Symplectic $C_infty$-algebras

SYMPLECTIC C ∞ -ALGEBRAS ALAST AIR HAMIL TON AND A NDREY LAZAREV Abstra ct. In this pap er w e show that a strongly homotopy comm utative (or C ∞ -) algebra with an in v ariant inner pro duct on its cohomology can b e un iquely extend ed to a symplectic C ∞ - algebra (an ∞ -generalisatio n of a commutativ e F rob enius algebra in trod uced b y Kon tsev ich). This result relies on t he algebraic Ho dge decomp osition of th e cyclic Ho chsc hild cohomology of a C ∞ -algebra and does not generalize to algebras o ver other op erads. Contents 1. In tro duction 1 1.1. Notation and conv ent ions 2 2. Noncomm u tativ e geometry 3 2.1. F ormal noncommutat iv e geometry 3 2.2. F ormal noncommutat iv e symplectic geometry 4 3. C ∞ -algebras and related cohomology theories 5 3.1. Main deﬁnitions 5 3.2. Cohomology theories 6 3.3. Symplectic C ∞ -algebras 8 4. Finite lev el structures and Ob struction theory 10 4.1. C n -structures 10 4.2. Symplectic C ∞ -structures 15 4.3. Unital Symplectic C ∞ -structures 18 5. The Main T heorem: a C orresp ondence b et ween C ∞ and Symplectic C ∞ -structures 21 5.1. Main theorem I: the n on-unital case 21 5.2. Main Theorem I I: the un ital case 25 References 26 1. Introduction The n otion of a strongly homotopy asso ciativ e algebra or an A ∞ -algebra was introdu ced in [20] f or the pur p oses of studying H -sp aces. It was originally d eﬁned via a system of h igher m ultiplicatio n maps sa tisfying a series o f co mplicated relati ons. It is an a lgebra ov er the op erad of planar trees; other op erads giv e r ise to v arious other ﬂa v ours of strongly homotop y algebras, cf. [15]. In this pap er we are concerned chieﬂy with the homotop y in v arian t generalisat ion of comm u tativ e algebras, the so-called C ∞ -algebras. More pr ecisely , w e shall study symple ctic C ∞ - algebras; this is a s trong homotop y analogue of a commutativ e F rob enius algebra. Sym plectic inﬁnity- algebras (also called cyclic inﬁn ity algebras or inﬁn it y algebras with an inv ariant in n er pro du ct) w ere intro d uced in [13] and [12] and were sh o wn to hav e a close relation with graph homology an d therefore to the in tersection theory on the modu li spaces of complex curv es and inv ariants of diﬀerentia ble manifolds. A short, informal introdu ction to graph homology is con tained in [22]; a more sub stan tial account is in [2]. T he connections b et wee n graph complexes and inﬁnit y algebras are studied in [6] and [7]. 2000 Mathematics Subje ct Class iﬁc ation. 13D03, 1 3D10, 46 L87, 55P62. Key wor ds and phr ases. Inﬁ n it y-algebra, cyclic cohomology , Harris on cohomology , symplectic structure, Hodge decomp osition. 1 Our main theorem here states a unital C ∞ -algebra with the structure of a F rob enius alge bra on its cohomology is we akly equiv alen t to a symplectic C ∞ -algebra which is uniquely determined up to homotop y . This is a rather surprisin g phenomenon wh ic h do es n ot hold in the con text of other inﬁn it y-algebras. One application of th is result is that the cohomolog y algebra of a simply-connected manifold supp orts the structure of a s ymplectic C ∞ -algebra whic h leads to a manif estly homotop y inv arian t constru ction of string top ology-t yp e op er ations, cf. [8, 10]. Another p oten tial app lication is to the study of the deriv ed category of coheren t sh ea ve s on a complex manifold; note that according to [1 ] the latter could be realised a s a certain lo calisation of the catego ry of d iﬀerential- graded mo dules ov er th e Dolb eault algebra, wh ic h sup p orts an in v arian t inner pr o duct in the Calabi-Y au case. The basic fact whic h leads to th is result is that the cohomology theory for C ∞ -algebras, kno wn as Ha rrison c ohomology , essen tially coincides with the cohomology t heory f or symplectic C ∞ -algebras, called cyclic Harrison cohomolog y . This resu lt in turn follo ws from the algebraic Ho dge d ecomp osition for the cyclic Hochsc hild cohomology of a C ∞ -algebra and is pro v ed in the author’s p ap er [9]. W e s hall u se the resu lts of this pap er extensiv ely . T o p ro v e the main theorem w e dev elop an obstru ction theory f or lifting th e symp lectic C n - structures and C n -morphisms which is of indep end en t interest. T he pro of is carried out b y com bin ing this obs tr uction th eory with the Ho dge decomp osition for cyclic cohomology . The pap er is organised as f ollo ws. In Section 2 w e in tro duce the basics of noncomm u tativ e geometry . Section 3 d iscu sses C ∞ -algebras, with and w ithout an in v arian t inner pro duct, and the asso ciated cohomology theories. T h ese sections are merely recollect ions of the main resu lts of [9] and conta in almost no pro ofs. Section 4 deals with th e obstruction theory for C n -structures and morp hisms, in b oth the symplectic and the nonsym p lectic cont exts. In Section 5 we p ro v e the t heorem men tioned ab o v e on the r elatio nship b et wee n symplectic and n onsymplectic C ∞ -algebras. T he cases of unital and non-unital symplectic C ∞ -algebras diﬀer slig h tly and w e treat eac h of them in their o wn detail. 1.1. Notation and con v en tions. W e adopt the n otatio n an d conv ent ions of [9]. Thr oughout the pap er our ground ring K will b e an ev enly graded comm u tativ e ring con taining the ﬁeld Q . K -algebras and K -mo dules will simply b e called algebras and mo du les. W e will assume that all of our K -mo dules are obtained from k -ve ctor spaces b y extension of scalars where k = Q or an y other sub ﬁ eld of K . All of our tensors will b e tak en ov er the ground ring K unless stated otherwise. Giv en a graded mo du le V we deﬁn e the tensor algebra T V by T V := K ⊕ V ⊕ V ⊗ 2 ⊕ . . . ⊕ V ⊗ n ⊕ . . . . W e deﬁne the free Lie algebra LV as the Lie su balgebra of th e commutato r algebra T V whic h consists of linear com binations of Lie monomials. W e will u se b to denote completion. Giv en a p roﬁnite graded mo du le V w e can deﬁne the completed (or pro-free) versions T V and L V . F or instance th e p ro-free asso ciativ e algebra w ould b e b T V := ∞ Y i =0 V ˆ ⊗ i where ˆ ⊗ denotes the completed tensor pro d uct. The pro-free Lie algebra b LV is the Lie subal- gebra of b T V consisting of all conv ergen t (p ossibly u ncoun tably) in ﬁnite linear com b inations of Lie monomials. Giv en a pr oﬁnite graded mo dule V , the Lie algebra consisting of all c ontinuous deriv ations ξ : b LV → b LV will b e denoted by Der( b LV ). In order to emph asise our use of geometrical i deas in this pap er we will us e the term ‘vect or ﬁeld’ s y n on ymously with ‘con tin u ou s deriv ation’. The group consisting of all inv ertible c ontinuous Lie algebra homomorphisms φ : b LV → b LV 2 will b e denoted by Aut( b LV ). Again, in ord er to emphasise the geometrical approac h w e will call an ‘in ve rtible cont in uous homomorph ism ’ o f formal graded comm utative , asso ciativ e or Lie algebras a ‘diﬀeomorph ism’. Giv en a proﬁnite graded mo d ule V we can place a grading on b T V which is diﬀeren t from the grading wh ic h is naturally inherited from V . W e say an elemen t x ∈ b T V has homogeneous or der n if x ∈ V ˆ ⊗ n . This grading naturally descends to the su bmo du le b LV . W e say a contin uous end omorphism (linear map) f : b LV → b LV has homogeneous or der n if it tak es any elemen t of ord er i to an element of order i + n − 1. An y v ector ﬁ eld ξ ∈ De r( b LV ) could b e written in the form (1.1) ξ = ξ 1 + ξ 2 + ξ 3 + . . . + ξ n + . . . , where ξ i is a vec tor ﬁeld of order i . Lik ewise any diﬀeomorphism φ ∈ Aut( b LV ) could b e wr itten in the form φ = φ 1 + φ 2 + φ 3 + . . . + φ n + . . . , where φ i is an endomorph ism of order i . W e call φ a p ointe d diﬀe omorphism if φ 1 = id. Our conv en tion will b e to alwa ys w ork w ith cohomolo gically graded ob jects and w e conse- quen tly deﬁne the su s p ension of a graded mo dule V as Σ V where Σ V i := V i +1 . W e d eﬁ ne the desusp ension of V as Σ − 1 V where Σ − 1 V i := V i − 1 . Th e term ‘diﬀeren tial graded algebra’ w ill b e abbreviated as ‘DGA’. Giv en a graded mo du le V , w e d en ote the graded K -linear dual Hom K ( V , K ) b y V ∗ . F or the sake of clarit y , w hen we write Σ V ∗ w e mean the graded mo du le Hom K (Σ V , K ). 2. Noncom mut a tive geometr y 2.1. F ormal noncomm utative geometry . This section revie ws th e basics of formal n oncom- m utativ e Lie calculus including the de Rham complex, the Lie deriv ativ e along a vec tor ﬁeld and related constru ctions. Deﬁnition 2.1. Let W b e a free graded mo d ule. Consid er the mo dule b T ( W ∗ ) ˆ ⊗ b L ( W ∗ ) and write x ˆ ⊗ y = x ˆ ⊗ dy . Then the mo dule of Lie 1-forms Ω 1 ( W ) is deﬁned as the quotient of b T ( W ∗ ) ˆ ⊗ b L ( W ∗ ) b y the relations x ˆ ⊗ d [ y , z ] = xy ˆ ⊗ dz − ( − 1) | z || y | xz ˆ ⊗ dy . This is a left b L ( W ∗ )-mo dule via the action x · ( y ˆ ⊗ dz ) := xy ˆ ⊗ dz . Deﬁnition 2.2. L et W b e a fr ee graded mo dule. The modu le of Lie f orm s Ω • ( W ) is deﬁn ed as Ω • ( W ) := b L ( W ∗ ) ⋉ b L (Σ − 1 Ω 1 ( W )) where the action of b L ( W ∗ ) on b L (Σ − 1 Ω 1 ( W )) is the restriction of the standard action of b L ( W ∗ ) on b T (Σ − 1 Ω 1 ( W )) to the Lie subalgebra of L ie monomials. The m ap d : b L ( W ∗ ) → Ω 1 ( W ) lifts uniquely to give Ω • ( W ) the stru cture of a formal DGLA. R emark 2.3 . This construction is functorial; i.e. give n fr ee graded mo dules V and W and a homomorphism of L ie algebras φ : b L ( V ∗ ) → b L ( W ∗ ) (in particular, an y linear map from W to V giv es rise to such a map), there is a un ique map φ ∗ : Ω • ( V ) → Ω • ( W ) extendin g φ to a homomorphism of d iﬀeren tial graded Lie algebras. Deﬁnition 2.4. Let W b e a fr ee graded mo dule and let ξ : b L ( W ∗ ) → b L ( W ∗ ) b e a vec tor ﬁ eld: 3 (i) W e can deﬁne a vect or ﬁeld L ξ : Ω • ( W ) → Ω • ( W ), called the L ie deriv ativ e, by the form ulae; L ξ ( x ) := ξ ( x ) , x ∈ b L ( W ∗ ); L ξ ( dx ) := ( − 1) | ξ | d ( ξ ( x )) , x ∈ b L ( W ∗ ) . (ii) W e can deﬁne a v ector ﬁeld i ξ : Ω • ( W ) → Ω • ( W ) of bidegree ( − 1 , | ξ | ), calle d the con traction op erator, by the formulae ; i ξ ( x ) := 0 , x ∈ b L ( W ∗ ); i ξ ( dx ) := ξ ( x ) , x ∈ b L ( W ∗ ) . These op erators satisfy certain imp ortan t identitie s wh ic h are su mmarised by the follo wing lemma. Lemma 2.5. L et V and W b e fr e e gr ade d mo dules. L e t ξ , γ : b L ( V ∗ ) → b L ( V ∗ ) b e ve ctor ﬁelds and let φ : b L ( V ∗ ) → b L ( W ∗ ) b e a diﬀe omorph ism, then we have the fol lowing identities: (i) L ξ = [ i ξ , d ] . (ii) [ L ξ , i γ ] = i [ ξ ,γ ] . (iii) L [ ξ ,γ ] = [ L ξ , L γ ] . (iv) [ i ξ , i γ ] = 0 . (v) [ L ξ , d ] = 0 . (vi) L φξ φ − 1 = φ ∗ L ξ φ ∗− 1 . (vii) i φξ φ − 1 = φ ∗ i ξ φ ∗− 1 .  