(3.2) ∆ µ n = (cid:2) ( ξ ⊗ ξ ) ∆ P (cid:0) e n − (cid:1)(cid:3) σ ,n ∆ ⊗ n , which define the compatibility of µ n and ∆.Structure relations in more general A ∞ -bialgebras of the form( H, ∆ m , µ n ) m,n ≥ are similar in spirit and formulated in [Umb08]. Special casesof the form ( H, ∆ , ∆ n , µ ) with a single ∆ n were studied by H.J. Baues in the case n = 3 [Bau98] and by A. Berciano and myself in cases n ≥ p is an odd prime and n ≥
3, these particular structures appear as tensor factors ofthe mod p homology of an Eilenberg-Mac Lane space of type K ( Z , n ).Dually, A ∞ -bialgebras ( H, ∆ , µ, µ n ) with a single µ n have a strictly associativemultiplication µ and ξ ⊗ ξ acts exclusively on the primitive terms of ∆ P for lacunaryreasons, in which case relation (3.2) reduces to(3.3) ∆ µ n = ( f n ⊗ µ n + µ n ⊗ f n ) σ ,n ∆ ⊗ n , where f n = µ ( µ ⊗ · · · (cid:0) µ ⊗ ⊗ n − (cid:1) . The first example of this particular structurenow follows. Example 1.
Let H be the primitively generated bialgebra Λ ( x, y ) with | x | = 1 , | y | =2 , and µ n (cid:0) x i y p | · · · | x i n y p n (cid:1) = (cid:26) y p + ··· + p n +1 , i · · · i n = 1 and p k ≥ , otherwise . One can easily check that H is an A ∞ -algebra, and a straightforward calculationtogether with the identity (cid:18) p + · · · + p n + 1 i (cid:19) = X s + ··· + s n = i − (cid:18) p s (cid:19) · · · (cid:18) p n s n (cid:19) + X s + ··· + s n = i (cid:18) p s (cid:19) · · · (cid:18) p n s n (cid:19) verifies relation (3.3). IGHER HOMOTPY HOPF ALGEBRA FOUND: A TEN YEAR RETROSPECTIVE 5
The second important role played by ∆ P is evident in A ∞ -bialgebras in which ω n,m is non-trivial for some m, n > . Just as an A ∞ -algebra structure on H isencoded operadically, an A ∞ -bialgebra structure on H is encoded matradically bya family of chain maps (cid:8) ε : C ∗ ( KK n,m ) → Hom − m − n ( H ⊗ m , H ⊗ n ) (cid:9) over contractible polytopes KK = ⊔ m,n ≥ KK n,m , called matrahedra , with singletop dimensional cells e m + n − such that ε (cid:0) e m + n − (cid:1) = ω n,m . Note that KK n, = KK ,n is the associahedron K n [SU08a]. Let M = { M n,m = Hom ( H ⊗ m , H ⊗ n ) } and let Θ = { θ nm = ω n,m } . The A ∞ -bialgebra matrad H ∞ is realized by C ∗ ( KK )and is a proper submodule of the free PROP M generated by Θ . The matrad product γ on H ∞ is defined in terms of ∆ P , and a monomial α in the free PROP M is acomponent of a structure relation if and only if α ∈ H ∞ .More precisely, in [Mar06] M. Markl defined the submodule S of special elements in PROP M whose additive generators are monomials α expressed as “elementaryfractions”(3.4) α = α y p · · · α y q p α qx · · · α qx p in which α qx i and α y j p are additive generators of S and the j th output of α qx i is linkedto the i th input of α y j p (here juxtaposition denotes tensor product). Representing θ nm graphically as a double corolla (see Figure 1), a general decomposable α isrepresented by a connected non-planar graph in which the generators appear inorder from left-to-right (see Figure 2). The matrad H ∞ is a proper submodule of S and the matrad product γ agrees with the restriction of Markl’s fraction productto H ∞ . θ nm = nm Figure 1.The diagonal ∆ P acts as a filter and admits certain elementary fractions asadditive generators of H ∞ . In dimensions 0 and 1 , the diagonal ∆ P is expressedgraphically in terms of up-rooted planar rooted trees (with levels) by∆ P ( ) = ⊗ and ∆ P ( ) = ⊗ + ⊗ . Define ∆ (0) P = 1; for each k ≥ , define ∆ ( k ) P = (cid:0) ∆ P ⊗ ⊗ k − (cid:1) ∆ ( k − P and view eachcomponent of ∆ ( k ) P ( θ q ) as a ( q − P q − ) × k +1 , andsimilarly for ∆ ( k ) P ( θ q ).The elements θ , θ , and θ generate two elementary fractions in M , each ofdimension zero, namely, α = and α = . RONALD UMBLE Define ∂ (cid:0) θ (cid:1) = α + α , and label the edge and vertices of the interval KK , by θ , α and α , respectively. Continuing inductively, the elements θ , θ , θ , θ ,α , and α generate 18 fractions in M , – one in dimension 2, nine in dimension 1and eight in dimension 0. Of these, 14 label the edges and vertices of the heptagon KK , . Since the generator θ must label the 2-face, we wish to discard the 2-dimensional decomposable e =and the appropriate components of its boundary. Note that e is a square whoseboundary is the union of four edges(3.5)Of the five fractions pictured above, only the first two in (3.5) have numeratorsand denominators that are components of ∆ ( k ) P ( P ) (numerators are components of∆ (1) P ( θ ) and denominators are exactly ∆ (2) P ( θ )). Our selection rule admits onlythese two particular fractions, leaving seven 1-dimensional generators to label theedges of KK , (see Figure 2). Now linearly extend the boundary map ∂ to theseven admissible 1-dimensional generators and compute the seven 0-dimensionalgenerators labeling the vertices of KK , . Since the 0-dimensional generatoris not among them, we discard it.Figure 2: The matrahedron KK , . Subtleties notwithstanding, this process continues indefinitely and produces H ∞ as an explicit free resolution of the bialgebra matrad H = (cid:10) θ , θ , θ (cid:11) in the cat-egory of matrads. We note that in [Mar06], M. Markl makes arbitrary choices(independent of our selection rule) to construct the polytopes B nm = KK n,m for IGHER HOMOTPY HOPF ALGEBRA FOUND: A TEN YEAR RETROSPECTIVE 7 m + n ≤
6. In this range, it is enough to consider components of the diagonal ∆ K on the associahedra.We conclude this section with a brief review of our diagonals ∆ P and ∆ K (upto sign); for details see [SU04]. Alternative constructions of ∆ K were subsequentlygiven by Markl and Shnider [MS06] and J.L. Loday [Lod07] (in this volume). Let n = { , , . . . , n } , n ≥ . A matrix E with entries from { } ∪ n is a step matrix if: • Each element of n appears as an entry of E exactly once. • The elements of n in each row and column of E form an increasing con-tiguous block. • Each diagonal parallel to the main diagonal of E contains exactly one ele-ment of n .Right-shift and down-shift matrix transformations, which include the identity(a trivial shift), act on step matrices and produce derived matrices. Let a = A | A | · · · | A p and b = B q | B q − | · · · | B be partitions of n . The pair a × b is an( p, q ) -complementary pair (CP) if B i and A j are the rows and columns of a q × p derived matrix. Since faces of P n are indexed by partitions of n, and CPs are inone-to-one correspondence with derived matrices, each CP is identified with someproduct face of P n × P n . Definition 1.
Define ∆ P ( e ) = e ⊗ e . Inductively, having defined ∆ P on C ∗ ( P k +1 ) for all ≤ k ≤ n − , define ∆ P on C n ( P n +1 ) by ∆ P ( e n ) = X ( p,q ) -CPs u × vp + q = n +2 ± u ⊗ v, and extend multiplicatively to all of C ∗ ( P n +1 ) . The diagonal ∆ p induces a diagonal ∆ K on C ∗ ( K ). Recall that faces of P n incodimension k are indexed by planar rooted trees with n + 1 leaves and k + 1 levels(PLTs), and forgetting levels defines the cellular projection θ : P n → K n +1 givenby A. Tonks [Ton97]. Thus faces of P n indexed by PLTs with multiple nodes inthe same level degenerate under θ , and corresponding generators lie in the kernelof the induced map θ : C ∗ ( P n ) → C ∗ ( K n +1 ). The diagonal ∆ K is given by∆ K θ = ( θ ⊗ θ )∆ P . Deformations of DG Bialgebras as A ( n ) -Bialgebras The discussion above provides the context to appreciate the extent to which G-Sdeformation theory motivates the notion of an of A ∞ -bialgebra. We describe thismotivation in this section. In retrospect, the bi(co)module structure encoded in theG-S differentials controls some (but not all) of the A ∞ -bialgebra structure relations.For example, all structure relation in A ∞ -bialgebras of the form ( H, d, µ, ∆ , µ n ) arecontrolled except(4.1) n − X i =0 ( − i ( n +1) µ n (cid:0) ⊗ i ⊗ µ n ⊗ ⊗ n − i − (cid:1) = 0 , which measures the interaction of µ n with itself. Nevertheless, such structuresadmit an A ( n )-algebra substructure and their single higher order operation µ n iscompatible with ∆ . Thus we refer to such structures here as
