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Acta Mathematica Universitatae Comenianiae (Bratislava) 62 (1993), 17–49

arXiv:math/9207209v1 [math.QA] 1 Jul 1992Acta Mathematica Universitatae Comenianiae (Bratislava) 62 (1993), 17–49THE FR¨OLICHER-NIJENHUIS BRACKETIN NON COMMUTATIVE DIFFERENTIAL GEOMETRYAndreas CapAndreas KrieglPeter W. MichorJiˇr´ı VanˇzuraInstitut f¨ur Mathematik, Universit¨at Wien,Strudlhofgasse 4, A-1090 Wien, Austria.Mathematical Institute ofthe ˇCSAV, department Brno,Mendelovo n´am. 1, CS 662 82 Brno, CzechoslovakiaTable of contentsIntroduction.

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Convenient vector spaces. .

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Non-commutative differential forms. .

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.73. Some related questions.

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The calculus of Fr¨olicher and Nijenhuis. .

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Distributions and integrability. .

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Bundles and connections. .

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Polyderivations and the Schouten-Nijenhuis bracket. .

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. .29IntroductionThere seems to be an emerging theory of non-commutative differential geometry.In the beginning the ideas of non-commutative geometry and of non-commutativetopology were intended as tools for attacking problems in topology, in particularthe Novikov conjecture and, more generally, the Baum-Connes conjecture.

Lateron, often motivated by physics, one tended to consider ‘non-commutative spaces’Key words and phrases. Non-commutative geometry, Fr¨olicher-Nijenhuis bracket, K¨ahler dif-ferentials, graded differential algebras.Supported by Project P 7724 PHY of ‘Fonds zur F¨orderung der wissenschaftlichen Forschung’Typeset by AMS-TEX1

2CAP, KRIEGL, MICHOR, VANˇZURAas basic structures and to study them in their own right. This is also the point ofview we adopt in this paper.

We carry over to a quite general non-commutativesetting some of the basic tools of differential geometry. From the very beginning weuse the setting of convenient vector spaces developed by Fr¨olicher and Kriegl.

Thereasons for this are the following: If the non-commutative theory should containsome version of differential geometry, a manifold M should be represented by thealgebra C∞(M, R) of smooth functions on it. The simplest considerations of groups(and quantum groups begin to play an important role now) need products, andC∞(M ×N, R) is a certain completion of the algebraic tensor product C∞(M, R)⊗C∞(N, R).

Now the setting of convenient vector spaces offers in its multilinearversion a monoidally closed category, i.e. there is an appropriate tensor productwhich has all the usual (algebraic) properties with respect to bounded multilinearmappings.

So multilinear algebra is carried into this kind of functional analysiswithout loss.Moreover convenient spaces are the best realm for differentiationwhich we need in section 6 to treat a non-commutative version of principal bundles.We note that all results of this paper also hold in a purely algebraic setting:Just equip each vector space with the finest locally convex topology, then all linearmappings are bounded. They even remain valid if we take a commutative ring ofcharacteristic ̸= 2, 3 instead of the ground field.In the first section we give a short description of the setting of convenient spaceselaborating those aspects which we will need later.Then we repeat the usualconstruction of non-commutative differential forms for convenient algebras in thesecond section.There we consider triples (A, ΩA∗, d), where (ΩA∗, d) is a gradeddifferential algebra with ΩA0 = A and ΩAn = 0 for negative n. Such a triple is called aquasi resolution of A in the book [Karoubi, 1987].

See in particular [Dubois-Violette,1988] who studies the action of the Lie algebra of all derivations on ΩA∗. We willcall (ΩA∗, d) a differential algebra for A.

A universal construction of such an algebraΩA∗for a commutative algebra A is described in [Kunz, 1986], where it is called thealgebra of K¨ahler differentials, since apparently this notion was proposed for the firsttime by [K¨ahler, 1953]. The first ones to subsume the theory of K¨ahler differentialsover a regular affine variety under standard homological algebra were [Hochschild,Kostant, Rosenberg, 1962].

We present below a non-commutative version of theconstruction of Kunz, since we will need more information. This is the constructionof [Karoubi, 1982, 1983] which is also used in [Connes, 1985].

Connes’ contributionsstarted the general interest in non-commutative differential geometry. He describedthe Chern character in K-homology coming from Fredholm modules and used theuniversal differential forms as a tool for describing the cyclic cohomology of analgebra.Next we show that the bimodule Ωn(A) represents the functor of the normal-ized Hochschild n-cocyles; this is in principle contained in [Connes, 1985].