Deﬁnition 2.6. Let W b e a fr ee graded mo d ule. The de Rham complex D R • ( W ) is deﬁned as the quotient of Ω • ( W ) ˆ ⊗ Ω • ( W ) b y the relations x ˆ ⊗ y = ( − 1) | x || y | y ˆ ⊗ x ; x, y ∈ Ω • ( W ); x ˆ ⊗ [ y , z ] = [ x, y ] ˆ ⊗ z ; x, y , z ∈ Ω • ( W ) . The d iﬀeren tial d : D R • ( W ) → D R • ( W ) is induced by th e d iﬀeren tial d : Ω • ( W ) → Ω • ( W ) b y sp ecifying d ( x ˆ ⊗ y ) := dx ˆ ⊗ y + ( − 1) | x | x ˆ ⊗ dy ; x, y ∈ Ω • ( W ) . R emark 2.7 . It follo w s from Remark 2.3 th at the constru ction of the d e Rh am complex is also functorial; that is to sa y that if φ : b L ( V ∗ ) → b L ( W ∗ ) is a homomorphism of algebras then it induces a morph ism of complexes φ ∗ : D R • ( V ) → D R • ( W ) deﬁn ed by the f orm ula φ ∗ ( x ˆ ⊗ y ) := φ ∗ ( x ) ˆ ⊗ φ ∗ ( y ); x, y ∈ Ω • ( V ) . F urthermore the Lie and con traction op er ators L ξ , i ξ : Ω • ( V ) → Ω • ( V ) d eﬁned by Deﬁnition 2.4 factor naturally through Lie and con tr action op erators L ξ , i ξ : D R • ( V ) → D R • ( V ), w h ic h b y an abu se of notation, we d enote by the same letters. Th e Lie op erator is giv en by the form ula L ξ ( x ˆ ⊗ y ) := L ξ ( x ) ˆ ⊗ y + ( − 1) | x | x ˆ ⊗ L ξ ( y ); x, y ∈ Ω • ( V ) . The contract ion op erator is deﬁned in the same w a y . 2.2. F ormal noncomm utative symplectic geometry. W e w ill b egin by r ecalling the basic terminology of symp lectic geometry . Deﬁnition 2.8. Let W b e a free graded mo dule. A homogeneous 2-form ω ∈ D R 2 ( W ) is nondegenerate if and only if the map Φ : Der( b L ( W ∗ )) → D R 1 ( W ) of degree | ω | deﬁ ned by the form ula (2.1) Φ( ξ ) := i ξ ( ω ) , ξ ∈ Der( b L ( W ∗ )) is bij ectiv e. F urthermore, w e say that ω is a sym plectic f orm if it is also closed (i.e. dw = 0). 4 W e say a 2-form ω ∈ D R 2 ( W ) is a c onstan t 2-form if it can b e written in the form ω = P i dy i dz i for some fu nctions y i , z i ∈ W ∗ . Prop osition 2.9. L et W b e a fr e e gr ade d mo dule. Ther e exists a map κ : { ω ∈ D R 2 ( W ) : ω is c onstant } → (Λ 2 W ) ∗ deﬁne d by the formula κ ( dxdy ) := ( − 1) x [ x ⊗ y − ( − 1) xy y ⊗ x ] which pr ovides an isomor phism b etwe en c onstant 2-forms and skew-symmetric biline ar forms on W . F u rthermor e, a c onstant 2-form ω ∈ D R 2 ( W ) is nonde gener ate if and only if the c orr esp onding biline ar form h− , −i := κ ( ω ) is nonde gener ate.  Deﬁnition 2.10. Let V and W b e f ree graded mo du les and let ω ∈ D R 2 ( V ) and ω ′ ∈ D R 2 ( W ) b e homogeneous s y m plectic forms: (i) W e s a y a vec tor ﬁeld ξ : b L ( V ∗ ) → b L ( V ∗ ) is a symple ctic ve ctor ﬁeld if L ξ ( ω ) = 0. (ii) W e say a diﬀeomorphism φ : b L ( V ∗ ) → b L ( W ∗ ) is a symple ctomor phism if φ ∗ ( ω ) = ω ′ . W e ha v e the f ollo wing simple pr op osition ab out symp lectic vec tor ﬁelds which can b e found in [4 ]: Prop osition 2.11. L et W b e a fr e e gr ade d mo dule and let ω ∈ D R 2 ( W ) b e a symple ctic form. Then the map Φ deﬁne d by e qu ation (2 .1) induc es a one-to-one c orr esp ondenc e b etwe en symplectic ve ctor ﬁelds and closed 1-forms: Φ : { ξ ∈ Der ( b L ( W ∗ )) : L ξ ( ω ) = 0 } → { α ∈ D R 1 ( W ) : dα = 0 } .  3. C ∞ -algebras and rel a t ed cohomology theories In this s ection we consider the n otion of a C ∞ -algebra and that of a symp lectic C ∞ -algebra. Our presenta tion w ill b e based on the n oncomm u tativ e Lie geometry review ed in the previous section. 3.1. Main deﬁnitions. Deﬁnition 3.1. Let V b e a free graded mo du le. A C ∞ -structure on V is a v ector ﬁeld m : b L Σ V ∗ → b L Σ V ∗ of degree one, su c h that m 2 = 0. Deﬁnition 3.2. Let V and U b e free graded mo du les. Let m and m ′ b e C ∞ -structures on V and U resp ectiv ely . A C ∞ -morphism f r om V to U is a con tin uou s algebra h omomorphism φ : b L Σ U ∗ → b L Σ V ∗ of degree zero su c h that φ ◦ m ′ = m ◦ φ . Deﬁnition 3.3. Let V b e a free graded mo du le: (1) W e say that a C ∞ -structure m : b L Σ V ∗ → b L Σ V ∗ is unital if ther e is a distinguished elemen t 1 ∈ V (the unit) of degree zero which can b e extended to a basis 1 , { x i } i ∈ I of V su ch that m h as the form; (3.1) m = A ( t ) ∂ τ + X i ∈ I B i ( t ) ∂ t i + ad τ − τ 2 ∂ τ , where τ , t := τ , { t i } i ∈ I is the top ological basis of Σ V ∗ whic h is dual to the basis Σ1 , { Σ x i } i ∈ I of Σ V and A ( t ) , { B i ( t ) } i ∈ I are Lie p o w er series in t . In this case w e sa y that the C ∞ -algebra V is u nital or that it has a unit. 5 (2) Supp ose that V and U are tw o unital C ∞ -algebras. W e sa y that a C ∞ -morphism φ : b L Σ U ∗ → b L Σ V ∗ is un ital if φ has the form ; φ ( τ ′ ) = τ + A ( t ) , φ ( t ′ i ) = B i ( t ); where τ , t an d τ ′ , t ′ are the top ologica l bases of Σ V ∗ and Σ U ∗ whic h are dual to the bases Σ1 V , { Σ x i } i ∈ I and Σ1 U , { Σ x ′ j } j ∈ J of Σ V and Σ U resp ectiv ely . 3.2. Cohomology theories. There are cohomology theories asso ciated to C ∞ -algebras whic h are the h ome for v arious obstruction groups. These theories are called the Harrison c ohomolo gy theories. In the case of a u nital C ∞ -algebra, the relev ant cohomology th eory is called normalise d Harrison cohomology . 3.2.1. Unnormalise d the ories. Deﬁnition 3.4. Let V b e a fr ee graded mo du le. Let m : b L Σ V ∗ → b L Σ V ∗ b e a C ∞ -structure. The Harrison complex of the C ∞ -algebra V with co eﬃcien ts in V is deﬁned on the mo du le consisting of all vec tor ﬁelds on b L Σ V ∗ : C • Harr ( V , V ) := Σ − 1 Der( b L Σ V ∗ ) . The d iﬀeren tial d : C • Harr ( V , V ) → C • Harr ( V , V ) is give n by d ( ξ ) := [ m, ξ ] , ξ ∈ Der( b L Σ V ∗ ) . The Harrison cohomology of V with co eﬃcien ts in V is deﬁn ed as the cohomolo gy of the complex C • Harr ( V , V ) and denoted by H • Harr ( V , V ). R emark 3.5 . The pur p ose of the desusp en s ions in the ab o v e deﬁ nition is to mak e the grading consisten t with th e classical grading on these cohomolog y theories. Deﬁnition 3.6. Let V b e a free graded m o dule and m : b L Σ V ∗ → b L Σ V ∗ b e a C ∞ -structure. The Harrison complex of the C ∞ -algebra V with coeﬃcien ts in V ∗ is deﬁned on the mo du le consisting of all 1-forms: C • Harr ( V , V ∗ ) := Σ D R 1 (Σ V ) . The d iﬀeren tial on this complex is the (su sp ension of th e) Lie op erator of the vecto r ﬁ eld m ; L m : D R 1 (Σ V ) → D R 1 (Σ V ) . The Harrison cohomology of V with co eﬃcien ts in V ∗ is deﬁn ed as the cohomology of th e complex C • Harr ( V , V ∗ ) and denoted by H • Harr ( V , V ∗ ). W e no w deﬁne the app ropriate cyclic cohomolog y theory; it is closely related to symplectic structures as we will see later on. Deﬁnition 3.7. Let V b e a fr ee graded mo du le. Let m : b L Σ V ∗ → b L Σ V ∗ b e a C ∞ -structure. The cyclic Harrison complex of the C ∞ -algebra V is deﬁned on the mo du le consisting of all 0-forms: C C • Harr ( V ) := Σ D R 0 (Σ V ) . The d iﬀeren tial on this complex is the (su sp ension of th e) Lie op erator of the vecto r ﬁ eld m ; L m : D R 0 (Σ V ) → D R 0 (Σ V ) . The cyclic Harrison cohomology of V is deﬁ n ed as the cohomology of the complex C C • Harr ( V ) and is denoted by H C • Harr ( V ). The m ain result on the cyclic versus non-cyclic Harrison cohomology of C ∞ -algebras is that they essen tially coincide, as p ro v ed in [9]. W e will u s e the follo wing v ersion of this result con- tained in the cited reference, see also [14]. Note that the noncommutati v e de Rham diﬀerentia l induces a map I : C C • Harr ( V ) → C • Harr ( V , V ∗ ). If V is a strictly graded commutativ e algebra 6 then b oth C C • Harr ( V ) and C • Harr ( V , V ∗ ) are bigraded ; we s ay an elemen t h as homogeneous bid e- gree ( i, j ) if it has order i and d egree j , wh ere a constant 1-form or function is considered to ha v e order zero. Theorem 3.8. L et V b e a unital strictly gr ade d c ommutative algebr a. The map I : H C i +1 ,j Harr ( V ) → H ij Harr ( V , V ∗ ) is; (i) a monom orphism if i = 1 , (ii) an e pimorphism if i = 2 , (iii) an isomorphism if i ≥ 3 .  3.2.2. Normalise d the ories. W e will n o w p ro vide the analogues of the d eﬁ nitions fr om th e p re- vious subsection for the case of unital C ∞ -algebras. Let V b e a unital C ∞ -algebra and let τ , t b e a top ological b asis of Σ V ∗ where τ is dual to the un it 1 ∈ V . W e say a ve ctor ﬁeld ξ ∈ Der ( b L Σ V ∗ ) is norm alised if it has the form m = A ( t ) ∂ τ + X i ∈ I B i ( t ) ∂ t i . W e will den ote the Lie sub algebra of Der( b L Σ V ∗ ) consisting of all norm alised v ector ﬁelds by Der( b L Σ V ∗ ). Deﬁnition 3.9. Let V b e a un ital C ∞ -algebra. The subs pace of C • Harr ( V , V ) giv en b y C • Harr ( V , V ) := Σ − 1 Der( b L Σ V ∗ ) forms a sub complex of C • Harr ( V , V ) whic h w e refer to as th e sub complex of normalised vec tor ﬁelds. No w consider the m o dule of 1-forms D R 1 (Σ V ) . W e sa y a 1-form α ∈ D R 1 (Σ V ) is normalised if it has the form α = A ( t ) ˆ ⊗ dτ + X i ∈ I B i ( t ) ˆ ⊗ dt i . W e will d enote the su bspace of D R 1 (Σ V ) consisting of all n ormalised 1-forms by D R 1 (Σ V ) . Deﬁnition 3.10. Let V b e a unital C ∞ -algebra. The subs pace of C • Harr ( V , V ∗ ) giv en by C • Harr ( V , V ∗ ) := Σ D R 1 (Σ V ) . forms a s u b complex of C • Harr ( V , V ∗ ) which w e refer to as the su b complex of normalised Lie 1-forms. Finally , consider the mod ule of 0-forms D R 0 (Σ V ) . W e sa y a 0-form α ∈ D R 0 (Σ V ) is nor- malised if it has the form α = X i ∈ I B i ( t ) ˆ ⊗ t i . W e will d enote the su bspace of D R 0 (Σ V ) consisting of all n ormalised 0-forms by D R 0 (Σ V ) . Deﬁnition 3.11. Let V b e a unital C ∞ -algebra. The subs pace of C C • Harr ( V ) given by C C • Harr ( V ) := Σ h D R 0 (Σ V ) i forms a sub complex of C C • Harr ( V ), which we refer to as the su b complex of normalised Lie 0-forms. 