Hopf A ( n ) -algebras. RONALD UMBLE General G-S deformations of DGBs, referred to here as quasi- A ( n ) -bialgebras, are“partial” A ( n )-bialgebras in the sense that all structure relations involving multiplehigher order operations are out of control.4.1. A ( n ) -Algebras and Their Duals. The signs in the following definition weregiven in [SU04] and differ from those given by Stasheff in [Sta63]. We note that ei-ther choice of signs induces an oriented combinatorial structure on the associahedra,and these structures are are equivalent. Let n ∈ N ∪ {∞} . Definition 2. An A ( n )-algebra is a graded module A together with structure maps { µ k ∈ Hom − k (cid:0) A ⊗ k , A (cid:1) } ≤ k In [GS92], M. Gerstenhaber and S. D.Schack defined the cohomology of an ungraded bialgebra by joining the dual coho-mology theories of G. Hochschild [HKR62] and P. Cartier [Car55]. This construc-tion was given independently by A. Lazarev and M. Movshev in [LM91]. The G-Scohomology of H reviewed here is a straight-forward extension to the graded caseand was constructed in [LM96] and [Umb96].Let ( H, d, µ, ∆) be a connected DGB. We assume | d | = 1 , although one couldassume | d | = − i ≥ , the i -fold bicomodule tensor power of H is the H -bicomodule H ⊗ i = ( H ⊗ i , λ i , ρ i ) with left and right coactions given by λ i = (cid:2) µ ( µ ⊗ · · · ( µ ⊗ ⊗ i − ) ⊗ ⊗ i (cid:3) σ ,i ∆ ⊗ i and ρ i = (cid:2) ⊗ i ⊗ µ (1 ⊗ µ ) · · · (1 ⊗ i − ⊗ µ ) (cid:3) σ ,i ∆ ⊗ i .When f : H ⊗ i → H ⊗∗ , there is the composition(1 ⊗ f ) λ i = (cid:2) µ ( µ ⊗ · · · ( µ ⊗ ⊗ i − ) ⊗ f (cid:3) σ ,i ∆ ⊗ i .Dually, for each j ≥ , the j -fold bimodule tensor power of H is the H -bimodule H ⊗ j = ( H ⊗ j , λ j , ρ j ) with left and right actions given by IGHER HOMOTPY HOPF ALGEBRA FOUND: A TEN YEAR RETROSPECTIVE 9 λ j = µ ⊗ j σ j, (cid:2) (∆ ⊗ ⊗ j − ) · · · (∆ ⊗ 1) ∆ ⊗ ⊗ j (cid:3) and ρ j = µ ⊗ j σ j, (cid:2) ⊗ j ⊗ (1 ⊗ j − ⊗ ∆) · · · (1 ⊗ ∆) ∆ (cid:3) .When g : H ⊗∗ → H ⊗ j , there is the composition λ j (1 ⊗ g ) = µ ⊗ j σ j, (cid:2) (∆ ⊗ ⊗ j − ) · · · (∆ ⊗ 1) ∆ ⊗ g (cid:3) .Let k be a field. Extend d, µ and ∆ to k [[ t ]]-linear maps and obtain a k [[ t ]]-DGB H = ( H [[ t ]] , d, µ, ∆) . We wish to deform H as an A ( n )-structure of the form H t = (cid:16) H [[ t ]] , d t = ω , t , µ t = ω , t , ∆ t = ω , t , ω j,it (cid:17) i + j = n +1 where ω j,it = P ∞ k =0 t k ω j,ik ∈ Hom − i − j (cid:16) H ⊗ i , H ⊗ j (cid:17) , ω , = d, ω , = µ, ω , = ∆ , and ω j,i = 0.Deformations of H are controlled by the G-S n - complex, which we now review.For k ≥ , let d ( k ) = k − P i =0 ⊗ i ⊗ d ⊗ ⊗ k − i − ∂ ( k ) = k − P i =0 ( − i ⊗ i ⊗ µ ⊗ ⊗ k − i − δ ( k ) = k − P i =0 ( − i ⊗ i ⊗ ∆ ⊗ ⊗ k − i − . These differentials induce strictly commuting differentials d, ∂, and δ on the tri-graded module { Hom p ( H ⊗ i , H ⊗ j ) } , which act on an element f in tridegree ( p, i, j )by d ( f ) = d ( j ) f − ( − p f d ( i ) ∂ ( f ) = λ j (1 ⊗ f ) − f ∂ ( i ) − ( − i ρ j ( f ⊗ δ ( f ) = (1 ⊗ f ) λ i − δ ( j ) f − ( − j ( f ⊗ ρ i . The submodule of total G-S r -cochains on H is C rGS ( H, H ) = L p + i + j = r +1 Hom p ( H ⊗ i , H ⊗ j )and the total differential D on a cochain f in tridegree ( p, i, j ) is given by D ( f ) = h ( − i + j d + ∂ + ( − i δ i ( f ) ,where the sign coefficients are chosen so that (1) D = 0 , (2) structure relations(ii) and (iii) in Definition 2 hold and (3) the restriction of D to the submodule of r -cochains in degree p = 0 agrees with the total (ungraded) G-S differential. The G-S cohomology of H with coefficients in H is given by H ∗ GS ( H, H ) = H ∗ { C rGS ( H, H ) , D } . Identify Hom p ( H ⊗ i , H ⊗ j ) with the point ( p, i, j ) in R . Then the G-S n - complex is that portion of the G-S complex in the region x ≥ − n and the submodule oftotal r -cochains in the n -complex is C rGS ( H, H ; n ) = L p = r − i − j +1 ≥ − n Hom p ( H ⊗ i , H ⊗ j ) (a 2-cocycle in the 3-complex appears in Figure 3). The G-S n - cohomology of H with coefficients in H is given by H ∗ GS ( H, H ; n ) = H ∗ { C rGS ( H, H ; n ) ; D } .Note that a general 2-cocycle α has a component of tridegree (3 − i − j, i, j ) for each i and j in the range 2 ≤ i + j ≤ n + 1. Thus α has n ( n + 1) / α determine an infinitesimal deformation , i.e., the component ω j,i in tridegree(3 − i − j, i, j ) defines the first order approximation ω j,i + tω j,i of the structure map ω j,it in H t .For simplicity, consider the case n = 3 . Each of the ten homogeneous compo-nents of the deformation equation D ( α ) = 0 produces the infinitesimal form of onestructure relation (see below). In particular, a deformation H t with structure maps { ω ,it } ≤ i ≤ is a simple A (3)-algebra and a deformation H t with structure maps { ω j, t } ≤ j ≤ is a simple A (3)-coalgebra.Figure 3. The 2-cocycle d + µ + ∆ + µ + ω + ∆ . For notational simplicity, let µ t = ω , t , ω t = ω , t and ∆ t = ω , , and considera deformation of ( H, d, µ, ∆) as a “quasi- A (3)-structure.” Then • d t = d + td + t d + · · ·• µ t = µ + tµ + t µ + · · ·• ∆ t = ∆ + t ∆ + t ∆ + · · ·• µ t = tµ + t µ + · · ·• ω t = tω + t ω + · · ·• ∆ t = t ∆ + t ∆ + · · · and d + µ + ∆ + µ + ω + ∆ is a total 2-cocycle (see Figure 3). Equatingcoefficients in D (cid:0) d + µ + ∆ + µ + ω + ∆ (cid:1) = 0 gives IGHER HOMOTPY HOPF ALGEBRA FOUND: A TEN YEAR RETROSPECTIVE 11 . d ( d ) = 0 6 . ∂ (cid:0) µ (cid:1) = 02 . d ( µ ) − ∂ ( d ) = 0 7 . δ (cid:0) ∆ (cid:1) = 03 . d (∆ ) + δ ( d ) = 0 8 . d ( ω ) + ∂ (∆ ) + δ ( µ ) = 04 . d (cid:0) µ (cid:1) + ∂ ( µ ) = 0 9 . ∂ (cid:0) ∆ (cid:1) + δ ( ω ) = 05 . d (cid:0) ∆ (cid:1) − δ (∆ ) = 0 10 . ∂ ( ω ) − δ (cid:0) µ (cid:1) = 0 . Requiring (cid:0) H, d t , µ t , µ t (cid:1) and (cid:0) H, d t , ∆ t , ∆ t (cid:1) to be simple A (3)-(co)algebras tellsus that relations (1) - (7) are linearizations of Stasheff’s strict A (4)-(co)algebrarelations, and relation (8) is the linearization of the Hopf relation relaxed up tohomotopy. Since µ t , ω t and ∆ t have no terms of order zero, relations (9) and (10)are the respective linearizations of new relations (9) and (10) below. Thus we obtainthe following structure relations in H t :1 . d t = 02 . d t µ t = µ t ( d t ⊗ ⊗ d t )3 . ∆ t d t = ( d t ⊗ ⊗ d t ) ∆ t . d t µ t + µ t ( d t ⊗ ⊗ ⊗ d t ⊗ ⊗ ⊗ d t ) = µ t (1 ⊗ µ t ) − µ t ( µ t ⊗ . ( d t ⊗ ⊗ ⊗ d t ⊗ ⊗ ⊗ d t ) ∆ t + ∆ t d t = (∆ t ⊗ 1) ∆ t − (1 ⊗ ∆ t ) ∆ t . µ t ( µ t ⊗ ⊗ − ⊗ µ t ⊗ ⊗ ⊗ µ t ) = µ t (cid:0) µ t ⊗ ⊗ µ t (cid:1) . (∆ t ⊗ ⊗ − ⊗ ∆ t ⊗ ⊗ ⊗ ∆ t ) ∆ t = (cid:0) ∆ t ⊗ ⊗ ∆ t (cid:1) ∆ t . ( d t ⊗ ⊗ d t ) ω t + ω t ( d t ⊗ ⊗ d t ) = ∆ t µ t − ( µ t ⊗ µ t ) σ , (∆ t ⊗ ∆ t )9 . ( µ t ⊗ ω t ) σ , (∆ t ⊗ ∆ t ) − (∆ t ⊗ − ⊗ ∆ t ) ω t − ( ω t ⊗ µ t ) σ , (∆ t ⊗ ∆ t )= ∆ t µ t − µ ⊗ t σ , (cid:2) (∆ t ⊗ 1) ∆ t ⊗ ∆ t + (cid:0) ∆ t ⊗ (1 ⊗ ∆ t ) ∆ t (cid:1)(cid:3) . ( µ t ⊗ µ t ) σ , (∆ t ⊗ ω t ) − ω t ( µ t ⊗ − ⊗ µ t ) − ( µ t ⊗ µ t ) σ , ( ω t ⊗ ∆ t )= (cid:2) µ t ( µ t ⊗ ⊗ µ t + µ t ⊗ µ t (1 ⊗ µ t ) (cid:3) σ , ∆ ⊗ t − ∆ t µ t . By dropping the formal deformation parameter t, we obtain the structure relationsin a quasi-simple A (3)-bialgebra.The first non-operadic example of a general A ∞ -bialgebra appears here as aquasi-simple A (3)-bialgebra and involves a non-trivial operation ω = ω , . Thesix additional relations satisfied by A ∞ -bialgebras of this particular form will beverified