In thethird section we introduce the non-commutative version of the Fr¨olicher-Nijenhuisbracket by investigating all bounded graded derivations of the algebra of differ-

NON COMMUTATIVE FR ¨OLICHER-NIJENHUIS BRACKET3ential forms. This bracket is then used to formulate the concept of integrabilityand involutiveness for distributions and to indicate a route towards a theorem ofFrobenius (the central result of usual differential geometry, if there is one).

Thisis then used to discuss bundles and connections in the non-commutative settingand to go some steps towards a non-commutative Chern-Weil homomorphism. Inthe final section we give a brief description of the non-commutative version of theSchouten-Nijenhuis bracket and describe Poisson structures.This work was ignited by a very stimulating talk of Max Karoubi in ˇCeskySternberk in June 1989, and we want to thank him for that.

4CAP, KRIEGL, MICHOR, VANˇZURA1. Convenient vector spaces1.1.

The traditional differential calculus works well for Banach spaces. For moregeneral locally convex spaces a whole flock of different theories were developed, eachof them rather complicated and none really convincing.

The main difficulty is thatthe composition of linear mappings stops to be jointly continuous at the level ofBanach spaces, for any compatible topology. This was the original motivation forthe development of a whole new field within general topology, convergence spaces.Then in 1982, Alfred Fr¨olicher and Andreas Kriegl presented independently thesolution to the quest for the right differential calculus in infinite dimensions.

Theyjoined forces in the further development of the theory and the (up to now) finaloutcome is the book [Fr¨olicher, Kriegl, 1988].The appropriate spaces for this differential calculus are the convenient vectorspaces mentioned above. In addition to their importance for differential calculusthese spaces form a category with very nice properties.In this section we will sketch the basic definitions and the most important resultsconcerning convenient vector spaces and Fr¨olicher-Kriegl calculus.

All locally convexspaces will be assumed to be Hausdorff.1.2. The c∞-topology.

Let E be a locally convex vector space. A curve c : R →Eis called smooth or C∞if all derivatives exist (and are continuous) - this is a conceptwithout problems.

Let C∞(R, E) be the space of smooth curves. It can be shownthat C∞(R, E) does not depend on the locally convex topology of E, only on itsassociated bornology (system of bounded sets).The final topologies with respect to the following sets of mappings into E coincide:(1) C∞(R, E).

(2) Lipschitz curves (so that { c(t)−c(s)t−s: t ̸= s} is bounded in E). (3) {EB →E : B bounded absolutely convex in E}, where EB is the linearspan of B equipped with the Minkowski functional pB(x) := inf{λ > 0 : x ∈λB}.

(4) Mackey-convergent sequences xn →x (there exists a sequence 0 < λn ր ∞with λn(xn −x) bounded).This topology is called the c∞-topology on E and we write c∞E for the resultingtopological space.In general (on the space D of test functions for example) itis finer than the given locally convex topology; it is not a vector space topology,since addition is no longer jointly continuous. The finest among all locally convextopologies on E which are coarser than the c∞-topology is the bornologification ofthe given locally convex topology.

If E is a Fr´echet space, then c∞E = E.1.3. Convenient vector spaces.

Let E be a locally convex vector space. E issaid to be a convenient vector space if one of the following equivalent conditions issatisfied (called c∞-completeness):(1) Any Mackey-Cauchy-sequence (so that (xn −xm) is Mackey convergent to

NON COMMUTATIVE FR ¨OLICHER-NIJENHUIS BRACKET50) converges. (2) If B is bounded closed absolutely convex, then EB is a Banach space.

(3) Any Lipschitz curve in E is locally Riemann integrable. (4) For any c1 ∈C∞(R, E) there is c2 ∈C∞(R, E) with c1 = c′2 (existence ofantiderivative).Obviously c∞-completeness is weaker than sequential completeness so any se-quentially complete locally convex vector space is convenient.

From 1.2.4 one easilysees that c∞-closed linear subspaces of convenient vector spaces are again con-venient.We always assume that a convenient vector space is equipped with itsbornological topology.1.4. Lemma.

Let E be a locally convex space. Then the following properties areequivalent:(1) E is c∞-complete.

(2) If f : R →E is scalarwise Lipk, then f is Lipk, for k > 1. (3) If f : R →E is scalarwise C∞then f is differentiable at 0.