7 3.3. Symplectic C ∞ -algebras. W e w ill no w recall (cf. [11] and [13]) the deﬁnition of a C ∞ - algebra with an inv ariant in ner p ro duct. W e will call it a symple ctic C ∞ -algebra in view of its in terpretation as a formal symplectic s up ermanifold together with an o d d Hamiltonian v ector ﬁeld. W e will also deﬁne the appropr iate cohomology theory in th e sym p lectic con text and show that it essen tially reduces to the cyclic theory . Note that the corresp onding results also hold for A ∞ -algebras (whic h we do not consider in the p resen t pap er), see [17]. Recall that C ∞ -structures on a free graded mo dule V can b e describ ed in terms of s y s tems of maps ˇ m i : V ⊗ i → V , i ≥ 1 satisfying certain h igher homotopy axioms: Deﬁnition 3.12. Let V b e a free grad ed mo d ule of ﬁnite rank and let h− , − i : V ⊗ V → K b e an inn er pr o duct on V . Let ˇ m i : V ⊗ i → V , i ≥ 1 b e a system of maps determining a C ∞ -structure on V . W e say that this structur e is inv ariant with r esp ect to the inner pr o duct h− , −i if the follo wing iden tit y holds for all x 0 , . . . , x n ∈ V , (3.2) h ˇ m n ( x 1 , . . . , x n ) , x 0 i = ( − 1) n + | x 0 | ( | x 1 | + ... + | x n | ) h ˇ m n ( x 0 , . . . , x n − 1 ) , x n i and w e say that V is a C ∞ -algebra with an inv ariant in n er pro d uct. The follo wing result is chec k ed b y a straigh tforw ard calculation taking place in the completed tensor algebra of Σ V ∗ : Prop osition 3.13. L et V b e a fr e e gr ade d mo dule of ﬁnite r ank. L et (3.3) ˇ m i : V ⊗ i → V , i ≥ 1 b e a C ∞ -structur e on V and let h− , − i : V ⊗ V → K b e an nonde gener ate skew-symmetric biline ar form on V . Consider the ve ctor ﬁeld m ∈ Der( b L Σ V ∗ ) c orr esp onding to the C ∞ -structur e and c onsider the symple c tic f orm ω ∈ D R 2 (Σ V ) c orr esp onding to the inner pr o duct h− , −i . The C ∞ -structur e (3.3) is invariant with r esp e ct to the inner pr o duct h− , −i if and only if L m ω = 0 .  This resu lt motiv ates the follo w in g deﬁnition of a symple ctic C ∞ -algebra: Deﬁnition 3.14. Let V b e a fr ee graded mo dule of ﬁ nite rank. A symple ctic C ∞ -structure on V is a symp lectic form ω ∈ D R 2 (Σ V ) together with a symple ctic vec tor ﬁeld m : b L Σ V ∗ → b L Σ V ∗ of degree one, su c h that m 2 = 0. Deﬁnition 3.15. Let V and U b e fr ee graded mo dules of ﬁnite ranks. Let ( m, ω ) and ( m ′ , ω ′ ) b e symple ctic C ∞ -structures on V and U r esp ectiv ely . A symple ctic C ∞ -morphism fr om V to U is a conti n uous algebra homomorph ism φ : b L Σ U ∗ → b L Σ V ∗ of degree zero su c h that φ ◦ m ′ = m ◦ φ and φ ∗ ( ω ′ ) = ω . W e h a ve the f ollo wing us eful lemmas regarding th e cohomology of a symp lectic C ∞ -algebra: Lemma 3.16. L et ( V , m, ω ) b e a symplectic C ∞ -algebr a. The map Φ deﬁne d by e quation (2.1) is an isomorphism of c omplexes: Φ : C • Harr ( V , V ) → Σ | ω |− 2 C • Harr ( V , V ∗ ) . 8 Pr o of. By Deﬁn ition 2.8 th e map Φ m ust b e bij ectiv e. It on ly remains to prov e that this map comm u tes with the diﬀerent ials. This will follo w from Lemma 2.5. Let ξ ∈ Der( b L Σ V ∗ ) b e a v ecto r ﬁ eld, then Φ(ad m ( ξ )) = i [ m,ξ ] ( ω ) = [ L m , i ξ ]( ω ) , = L m i ξ ( ω ) = L m Φ( ξ ) .  W e n eed an analogue of Lemma 3.16 in the con text of the cyclic theory: Lemma 3.17. L et ( V , m, ω ) b e a symplectic C ∞ -algebr a. The Lie sub algebr a of C • Harr ( V , V ) :=  Σ − 1 Der( b L Σ V ∗ ) , ad m  c onsisting of al l s ymplectic ve ctor ﬁelds f orms a sub c omplex denote d b y S C • Harr ( V , V ) . F urther- mor e , ther e is an isomor phism Υ : Σ | ω |− 2 C C • Harr ( V ) → S C • Harr ( V , V ) deﬁne d by the formula, (3.4) Υ( α ) := ( − 1) | α | Φ − 1 d ( α ); wher e Φ is the map deﬁne d by e quation (2.1) . Pr o of. Since the C ∞ -structure m is a symp lectic v ector ﬁ eld, the Lie sub algebra of symplectic v ecto r ﬁ elds do es indeed form a su b complex as claimed. Lemma 2.5 part (v) and Lemma 3.16 tell us th at Υ is a map of complexes, i.e. it resp ects the diﬀerential s. It remains to prov e that Υ is b ijectiv e; this follo ws f r om Prop osition 2.11 and the (formal) Poincar ´ e lemma.  W e will also need one for the normalised cyclic th eory: Lemma 3.18. L et ( V , m, ω ) b e a u nital sym p lectic C ∞ -algebr a whose symple ctic form ω is constan t and denote the sub c omplex of C • Harr ( V , V ) c onsisting of al l normalised symplectic ve ctor ﬁelds by S C • Harr ( V , V ) . The map Υ deﬁne d by e quation (3.4) r estricts to a map ¯ Υ : Σ | ω |− 2 C C • Harr ( V ) → S C • Harr ( V , V ) which is also an isomorphism. Pr o of. It is clear that the map Φ of Lemma 3. 16 restricts to an isomorph ism b etw een the sub complex of normalised v ector ﬁ elds C • Harr ( V , V ) and the sub complex of normalised 1-forms C • Harr ( V , V ∗ ); hence, it remains to s ho w that an y closed norm alised 1-form is the b ound ary of a normalised 0-form. Let τ , t b e a top ological basis of Σ V ∗ and let Σ ¯ V ∗ b e th e su mmand generated b y the t i . It follo ws f r om the resu lts of [9] that there exists the follo wing commutat iv e diagram: D R 0 (Σ V ) d / /  _ l   D R 1 (Σ V ) d / /  _ l   D R 2 closed (Σ V ) ζ   Q ∞ n =2 [(Σ ¯ V ∗ ) ˆ ⊗ n ] Z /n Z N / / O O O O Q ∞ n =2 (Σ ¯ V ∗ ) ˆ ⊗ ( n − 1) ˆ ⊗ Σ V ∗ 1 − z / / O O O O Q ∞ n =2 (Σ V ∗ ) ˆ ⊗ n The leftmost ve rtical sp litting is giv en by equation (8.2). Th e middle v ertical sp litting is giv en b y equation (7.6). T h e map l is giv en b y equation (3.5). T he map ζ is d eﬁ ned b y Th eorem 3.5. The fact that the t w o vertica l s p littings commute f ollo ws from Corollary 8.2 (i). Th e map N is the norm m ap and z is a cyclic p ermutatio n. Since the b ottom ro w is clearly exact, the Lemma no w follo ws.  9 4. Finite level st ructures and Obs truction theor y 4.1. C n -structures. In this section we deﬁne C n -algebras and discus s the problem of lifting a C n -structure to a C n +1 -structure. The obstru ction theory of n -algebras is in principle known to the exp erts, but it is hard to ﬁnd an explicit reference in the literature, esp ecially in the C n -algebra case. T he pr ecursor for this t yp e of obstruction th eory is th e semin al w ork of A. Robinson, [18 ], although the main ideas go back to early wo rk on algebraic deformation th eory; [3], [1 6]. In the context of unstable h omotop y theory , somewhat similar constructions w ere emplo yed by Stasheﬀ in [21]. W e shall only consider the case of minimal C n -structures as in this case it is p ossible to giv e a conv enien t inte rpretation of obstructions in cohomologica l terms. Let V b e a free graded mo du le. An y ve ctor ﬁeld m ∈ Der( b L Σ V ∗ ) could b e written in the form m = m 1 + m 2 + . . . + m n + . . . where m i is a v ector ﬁ eld of order i . Sin ce we often need to work mo d ulo the endomorphisms of b L Σ V ∗ of order ≥ n , we will denote the mo d u le of suc h end omorphisms by ( n ). 4.1.1. Obstruction the ory for C n -algebr a structur es. In this sub s ection we will dev elop the ob- struction theory f or C n -algebra structures. Let us b egin with a deﬁn ition: Deﬁnition 4.1. Let V b e a free graded mo dule. A minimal C n -structure ( n ≥ 3) on V is a v ecto r ﬁ eld m ∈ Der( b L Σ V ∗ ) of degree one which has th e f orm m = m 2 + . . . + m n − 1 , m i has ord er i and satisﬁes the condition m 2 = 0 mo d ( n + 1). Let m and m ′ b e t w o minimal C n -structures on V . W e say m and m ′ are equiv alen t if there is a diﬀeomorph ism φ ∈ Aut( b L Σ V ∗ ) of the form (4.1) φ = id + φ 2 + φ 3 + . . . + φ k + . . . where φ i is an endomorph ism of order i , su c h that φ ◦ m ◦ φ − 1 = m ′ mo d ( n ) . R emark 4.2 . Recall that we refer to diﬀeomorphism s of the form (4.1) as p ointe d diﬀe omor- phisms . R emark 4.3 . Note that this d eﬁnition is s ligh tly at o dds with the deﬁnition of n -algebras (sp ecif- ically A n -algebras) giv en by Stasheﬀ in [19 ] and [20]. Every minimal C n -algebra is a C n -algebra under Stasheﬀ ’s deﬁn ition, h o wev er, giv en a minimal C n -structure m = m 2 + . . . + m n − 1 and an arbitrary vecto r ﬁeld m n of ord er n , m ′ := m + m n is a C n -algebra un d er Stasheﬀ ’s deﬁnition whic h is ob viously not a minimal C n -algebra. This distinction will b e necessary in order to dev elop an obstr uction theory . R emark 4.4 . Clearly if m = m 2 + . . . + m n − 1 is a minimal C n -algebra then m 2 determines th e structure of a strictly graded comm utativ e algebra on the underlying free graded mo du le V whic h w e will call the un derlying commutati v e alge bra. Ob serv e that t w o equiv alen t minimal C n -structures ha v e the same und erlying comm u tativ e algebra. Since the underlyin g algebra A := ( V , m 2 ) is a strictl y graded co mm utativ e algebra, C • Harr ( A, A ) can b e giv en a bigrading: we sa y a ve ctor ﬁ eld ξ ∈ C • Harr ( A, A ) has bidegree ( i, j ) if it is a vect or ﬁeld of order i and has degree j as an elemen t in the proﬁn ite graded mo du le Σ − 1 Der( b L Σ V ∗ ). The d iﬀeren tial on C • Harr ( A, A ) then has bidegree (1 , 1). It will b e useful to introd uce the follo wing deﬁn ition: Deﬁnition 4.5. L et A := ( V , µ 2 ) b e a s trictly graded commutativ e algebra, then the mo duli space of minimal C n -structures on V ﬁxin g µ 2 is denoted by C n ( A ) and deﬁn ed as th e quotien t of the set { m : b L Σ V ∗ → b L Σ V ∗ : m is a minimal C n -structure and m 2 = µ 2 } b y the equiv alence r elatio n deﬁned in Deﬁnition 4.1. 