in the next section. Example 2. Let H be the primitively generated bialgebra Λ ( x, y ) with | x | = 1 , | y | = 2 , trivial differential, and ω : H ⊗ → H ⊗ given by ω ( a | b ) = x | y + y | x, a | b = y | yx | x, a | b ∈ { x | y, y | x } , otherwise . Then (∆ ⊗ − ⊗ ∆) ω ( y | y ) = (∆ ⊗ − ⊗ ∆) ( x | y + y | x ) = 1 | x | y + 1 | y | x − x | y | − y | x | µ ⊗ ω − ω ⊗ µ ) (1 | | y | y + y | | | y + 1 | y | y | y | y | | 1) =( µ ⊗ ω − ω ⊗ µ ) σ , (∆ ⊗ ∆) ( y | y ); and similar calculations show agreement on x | y and y | x and verifies relation (9). To verify relation (10), note that ω ( µ ⊗ − ⊗ µ )and ( µ ⊗ µ ) σ , (∆ ⊗ ω − ω ⊗ ∆) are supported on the subspace spanned by B = { | y | y, y | y | , | x | y, x | y | , | y | x, y | x | } ,and it is easy to check agreement on B. Finally, note that ( H, µ, ∆ , ω ) can berealized as the linear deformation ( H [[ t ]] , µ, ∆ , tω ) | t =1 . A ∞ -Bialgebras in Perspective Although G-S deformation cohomology motivates the notion of an A ∞ -bialgebra,G-S deformations of DGBs are less constrained than A ∞ -bialgebras and conse-quently fall short of the mark. To indicate of the extent of this shortfall, let usidentify those structure relations that fail to appear via deformation cohomologybut must be verified to assert that Example 2 is an A ∞ -bialgebra.As mentioned above, structure relations in a general A ∞ -bialgebra arise from thehomogeneous components of a square-zero differential on some universal complex.So to begin, let us construct the particular complex that determines the structurerelations in an A ∞ -bialgebra of the form (cid:0) H, d, µ, ∆ , ω , (cid:1) . Given arbitrary maps d = ω , , µ = ω , , ∆ = ω , , and ω , with ω j,i ∈ Hom − i − j (cid:0) H ⊗ i , H ⊗ j (cid:1) , consider ω = P ω j,i . Freely extend • d as a linear map ( H ⊗ p ) ⊗ q → ( H ⊗ p ) ⊗ q for each p, q ≥ , • d + ∆ as a derivation of T a H , • d + µ as a coderivation of T c H , • ∆ + ω , as an algebra map T a H → T a (cid:0) H ⊗ (cid:1) , and • µ + ω , as a coalgebra map T c (cid:0) H ⊗ (cid:1) → T c H .The biderivative ω is the sum of these free extensions.Note that in this restricted setting, relation (10) in Definition 2 reduces to( µ ⊗ µ ) ⊚ (cid:0) ∆ ⊗ ω , − ω , ⊗ ∆ (cid:1) = ω , ⊚ ( µ ⊗ − ⊗ µ ).Factors µ ⊗ ⊗ µ are components of d + µ ; factors ∆ ⊗ ω , and ω , ⊗ ∆ arecomponents of ∆ + ω , ; and the factor µ ⊗ µ is a component of µ + ω , . •• ❅❅❅❅❅■ H ⊗ j H ⊗ i ω j,i Figure 4. The initial map ω j,i .To picture this, identify the isomorphic modules ( H ⊗ p ) ⊗ q ≈ ( H ⊗ q ) ⊗ p with thepoint ( p, q ) ∈ N and picture the initial map ω j,i : H ⊗ i → H ⊗ j as a “transgressive” IGHER HOMOTPY HOPF ALGEBRA FOUND: A TEN YEAR RETROSPECTIVE 13 arrow from ( i, 1) to (1 , j ) (see Figure 4). Components of the various free extensionsabove are pictured as arrows that initiate or terminate on the axes. For example,the vertical arrow ∆ ⊗ ∆ , the short left-leaning arrow ∆ ⊗ ω , − ω , ⊗ ∆ and the longleft-leaning arrow ω , ⊗ ω , in Figure 5 represent components of ∆ + ω , . Sincewe are only interested in transgressive quadratic ⊚ -compositions, it is sufficientto consider the components of ω pictured in Figure 5. Quadratic compositionsalong the x -axis correspond to relations (1), (2), (4) and (6) in Definition 2; thosein the square with its diagonal correspond to relation (8); those in the verticalparallelogram correspond to relation (9); and those in the horizontal parallelogramcorrespond to relation (10).Figure 5. Components of ω when ω = d + µ + ∆ + ω , . The following six additional relations are not detected by deformation cohomologybecause the differentials only detect the interactions between ω and (deformationsof) d, µ, and ∆ induced by the underlying bi(co)module structure:11. ( µ ⊗ ω − ω ⊗ µ ) σ , (∆ ⊗ ω − ω ⊗ ∆) = 0;12. ( µ ⊗ µ ) σ , ( ω ⊗ ω ) = 0;13. ( ω ⊗ ω ) σ , (∆ ⊗ ∆) = 0;14. ( µ ⊗ ω − ω ⊗ µ ) σ , ( ω ⊗ ω ) = 0;15. ( ω ⊗ ω ) σ , (∆ ⊗ ω − ω ⊗ ∆) = 0;16. ( ω ⊗ ω ) σ , ( ω ⊗ ω ) = 0 . Definition 3. Let H be a k -module together with and a family of maps (cid:8) d = ω , , µ = ω , , ∆ = ω , , ω , (cid:9) with ω j,i ∈ Hom − i − j (cid:0) H ⊗ i , H ⊗ j (cid:1) , and let ω = P ω j,i . Then (cid:0) H, d, µ, ∆ , ω , (cid:1) is an A ∞ -bialgebra if ω ⊚ ω = 0 . Example 3. Continuing Example 2, verification of relations (11) - (16) aboveis straightforward and follows from the fact that σ , ( y | x | x | y ) = − y | x | x | y. Thus ( H, µ, ∆ , ω ) is an A ∞ -bialgebra with non-operadic structure. Let H be a graded module and let (cid:8) ω j,i : H ⊗ i → H ⊗ j (cid:9) i,j ≥ be an arbitraryfamily of maps. Given a diagonal ∆ P on the permutahedra and the notion ofa ∆ P -(co)derivation, one continues the procedure described above to obtain thegeneral biderivative defined in [SU05]. And as above, the general A ∞ -bialgebrastructure relations are the homogeneous components of ω ⊚ ω = 0.For example, consider an A ∞ -bialgebra ( H, µ, ∆ , ω j,i ) with exactly one higherorder operation ω j,i , i + j ≥ . When constructing ω , we extend µ as a coderivation,identify the components of this extension in Hom (cid:0) H ⊗ i , H ⊗ j (cid:1) with the vertices ofthe permutahedron P i + j − , and identify ω j,i with its top dimensional cell. Since µ, ∆ and ω j,i are the only operations in H , all compositions involving these operationshave degree 0 or 3 − i − j , and k -faces of P i + j − in the range 0 < k < i + j − ω j,i as a ∆ P -coderivation only involvesthe primitive terms of ∆ ( P i + j − ), and the components of this extension are termsin the expression δ (cid:0) ω j,i (cid:1) . Indeed, whenever ω j,i and its extension are compatiblewith the underlying DGB structure, the relation δ (cid:0) ω j,i (cid:1) = 0 is satisfied. Dually, wehave ∂ (cid:0) ω j,i (cid:1) = 0 whenever ω j,i and its extension as a ∆ P -derivation are compatiblewith the underlying DGB structure. These structure relations can be expressed ascommutative diagrams in the integer lattice N (see Figures 7 and 8 below). Definition 4. Let n ≥ . A simple Hopf A ( n ) -algebra is a tuple ( H, d, µ, ∆ ,µ n ) with the following properties: 1. ( H, d, ∆) is a coassociative DGC; 2. ( H, d, µ, µ n ) is an A ( n ) -algebra; and 3. ∆ µ n = [ µ ( µ ⊗ · · · (cid:0) µ ⊗ ⊗ n − (cid:1) ⊗ µ n + µ n ⊗ µ (1 ⊗ µ ) · · · (cid:0) ⊗ n − ⊗ µ (cid:1) ] σ ,n ∆ ⊗ n . A simple Hopf A ∞ -algebra ( H, d, µ, ∆ , µ n ) is a simple Hopf A ( n ) -algebra satisfyingthe relation in offset (4.1) above. There are the completely dual notions of a simpleHopf A ( n ) -coalgebra and a simple Hopf A ∞ -coalgebra. General Hopf A ∞ -(co)algebras were defined by A. Berciano and this author in[BU08]; A ∞ -bialgebras with operations exclusively of the forms ω j, and ω ,i , called special A ∞ -bialgebras, were considered by this author in [Umb08].Simple Hopf A ( n )-algebras are especially interesting because their structurerelations can be controlled by G-S deformation theory. In fact, if n ≥ H t = ( H [[ t ]] , d t , µ t , ∆ t , µ nt ) is a deformation, then µ nt = tµ n + t µ n + · · · has noterm of order zero. Consequently, if D ( µ n ) = 0 , then tµ n automatically satisfies therequired structure relations in a simple Hopf A ( n )-algebra and ( H [[ t ]] , d, µ, ∆ , tµ n )is a linear deformation of H as a simple Hopf A ( n )-algebra. This proves: Theorem 1. If ( H, d, µ, ∆) is a DGB and µ n ∈ Hom − n ( H ⊗ n , H ) , n ≥ , is a2-cocycle, then ( H [[ t ]] , d, µ, ∆ , tµ n ) is a linear deformation of H as a simple Hopf A ( n ) -algebra. IGHER HOMOTPY HOPF ALGEBRA FOUND: A TEN YEAR RETROSPECTIVE 15 Figure 7. The structure relation ∂ (cid:0) ω j,i (cid:1) = 0 . Figure 8. The structure relation δ (cid:0) ω j,i (cid:1) = 0 . I am grateful to Samson Saneblidze and Andrey Lazarev for their helpful sugges-tions on early drafts of this paper, and to the referee, the editors, and Jim Stashefffor their assistance with the final draft. I wish to thank Murray and Jim for theirencouragement and support of this project over the years and I wish them bothmuch happiness and continued success. References [Bau98] Baues, H.J.: The cobar construction as a Hopf algebra and the Lie differential. Invent.Math., , 467–489 (1998)[BU08] Berciano, A., Umble, R.: Some naturally occurring examples of A ∞ -bialgebras. PreprintarXiv.0706.0703.