(4) If f : R →E is scalarwise C∞then f is C∞.Here a mapping f : R →E is called Lipk if all partial derivatives up to order kexist and are Lipschitz, locally on R. f scalarwise C∞means that λ ◦f is C∞forall continuous linear functionals on E.This lemma says that on a convenient vector space one can recognize smoothcurves by investigating compositions with continuous linear functionals.1.5. Smooth mappings.

Let E and F be locally convex vector spaces. A mappingf : E →F is called smooth or C∞, if f ◦c ∈C∞(R, F) for all c ∈C∞(R, E); sof∗: C∞(R, E) →C∞(R, F) makes sense.

Let C∞(E, F) denote the space of allsmooth mappings from E to F.For E and F finite dimensional this gives the usual notion of smooth mappings:this has been first proved in [Boman, 1967]. Constant mappings are smooth.

Mul-tilinear mappings are smooth if and only if they are bounded. Therefore we denoteby L(E, F) the space of all bounded linear mappings from E to F.1.6.

Lemma. For any locally convex space E there is a convenient vector space˜E called the completion of E and a bornological embedding i : E →˜E, which ischaracterized by the property that any bounded linear map from E into an arbitraryconvenient vector space extends to ˜E.1.7.

As we will need it later on we describe the completion in a special situation:Let E be a locally convex space with completion i : E →˜E, f : E →E a boundedprojection and ˜f : ˜E →˜E the prolongation of i ◦f. Then ˜f is also a projection and˜f( ˜E) = ker(Id −˜f) is a c∞-closed and thus convenient linear subspace of ˜E.

Usingthat f(E) is a direct summand in E one easily shows that ˜f( ˜E) is the completion

6CAP, KRIEGL, MICHOR, VANˇZURAof f(E). This argument applied to Id −f shows that ker( ˜f) is the completion ofker(f).1.8.

Structure on C∞(E, F). We equip the space C∞(R, E) with the bornologifi-cation of the topology of uniform convergence on compact sets, in all derivatives sep-arately.

Then we equip the space C∞(E, F) with the bornologification of the initialtopology with respect to all mappings c∗: C∞(E, F) →C∞(R, F), c∗(f) := f ◦c,for all c ∈C∞(R, E).1.9. Lemma.

For locally convex spaces E and F we have:(1) If F is convenient, then also C∞(E, F) is convenient, for any E. The spaceL(E, F) is a closed linear subspace of C∞(E, F), so it is convenient also. (2) If E is convenient, then a curve c : R →L(E, F) is smooth if and only ift 7→c(t)(x) is a smooth curve in F for all x ∈E.1.10.

Theorem. The category of convenient vector spaces and smooth mappingsis cartesian closed.

So we have a natural bijectionC∞(E × F, G) ∼= C∞(E, C∞(F, G)),which is even a diffeomorphism.Of course this statement is also true for c∞-open subsets of convenient vectorspaces.1.11. Corollary.

Let all spaces be convenient vector spaces. Then the followingcanonical mappings are smooth.ev : C∞(E, F) × E →F,ev(f, x) = f(x).ins : E →C∞(F, E × F),ins(x)(y) = (x, y).

()∧: C∞(E, C∞(F, G)) →C∞(E × F, G),ˆf(x, y) = f(x)(y). ()∨: C∞(E × F, G) →C∞(E, C∞(F, G)),ˇg(x)(y) = g(x, y).comp : C∞(F, G) × C∞(E, F) →C∞(E, G)C∞(,) : C∞(F, F ′) × C∞(E′, E) →C∞(C∞(E, F), C∞(E′, F ′))(f, g) 7→(h 7→f ◦h ◦g)Y:YC∞(Ei, Fi) →C∞(YEi,YFi)1.12.

Theorem. Let E and F be convenient vector spaces.

Then the differentialoperatord : C∞(E, F) →C∞(E, L(E, F)),df(x)v := limt→0f(x + tv) −f(x)t,

NON COMMUTATIVE FR ¨OLICHER-NIJENHUIS BRACKET7exists and is linear and bounded (smooth). Also the chain rule holds:d(f ◦g)(x)v = df(g(x))dg(x)v.1.13.

The category of convenient vector spaces and bounded linear maps is com-plete and cocomplete, so all categorical limits and colimits can be formed.Inparticular we can form products and direct sums of convenient vector spaces.For convenient vector spaces E1, . .

. ,En and F we can now consider the spaceof all bounded n-linear maps, L(E1, .

. ., En; F), which is a closed linear subspaceof C∞(Qni=1 Ei, F) and thus again convenient.