10 W e w ill now describ e the appropriate term in ology whic h is necessary in discussin g extensions of C n -structures to structures of higher ord er: Deﬁnition 4.6. L et V b e a fr ee graded mo du le and let m = m 2 + . . . + m n − 1 b e a minimal C n -structure on V . W e sa y that m is an extendable C n -structure if there exists a v ector ﬁeld m n ∈ Der ( b L Σ V ∗ ) of order n and degree one su c h that m + m n is a C n +1 -structure on V and w e call m n an extension of m . Let m n and m ′ n b e tw o extensions of m . W e s ay that m n and m ′ n are equiv alent if ther e exists a d iﬀeomorphism φ ∈ Aut( b L Σ V ∗ ) of the form φ = id + φ n − 1 + φ n + . . . + φ n + k + . . . where φ i is an endomorph ism of order i , su c h that (4.2) φ ◦ ( m + m n ) ◦ φ − 1 = m + m ′ n mo d ( n + 1 ) . The qu otien t of the set of all extensions of m b y this equiv alence relation will b e denoted by E n ( m ). R emark 4.7 . Let m n and m ′ n b e tw o equiv alen t extensions of m , then th er e is a diﬀeomorphism φ satisfying (4.2) and a ve ctor ﬁeld ξ of order n − 1 and degree zero such that φ has the form φ = id + ξ + endomorphism s of ord er ≥ n . F rom this we conclude that m ′ n = m n + [ ξ , m 2 ]. Con v ersely if there exists a ve ctor ﬁeld ξ of order n − 1 and d egree zero su c h that m ′ n = m n + [ ξ , m 2 ] then the diﬀeomorphism φ := exp( ξ ) satisﬁes (4.2). This means that t w o extensions of m are equiv alen t if and only if their diﬀerence is a Harrison cob oun dary . Deﬁnition 4.8. Let V b e a free graded m o dule and let m = m 2 + . . . + m n − 1 b e a C n -structure on V . W e deﬁn e the v ector ﬁ eld Ob s( m ) of order n + 1 and d egree 2 by (4.3) Obs( m ) := 1 2 X i + j = n +2 3 ≤ i,j ≤ n − 1 [ m i , m j ] . W e will n o w formulate and pro v e a result analogous to theorems 7.14 and 7.15 of [5]: Theorem 4.9. L et A := ( V , m 2 ) b e a strictly gr ade d c ommutative algebr a. F or al l n ≥ 3 (4.3) induc es a map C n ( A ) → H n +1 , 3 Harr ( A, A ) , m 7→ Obs( m ); which we wil l denote by Obs n . The kernel of this map c onsists of pr e ci sely those C n -structur es which ar e extendable: { m ∈ C n ( A ) : m is extendable } = ker(Obs n ) . Pr o of. First of all let us s ho w that giv en an y C n -structure m = m 2 + . . . + m n − 1 on V , Obs( m ) is a co cycle in C • Harr ( A, A ). Let us d eﬁne the vecto r ﬁeld m by the formula m := m 3 + . . . + m n − 1 so that we ha v e m = m 2 + m . By the deﬁn ition of Obs( m ) (cf. equ ation (4.3)) we ha v e the equalit y m 2 = Obs( m ) mo d ( n + 2 ) . It follo ws that [ m 2 , O bs( m )] = 1 2 [ m 2 , [ m, m ]] = [[ m 2 , m ] , m ] mo d ( n + 3) . 11 Since A := ( V , m 2 ) is a strictly graded comm utativ e algebra, we kno w th at [ m 2 , m 2 ] = 0. In addition, since m is a C n -structure w e kn ow that [ m 2 , m ] + 1 2 [ m, m ] = 0 mo d ( n + 1 ) . F rom these observ ations it follo ws th at [ m 2 , O bs( m )] = [[ m 2 , m ] , m ] = [[ m 2 , m ] , m ] = − 1 2 [[ m, m ] , m ] = 0 m o d ( n + 3) , where the last equality follo ws from the Jacobi iden tit y , hence O b s( m ) is a co cycle as claimed. Next we need to sho w that if m = m 2 + m 3 + . . . + m n − 1 and m ′ = m 2 + m ′ 3 + . . . + m ′ n − 1 are t w o equiv alent C n -structures on V then Ob s( m ) and Obs( m ′ ) are cohomologous Harrison co cycles. Since m and m ′ are equiv alen t there exists a p ointe d diﬀeomorphism φ ∈ Au t( b L Σ V ∗ ) suc h that m ′ = φ ◦ m ◦ φ − 1 mo d ( n ) . This means ther e exists a v ector ﬁeld ξ of order n and degree 1 such that m ′ = φ ◦ m ◦ φ − 1 + ξ mo d ( n + 1) . Recall again that by the deﬁnition of Ob s( m ) we ha v e the equalit y m 2 = Obs( m ) mo d ( n + 2) and lik ewise w e ha v e the s ame equalit y for m ′ . No w from the calculation m ′ 2 = φ ◦ m 2 ◦ φ − 1 + [ m 2 , ξ ] mo d ( n + 2 ) , = Obs( m ) + [ m 2 , ξ ] mo d ( n + 2 ); w e conclude that Ob s( m ′ ) = Obs( m ) + [ m 2 , ξ ]. So far we hav e p ro ven that the map Ob s n is we ll d eﬁned. I n ord er to ﬁ n ish the pro of we need to sh o w that a C n -structure m = m 2 + . . . + m n − 1 on V is extend able if and only if Obs( m ) is cohomologo us to zero. The r eason that this is true is b ecause for an y ve ctor ﬁeld m n of order n and d egree 1 the follo wing iden tit y holds: (4.4) ( m + m n ) 2 = Obs( m ) + [ m 2 , m n ] mo d ( n + 2) .  The n ext resu lt, also reminiscent of deformation theory , analyses d iﬀerent extensions of a C n -structure in terms of Harrison cohomology: Theorem 4.10. L et A := ( V , m 2 ) b e a strictly gr ade d c ommutative algebr a and let m ∈ C n ( A ) b e an extendable C n -structur e, then H n, 2 Harr ( A, A ) acts fr e ely and tr ansitively on E n ( m ) : H n, 2 Harr ( A, A ) × E n ( m ) → E n ( m ) , ( ξ n , m n ) 7→ m n + ξ n . Pr o of. By equation (4.4) w e see that m n is an extension of m if and only if (4.5) [ m 2 , m n ] = − Ob s( m ) . This means that if m n is an extension of m and ξ n is a Harrison cocycle then m n + ξ n is an extension of m . F urthermore if m n and m ′ n are equiv alent extensions and ξ n and ξ ′ n are cohomologo us co cycles then by Remark 4.7, m n + ξ n and m ′ n + ξ ′ n are equiv alen t extensions of m , therefore the ab o ve action is well d eﬁned. Condition (4.5 ) sh o ws us that if m n and m ′ n are tw o extensions of m then m n − m ′ n is a Harrison co cycle and hence the ab o ve action is transitiv e. F u rthermore if m n is an extension of m and ξ n is a Harrison co cycle s uc h that m n and m n + ξ n are equiv alen t extensions then by Remark 4.7 , ξ n is a Harrison cob oundary , thus the ab o v e action is free.  12 4.1.2. Obstruction the ory for C n -algebr a morphisms. In this subsection w e will dev elop the obstruction theory f or morph isms b et we en t wo C n -algebras. Deﬁnition 4.11. Let V b e a free graded mo dule and let m and m ′ b e t wo minimal C n - structures on V . A minimal C n -morphism fr om m to m ′ is a diﬀeomorphism φ ∈ Aut( b L Σ V ∗ ) of degree zero su c h that φ ◦ m = m ′ ◦ φ mo d ( n ) . Let φ and φ ′ b e t wo s u c h min im al C n -morphisms. W e s ay that φ and φ ′ are h omotopic if there exists a ve ctor ﬁeld η of degree − 1 su c h that φ = φ ′ ◦ exp([ m, η ]) mo d ( n − 1) . R emark 4.12 . There is a Lie p ow er series p ( x, y ) := x + y + 1 2 [ x, y ] + . . . b elonging to the pro-free Lie algebra on t w o generators x an d y su c h that for t w o v ector ﬁelds γ and ξ of degree zero exp( p ( x, y )) = exp( x ) ◦ exp( y ) . It follo ws that homotop y is an equiv alence relation. W e will n o w introdu ce the mo du li space of C n -morphisms: Deﬁnition 4.13. Let V b e a free graded mod ule a nd let m and m ′ b e t wo minimal C n -structures on V with the same underlying algebra, that is to sa y that m 2 = m ′ 2 . The mo d uli space of minimal C n -morphisms f rom m to m ′ is denoted by M n ( m ; m ′ ) and d eﬁned as the qu otien t of the set { φ : b L Σ V ∗ → b L Σ V ∗ : φ is a p ointe d minimal C n -morphism f r om m to m ′ } b y the homotopy equiv alence relation deﬁned in Deﬁnition 4.11. Let m = m 2 + . . . + m n b e a C n +1 -structure on a free graded m o dule V . W e deﬁne the corresp onding C n -structure ¯ m on V as ¯ m := m 2 + . . . + m n − 1 . No w we introd u ce the terminology dealing with extensions of C n -morphisms: Deﬁnition 4.14. Let V b e a free graded m o dule and let m and m ′ b e tw o m inimal C n +1 - structures on V with the s ame und erlying algebra. W e sa y that a p oin ted C n -morphism φ from ¯ m to ¯ m ′ is extendable if th er e exists a vect or ﬁeld γ ∈ Der( b L Σ V ∗ ) of order n − 1 and degree zero su c h that exp( γ ) ◦ φ is a C n +1 -morphism f r om m to m ′ and w e call γ an extension of φ . Let γ and γ ′ b e t w o extensions of φ . W e sa y that γ and γ ′ are equiv alen t if there exists a v ecto r ﬁ eld η of order n − 2 and degree − 1 su c h that (4.6) exp( γ ) ◦ φ = exp( γ ′ ) ◦ φ ◦ exp([ m, η ]) mo d ( n ) . The quotien t of the s et of all extensions of φ by this equiv alence r elation will b e denoted b y E M n ( φ : m, m ′ ). R emark 4.15 . Let γ and γ ′ b e tw o extensions of φ and let η b e a vecto r ﬁ eld of order n − 2 and degree − 1, then equation (4.6) is satisﬁed if and only if γ = γ ′ + [ m 2 , η ] . It f ollo ws that t wo extensions are equiv alen t if and only if their diﬀerence is a Harrison cob ound- ary in the Harrison complex of the underlying algebra A := ( V , m 2 ). Next we will deﬁne the appropr iate obs truction to the extension of C n -morphisms: Deﬁnition 4.16. Let V b e a free graded m o dule and let m and m ′ b e tw o m inimal C n +1 - structures on V with the same underlying algebra. Let φ b e a p ointed C n -morphism f rom ¯ m to ¯ m ′ . W e deﬁne the ve ctor ﬁeld obs( φ ) of order n and degree 1 by (4.7) obs( φ ) := φ ◦ m ◦ φ − 1 − m ′ mo d ( n + 1) . 13 W e no w ha v e all the termin ology in place to formulate the analogues of theorems 4.9 and 4.10: Theorem 4.17. L et A := ( V , m 2 ) b e a strictly gr ade d c ommutative algebr a and let m and m ′ b e two minimal C n +1 -structur es ( n ≥ 3 ) on V whose underlying algebr a i s A , then (4.7 ) induc es a map M n ( ¯ m ; ¯ m ′ ) → H n, 2 Harr ( A, A ) , φ 7→ obs( φ ); which we wil l denote by obs n . The kernel of this map c onsists of pr e cisely those C n -morphisms which ar e extendable: { φ ∈ M n ( ¯ m ; ¯ m ′ ) : φ is extendable } = ke r(obs n ) . Pr o of. Let φ b e a p ointe d C n -morphism from ¯ m to ¯ m ′ , then by Th eorem 4.10 obs( φ ) is a co cycle in C • Harr ( A, A ) and φ is extendable if and only if obs( φ ) is cohomologo us to zero. W e need only sho w that if φ ′ is another p oin ted C n -morphism from ¯ m to ¯ m ′ whic h is homotopic to φ then obs( φ ) and obs ( φ ′ ) are cohomologous co cycles. Since φ and φ ′ are homotopic there exists a v ector ﬁeld γ of ord er n − 1 and degree 0 and a v ecto r ﬁ eld η of degree − 1 suc h th at φ = exp( γ ) ◦ φ ′ ◦ exp([ m, η ]) mo d ( n ) . Note that f or any ve ctor ﬁeld η of d egree − 1, exp([ m, η ]) ◦ m ◦ exp( − [ m, η ]) = exp (ad([ m, η ])) [ m ] = m mo d ( n + 2) . W e use these facts to demonstr ate that the relev an t obstructions are cohomologous: obs( φ ) = φ ◦ m ◦ φ − 1 − m ′ mo d ( n + 1) , = exp( γ ) ◦ φ ′ ◦ exp ([ m, η ]) ◦ m ◦ exp( − [ m, η ]) ◦ φ ′− 1 ◦ exp( − γ ) − m ′ mo d ( n + 1) , = φ ′ ◦ m ◦ φ ′− 1 + [ γ , m 2 ] − m ′ mo d ( n + 1 ) , = obs( φ ′ ) + [ γ , m 2 ] mo d ( n + 1) .  