[Car55] Cartier, P.: Cohomologie des Coalgebras. In: S´eminaire Sophus Lie, Expos´e 5, (1955-56)[CS04] Chas, M., Sullivan, D.: Closed string operators in topology leading to Lie bialgebras andhigher string algebra. The legacy of Niels Henrik Abel, 771–784, Springer, Berlin, (2004)[Ger63] Gerstenhaber, M.: The cohomology structure of an associative ring. Ann. of Math., (2)267–288 (1963)[GS92] Gerstenhaber, M., Schack, S.D.: Algebras, bialgebras, quantum groups, and algebraicdeformations. In: Contemporary Math. 134. AMS, Providence (1992)[God08] Godin, V.: Higher String topology operations. Preprint arXiv:0711.4859.[Gug82] Gugenheim, V.K.A.M.: On a perturbation theory for the homology of the loop space. J.Pure Appl. Algebra, (6) 1241–1261 (1992)[Mac67] MacLane, S.: Homology. Springer-Verlag, Berlin/New York (1967)[Mar06] Markl, M.: A resolution (minimal model) of the PROP for bialgebras. J. Pure and Appl.Algebra, W -construction and products of A ∞ -algebras. Trans. AMS, (6) 2353–2372 (2006)[MSS02] Markl, M., Shnider, S., Stasheff, J.: Operads in Algebra, Topology and Physics. Mathe-matical Surveys and Monographs , AMS, Providence (2002)[McC98] McCleary, J. (ed): Higher Homotopy Structures in Topology and Mathematical Physics.Contemporary Mathematics , AMS, Providence (1998)[May72] May, J.P.: Geometry of Iterated Loop Spaces. SLNM , Springer, Berlin (1972)[Pir02] Pirashvili, T.: On the PROP coresponding to bialgebras. Cah. Topol. G´eom. Diff´er.Cat´eg., (3) 221–239 (2002)[San96] Saneblidze, S.: The formula determining an A ∞ -coalgebra structure on a free algebra.Bull. Georgian Acad. Sci., (1) 363–411 (2004)[SU05] ————–: The biderivative and A ∞ -bialgebras. J. Homology, Homotopy and Appl., (2)161–177 (2005) IGHER HOMOTPY HOPF ALGEBRA FOUND: A TEN YEAR RETROSPECTIVE 17 [SU08a] ————–: Matrads, matrahedra and A ∞ -bialgebras. J. Homology, Homotopy and Appl.,to appear. Preprint math.AT/0508017.[SU08b] ————–: The category of A ∞ -bialgebras. In preparation.[Sho03] Shoikhet, B.: The CROCs, non-commutative deformations, and (co)associative bialge-bras. Preprint math.QA/0306143.[Sta63] Stasheff, J.: Homotopy associativity of H -spaces I, II. Trans. AMS A ∞ -bialgebras. J. Mathematical Sciences, toappear. Preprint arXiv:math.AT/0506446.[Val04] Vallette, B.: Koszul duality for PROPs. Trans. AMS, Department of Mathematics, Millersville University of Pennsylvania, Millersville,PA. 17551 E-mail address : [email protected] r X i v : . [ m a t h . A T ] D ec HIGHER HOMOTOPY HOPF ALGEBRAS FOUND: A TENYEAR RETROSPECTIVE RONALD UMBLE Abstract. At the 1996 conference honoring Jim Stasheff in the year of his60th birthday, I initiated the search for A ∞ -bialgebras in a talk entitled “InSearch of Higher Homotopy Hopf Algebras.” The idea in that talk was to thinkof a DG bialgebra as some (unknown) higher homotopy structure with trivialhigher order structure and apply a graded version of Gerstenhaber and Schack’sbialgebra deformation theory. Indeed, deformation cohomology, which detectssome (but not all) A ∞ -bialgebra structure, motivated the definition given byS. Saneblidze and myself in 2004. To Murray Gerstenhaber and Jim Stasheff Introduction In a preprint dated June 14, 2004, Samson Saneblidze and I announced the def-inition of A ∞ -bialgebras [SU05], marking approximately six years of collaborationthat continues to this day. Unknown to us at the time, A ∞ -bialgebras are ubiqui-tous and fundamentally important. Indeed, over a field F , the bialgebra structureon the singular chains of a loop space Ω X pulls back along a quasi-isomorphism g : H ∗ (Ω X ; F ) → C ∗ (Ω X ) to an A ∞ -bialgebra structure on homology that isunique up to isomorphism [SU11].Many have tried unsuccessfully to define A ∞ -bialgebras. The illusive ingredientin the definition turned out to be an explicit diagonal ∆ P on the permutahedra P = ⊔ n ≥ P n , the first construction of which was given by S. Saneblidze and myselfin [SU04]. This paper is an account of the historical events leading up to thediscovery of A ∞ -bialgebras and the truly remarkable role played by ∆ P in thisregard. Although the ideas and examples presented here are quite simple, theyrepresent and motivate general theory in [SU04], [SU05], [SU10], and [SU11].Through their work in the theory of PROPs and the related area of infinity Liebialgebras, many authors have contributed indirectly to this work, most notably M.Chas and D. Sullivan [CS04], J-L. Loday [Lod08], M. Markl [Mar06], T. Pirashvili[Pir02], and B. Vallette [Val04]; for extensive bibliographies see [Sul07] and [Mar08].Several new results spin off of this discussion and are included here: Example1 in Section 3 introduces the first example of a bialgebra H endowed with an A ∞ -algebra structure that is compatible with the comultiplication. Example 2 inSection 4, introduces the first example of a “non-operadic” A ∞ -bialgebra with anon-trivial operation ω , : H ⊗ → H ⊗ . And in Section 5 we prove Theorem Date : December 1, 2010.1991 Mathematics Subject Classification. Key words and phrases. A ∞ -bialgebra, biassociahedron, matrad, operad, permutahedron . This research funded in part by a Millersville University faculty research grant. Given a DG bialgebra ( H, d, µ, ∆) and a Gerstenhaber-Schack 2 -cocycle µ n ∈ Hom − n ( H ⊗ n , H ) , n ≥ , let H = ( H [[ t ]] , d, µ, ∆) . Then ( H [[ t ]] , d, µ, ∆ , tµ n ) isa linear deformation of H as a Hopf A ( n )- algebra. The Historical Context Two papers with far-reaching consequences in algebra and topology appearedin 1963. In [Ger63] Murray Gerstenhaber introduced the deformation theory ofassociative algebras and in [Sta63] Jim Stasheff introduced the notion of an A ( n )-algebra. Although the notion of what we now call a “non-Σ operad” appears inboth papers, this connection went unnoticed until after Jim’s visit to the Universityof Pennsylvania in 1983. Today, Gerstenhaber’s deformation theory and Stasheff’shigher homotopy algebras are fundamental tools in algebra, topology and physics.An extensive bibliography of applications appears in [MSS02].By 1990, techniques from deformation theory and higher homotopy structureshad been applied by many authors, myself included [Umb89], [LU92], to classifyrational homotopy types with a fixed cohomology algebra. And it seemed rea-sonable to expect that rational homotopy types with a fixed Pontryagin algebra H ∗ (Ω X ; Q ) could be classified in a similar way. Presumably, such a theory wouldinvolve deformations of DG bialgebras (DGBs) as some higher homotopy structurewith compatible A ∞ -algebra and A ∞ -coalgebra substructures, but the notion ofcompatibility was not immediately clear and an appropriate line of attack seemedillusive. But one thing was clear: If we apply a graded version of Gerstenhaberand Schack’s (G-S) deformation theory [GS92], [LM91], [LM96], [Umb96] and de-form a DGB H as some (unknown) higher homotopy structure, new operations ω j,i : H ⊗ i → H ⊗ j appear and their interactions with the deformed bialgebra oper-ations are partially detected by the differentials. While this is but one small pieceof a very large puzzle, it gave us a clue.During the conference honoring Jim Stasheff in the year of his 60th birthday,held at Vassar College in June 1996, I discussed this particular clue in a talk entitled“ In Search of Higher Homotopy Hopf Algebras ” ([McC98] p. xii). Although G-Sdeformations of DGBs are less constrained than the A ∞ -bialgebras known today,they motivated the definition announced eight years later.Following the Vassar conference, forward progress halted. Questions of struc-tural compatibility seemed mysterious and inaccessible. Then in 1998, Jim Stasheffran across some related work by S. Saneblidze [San96], of the A. Razmadze Mathe-matical Institute in Tbilisi, and suggested that I get in touch with him. Thus beganour long and fruitful collaboration. Over the months that followed, Saneblidze ap-plied techniques of homological perturbation theory to solve the aforementionedclassification problem [San99], but the higher order structure in the limit is implicitand the structure relations are inaccessible. In retrospect, this is not surprisingas explicit structure relations require explicit combinatorial diagonals ∆ P on thepermutahedra P = ⊔ n ≥ P n and ∆ K on the associahedra K = ⊔ n ≥ K n . But suchdiagonals are difficult to construct and were unknown to us at the time. Indeed,one defines the tensor product of A ∞ -algebras in terms of ∆ K , and the search fora construction of ∆ K had remained a long-standing problem in the theory of op-erads. We announced our construction of ∆ K in 2000 [SU00]; our construction of∆ P followed a year or two later (see [SU04]). IGHER HOMOTOPY HOPF ALGEBRAS FOUND: A TEN YEAR RETROSPECTIVE 3 Two Important Roles For ∆ P The diagonal ∆ P plays two fundamentally important roles in the theory of A ∞ -bialgebras. First, one builds the structure relations from components of (co)freeextensions of initial maps as higher (co)derivations with respect to ∆ P , and second,∆ P specifies exactly which of these components to use.To appreciate the first of these roles, recall the following definition given byStasheff in his seminal work on A ∞ -algebras in 1963 [Sta63]: Let A be a gradedmodule, let { µ i ∈ Hom i − (cid:0) A ⊗ i , A (cid:1) } n ≥ be an arbitrary family of maps, and let d be the cofree extension of Σ µ i as a coderivation of the tensor coalgebra T c A (witha shift in dimension). Then ( A, µ i ) is an A ∞ -algebra if d = 0; when this occurs,the universal complex ( T c A, d ) is called the tilde-bar construction and the structurerelations in A are the homogeneous components of d = 0. Similarly, let H be agraded module and let { ω j,i ∈ Hom − i − j (cid:0) H ⊗ i , H ⊗ j (cid:1) } i,j ≥ , be an arbitrary familyof maps. When (cid:0) H, ω j,i (cid:1) is an A ∞ -bialgebra, the map ω = Σ ω j,i uniquely extendsto its biderivative d ω ∈ End (cid:0) T H ⊕ T ( H ⊗ ) ⊕ · · · (cid:1) , which is the sum of various(co)free extensions of various subfamilies of (cid:8) ω j,i (cid:9) as ∆ P -(co)derivations ([SU05]).And indeed, the structure relations in H are the homogeneous components of d ω = 0with respect to an appropriate composition product.To demonstrate the spirit of this, consider a free graded module H of finitetype and an (arbitrary) map ω = µ + µ + ∆ with components µ : H ⊗ → H,µ : H ⊗ → H, and ∆ : H → H ⊗ . Extend ∆ as a coalgebra map ∆ : T c H → T c (cid:0) H ⊗ (cid:1) , extend µ + µ as a coderivation d : T c H → T c H , and extend µ asan algebra map µ : T a (cid:0) H ⊗ (cid:1) → T a H. Finally, extend ( µ ⊗ µ and (1 ⊗ µ ) µ as algebra maps f, g : T a (cid:0) H ⊗ (cid:1) → T a H , and extend µ as an ( f, g )-derivation µ : T a (cid:0) H ⊗ (cid:1) → T a H. The components of the biderivative in d + µ + µ + ∆ ∈ L p,q,r,s ≥ Hom (cid:16) ( H ⊗ p ) ⊗ q , ( H ⊗ r ) ⊗ s (cid:17) determine the structure relations. Let σ r,s : ( H ⊗ r ) ⊗ s → ( H ⊗ s ) ⊗ r denote thecanonical permutation of tensor factors and define a composition product ⊚ onhomogeneous components A and B of d + µ + µ + ∆ by A ⊚ B = (cid:26) A ◦ σ r,s ◦ B, if defined0 , otherwise.When A ⊚ B is defined, ( H ⊗ r ) ⊗ s is the target of B , and ( H ⊗ s ) ⊗ r is the source of A. Then (cid:0) H, µ, µ , ∆ (cid:1) is an A ∞ -infinity bialgebra if d ω ⊚ d ω = 0 . Note that ∆ µ and( µ ⊗ µ ) σ , (∆ ⊗ ∆) are the homogeneous components of d ω ⊚ d ω in Hom (cid:0) H ⊗ , H ⊗ (cid:1) ;consequently, d ω ⊚ d ω = 0 implies the Hopf relation∆ µ = ( µ ⊗ µ ) σ , (∆ ⊗ ∆) . Now if ( H, µ, ∆) is a bialgebra, the operations µ t , µ t , and ∆ t in a G-S deformationof H satisfy(3.1) ∆ t µ t = (cid:2) µ t ( µ t ⊗ ⊗ µ t + µ t ⊗ µ t (1 ⊗ µ t ) (cid:3) σ , ∆ ⊗ t and the homogeneous components of d ω ⊚ d ω = 0 in Hom (cid:0) H ⊗ , H ⊗ (cid:1) are exactlythose in (3.1). So this is encouraging.Recall that the permutahedron P is a point 0 and P is an interval 01. In thesecases ∆ P agrees with the Alexander-Whitney diagonal on the simplex:∆ P (0) = 0 ⊗ P (01) = 0 ⊗ 01 + 01 ⊗ . RONALD UMBLE If X is an n -dimensional cellular complex, let C ∗ ( X ) denote the cellular chains of X. When X has a single top dimensional cell, we denote it by e n . An A ∞ -algebrastructure { µ n } n ≥ on H is encoded operadically by a family of chain maps { ξ : C ∗ ( P n − ) → Hom ( H ⊗ n , H ) } , which factor through the map θ : C ∗ ( P n − ) → C ∗ ( K n ) induced by cellular projec-tion P n − → K n given by A. Tonks [Ton97] and satisfy ξ (cid:0) e n − (cid:1) = µ n . The factthat ( ξ ⊗ ξ ) ∆ P (cid:0) e (cid:1) = µ ⊗ µ and( ξ ⊗ ξ ) ∆ P (cid:0) e (cid:1) = µ ( µ ⊗ ⊗ µ + µ ⊗ µ (1 ⊗ µ )are components of µ and µ suggests that we extend a given µ n as a higher deriva-tion µ n : T a ( H ⊗ n ) → T a H with respect to ∆ P . Indeed, an A ∞ -bialgebra of theform ( H, ∆ , µ n ) n ≥ is defined in terms of the usual A ∞ -algebra relations togetherwith the relations(3.2) ∆ µ n = (cid:2) ( ξ ⊗ ξ ) ∆ P (cid:0) e n − (cid:1)(cid:3) σ ,n ∆ ⊗ n , which define the compatibility of µ n and ∆.Structure relations in more general A ∞ -bialgebras of the form( H, ∆ m , µ n ) m,n ≥ are similar in spirit and formulated in [Umb06]. Special casesof the form ( H, ∆ , ∆ n , µ ) with a single ∆ n were studied by H.J. Baues in the case n = 3 [Bau98] and by A. Berciano and myself with n ≥ p isan odd prime and n ≥ 3, these particular structures appear as tensor factors of themod p homology of an Eilenberg-Mac Lane space of type K ( Z , n ).Dually, A ∞ -bialgebras ( H, ∆ , µ, µ n ) with a single µ n have a coassociative co-multiplication ∆ , an associative multiplication µ, and ξ ⊗ ξ acts exclusively on theprimitive terms of ∆ P for lacunary reasons, in which case relation (3.2) reduces to(3.3) ∆ µ n = ( f n ⊗ µ n + µ n ⊗ f n ) σ ,n ∆ ⊗ n , where f n = µ ( µ ⊗ · · · (cid:0) µ ⊗ ⊗ n − (cid:1) . The first example of this particular structurenow follows. Example 1. Let H be the primitively generated bialgebra Λ ( x, y ) with | x | = 1 , | y | =2 , and µ n (cid:0) x i y p | · · · | x i n y p n (cid:1) = (cid:26) y p + ··· + p n +1 , i · · · i n = 1 and p k ≥ , otherwise . One can easily check that H is an A ∞ -algebra, and a straightforward calculationtogether with the identity (cid:18) p + · · · + p n + 1 i (cid:19) = P s + ··· + s n = i − (cid:18) p s (cid:19) · · · (cid:18) p n s n (cid:19) + P s + ··· + s n = i (cid:18) p s (cid:19) · · · (cid:18) p n s n (cid:19) verifies relation (3.3). The second important role played by ∆ P is evident in A ∞ -bialgebras in which ω n,m is non-trivial for some m, n > . Just as an A ∞ -algebra structure on H isencoded operadically, an A ∞ -bialgebra structure on H is encoded matradically bya family of chain maps (cid:8) ε : C ∗ ( KK n,m ) → Hom − m − n ( H ⊗ m , H ⊗ n ) (cid:9) IGHER HOMOTOPY HOPF ALGEBRAS FOUND: A TEN YEAR RETROSPECTIVE 5 over contractible polytopes