It can be shown that multilinearmaps are bounded if and only if they are partially bounded, i.e. bounded in eachcoordinate and that there is a natural isomorphism (of convenient vector spaces)L(E1, .

. ., En; F) ∼= L(E1, .

. .

, Ek; L(Ek+1, . .

., En; F))1.14.Theorem. On the category of convenient vector spaces there is a uniquetensor product ˜⊗which makes the category symmetric monoidally closed, i.e.

thereare natural isomorphisms of convenient vector spacesL(E1; L(E2, E3)) ∼= L(E1 ˜⊗E2, E3),E1 ˜⊗E2 ∼= E2 ˜⊗E1,E1 ˜⊗(E2 ˜⊗E3) ∼= (E1 ˜⊗E2)˜⊗E3,E ˜⊗R ∼= E.The tensor product can be constructed as follows: On the algebraic tensor prod-uct put the finest locally convex topology such that the canonical bilinear map fromthe product into the tensor product is bounded and then take the completion ofthis space.1.15. Remarks.

Note that the conclusion of theorem 1.10 is the starting pointof the classical calculus of variations, where a smooth curve in a space of functionswas assumed to be just a smooth function in one variable more.If one wants theorem 1.10 to be true and assumes some other obvious properties,then the calculus of smooth functions is already uniquely determined.There are, however, smooth mappings which are not continuous. This is un-avoidable and not so horrible as it might appear at first sight.

For example theevaluation E × E′ →R is jointly continuous if and only if E is normable, but it isalways smooth. Clearly smooth mappings are continuous for the c∞-topology.For Fr´echet spaces smoothness in the sense described here coincides with thenotion C∞cof [Keller, 1974].

This is the differential calculus used by [Michor, 1980],[Milnor, 1984], and [Pressley, Segal, 1986].2. Non-commutative differential forms2.1.

Axiomatic setting for the algebra of differential forms. Throughoutthis section we assume that A is a convenient algebra, i.e.

A is a convenient vector

8CAP, KRIEGL, MICHOR, VANˇZURAspace together with a bounded bilinear associative multiplication A × A →A.Moreover we assume that A has a unit 1. We consider now a graded associativeconvenient algebra ΩA∗= Lp≥0 ΩAp where ΩA0 = A and each ΩAp is a convenientvector space, with a bounded bilinear product : ΩAp ×ΩAq →ΩAp+q, such that there isa bounded linear mapping d = dp : ΩAp →ΩAp+1 with d2 = 0 and d(ωpωq) = dωpωq +(−1)pωpdωq for all ωp ∈ΩAp and ωq ∈ΩAq .

This mapping is called the differentialof ΩA∗.Note that we do not assume that the product is graded commutative:ωpωq ̸= (−1)pqωqωp in general.Let [ΩA∗, ΩA∗]r be the locally convex closure of the subspace generated by allgraded commutators [ωp, ωq] := ωpωq −(−1)pqωqωp with p + q = r. We put ¯ΩAr :=ΩAr /[ΩA∗, ΩA∗]r and we let T : ΩAr →¯ΩAr be the projection which will be called thegraded trace of ΩA∗.Since we have d([ωp, ωq]) = [dωp, ωq] + (−1)p[ωp, dωq], the differential passes to¯ΩA∗and still satisfies d2 = 0. The separated homology of this quotient complex iscalled the non-commutative De Rham homology of ΩA∗or of A, if ΩA∗is clear.

Wedenote it byH ¯ΩAp = ¯HAp = ker(d : ¯ΩAp →¯ΩAp+1)/im(d : ¯ΩAp−1 →¯ΩAp ).2.2. Derivations.

Let M be a convenient bimodule over the convenient algebraA, i.e. M is a convenient vector space together with two bounded homomorphismsof unital algebras λ : A →L(M, M) and ρ : Aop →L(M, M), where Aop denotesthe opposite algebra to A, such that for a, b ∈A we have λ(a) ◦ρ(b) = ρ(b) ◦λ(a).We will write am for λ(a)(m) and ma for ρ(a)(m).

This definition is equivalentto having bounded bilinear maps λ : A × M →M and ρ : M × A →M, whichsatisfy the usual axioms. A (bounded) derivation of A in M is a bounded linearmapping D : A →M such that D(ab) = D(a)b + aD(b) for all a, b ∈A.

We denoteby Der(A; M) the vector space of all derivations of A into M. This is obviouslya closed linear subspace of L(A, M) and thus a convenient vector space. If A iscommutative, then Der(A; M) is again an A-module.The vector space Der(A; A) is a convenient Lie algebra where the bracket is thecommutator.