Theorem 4.18. L et A := ( V , m 2 ) b e a strictly gr ade d c ommutative algebr a and let m and m ′ b e two minimal C n +1 -structur es on V whose underlying algebr a is A . L et φ ∈ M n ( ¯ m ; ¯ m ′ ) b e an extendable C n -morphism, then H n − 1 , 1 Harr ( A, A ) acts fr e ely and tr ansitively on E M n ( φ : m, m ′ ) : H n − 1 , 1 Harr ( A, A ) × E M n ( φ : m, m ′ ) → E M n ( φ : m, m ′ ) , ( ξ , γ ) 7→ γ + ξ . Pr o of. A v ector ﬁeld γ of ord er n − 1 and degree zero is an extension of φ if an d only if (4.8) [ m 2 , γ ] = obs ( φ ) . It follo ws that if γ is an extension of φ and ξ is a Harrison co cycle th en γ + ξ is an extension of φ . Moreo v er if γ and γ ′ are equiv alen t extensions and ξ and ξ ′ are cohomologous co cycles, then by Remark 4.15 γ + ξ and γ ′ + ξ ′ are equiv alen t extensions; hence the ab o v e action is well deﬁned. If γ and γ ′ are t w o extensions of φ then by (4.8) their d iﬀerence is a co cycle and therefore the ab o v e action is transitive. If γ is an extension of φ and ξ is a co cycle such that γ and γ + ξ are equiv alen t extensions, th en b y Remark 4.15 ξ is a cobou n dary and h ence the ab o ve action is fr ee.  14 4.2. Symplectic C ∞ -structures. In this section w e dev elop the obstr uction th eory in the symplectic conte xt whic h is parallel to our treatmen t in s ection 4. In this section we shall only consider symple ctic C n -algebras with c onstant symp lectic f orms. This is not r eally a signiﬁcant restriction how ev er since the Darb oux theorem , cf. [9] tells us that an y symplectic C n -algebra is isomorphic to one of this form. It follo ws from Prop osition 3.13 that a strictly graded commutativ e symple c tic C ∞ -algebra A := ( V , m 2 , ω ) is the same thing as a commutat iv e F rob enius algebra and we hereafter refer to them as suc h. Recall that giv en a symple ctic C ∞ -algebra A := ( V , m, ω ), Lemma 3.17 giv es u s an isomor- phism Υ b et wee n the (sh ifted) cyclic Harrison complex Σ | ω |− 2 C C • Harr ( A ) and the su b complex of C • Harr ( A, A ) consisting of all sym plectic vec tor ﬁelds whic h we denoted by S C • Harr ( A, A ). F ur- thermore if A is in fact a strictly graded commutat iv e F rob enius algebra then Υ resp ects the natural b igrading in duced on the cohomology . Υ : C C n +1 ,j + | ω |− 2 Harr ( A ) → S C nj Harr ( A, A ) . Recall that giv en a f r ee graded mo d u le V , the mo d ule consisting of all con tin u ous endomor- phisms f : b L Σ V ∗ → b L Σ V ∗ of order ≥ n is denoted by ( n ). 4.2.1. Obstruction the ory for symple ctic C n -algebr a structur es. In this section w e will formulate the symple ctic an alogues of the theorems and deﬁnitions giv en in section 4.1.1. Deﬁnition 4.19. Let V b e a free graded mo d ule of ﬁnite rank. A minimal symple ctic C n - structure ( n ≥ 3) on V consists of a c onstant symplectic form ω ∈ D R 2 (Σ V ) and a symple ctic v ecto r ﬁ eld m ∈ Der( b L Σ V ∗ ) of degree one which has th e f orm m = m 2 + . . . + m n − 1 , m i has ord er i and satisﬁes the condition m 2 = 0 mo d ( n + 1). Let m and m ′ b e t wo minimal symplectic C n -structures on V . W e sa y m and m ′ are equiv alen t if there is a symple ctomorph ism of degree zero φ ∈ Aut( b L Σ V ∗ ) of the form (4.9) φ = id + φ 2 + φ 3 + . . . + φ k + . . . where φ i is an endomorph ism of order i and su c h that φ ◦ m ◦ φ − 1 = m ′ mo d ( n ) . R emark 4.20 . F ollo win g th e con v ention made in section 1.1 w e sh all call a sy m plectomorphism of the form (4.9) a p ointe d symple ctomorphism . Note that u nder the deﬁnition of equiv alence, it w ould b e imp ossible to hav e tw o equiv alent m inimal symplectic C n -structures on V whic h had diﬀerent symp lectic forms, h ence th e reason for the omission of the sym plectic forms in the deﬁnition. R emark 4.21 . Ob viously if m = m 2 + . . . + m n − 1 is a minimal symp lectic C n -algebra with symplectic form ω then A := ( V , m 2 , ω ) is a strictly graded commutat iv e F roben ius algebra whic h we will cal l th e underlying F rob enius alg ebra. Observ e that t w o equiv alen t minimal symplectic C n -structures ha v e the same und erlying F rob en iu s algebra. It will b e useful to introd uce the follo wing deﬁn ition: Deﬁnition 4.22. Let A := ( V , µ 2 , ω ) b e a strictly graded comm utativ e F roben iu s algebra, then the mo d u li space of min im al symple ctic C n -structures on V ﬁxing µ 2 and ω is denoted by S C n ( A ) and d eﬁ ned as the quotien t of the set { m : b L Σ V ∗ → b L Σ V ∗ : m is a minimal symple ctic C n -structure with r esp ect to ω and m 2 = µ 2 } b y the equiv alence r elatio n deﬁned in Deﬁnition 4.19. W e w ill n o w describ e the app r opriate terminology necessary in discu ssing extensions of sym- plectic C n -structures to str u ctures of higher order: 15 Deﬁnition 4.23. Let V b e a free graded mo d u le of ﬁnite rank and let ω ∈ D R 2 (Σ V ) b e a constan t s ymplectic form. W e sa y th at a minimal symplectic C n -structure m = m 2 + . . . + m n − 1 on V is extendable if there exists a symple ctic v ector ﬁeld m n ∈ Der( b L Σ V ∗ ) of ord er n such that m + m n is a symp lectic C n +1 -structure on V and we call m n an extension of m . Let m n and m ′ n b e tw o extensions of m . W e s ay that m n and m ′ n are equiv alent if ther e exists a symple ctomorphism φ ∈ Aut( b L Σ V ∗ ) of degree zero of the form φ = id + φ n − 1 + φ n + . . . + φ n + k + . . . where φ i is an endomorph ism of order i and su c h that φ ◦ ( m + m n ) ◦ φ − 1 = m + m ′ n mo d ( n + 1 ) . The qu otien t of the set of all extensions of m b y this equiv alence relation will b e denoted by S E n ( m ). W e no w identi fy th e appropriate obstr u ction to extending symple ctic C n -structures. W e will use the s ame notation as in Deﬁnition 4.8. W e will later justify this abuse of notation b y Lemma 5.3: Deﬁnition 4.24. Let V b e a free graded mo d u le of ﬁnite rank and let ω ∈ D R 2 (Σ V ) b e a constan t symplectic form. Let m = m 2 + . . . + m n − 1 b e a m inimal s ymplectic C n -structure on V . W e deﬁ ne the 0-form Obs( m ) of order n + 2 and degree 2 + | ω | by (4.10) Obs( m ) := 1 2 Υ − 1     X i + j = n +2 3 ≤ i,j ≤ n − 1 [ m i , m j ]     . W e can n o w formulate the symplectic analogues of theorems 4.9 and 4.10 . W e sh all omit the pro ofs since they are the same as the pro ofs of section 4 ve rbatim except that we must use the fact fr om Lemma 3.17 that Υ is an isomorph ism . Theorem 4.25. L et A := ( V , m 2 , ω ) b e a strictly gr ade d c ommutative F r ob enius algebr a. F or al l n ≥ 3 (4.10) induc es a map S C n ( A ) → H C n +2 , 1+ | ω | Harr ( A ) , m 7→ Obs( m ); which we wil l denote by O bs n . The kernel of this map c onsists of pr e cisely those symple ctic C n -structur es which ar e extendable: { m ∈ S C n ( A ) : m is extendable } = ker(Obs n ) .  Theorem 4.26. L et A := ( V , m 2 , ω ) b e a strictly gr ade d c ommutative F r ob enius algebr a and let m ∈ S C n ( A ) b e an extendable symple ctic C n -structur e, then H C n +1 , | ω | Harr ( A ) acts fr e ely and tr ansitively on S E n ( m ) : H C n +1 , | ω | Harr ( A ) × S E n ( m ) → S E n ( m ) , ( α, m n ) 7→ m n + Υ( α ) .  4.2.2. Obstruction the ory for symple ctic C n -algebr a morphisms. In this section w e will devel op the obstruction theory for morphism s b et w een t wo symple ctic C n -algebras. Again, the form u- lation is entirely analogous to that of section 4.1.2 . Deﬁnition 4.27. Let V b e a free graded mo d u le of ﬁnite rank and let ω ∈ D R 2 (Σ V ) b e a constan t symplectic form. Let m and m ′ b e t w o minimal symple ctic C n -structures on V . A 16 minimal symple ctic C n -morphism from m to m ′ is a symple c tomorp hism φ ∈ Aut( b L Σ V ∗ ) of degree zero su ch that φ ◦ m = m ′ ◦ φ mo d ( n ) . Let φ and φ ′ b e t wo suc h minimal symplectic C n -morphisms. W e sa y φ and φ ′ are homotopic if there exists a symple ctic v ector ﬁ eld η of d egree − 1 suc h th at φ = φ ′ ◦ exp([ m, η ]) mo d ( n − 1) . W e will n o w introdu ce the mo du li space of symple ctic C n -morphisms: Deﬁnition 4.28. Let V b e a free graded mo d u le of ﬁnite rank and let ω ∈ D R 2 (Σ V ) b e a constan t sy m plectic f orm . Let m and m ′ b e t wo m inimal sy m plectic C n -structures on V w ith the same und erlying F rob eniu s algebra (i.e. m 2 = m ′ 2 ). Th e m o duli s pace of minimal symple c tic C n -morphisms f rom m to m ′ is denoted by S M n ( m ; m ′ ) and deﬁned as th e quotien t of the set { φ : b L Σ V ∗ → b L Σ V ∗ : φ is a m in imal p ointe d symple ctic C n -morphism f r om m to m ′ } b y the homotopy equiv alence relation deﬁned in Deﬁnition 4.27. Let m = m 2 + . . . + m n b e a symplectic C n +1 -structure on a free graded mo du le V . Recall that we deﬁ ne the corresp ond in g s ymplectic C n -structure ¯ m on V as ¯ m := m 2 + . . . + m n − 1 . No w we will introd uce the terminology dealing with extensions of symp lectic C n -morphisms: Deﬁnition 4.29. Let V b e a free graded mo d u le of ﬁnite rank and let ω ∈ D R 2 (Σ V ) b e a constan t symplectic f orm. L et m and m ′ b e t wo minimal symp lectic C n +1 -structures on V with the same und erlying F rob enius algebra. W e say th at a p oin ted symplectic C n -morphism φ from ¯ m to ¯ m ′ is extendable if th er e exists a symple ctic vec tor ﬁ eld γ of order n − 1 and degree zero suc h that exp( γ ) ◦ φ is a symp lectic C n +1 -morphism fr om m to m ′ and we call γ an extension of φ . Let γ and γ ′ b e t w o extensions of φ . W e sa y that γ and γ ′ are equiv alen t if there exists a symple c tic v ecto r ﬁ eld η of order n − 2 and degree − 1 such that exp( γ ) ◦ φ = exp( γ ′ ) ◦ φ ◦ exp([ m, η ]) mo d ( n ) . The quotien t of the s et of all extensions of φ by this equiv alence r elation will b e denoted b y S E M n ( φ : m, m ′ ). Next w e will deﬁne the appr opriate obstruction to the extension of symple ctic C n -morphisms. W e shall use the same notation as in Deﬁnition 4.16. This abuse of notation will later b e justiﬁed b y Lemma 5.4. Deﬁnition 4.30. Let V b e a free graded mo d u le of ﬁnite rank and let ω ∈ D R 2 (Σ V ) b e a constan t symplectic f orm. L et m and m ′ b e t wo minimal symp lectic C n +1 -structures on V with the same u nderlying F rob enius algebra. Let φ b e a p ointe d symp lectic C n -morphism from ¯ m to ¯ m ′ . W e deﬁne the 0-form obs( φ ) of ord er n + 1 and degree 1 + | ω | by (4.11) obs( φ ) := Υ − 1  φ ◦ m ◦ φ − 1 − m ′  mo d ( n + 2 ) . W e will no w pr o ve the s y m plectic analogues of theorems 4.17 and 4.18. W e s h all omit the pro ofs since they are the same as those of section 4 v erbatim except that w e m ust use th e f act from L emma 3.17 that Υ is an isomorp h ism. Theorem 4.31. L et A := ( V , m 2 , ω ) b e a strictly gr ade d c ommutative F r ob enius algebr a and let m and m ′ b e two minimal symple ctic C n +1 -structur es ( n ≥ 3) on V whose underlying F r ob enius algebr a is A , then (4.11) induc es a map S M n ( ¯ m ; ¯ m ′ ) → H C n +1 , | ω | Harr ( A ) , φ 7→ obs( φ ); 17 which we wil l denote by obs n . The kernel of this map c onsists of pr e cisely th ose symple ctic C n -morphisms which ar e extendable: { φ ∈ S M n ( ¯ m ; ¯ m ′ ) : φ is extendable } = ke r(obs n ) .  Theorem 4.32. L et A := ( V , m 2 , ω ) b e a strictly gr ade d c ommutative F r ob enius algebr a and let m and m ′ b e two minimal symple ctic C n +1 -structur es on V whose underlying F r ob eni u s algebr a is A . L et φ ∈ S M n ( ¯ m ; ¯ m ′ ) b e an extendable symple ctic C n -morphism, then H C n, | ω |− 1 Harr ( A ) acts fr e ely and tr ansitively on S E M n ( φ : m, m ′ ) : H C n, | ω |− 1 Harr ( A ) × S E M n ( φ : m, m ′ ) → S E M n ( φ : m, m ′ ) , ( α, γ ) 7→ γ + Υ( α ) .  4.3. Unital Symplectic C ∞ -structures. In this section we will d evelo p th e obs truction the- ory for unital symple ctic C ∞ -structures. Here it is our int en tion to b e as br ief as p ossible as the theory is essen tially the same as the th eories describ ed in the previous tw o sections. Again, w e shall only consider un ital symp lectic C n -algebras with constan t symplectic form s . Giv en a u nital strictly graded comm utativ e F rob eniu s algebra A := ( V , m 2 , ω ), Lemm a 3.18 giv es us an isomorph ism ¯ Υ : C C n +1 ,j + | ω |− 2 Harr ( A ) → S C nj Harr ( A, A ) b et w een the (shifted) n orm alised cyclic Harrison complex and the s ub complex of C • Harr ( A, A ) consisting of all norm alised symplectic vec tor ﬁelds. 4.3.1. Obstruction the ory for unital symple ctic C n -algebr a structur es. Deﬁnition 4.33. Let V b e a free graded mo dule of ﬁn ite rank with basis 1 , x 1 , . . . , x d . A minimal unital symple ctic C n -structure ( n ≥ 3) on V consists of a c onstant symplectic form ω ∈ D R 2 (Σ V ) and a symple ctic ve ctor ﬁeld m ∈ Der( b L Σ V ∗ ) of degree one which has th e f orm m = m 2 + . . . + m n − 1 , m i has ord er i and satisﬁes the condition m 2 = 0 mo d ( n + 1). F urthermore, we imp ose the condition that for all i ≥ 3, m i is a normalise d ve ctor ﬁ eld and that m 2 has the form m 2 = ad τ − 1 2 [ τ , τ ] ∂ τ + X 1 ≤ i,j,k ≤ d a k ij [ t i , t j ] ∂ t k , a k ij ∈ K . Let m and m ′ b e t w o minimal unital sy m plectic C n -structures on V . W e sa y m and m ′ are equiv alen t if there is a unital symple ctomorphism of degree zero φ ∈ Aut( b L Σ V ∗ ) of the form φ = id + φ 2 + φ 3 + . . . + φ k + . . . where φ i is an endomorph ism of order i and su c h that φ ◦ m ◦ φ − 1 = m ′ mo d ( n ) . It will b e useful to introd uce the follo wing deﬁn ition: Deﬁnition 4.34. Let A := ( V , µ 2 , ω ) b e a un ital s tr ictly graded c omm utativ e F rob enius algebra, then the modu li sp ace of minimal unital symple ctic C n -structures on V ﬁxing µ 2 and ω is denoted b y U S C n ( A ) and d eﬁ ned as the quotien t of the set { m : b L Σ V ∗ → b L Σ V ∗ : m is a minimal unital symple ctic C n -structure and m 2 = µ 2 } b y the equiv alence r elatio n deﬁned in Deﬁnition 4.33. 18 Deﬁnition 4.35. Let V b e a free graded mo d u le of ﬁnite rank and let ω ∈ D R 2 (Σ V ) b e a constan t symplectic form. W e sa y that a minim al un ital symplectic C n -structure m = m 2 + . . . + m n − 1 on V is extendable if there exists a normalise d symple ctic vecto r ﬁeld m n ∈ Der( b L Σ V ∗ ) of order n su ch that m + m n is a unital symp lectic C n +1 -structure on V and w e call m n an extension of m . Let m n and m ′ n b e tw o extensions of m . W e s ay that m n and m ′ n are equiv alent if ther e exists a u ni tal symple ctomorp hism φ ∈ Aut( b L Σ V ∗ ) of degree zero of the f orm φ = id + φ n − 1 + φ n + . . . + φ n + k + . . . where φ i is an endomorph ism of order i and su c h that φ ◦ ( m + m n ) ◦ φ − 1 = m + m ′ n mo d ( n + 1 ) . The qu otien t of the set of all extensions of m b y this equiv alence relation will b e denoted by U S E n ( m ). Deﬁnition 4.36. Let V b e a free graded mo d u le of ﬁnite rank and let ω ∈ D R 2 (Σ V ) b e a constan t symp lectic form. Let m = m 2 + . . . + m n − 1 b e a minimal un ital symplectic C n -structure on V . W e deﬁn e the normalised 0-form Obs( m ) of order n + 2 and degree 2 + | ω | by (4.12) Obs( m ) := 1 2 ¯ Υ − 1     X i + j = n +2 3 ≤ i,j ≤ n − 1 [ m i , m j ]     . Theorem 4.37. L et A := ( V , m 2 , ω ) b e a unital strictly gr ade d c ommutative F r ob eniu s algebr a. F or al l n ≥ 3 (4.12) induc es a map U S C n ( A ) → H C n +2 , 1+ | ω | Harr ( A ) , m 7→ Obs( m ); which we wil l denote by Obs n . The kernel of this map c onsists of pr e cisely those unital symple ctic C n -structur es which ar e extendable: { m ∈ U S C n ( A ) : m is extendable } = ker(Obs n ) .  Theorem 4.38. L et A := ( V , m 2 , ω ) b e a unital strictly gr ade d c ommutat ive F r ob enius algebr a and let m ∈ U S C n ( A ) b e an extendable unital symple ctic C n -structur e, then H C n +1 , | ω | Harr ( A ) acts fr e ely and tr ansitively on U S E n ( m ) : H C n +1 , | ω | Harr ( A ) × U S E n ( m ) → U S E n ( m ) , ( α, m n ) 7→ m n + ¯ Υ( α ) .  4.3.2. Obstruction the ory for unital symple ctic C n -algebr a morphisms. Deﬁnition 4.39. Let V b e a free grad ed mo dule of ﬁnite rank with basis 1 , x 1 , . . . , x d and let ω ∈ D R 2 (Σ V ) b e a constan t symplectic form . Let m and m ′ b e t wo minimal unital symple ctic C n -structures on V . A minimal unital sympl e ctic C n -morphism from m to m ′ is a symple cto- morphism φ ∈ Aut( b L Σ V ∗ ) of degree zero suc h th at φ ◦ m = m ′ ◦ φ mod ( n ) and satisfying th e condition φ ( τ ) = τ + A ( t ) , φ ( t i ) = t i + B i ( t ); where A ( t ) , { B i ( t ) } d i =1 are Lie p o wer series in the v ariables t i . 19 Let φ and φ ′ b e t wo such m inimal unital symplectic C n -morphisms. W e sa y φ and φ ′ are homotopic if th ere exists a normalise d symple ctic v ector ﬁ eld η of degree − 1 suc h th at φ = φ ′ ◦ exp([ m, η ]) mo d ( n − 1) . W e will n o w introdu ce the mo du li space of u nital symple ctic C n -morphisms: Deﬁnition 4.40. Let V b e a free graded mo d u le of ﬁnite rank and let ω ∈ D R 2 (Σ V ) b e a constan t symplectic form. L et m and m ′ b e tw o m inimal unital s ymplectic C n -structures on V with the same und er lyin g F rob eniu s algebra. Th e mo duli space of m inimal unital symple ctic C n -morphisms f r om m to m ′ is denoted b y U S M n ( m ; m ′ ) and deﬁned as the qu otien t of the set { φ : b L Σ V ∗ → b L Σ V ∗ : φ is a m in imal p ointe d unital symple c tic C n -morphism f r om m to m ′ } b y the homotopy equiv alence relation deﬁned in Deﬁnition 4.39. Let m = m 2 + . . . + m n b e a unital symp lectic C n +1 -structure on a free graded mo dule V . Recall th at we deﬁn e th e corresp onding un ital symp lectic C n -structure ¯ m on V as ¯ m := m 2 + . . . + m n − 1 . Deﬁnition 4.41. Let V b e a free graded mo d u le of ﬁnite rank and let ω ∈ D R 2 (Σ V ) b e a constan t symplectic form. Let m and m ′ b e tw o m inimal unital s ymplectic C n +1 -structures on V with the same un derlying F rob enius algebra. W e sa y that a p ointed u nital symplectic C n -morphism φ from ¯ m to ¯ m ′ is extendable if there exists a normal ise d symple ctic ve ctor ﬁeld γ of order n − 1 and degree zero suc h th at exp( γ ) ◦ φ is a unital symplectic C n +1 -morphism from m to m ′ and we call γ an extension of φ . Let γ and γ ′ b e t w o extensions of φ . W e sa y that γ and γ ′ are equiv alen t if there exists a normalise d symple ctic vect or ﬁeld η of order n − 2 and d egree − 1 su c h that exp( γ ) ◦ φ = exp( γ ′ ) ◦ φ ◦ exp([ m, η ]) mo d ( n ) . The quotien t of the s et of all extensions of φ by this equiv alence r elation will b e denoted b y U S E M n ( φ : m, m ′ ). Deﬁnition 4.42. Let V b e a free graded mo d u le of ﬁnite rank and let ω ∈ D R 2 (Σ V ) b e a constan t symplectic form. Let m and m ′ b e tw o m inimal unital s ymplectic C n +1 -structures on V with the same un derlying F rob enius algebra. Let φ b e a p oin ted unital symplectic C n - morphism from ¯ m to ¯ m ′ . W e deﬁ n e the normalised 0-form obs( φ ) of ord er n + 1 and degree 1 + | ω | by (4.13) obs( φ ) := ¯ Υ − 1  φ ◦ m ◦ φ − 1 − m ′  mo d ( n + 2 ) . Theorem 4.43. L et A := ( V , m 2 , ω ) b e a unital strictly gr ade d c ommutat ive F r ob enius algebr a and let m and m ′ b e two minimal unital symple ctic C n +1 -structur es ( n ≥ 3) on V whose underlying F r ob enius algebr a is A , then (4.13) induc es a map U S M n ( ¯ m ; ¯ m ′ ) → H C n +1 , | ω | Harr ( A ) , φ 7→ obs( φ ); which we wil l denote by obs n . The kernel of this map c onsists of pr e cisely those unital symple ctic C n -morphisms which ar e extendable: { φ ∈ U S M n ( ¯ m ; ¯ m ′ ) : φ is extendable } = ke r(obs n ) .  