KK = ⊔ m,n ≥ KK n,m , called biassociahedra , with singletop dimensional cells e m + n − such that ε (cid:0) e m + n − (cid:1) = ω n,m . Note that KK n, = KK ,n is the associahedron K n [SU10]. Let M = { M n,m = Hom ( H ⊗ m , H ⊗ n ) } andlet Θ = { θ nm = ω n,m } . The A ∞ -bialgebra matrad H ∞ is realized by C ∗ ( KK ) andis a proper submodule of the free PROP M generated by Θ . The matrad product γ on H ∞ is defined in terms of ∆ P , and a monomial α in the free PROP M is acomponent of a structure relation if and only if α ∈ H ∞ .More precisely, in [Mar06] M. Markl defined the submodule S of special elements in PROP M whose additive generators are monomials α expressed as “elementaryfractions”(3.4) α = α y p · · · α y q p α qx · · · α qx p in which α qx i and α y j p are additive generators of S and the j th output of α qx i is linkedto the i th input of α y j p (here juxtaposition denotes tensor product). Representing θ nm graphically as a double corolla (see Figure 1), a general decomposable α isrepresented by a connected non-planar graph in which the generators appear inorder from left-to-right (see Figure 2). The matrad H ∞ is a proper submodule of S and the matrad product γ agrees with the restriction of Markl’s fraction productto H ∞ . θ nm = nm Figure 1.The diagonal ∆ P acts as a filter and admits certain elementary fractions asadditive generators of H ∞ . In dimensions 0 and 1 , the diagonal ∆ P is expressedgraphically in terms of up-rooted planar rooted trees (with levels) by∆ P ( ) = ⊗ and ∆ P ( ) = ⊗ + ⊗ . Define ∆ (0) P = 1; for each k ≥ , define ∆ ( k ) P = (cid:0) ∆ P ⊗ ⊗ k − (cid:1) ∆ ( k − P and view eachcomponent of ∆ ( k ) P ( θ q ) as a ( q − P q − ) × k +1 , andsimilarly for ∆ ( k ) P ( θ q ).The elements θ , θ , and θ generate two elementary fractions in M , each ofdimension zero, namely, α = and α = . Define ∂ (cid:0) θ (cid:1) = α + α , and label the edge and vertices of the interval KK , by θ , α and α , respectively. Continuing inductively, the elements θ , θ , θ , θ ,α , and α generate 18 fractions in M , – one in dimension 2, nine in dimension 1and eight in dimension 0. Of these, 14 label the edges and vertices of the heptagon KK , . Since the generator θ must label the 2-face, we wish to discard the 2-dimensional decomposable RONALD UMBLE e =and the appropriate components of its boundary. Note that e is a square whoseboundary is the union of four edges(3.5)Of the five fractions pictured above, only the first two in (3.5) have numeratorsand denominators that are components of ∆ ( k ) P ( P ) (numerators are components of∆ (1) P ( θ ) and denominators are exactly ∆ (2) P ( θ )). Our selection rule admits onlythese two particular fractions, leaving seven 1-dimensional generators to label theedges of KK , (see Figure 2). Now linearly extend the boundary map ∂ to theseven admissible 1-dimensional generators and compute the seven 0-dimensionalgenerators labeling the vertices of KK , . Since the 0-dimensional generatoris not among them, we discard it.Figure 2: The biassociahedron KK , . Subtleties notwithstanding, this process continues indefinitely and produces H ∞ as an explicit free resolution of the bialgebra matrad H = (cid:10) θ , θ , θ (cid:11) in the cat-egory of matrads. We note that in [Mar06], M. Markl makes arbitrary choices(independent of our selection rule) to construct the polytopes B nm = KK n,m for m + n ≤ 6. In this range, it is enough to consider components of the diagonal ∆ K on the associahedra.We conclude this section with a brief review of our diagonals ∆ P and ∆ K (upto sign); for details see [SU04]. Alternative constructions of ∆ K were subsequentlygiven by Markl and Shnider [MS06] and J.L. Loday [Lod10] (in this volume). Let n = { , , . . . , n } , n ≥ . A matrix E with entries from { } ∪ n is a step matrix if: IGHER HOMOTOPY HOPF ALGEBRAS FOUND: A TEN YEAR RETROSPECTIVE 7 • Each element of n appears as an entry of E exactly once. • The elements of n in each row and column of E form an increasing con-tiguous block. • Each diagonal parallel to the main diagonal of E contains exactly one ele-ment of n .Right-shift and down-shift matrix transformations, which include the identity (atrivial shift), act on step matrices and produce derived matrices. Given a q × p integer matrix M = ( m ij ) , choose proper subsets S i ⊂ { non-zeroentries in row i } and T j ⊂ { non-zero entries in column j } , and define down-shift and right-shift operations D S i and R T j on M as follows: • If S i = ∅ , max row( i + 1) < min S i = m ij , and m i +1 ,k = 0 for all k ≥ j ,then D S i M is the matrix obtained from M by interchanging each m ik ∈ S i with m i +1 ,k ; otherwise D S i M = M. • If T j = ∅ , max col( j + 1) < min T j = m ij , and m k,j +1 = 0 for all k ≥ i, then R T j M is the matrix obtained from M by interchanging each m k,j ∈ T j with m k,j +1 ; otherwise R T j M = M. Given a q × p step matrix E together with subsets S , . . . , S q and T , . . . , T p asabove, there is the derived matrix R T p · · · R T R T D S q · · · D S D S E. In particular, step matrices are derived matrices under the trivial action with S i = T j = ∅ for all i, j .Let a = A | A | · · · | A p and b = B q | B q − | · · · | B be partitions of n . The pair a × b is an ( p, q ) -complementary pair (CP) if B i and A j are the rows and columnsof a q × p derived matrix. Since faces of P n are indexed by partitions of n, and CPsare in one-to-one correspondence with derived matrices, each CP is identified withsome product face of P n × P n . Definition 1. Define ∆ P ( e ) = e ⊗ e . Inductively, having defined ∆ P on C ∗ ( P k +1 ) for all ≤ k ≤ n − , define ∆ P on C n ( P n +1 ) by ∆ P ( e n ) = X ( p,q ) -CPs u × vp + q = n +2 ± u ⊗ v, and extend multiplicatively to all of C ∗ ( P n +1 ) . The diagonal ∆ P induces a diagonal ∆ K on C ∗ ( K ). Recall that faces of P n incodimension k are indexed by planar rooted trees with n + 1 leaves and k + 1 levels(PLTs), and forgetting levels defines the cellular projection θ : P n → K n +1 givenby A. Tonks [Ton97]. Thus faces of P n indexed by PLTs with multiple nodes inthe same level degenerate under θ , and corresponding generators lie in the kernelof the induced map θ : C ∗ ( P n ) → C ∗ ( K n +1 ). The diagonal ∆ K is given by∆ K θ = ( θ ⊗ θ )∆ P . Deformations of DG Bialgebras as A ( n ) -Bialgebras The discussion above provides the context to appreciate the extent to which G-Sdeformation theory motivates the notion of an of A ∞ -bialgebra. We describe thismotivation in this section. In retrospect, the bi(co)module structure encoded in theG-S differentials controls some (but not all) of the A ∞ -bialgebra structure relations. RONALD UMBLE For example, all structure relation in A ∞ -bialgebras of the form ( H, d, µ, ∆ , µ n ) arecontrolled except(4.1) n − X i =0 ( − i ( n +1) µ n (cid:0) ⊗ i ⊗ µ n ⊗ ⊗ n − i − (cid:1) = 0 , which measures the interaction of µ n with itself. Nevertheless, such structuresadmit an A ( n )-algebra substructure and their single higher order operation µ n iscompatible with ∆ . Thus we refer to such structures here as Hopf A ( n ) -algebras. General G-S deformations of DGBs, referred to here as quasi- A ( n ) -bialgebras, are“partial” A ( n )-bialgebras in the sense that all structure relations involving multiplehigher order operations are out of control.4.1. A ( n ) -Algebras and Their Duals. The signs in the following definition weregiven in [SU04] and differ from those given by Stasheff in [Sta63]. We note that ei-ther choice of signs induces an oriented combinatorial structure on the associahedra,and these structures are are equivalent. Let n ∈ N ∪ {∞} . Definition 2. An A ( n ) -algebra is a