It is an A-module if and only if A is commutative.2.3. The algebra of dual numbers of a convenient algebra A with respect toa convenient A-bimodule M is the semidirect product AⓈM, i.e.

the convenientvector space A × M with the bounded bilinear multiplication (a1, m1)(a2, m2) :=(a1a2, a1m2 + m1a2). This is an associative convenient algebra with unit (1, 0).2.4.

Lemma. The bounded derivations from A into the A-bimodule M correspondexactly to the bounded algebra homomorphisms ϕ : A →AⓈM satisfying pr1 ◦ϕ =IdA.□

NON COMMUTATIVE FR ¨OLICHER-NIJENHUIS BRACKET92.5. Universal derivations.

A bounded derivation D : A →M into a bimoduleM is called universal if the following holds:For any bounded derivation D′ : A →N into a convenient A-bimodule Nthere is a unique bounded A-bimodule homomorphism Φ : M →N suchthat D′ = Φ ◦D.Of course for any two universal derivations D1 : A →M1 and D2 : A →M2 thereis a unique A-bimodule isomorphism Φ : M1 →M2 such that D2 = Φ ◦D1. So auniversal derivation is unique up to canonical isomorphism.Lemma.

For every convenient algebra A there exists a universal derivation whichwe denote by d : A →Ω1(A).Proof. First we define an A-bimodule structure on A˜⊗A as follows: Let (a, b) 7→a ⊗b : A × A →A˜⊗A be the canonical bilinear map.Now consider the map¯λ : A →L(A × A, A˜⊗A) defined by ¯λ(a)(b, c) := ab ⊗c.

Obviously the map ¯λ hasvalues in the space L(A, A; A˜⊗A) of bilinear maps and thus we can compose it withthe isomorphisms of 1.13 and 1.14 to get λ : A →L(A˜⊗A, A˜⊗A) which is easilyseen to be an algebra homomorphism. Similarly we define ρ : A →L(A˜⊗A, A˜⊗A)using ¯ρ(a)(b, c) := b ⊗ca.The multiplication on A induces a bounded linear map µ : A˜⊗A →A which is anA-bimodule homomorphism by associativity.

Thus Ω1(A) := ker(µ) is a convenientA-bimodule.Next we define d : A →Ω1(A) by d(a) := 1⊗a−a⊗1. Obviously d is a boundedderivation.To see that this derivation is universal let D : A →M be a bounded derivationfrom A into a convenient A-bimodule M. Let ¯Φ : A × A →M be the map definedby ¯Φ(a, b) := aD(b).

Then ¯Φ is obviously bilinear and bounded and thus it inducesa bounded linear map Φ : A˜⊗A →M, whose restriction to Ω1(A) we also denoteby Φ. As any derivation vanishes on 1 we get:(Φ ◦d)(a) = Φ(1 ⊗a −a ⊗1) = 1D(a) −aD(1) = D(a)So it remains to show that Φ is a bimodule homomorphism.

For a, b, c ∈A we get:(Φ ◦λ(a))(b ⊗c) = Φ(ab ⊗c) = abD(c) = a(Φ(b ⊗c)) and thus Φ : A˜⊗A →M is ahomomorphism of left modules.On the other hand (Φ ◦ρ(a))(b ⊗c) = bD(ca) = (bD(c))a + bcD(a) and thuswe get the identity (Φ ◦ρ(a))(x) = (Φ(x))a + µ(x)D(a) for all x ∈A˜⊗A and soΦ : Ω1(A) →M is a homomorphism of right modules, too.□2.6. Corollary.

For an A-bimodule M the canonical linear mappingd∗: HomAA(Ω1(A), M) →Der(A; M)ϕ 7→ϕ ◦d

10CAP, KRIEGL, MICHOR, VANˇZURAis an isomorphism of convenient vector spaces, where Der(A; M) carries the struc-ture described in 2.2, while the space HomAA(Ω1(A), M) of bounded bimodule homo-morphisms is considered as a closed linear subspace of L(Ω1(A), M). In particularwe have HomAA(Ω1(A), A) ∼= Der(A; A).Proof.

Since d is bounded and linear so is d∗. In the proof of the lemma above we sawthat the inverse to d∗is given by mapping D to the prolongation of ℓ◦(Id×D), whereℓdenotes the left action of A on M and this map is easily seen to be bounded.□2.7.

Lemma. Let A be a convenient algebra, M a convenient right A-module andN a convenient left A-module.