Theorem 4.44. L et A := ( V , m 2 , ω ) b e a unital strictly gr ade d c ommutat ive F r ob enius algebr a and let m and m ′ b e two minimal unital symple ctic C n +1 -structur es on V whose underlying 20 F r ob enius algebr a i s A . L et φ ∈ U S M n ( ¯ m ; ¯ m ′ ) b e an extendable unital symple ctic C n -morphism, then H C n, | ω |− 1 Harr ( A ) acts fr e ely and tr ansitively on U S E M n ( φ : m, m ′ ) : H C n, | ω |− 1 Harr ( A ) × U S E M n ( φ : m, m ′ ) → U S E M n ( φ : m, m ′ ) , ( α, γ ) 7→ γ + ¯ Υ( α ) .  5. The Main Theorem : a Corre sponden ce be tween C ∞ and Symp lectic C ∞ -structures In th is section w e will pro ve our main result; that a sym p lectic C ∞ -algebra ( V , m, ω ) is uniquely d etermined by its underlying C ∞ -algebra ( V , m ) together with the structur e of a com- m utativ e F rob enius algebra on ( V , m 2 ). Th e p oint is that the obstruction theories f or symplectic and nonsymplectic C ∞ -algebras turn out to b e ‘isomorphic’ thanks to Theorem 3.8. 5.1. Main theorem I: the non-unital case. O n ce agai n, in this section we shall o nly consider symplectic C ∞ -algebras with constan t symplectic forms. It will b e necessary to int ro duce the follo wing deﬁnitions in ord er to formulate our m ain theorem later in this section: Deﬁnition 5.1. (i) Let A := ( V , µ 2 ) b e a strictly graded commutat iv e algebra, then the m o duli space of minimal C ∞ -structures on V ﬁxin g µ 2 is d en oted by C ∞ ( A ) and deﬁn ed as the quotien t of the set { m : b L Σ V ∗ → b L Σ V ∗ : m is a minimal C ∞ -structure and m 2 = µ 2 } b y the actio n under conjugation of the group G consisting of all diﬀeomorphisms φ ∈ Aut( b L Σ V ∗ ) of the form φ = id + φ 2 + φ 3 + . . . + φ k + . . . , where φ i is an end omorphism of order i . (ii) Let A := ( V , µ 2 , ω ) b e a strictly graded comm u tativ e F rob eniu s algebra, then the mo duli space of minimal symple ctic C ∞ -structures on V ﬁxing µ 2 and ω is denoted by S C ∞ ( A ) and deﬁned as the quotien t of the set { m : b L Σ V ∗ → b L Σ V ∗ : m is a minimal symple ctic C ∞ -structure and m 2 = µ 2 } b y the acti on under conjugation of the group G consisting of all symple ctomorphisms φ ∈ Aut( b L Σ V ∗ ) of the form φ = id + φ 2 + φ 3 + . . . + φ k + . . . , where φ i is an end omorphism of order i . Recall th at we call a d iﬀeomorphism (symplectomorphism) of th e form φ = id + φ 2 + φ 3 + . . . + φ k + . . . a p ointe d diﬀeomorphism (symplectomorphism). Deﬁnition 5.2. (i) Let V b e a free graded mo dule and let m and m ′ b e t w o minimal C ∞ -structures on V with th e s ame und erlying algebra. W e sa y that t wo C ∞ -morphisms φ and φ ′ from m to m ′ are homotopic if th er e exists a v ector ﬁeld η of degree − 1 suc h that φ = φ ′ ◦ exp([ m, η ]) . W e denote the mo du li space of C ∞ -morphisms from m to m ′ b y M ∞ ( m ; m ′ ) and deﬁne it as the quotien t of the set { φ : b L Σ V ∗ → b L Σ V ∗ : φ is a p ointe d C ∞ -morphism f r om m to m ′ } b y the homotopy equiv alence relation deﬁned ab o v e. 21 (ii) Let V b e a free graded mo dule of ﬁn ite ran k and let ω ∈ DR 2 (Σ V ) b e a constan t symplectic form. Let m and m ′ b e tw o minimal symple ctic C ∞ -structures on V with the same underlyin g F rob enius algebra. W e sa y that tw o symple ctic C ∞ -morphisms φ and φ ′ from m to m ′ are homotopic if th ere exists a symple ctic v ector ﬁeld η of degree − 1 su ch that φ = φ ′ ◦ exp([ m, η ]) . W e denote th e mo d uli space of symp le ctic C ∞ -morphisms from m to m ′ b y S M ∞ ( m ; m ′ ) and deﬁne it as the quotien t of the set { φ : b L Σ V ∗ → b L Σ V ∗ : φ is a p ointe d symple ctic C ∞ -morphism f r om m to m ′ } b y the homotopy equiv alence relation deﬁned ab o v e. Recall that in sections 4 and 4.2 w e deﬁn ed mo duli sp aces C n ( A ) and S C n ( A ). Giv en a strictly graded commutati v e F rob eniu s algebra A we can d eﬁ ne a map ι : S C n ( A ) → C n ( A ) whic h is induced b y the canonical inclusion of the Lie subalgebra of symplectic v ector ﬁ elds into the Lie alge bra Der( b L Σ V ∗ ) of all vec tor ﬁelds. T his map is well deﬁ ned b ecause the group of symplectomorphisms is a su bgroup of the group Aut( b L Σ V ∗ ) of all diﬀeomorphisms. T h is map is also d eﬁned for n = ∞ . Similarly if ( m, ω ) is a symplectic C n -structure on a f ree graded m o dule V of ﬁnite ran k then w e can d eﬁ ne a map ι : S E n ( m ) → E n ( m ) whic h is again indu ced by the canonical inclusion of the Lie subalgebra of s ymplectic ve ctor ﬁelds in to Der( b L Σ V ∗ ). The same applies to the mo duli spaces M n ( m ; m ′ ) and S M n ( m ; m ′ ) whic h w e also in tro d uced in s ections 4 and 4.2. Giv en a constan t symp lectic form ω ∈ D R 2 (Σ V ) and tw o sym plectic C n - structures m and m ′ with th e same und erlying F rob en iu s algebra we can deﬁ ne a map ι : S M n ( m ; m ′ ) → M n ( m ; m ′ ) whic h is induced b y the canonical inclusion of the subgroup of symplectomorphisms into the group Au t( b L Σ V ∗ ) of all diﬀeomorphisms. The fact that this map is w ell deﬁn ed mo dulo the homotop y equiv alence relations follo w s tautolo gically fr om the fact that the symplectic v ector ﬁelds are a Lie sub algebra of the Lie algebra of all vect or ﬁelds. Lik ewise th is m ap is also deﬁn ed for n = ∞ . Giv en a constant symp lectic form ω ∈ D R 2 (Σ V ) , t wo symplectic C n +1 -structures m and m ′ and a sy m plectic C n -morphism φ f rom ¯ m to ¯ m ′ w e can also deﬁn e a m ap ι : S E M n ( φ : m, m ′ ) → E M n ( φ : m, m ′ ) whic h is induced b y the canonical inclusion of the Lie sub algebra of symplectic vecto r ﬁelds in to Der( b L Σ V ∗ ). Again it is tautological to c hec k that this map is w ell deﬁned mo dulo the homotop y equiv alence r elatio ns. If A := ( V , m 2 , ω ) is a strictly graded comm utativ e F rob enius algebra then r ecall that there is a natural bigradin g on cohomolo gy . The isomorphisms of lemmas 3.16 and 3.17 resp ect th is bigrading. Th is giv es us the follo wing comm utativ e diagram for all j ∈ Z and all n ≥ 1: (5.1) H C n +1 ,j + | ω |− 2 Harr ( A ) I / / Ψ ) ) S S S S S S S S S S S S S S Υ   H n,j + | ω |− 2 Harr ( A, A ∗ ) S H nj Harr ( A, A )   / / H nj Harr ( A, A ) Φ O O where Φ is the isomorphism whic h is d eﬁned b y equation (2.1), Υ is the isomorph ism wh ic h is deﬁned by equation (3.4) and I is the map of Th eorem 3.8 whic h comes from the p erio dicit y exact sequence. T he map Ψ is deﬁned as Ψ := Φ − 1 ◦ I . An y action of the group H nj Harr ( A, A ) 22 could b e pulled bac k along Ψ to an action of the group H C n +1 ,j + | ω |− 2 Harr ( A ). W e w ill no w need the follo wing auxiliary lemmas in order to prov e our main result: Lemma 5.3. L et A := ( V , m 2 , ω ) b e a strictly gr ade d c ommutative F r ob enius algebr a: (i) F or al l n ≥ 3 the fol lowing diagr am c ommutes: C n ( A ) Obs n / / H n +1 , 3 Harr ( A, A ) S C n ( A ) ι O O Obs n / / H C n +2 , 1+ | ω | Harr ( A ) Ψ O O (ii) L et m ∈ S C n ( A ) b e an extendable symple ctic C n -structur e. The map ι : S E n ( m ) → E n ( m ) is H C n +1 , | ω | Harr ( A ) -e quivariant. Pr o of. This is a tautologi cal consequence of diagram (5.1).  Lemma 5.4. L et A := ( V , m 2 , ω ) b e a strictly gr ade d c ommutative F r ob enius algebr a and let m and m ′ b e two minimal symple ctic C n +1 -structur es on V whose underlying F r ob enius algebr a is A : (i) The fol lowing diagr am c ommutes: M n ( ¯ m ; ¯ m ′ ) obs n / / H n, 2 Harr ( A, A ) S M n ( ¯ m ; ¯ m ′ ) ι O O obs n / / H C n +1 , | ω | Harr ( A ) Ψ O O (ii) L et φ ∈ S M n ( ¯ m ; ¯ m ′ ) b e an extendable symple ctic C n -morphism. The map ι : S E M n ( φ : m, m ′ ) → E M n ( φ : m, m ′ ) is H C n, | ω |− 1 Harr ( A ) -e quivariant. Pr o of. Again this f ollo ws tautolog ically fr om diagram (5.1).  W e are now ready to formula te our main result: Theorem 5.5. L et A := ( V , m 2 , ω ) b e a strictly gr ade d unital c ommutative F r ob enius algebr a: (i) The map ι : S C ∞ ( A ) → C ∞ ( A ) is a bije ction. (ii) L et m and m ′ b e two minimal symple ctic C ∞ -structur es on V whose u nderlying F r ob enius algebr a is A . The map ι : S M ∞ ( m ; m ′ ) → M ∞ ( m ; m ′ ) is a surje ction. Pr o of. Let us b egin by pr oving that th e m ap ι : S C ∞ ( A ) → C ∞ ( A ) is s u rjectiv e. Let m = m 2 + m 3 + . . . + m n + . . . b e a minimal C ∞ -structure on V . W e will in ductiv ely construct a sequence of symple ctic v ector ﬁelds m ′ i , 3 ≤ i < ∞ and a sequence of vect or ﬁelds γ i , 2 ≤ i < ∞ , wher e m ′ i has order i and degree one and γ i has ord er i and degree zero, suc h that: (i) m ′ := m 2 + m ′ 3 + . . . + m ′ n + . . . is a minimal symple ctic C ∞ -structure. 