graded module A together with structure maps { µ k ∈ Hom − k (cid:0) A ⊗ k , A (cid:1) } ≤ k In [GS92], M. Gerstenhaber and S. D.Schack defined the cohomology of an ungraded bialgebra by joining the dual coho-mology theories of G. Hochschild [HKR62] and P. Cartier [Car55]. This construc-tion was given independently by A. Lazarev and M. Movshev in [LM91]. The G-Scohomology of H reviewed here is a straight-forward extension to the graded caseand was constructed in [LM96] and [Umb96].Let ( H, d, µ, ∆) be a connected DGB. We assume | d | = 1 , although one couldassume | d | = − i ≥ , the i -fold bicomodule tensor power of H is the H -bicomodule H ⊗ i = ( H ⊗ i , λ i , ρ i ) with left and right coactions given by IGHER HOMOTOPY HOPF ALGEBRAS FOUND: A TEN YEAR RETROSPECTIVE 9 λ i = (cid:2) µ ( µ ⊗ · · · ( µ ⊗ ⊗ i − ) ⊗ ⊗ i (cid:3) σ ,i ∆ ⊗ i and ρ i = (cid:2) ⊗ i ⊗ µ (1 ⊗ µ ) · · · (1 ⊗ i − ⊗ µ ) (cid:3) σ ,i ∆ ⊗ i .When f : H ⊗ i → H ⊗∗ , there is the composition(1 ⊗ f ) λ i = (cid:2) µ ( µ ⊗ · · · ( µ ⊗ ⊗ i − ) ⊗ f (cid:3) σ ,i ∆ ⊗ i .Dually, for each j ≥ , the j -fold bimodule tensor power of H is the H -bimodule H ⊗ j = ( H ⊗ j , λ j , ρ j ) with left and right actions given by λ j = µ ⊗ j σ j, (cid:2) (∆ ⊗ ⊗ j − ) · · · (∆ ⊗ 1) ∆ ⊗ ⊗ j (cid:3) and ρ j = µ ⊗ j σ j, (cid:2) ⊗ j ⊗ (1 ⊗ j − ⊗ ∆) · · · (1 ⊗ ∆) ∆ (cid:3) .When g : H ⊗∗ → H ⊗ j , there is the composition λ j (1 ⊗ g ) = µ ⊗ j σ j, (cid:2) (∆ ⊗ ⊗ j − ) · · · (∆ ⊗ 1) ∆ ⊗ g (cid:3) .Let k be a field. Extend d, µ and ∆ to k [[ t ]]-linear maps and obtain a k [[ t ]]-DGB H = ( H [[ t ]] , d, µ, ∆) . We wish to deform H as an A ( n )-structure of the form H t = (cid:16) H [[ t ]] , d t = ω , t , µ t = ω , t , ∆ t = ω , t , ω j,it (cid:17) i + j = n +1 where ω j,it = ∞ P k =0 t k ω j,ik ∈ Hom − i − j (cid:16) H ⊗ i , H ⊗ j (cid:17) , ω , = d, ω , = µ, ω , = ∆ , and ω j,i = 0.Deformations of H are controlled by the G-S n - complex, which we now review.For k ≥ , let d ( k ) = k − P i =0 ⊗ i ⊗ d ⊗ ⊗ k − i − ∂ ( k ) = k − P i =0 ( − i ⊗ i ⊗ µ ⊗ ⊗ k − i − δ ( k ) = k − P i =0 ( − i ⊗ i ⊗ ∆ ⊗ ⊗ k − i − . These differentials induce strictly commuting differentials d, ∂, and δ on the tri-graded module { Hom p ( H ⊗ i , H ⊗ j ) } , which act on an element f in tridegree ( p, i, j )by d ( f ) = d ( j ) f − ( − p f d ( i ) ∂ ( f ) = λ j (1 ⊗ f ) − f ∂ ( i ) − ( − i ρ j ( f ⊗ δ ( f ) = (1 ⊗ f ) λ i − δ ( j ) f − ( − j ( f ⊗ ρ i . The submodule of total G-S r -cochains on H is C rGS ( H, H ) = L p + i + j = r +1 Hom p ( H ⊗ i , H ⊗ j )and the total differential D on a cochain f in tridegree ( p, i, j ) is given by D ( f ) = h ( − i + j d + ∂ + ( − i δ i ( f ) ,where the sign coefficients are chosen so that (1) D = 0 , (2) structure relations(ii) and (iii) in Definition 2 hold and (3) the restriction of D to the submodule of r -cochains in degree p = 0 agrees with the total (ungraded) G-S differential. The G-S cohomology of H with coefficients in H is given by H ∗ GS ( H, H ) = H ∗ { C rGS ( H, H ) , D } . Identify Hom p ( H ⊗ i , H ⊗ j ) with the point ( p, i, j ) in R . Then the G-S n - complex is that portion of the G-S complex in the region x ≥ − n and the submodule oftotal r -cochains in the n -complex is C rGS ( H, H ; n ) = L p = r − i − j +1 ≥ − n Hom p ( H ⊗ i , H ⊗ j )(a 2-cocycle in the 3-complex appears in Figure 3). The G-S n - cohomology of H with coefficients in H is given by H ∗ GS ( H, H ; n ) = H ∗ { C rGS ( H, H ; n ) ; D } .Note that a general 2-cocycle α has a component of tridegree (3 − i − j, i, j ) for each i and j in the range 2 ≤ i + j ≤ n + 1. Thus α has n ( n + 1) / α determine an infinitesimal deformation , i.e., the component ω j,i in tridegree(3 − i − j, i, j ) defines the first order approximation ω j,i + tω j,i of the structure map ω j,it in H t .For simplicity, consider the case n = 3 . Each of the ten homogeneous compo-nents of the deformation equation D ( α ) = 0 produces the infinitesimal form of onestructure relation (see below). In particular, a deformation H t with structure maps { ω ,it } ≤ i ≤ is a simple A (3)-algebra and a deformation H t with structure maps { ω j, t } ≤ j ≤ is a simple A (3)-coalgebra.Figure 3. The 2-cocycle d + µ + ∆ + µ + ω + ∆ . For notational simplicity, let µ t = ω , t , ω t = ω , t and ∆ t = ω , , and considera deformation of ( H, d, µ, ∆) as a “quasi- A (3)-structure.” Then • d t = d + td + t d + · · ·• µ t = µ + tµ + t µ + · · ·• ∆ t = ∆ + t ∆ + t ∆ + · · ·• µ t = tµ + t µ + · · ·• ω t = tω + t ω + · · ·• ∆ t = t ∆ + t ∆ + · · · IGHER HOMOTOPY HOPF ALGEBRAS FOUND: A TEN YEAR RETROSPECTIVE 11 and d + µ + ∆ + µ + ω + ∆ is a total 2-cocycle (see Figure 3). Equatingcoefficients in D (cid:0) d + µ + ∆ + µ + ω + ∆ (cid:1) = 0 gives1 . d ( d ) = 0 6 . ∂ (cid:0) µ (cid:1) = 02 . d ( µ ) − ∂ ( d ) = 0 7 . δ (cid:0) ∆ (cid:1) = 03 . d (∆ ) + δ ( d ) = 0 8 . d ( ω ) + ∂ (∆ ) + δ ( µ ) = 04 . d (cid:0) µ (cid:1) + ∂ ( µ ) = 0 9 . ∂ (cid:0) ∆ (cid:1) + δ ( ω ) = 05 . d (cid:0) ∆ (cid:1) − δ (∆ ) = 0 10 . ∂ ( ω ) − δ (cid:0) µ (cid:1) = 0 . Requiring (cid:0) H, d t , µ t , µ t (cid:1) and (cid:0) H, d t , ∆ t , ∆ t (cid:1) to be simple A (3)-(co)algebras tellsus that relations (1) - (7) are linearizations of Stasheff’s strict A (4)-(co)algebrarelations, and relation (8) is the linearization of the Hopf relation relaxed up tohomotopy. Since µ t , ω t and ∆ t have no terms of order zero, relations (9) and (10)are the respective linearizations of new relations (9) and (10) below. Thus we obtainthe following structure relations in H t :1 . d t = 02 . d t µ t = µ t ( d t ⊗ ⊗ d t )3 . ∆ t d t = ( d t ⊗ ⊗ d t ) ∆ t . d t µ t + µ t ( d t ⊗ ⊗ ⊗ d t ⊗ ⊗ ⊗ d t ) = µ t (1 ⊗ µ t ) − µ t ( µ t ⊗ . ( d t ⊗ ⊗ ⊗ d t ⊗ ⊗ ⊗ d t ) ∆ t + ∆ t d t = (∆ t ⊗ 1) ∆ t − (1 ⊗ ∆ t ) ∆ t . µ t ( µ t ⊗ ⊗ − ⊗ µ t ⊗ ⊗ ⊗ µ t ) = µ t (cid:0) µ t ⊗ ⊗ µ t (cid:1) . (∆ t ⊗ ⊗ − ⊗ ∆ t ⊗ ⊗ ⊗ ∆ t ) ∆ t = (cid:0) ∆ t ⊗ ⊗ ∆ t (cid:1) ∆ t . ( d t ⊗ ⊗ d t ) ω t + ω t ( d t ⊗ ⊗ d t ) = ∆ t µ t − ( µ t ⊗ µ t ) σ , (∆ t ⊗ ∆ t )9 . ( µ t ⊗ ω t ) σ , (∆ t ⊗ ∆ t ) − (∆ t ⊗ − ⊗ ∆ t ) ω t − ( ω t ⊗ µ t ) σ , (∆ t ⊗ ∆ t )= ∆ t µ t − µ ⊗ t σ , (cid:2) (∆ t ⊗ 1) ∆ t ⊗ ∆ t + (cid:0) ∆ t ⊗ (1 ⊗ ∆ t ) ∆ t (cid:1)(cid:3) . ( µ t ⊗ µ t ) σ , (∆ t ⊗ ω t ) − ω t ( µ t ⊗ − ⊗ µ t ) − ( µ t ⊗ µ t ) σ , ( ω t ⊗ ∆ t )= (cid:2) µ t ( µ t ⊗ ⊗ µ t + µ t ⊗ µ t (1 ⊗ µ t ) (cid:3) σ , ∆ ⊗ t − ∆ t µ t . By dropping the formal deformation parameter t, we obtain the structure relationsin a quasi-simple A (3)-bialgebra.The first non-operadic example of an A ∞ -bialgebra appears here as a quasi-simple A (3)-bialgebra and involves a non-trivial operation ω = ω , . The six addi-tional relations satisfied by A ∞ -bialgebras of this particular form will be verified inthe next section. Example 2. Let H be the primitively generated bialgebra Λ ( x, y ) with | x | = 1 , | y | = 2 , trivial differential, and ω : H ⊗ → H ⊗ given by ω ( a | b ) = x | y + y | x, a | b = y | yx | x, a | b ∈ { x | y, y | x } , otherwise . Then (∆ ⊗ − ⊗ ∆) ω ( y | y ) = (∆ ⊗ − ⊗ ∆) ( x | y + y | x ) = 1 | x | y + 1 | y | x − x | y | − y | x | µ ⊗ ω − ω ⊗ µ ) (1 | | y | y + y | | | y + 1 | y | y | y | y | | 1) =( µ ⊗ ω − ω ⊗ µ ) σ , (∆ ⊗ ∆) ( y | y ); similar calculations show agreement on x | y and y | x and verifies relation (9). To verify relation (10), note that ω ( µ ⊗ − ⊗ µ )and ( µ ⊗ µ ) σ , (∆ ⊗ ω − ω ⊗ ∆) are supported on the subspace spanned by B = { | y | y, y | y | , | x | y, x | y | , | y | x, y | x | } ,and it is easy to check agreement on B. Finally, note that ( H, µ, ∆ , ω ) can berealized as the linear deformation ( H [[ t ]] , µ, ∆ , tω ) | t =1 . A ∞ -Bialgebras in Perspective Although G-S deformation cohomology motivates the notion of an A ∞ -bialgebra,G-S deformations