23 (ii) φ := . . . ◦ exp( γ n ) ◦ . . . ◦ exp( γ 3 ) ◦ exp( γ 2 ) is a C ∞ -morphism f r om m to m ′ . Let us assume that we hav e constructed a sequen ce of symple ctic v ector ﬁ elds m ′ 3 , . . . , m ′ n − 1 of degree one and a sequ ence of vect or ﬁelds γ 2 , . . . , γ n − 2 of degree zero, wh ere m ′ i and γ i ha v e order i , satisfying: (i) m ′ := m 2 + m ′ 3 + . . . + m ′ n − 1 is a minimal symple ctic C n -structure. (ii) φ := exp( γ n − 2 ) ◦ . . . ◦ exp( γ 2 ) is a minimal C n -morphism from ¯ m := m 2 + . . . + m n − 1 to m ′ . Note that the base case n = 3 is trivial. T he C n -structures ¯ m and m ′ represent the same class in C n ( A ), therefore by Lemma 5.3 and Theorem 4.9 we see that Ψ(Obs n ( m ′ )) = Obs n ( ι ( m ′ )) = Obs n ( ¯ m ) = 0 b ecause the C n -structure ¯ m is extendable. Theorem 3.8 now implies that Obs n ( m ′ ) = 0 and therefore by Theorem 4.25, m ′ is an extendable sy m plectic C n -structure (r ecall f r om diagram (5.1) th at Ψ was deﬁned in terms of the map I of Corollary 3.8). Consider the C ∞ -structure φ ◦ m ◦ φ − 1 ; th ere exists a vect or ﬁeld ˙ m n (not n ecessarily sym- plectic) of ord er n and degree one suc h that φ ◦ m ◦ φ − 1 = m ′ + ˙ m n mo d ( n + 1 ) and th erefore m ′ + ˙ m n is a C n +1 -structure on V . W e also know from the ab o ve argument that there exists a symple ctic v ector ﬁeld m ′ n of order n and degree one suc h that m ′ + m ′ n is a symple c tic C n +1 -structure. Let u s denote the corresp ond ing class of m ′ n in S E n ( m ′ ) by [ m ′ n ] and the corresp onding class of ˙ m n in E n ( m ′ ) by [ ˙ m n ]. By Theorem 4.10 there exists a cohomology class ξ n ∈ H n, 2 Harr ( A, A ) suc h that ι ([ m ′ n ]) + ξ n = [ ˙ m n ] . By Theorem 3.8 and Lemma 5.3 there exists a cohomology class α ∈ H C n +1 , | ω | Harr ( A ) su c h that ι ([ m ′ n ] + Υ( α )) = [ ˙ m n ] . W e see that by mo difying our choic e of symplectic vec tor ﬁeld m ′ n appropriately we can assu me that th ere exists a v ector ﬁeld γ n − 1 of order n − 1 and degree zero su ch that exp( γ n − 1 ) ◦ ( m ′ + ˙ m n ) ◦ exp( − γ n − 1 ) = m ′ + m ′ n mo d ( n + 1) . This completes the in d uctiv e step and prov es that the map ι : S C ∞ ( A ) → C ∞ ( A ) is s u rjectiv e. The pro of that the map ι : S M ∞ ( m ; m ′ ) → M ∞ ( m ; m ′ ) is surjectiv e pro ceeds in a similar fashion to the ab o v e pro of. W e shall outline the main id eas of th e p ro of and lea v e th e reader to p ro vid e th e details: (i) Giv en a C ∞ -morphism φ from m to m ′ the idea is to construct by in d uction, a sequence of symple ctic vect or ﬁelds γ ′ i , 2 ≤ i < ∞ of degree zero and a sequence of vect or ﬁelds η i , 1 ≤ i < ∞ of degree − 1, where γ ′ i and η i ha v e order i , such that: (a) φ ′ := . . . ◦ exp( γ ′ n ) ◦ . . . ◦ exp( γ ′ 3 ) ◦ exp( γ ′ 2 ) is a symple ctic C ∞ -morphism from m to m ′ . (b) φ = φ ′ ◦ [ . . . ◦ exp([ m, η n ]) ◦ . . . ◦ exp([ m, η 2 ]) ◦ exp([ m, η 1 ])] . (ii) A t the n th stage u s e Lemma 5.4, T h eorem 3.8 and theorems 4.17 and 4.31 to sho w that the obstruction to extending the giv en symplectic C n -morphism to the next lev el v anishes. 24 (iii) Use Lemma 5.4, Theorem 4.18 and b oth parts (ii) and (iii) of Corollary 3.8 to mo dify th is extended symplectic C n +1 -morphism to one whic h is homotop y equiv alen t to φ mo dulo ( n ). Finally we sh o w that the map ι : S C ∞ ( A ) → C ∞ ( A ) is injectiv e. Let m, m ′ ∈ S C ∞ ( A ) b e tw o symplectic C ∞ -structures and s upp ose that there exists a p ointe d C ∞ -morphism φ from m to m ′ , then by part (ii) of this theorem there exists a symple ctic p ointed C ∞ -morphism φ ′ from m to m ′ whic h is homotop y equiv alen t to φ .  R emark 5.6 . A natural question is w hether a result similar to Theorem 5.5 holds in the A ∞ and L ∞ con texts. The answer is no. The crucial p oin t on wh ic h the pro of of Theorem 5.5 tur n s is that the cyclic Harrison theory essential ly coincides w ith the (non cyclic) Harrison theory , cf. Theorem 3.8. T his fails badly for b oth the asso ciativ e and Lie cases. F or examp le, let g b e a semisimple Lie algebra whic h could b e considered as a sym p lectic L ∞ -algebra together with its Killing form. Then H i ( g , g ) = 0 unless i = 0; how ev er the cyclic theory H • ( g , K ) is not zero in dimensions ≥ 3. In the asso ciativ e case a relev an t counterexa mple is pr o vid ed b y the group ring of a ﬁnite nonab elian group G . Indeed, in this case H C • Hoch ( K [ G ]) is isomorph ic to the direct su m of copies of H C • Hoch ( K ) ∼ = K [ u ] where the summation is o v er all conjugacy classes of G . On the other hand H i Hoch ( K [ G ] , K [ G ]) = 0 for all i > 0. R emark 5. 7 . There is a constru ction, cf. [1 3] wh ic h associates to an y minimal symp lectic C ∞ - algebra a cycle in an appropriate version of the graph complex. Moreo ve r, t w o we akly equiv alent symplectic C ∞ -algebras giv e rise to homologous cycles. Th erefore, our result sh o ws that the corresp onding graph homology class only dep end s on the u nderlying C ∞ -algebra together with the s tructure of a graded comm u tativ e F rob enius algebra on its under lyin g mo dule. In par- ticular, a graph homology class could b e asso ciated to an y P oincar ´ e dualit y space. It w ou ld b e interesting to expr ess th is construction in classical homotop y theoretic terms, i.e. Massey pro du cts. 5.2. Main Theorem I I: the unital case. Once again, in th is sectio n we shall only consider symplectic C ∞ -algebras with constan t symplectic form s . Deﬁnition 5.8. Let A := ( V , µ 2 , ω ) b e a unital strictly graded comm u tativ e F rob enius algebra, then the m o duli space of minimal unital sym ple ctic C ∞ -structures on V ﬁx in g µ 2 and ω is denoted by U S C ∞ ( A ) and d eﬁ ned as the quotient of the set { m : b L Σ V ∗ → b L Σ V ∗ : m is a minimal unital symple ctic C ∞ -structure and m 2 = µ 2 } b y th e action under conju gation of the group G consisting of all symple ctomorphisms φ ∈ Aut( b L Σ V ∗ ) such that φ ( τ ) = τ + A ( t ) , φ ( t i ) = t i + B i ( t ); where A ( t ) , { B i ( t ) } d i =1 are Lie p o wer series in the v ariables t i consisting of terms of order ≥ 2. An element of G will b e referred to as a p ointe d unital symple ctomorp hism . Deﬁnition 5.9. Let V b e a free graded mo d ule of ﬁnite rank and let ω ∈ D R 2 (Σ V ) b e a constan t s y m plectic form. L et m and m ′ b e tw o minimal unital symple ctic C ∞ -structures on V with the same u nderlying F rob enius algebra. W e sa y that tw o unital symple ctic C ∞ -morphisms φ and φ ′ from m to m ′ are homotopic if there exists a normalise d symple ctic vec tor ﬁeld η of degree − 1 su c h that φ = φ ′ ◦ exp([ m, η ]) . W e denote the mod uli space of unital symple ctic C ∞ -morphisms from m to m ′ b y U S M ∞ ( m ; m ′ ) and deﬁne it as the quotien t of the set { φ : b L Σ V ∗ → b L Σ V ∗ : φ is a p ointe d u nital symple ctic C ∞ -morphism from m to m ′ } b y the homotopy equiv alence relation deﬁned ab ov e. 25 Giv en a un ital strictly graded comm utativ e F rob enius algebra A we can deﬁne a map ι : U S C ∞ ( A ) → S C ∞ ( A ) whic h is ju st deﬁned b y forgetting the u nital stru cture. Giv en a free graded m o dule V of ﬁn ite rank, a constan t symplectic form ω ∈ D R 2 (Σ V ) and t wo minimal unital symplectic C ∞ -structures m and m ′ with the same und er lyin g F rob enius algebra, w e can deﬁne a map ι : U S M ∞ ( m ; m ′ ) → S M ∞ ( m ; m ′ ) whic h is also deﬁn ed by forgetting the u nital structur e. W e are now ready to formula te p art I I of our main r esult: Theorem 5.10. L et A := ( V , m 2 , ω ) b e a connected unital strictly gr ade d c ommutative F r ob e- nius algebr a: (i) The map ι : U S C ∞ ( A ) → S C ∞ ( A ) is a bije ction. (ii) L et m and m ′ b e two minimal unital symple c tic C ∞ -structur es on V whose underlying F r ob enius algebr a is A . Th e map ι : U S M ∞ ( m ; m ′ ) → S M ∞ ( m ; m ′ ) is a surje ction. Pr o of. The pro of pro ceeds in exactly the same wa y as the pr o of of Theorem 5.5 b y m aking use of the obstruction theory describ ed in sections 4.2 and 4.3 and applying the result of [9] Prop osition 8.4 th at the map ι : H C ij Harr ( A ) → H C ij Harr ( A ) induced by th e inclusion is an isomorphism, except in b idegree ( i, j ) = (3 , 2). A p riori, the fact th at this map is not s urjectiv e in b idegree (3 , 2) might pr esen t a problem in applying an obstruction theory argumen t; h o wev er, sin ce the algebra A is connected, this implies that the degree of the symplectic form ω is less than or equal to 2. A careful analysis then reve als that none of the cohomolo gy classes relev an t to the obstruction theory lie in the critical b idegree (3 , 2); cf. theorems 4.37 , 4.38, 4.43 and 4.44.  Referen ces [1] J. Blo ck. Duali ty and e qui valenc e of mo dule c ate gories in nonc ommutative ge ometry I , arXiv:math .QA/05092 84 . [2] J. Conant, K . V ogtmann, On a the or em of Kontsevich. Algebr. Geom. T op ol. 3 ( 2003), 1167-1224. [3] M. Gerstenhab er, On the deformation of rings and algebr as. An n. of Math. (2) 79 1964 59-103. [4] V. Ginzburg, Nonc ommutative Sympl e ctic Ge ometry, Q ui ver V arieties, and Op er ads. Math. R es. Lett. V ol. 8, 2001, No. 3, pp . 377-400. [5] A. H amilton, On the Classiﬁc ation of Mo or e Algebr as and their Deformations. Homology , Homotopy and Applications, V ol. 6(1), (2004), 87-107. [6] A. Hamilton, A sup er-analo gue of Kontsevich’s the or em on gr aph homolo gy. Lett. Math. Phys. V ol. 76(1), 37-55, (2006). [7] A. Hamilton, A. Lazarev; Char acteristic classes of A ∞ -algebr as . arXiv:math.Q A/0608395 . [8] A. Hamilton, A. Lazarev; Symple ctic A ∞ -algebr as and string top olo gy op er ations . arXiv0:707.400 3 [9] A.Hamilton, A.Lazarev, Cohomolo gy the ories f or homotopy algebr as and nonc ommutative ge ometry . arXiv:0707 .4012 [10] A . 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Richardson, Cohomolo gy and deformations in gr ade d Lie algebr as. Bull. Amer. Math. Soc. 72 1966 1-29. [17] M. Penk av a, A. Schw arz, A ∞ algebr as and the c ohomolo gy of mo dul i sp ac es. Lie groups and Lie algebras: E. B. Dyn k in’s Seminar, 91-107, Amer. Math. S oc. T ransl. Ser. 2, 169, Amer. Math. So c., Providence, R I, 1995. [18] A . Robinson, Obstruction the ory and the strict asso ciativity of Mor ava K -the ories. Adv ances in homotopy theory (Cortona, 1988), 143-152, Lond on Math. So c. Lectu re Note Ser., 139, Cambridge U niv. Press, Cam- bridge, 1989. [19] J. Stasheﬀ, Homotopy Asso ciativity of H -Sp ac es I. T rans. A mer. Math. S oc. V ol. 108, 1963, No. 2, pp. 275-292. [20] J. St asheﬀ, Homotopy Asso ciativity of H -Sp ac es II . T rans. Amer. Math. Soc. V ol. 108, 1963, N o. 2, pp . 293-312. [21] J. Stasheﬀ, H -sp ac es fr om a homoto py p oint of view. Lectu re Notes in Mathematics, V ol. 161 S pringer-V erlag, Berlin-New Y ork 1970. [22] A . V oronov, Notes on universal algebr a. Graphs and patterns in mathematics and theoretical ph ysics, 81–103, Proc. Sy mp os. Pure Math., 73, A mer. Math. So c., Pro vidence, R I , 2005. Ma thema tics De p ar tment, U niversity of Leicester, Lei cester, England. LE1 7RH. E-mail addr ess : hamilton@mp im-bonn.m pg.de and al179@le.ac.uk 27

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