of DGBs are less constrained than A ∞ -bialgebras and fall short ofthe mark. To indicate of the extent of this shortfall, let us identify those structurerelations that fail to appear via deformation cohomology but must be verified toassert that Example 2 is an A ∞ -bialgebra.As mentioned above, structure relations in a general A ∞ -bialgebra arise from thehomogeneous components of the equation d ω ⊚ d ω = 0. So to begin, let us constructthe components of the biderivative d ω that determine the structure relations in an A ∞ -bialgebra of the form (cid:0) H, d, µ, ∆ , ω , (cid:1) . Given arbitrary maps d = ω , , µ = ω , , ∆ = ω , , and ω , with ω j,i ∈ Hom − i − j (cid:0) H ⊗ i , H ⊗ j (cid:1) , consider ω = P ω j,i . (Co)freely extend • d as a linear map ( H ⊗ p ) ⊗ q → ( H ⊗ p ) ⊗ q for each p, q ≥ , • d + ∆ as a derivation of T a H , • d + µ as a coderivation of T c H , • ∆ + ω , as a coalgebra map T c H → T c (cid:0) H ⊗ (cid:1) , and • µ + ω , as an algebra map T a (cid:0) H ⊗ (cid:1) → T a H .Note that in this restricted setting, relation (10) in Definition 2 reduces to( µ ⊗ µ ) ⊚ (cid:0) ∆ ⊗ ω , − ω , ⊗ ∆ (cid:1) = ω , ⊚ ( µ ⊗ − ⊗ µ ).Factors µ ⊗ ⊗ µ are components of d + µ ; factors ∆ ⊗ ω , and ω , ⊗ ∆ arecomponents of ∆ + ω , ; and the factor µ ⊗ µ is a component of µ + ω , . •• ❅❅❅❅❅■ H ⊗ j H ⊗ i ω j,i Figure 4. The initial map ω j,i . IGHER HOMOTOPY HOPF ALGEBRAS FOUND: A TEN YEAR RETROSPECTIVE 13 To picture this, identify the isomorphic modules ( H ⊗ p ) ⊗ q ≈ ( H ⊗ q ) ⊗ p with thepoint ( p, q ) ∈ N and picture the initial map ω j,i : H ⊗ i → H ⊗ j as a “transgressive”arrow from ( i, 1) to (1 , j ) (see Figure 4).Components of the various (co)free extensions above are pictured as arrows thatinitiate or terminate on the axes. For example, the vertical arrow ∆ ⊗ ∆ , the shortleft-leaning arrow ∆ ⊗ ω , − ω , ⊗ ∆ and the long left-leaning arrow ω , ⊗ ω , in Figure 5 represent components of ∆ + ω , .Figure 5. Components of d ω when ω = d + µ + ∆ + ω , . Since we are only interested in transgressive quadratic ⊚ -compositions, it issufficient to consider the components of d ω pictured in Figure 5. Quadratic compo-sitions along the x -axis correspond to relations (1), (2), (4) and (6) in Definition 2;those in the square with its diagonal correspond to relation (8); those in the verticalparallelogram correspond to relation (9); and those in the horizontal parallelogramcorrespond to relation (10).The following six additional relations are not detected by deformation cohomol-ogy because the differentials only detect the interactions between ω and (deforma-tions of) d, µ, and ∆ induced by the underlying bi(co)module structure:11. ( µ ⊗ ω − ω ⊗ µ ) σ , (∆ ⊗ ω − ω ⊗ ∆) = 0;12. ( µ ⊗ µ ) σ , ( ω ⊗ ω ) = 0;13. ( ω ⊗ ω ) σ , (∆ ⊗ ∆) = 0;14. ( µ ⊗ ω − ω ⊗ µ ) σ , ( ω ⊗ ω ) = 0;15. ( ω ⊗ ω ) σ , (∆ ⊗ ω − ω ⊗ ∆) = 0;16. ( ω ⊗ ω ) σ , ( ω ⊗ ω ) = 0 . Definition 3. Let H be a k -module together with and a family of maps (cid:8) d = ω , , µ = ω , , ∆ = ω , , ω , (cid:9) ,where ω j,i ∈ Hom − i − j (cid:0) H ⊗ i , H ⊗ j (cid:1) , and let ω = P ω j,i . Then (cid:0) H, d, µ, ∆ , ω , (cid:1) isan A ∞ -bialgebra if d ω ⊚ d ω = 0 . Example 3. Continuing Example 2, verification of relations (11) - (16) aboveis straightforward and follows from the fact that σ , ( y | x | x | y ) = − y | x | x | y. Thus ( H, µ, ∆ , ω ) is an A ∞ -bialgebra with non-operadic structure. Let H be a graded module and let (cid:8) ω j,i : H ⊗ i → H ⊗ j (cid:9) i,j ≥ be an arbitraryfamily of maps. Given a diagonal ∆ P on the permutahedra and the notion ofa ∆ P -(co)derivation, one continues the procedure described above to obtain thegeneral biderivative defined in [SU05]. And as above, the general A ∞ -bialgebrastructure relations are the homogeneous components of d ω ⊚ d ω = 0.For example, consider an A ∞ -bialgebra ( H, µ, ∆ , ω j,i ) with exactly one higherorder operation ω j,i , i + j ≥ . When constructing d ω , we extend µ as a coderivation,identify the components of this extension in Hom (cid:0) H ⊗ i , H ⊗ j (cid:1) with the vertices ofthe permutahedron P i + j − , and identify ω j,i with its top dimensional cell. Since µ, ∆ and ω j,i are the only operations in H , all compositions involving these operationshave degree 0 or 3 − i − j , and k -faces of P i + j − in the range 0 < k < i + j − ω j,i as a ∆ P -coderivation only involvesthe primitive terms of ∆ ( P i + j − ), and the components of this extension are termsin the expression δ (cid:0) ω j,i (cid:1) . Indeed, whenever ω j,i and its extension are compatiblewith the underlying DGB structure, the relation δ (cid:0) ω j,i (cid:1) = 0 is satisfied.Figure 6. The structure relation ∂ (cid:0) ω j,i (cid:1) = 0 . IGHER HOMOTOPY HOPF ALGEBRAS FOUND: A TEN YEAR RETROSPECTIVE 15 Figure 7. The structure relation δ (cid:0) ω j,i (cid:1) = 0 . Dually, we have ∂ (cid:0) ω j,i (cid:1) = 0 whenever ω j,i and its extension as a ∆ P -derivationare compatible with the underlying DGB structure. These structure relations canbe expressed as commutative diagrams in the integer lattice N (see Figures 6 and7). Definition 4. Let n ≥ . A Hopf A ( n ) -algebra is a tuple ( H, d, µ, ∆ ,µ n ) with the following properties: 1. ( H, d, ∆) is a coassociative DGC; 2. ( H, d, µ, µ n ) is an A ( n ) -algebra; and 3. ∆ µ n = [ µ ( µ ⊗ · · · (cid:0) µ ⊗ ⊗ n − (cid:1) ⊗ µ n + µ n ⊗ µ (1 ⊗ µ ) · · · (cid:0) ⊗ n − ⊗ µ (cid:1) ] σ ,n ∆ ⊗ n . A Hopf A ∞ -algebra ( H, d, µ, ∆ , µ n ) is a Hopf A ( n ) -algebra satisfying the rela-tion in offset (4.1) above. There are the completely dual notions of a Hopf A ( n ) -coalgebra and a Hopf A ∞ -coalgebra. Hopf A ∞ -(co)algebras were defined by A. Berciano and this author in [BU10],but with a different choice of signs. A ∞ -bialgebras with operations exclusively ofthe forms ω j, and ω ,i , called special A ∞ -bialgebras, were considered by this authorin [Umb06].Hopf A ( n )-algebras are especially interesting because their structure relationsare controlled by G-S deformation cohomology. In fact, if n ≥ H t =( H [[ t ]] , d t , µ t , ∆ t , µ nt ) is a deformation, then µ nt = tµ n + t µ n + · · · has no termof order zero. Consequently, if D ( µ n ) = 0 , then tµ n automatically satisfies therequired structure relations and ( H [[ t ]] , d, µ, ∆ , tµ n ) is a linear deformation of H as a Hopf A ( n )-algebra. Thus we have proved: Theorem 1. If ( H, d, µ, ∆) is a DGB and µ n ∈ Hom − n ( H ⊗ n , H ) , n ≥ , is a2-cocycle, then ( H [[ t ]] , d, µ, ∆ , tµ n ) is a linear deformation of H as a Hopf A ( n ) -algebra. I am grateful to Samson Saneblidze and Andrey Lazarev for their helpful sugges-tions on early drafts of this paper, and to the referee, the editors, and Jim Stashefffor their assistance with the final draft. I wish to thank Murray and Jim for theirencouragement and support of this project over the years and I wish them bothmuch happiness and continued success. References [Bau98] Baues, H.J.: The cobar construction as a Hopf algebra and the Lie differential. Invent.Math., , 467–489 (1998)[BU10] Berciano, A., Umble, R.: Some naturally occurring examples of A ∞ -bialgebras. J. PureAppl. Algebra (2010 in press). Preprint arXiv.0706.0703.[Car55] Cartier, P.: Cohomologie des Coalgebras. In: S´eminaire Sophus Lie, Expos´e 5, (1955-56)[CS04] Chas, M., Sullivan, D.: Closed string operators in topology leading to Lie bialgebrasand higher string algebra. The legacy of Niels Henrik Abel, 771–784, Springer, Berlin,(2004)[Ger63] Gerstenhaber, M.: The cohomology structure of an associative ring. 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AMS, Department of Mathematics, Millersville University of Pennsylvania, Millersville,PA. 17551 E-mail address ::