(1) There is a convenient vector space M ˜⊗AN and a bounded bilinear mapb : M × N →M ˜⊗AN, (m, n) 7→m ⊗A n such that b(ma, n) = b(m, an) forall a ∈A, m ∈M and n ∈N which has the following universal property: IfE is a convenient vector space and f : M × N →E is a bounded bilinearmap such that f(ma, n) = f(m, an) then there is a unique bounded linearmap ˜f : M ˜⊗AN →E with ˜f ◦b = f.(2) Let LA(M, N; E) denote the space of all bilinear bounded maps f : M ×N →E having the above property, which is a closed linear subspace ofL(M, N; E).Then we have an isomorphism of convenient vector spacesLA(M, N; E) ∼= L(M ˜⊗AN, E). (3) If B is another convenient algebra such that N is a convenient right B-module and such that the actions of A and B on N commute, then M ˜⊗ANis in a canonical way a convenient right B-module.

(4) If in addition P is a convenient left B-module then there is a natural iso-morphism of convenient vector spacesM ˜⊗A(N ˜⊗BP) ∼= (M ˜⊗AN)˜⊗BPProof. We construct M ˜⊗AN as follows: Let M ⊗N be the algebraic tensor productof M and N equipped with the (bornological) topology mentioned in 1.14 and let Vbe the locally convex closure of the subspace generated by all elements of the formma⊗n−m⊗an and define M ˜⊗AN to be the completion of M ⊗AN := (M ⊗N)/V .As M ⊗N has the universal property that bounded bilinear maps from M ×N intoarbitrary locally convex spaces induce bounded and hence continuous linear mapson M ⊗N, (1) is clear.

(2): By (1) the bounded linear map b∗: L(M ˜⊗AN, E) →LA(M, N; E) is abijection. Thus it suffices to show that its inverse is bounded, too.

From 1.14 weget a bounded linear map ϕ : L(M, N; E) →L(M ⊗N, E) which is inverse to themap induced by the canonical bilinear map. Now let Lann V (M ⊗N, E) be theclosed linear subspace of L(M ⊗N, E) consisting of all maps which annihilate V .Restricting ϕ to LA(M, N; E) we get a bounded linear map ϕ : LA(M, N; E) →Lann V (M ⊗N, E).

NON COMMUTATIVE FR ¨OLICHER-NIJENHUIS BRACKET11Let ψ : M ⊗N →M ⊗A N →M ˜⊗AN be the composition of the canonicalprojection with the inclusion into the completion. Then ψ induces a well definedlinear map ˆψ : Lann V (M ⊗N, E) →L(M ˜⊗AN, E) and ˆψ ◦ϕ is inverse to b∗.

So itsuffices to show that ˆψ is bounded.This is the case if and only if the associated map Lann V (M ⊗N, E)×(M ˜⊗AN) →E is bounded. This in turn is equivalent to boundedness of the associated mapM ˜⊗AN →L(Lann V (M ⊗N, E), E).

But this is just the prolongation to the com-pletion of the map M ⊗A N →L(Lann V (M ⊗N, E), E) which sends x to theevaluation at x and this map is clearly bounded. (3): Let ρ : Bop →L(N, N) be the right action of B on N and let Φ : LA(M ×N, M ˜⊗AN) ∼= L(M ˜⊗AN, M ˜⊗AN) be the isomorphism constructed in (2).Wedefine the right module structure on M ˜⊗AN as:Bopρ−→L(N, N)Id×.−−−→L(M × N, M × N)b∗−→−→LA(M, N; M ˜⊗AN)Φ−→L(M ˜⊗AN, M ˜⊗AN)This map is obviously bounded and easily seen to be an algebra homomorphism.

(4): Straightforward computations show that both spaces have the followinguniversal property: For a convenient vector space E and a trilinear map f : M ×N × P →E which satisfies f(ma, n, p) = f(m, an, p) and f(m, nb, p) = f(m, n, bp)there is a unique linear map prolonging f.□2.8. Homomorphisms of differential algebras.

Let ϕ : A →B be a homo-morphism of convenient algebras, let (ΩA, dA) be a differential algebra for A in thesense of 2.1, and let (ΩB, dB) be one for B.By a ϕ-homomorphism Φ : ΩA →ΩB we mean a bounded homomorphism ofgraded differential algebras such that Φ0 = ϕ : ΩA0 = A →B = ΩB0 .2.9. Theorem.

Existence of the universal graded differential algebra. Foreach convenient algebra A there is a convenient graded differential algebra (Ω(A), d)for A with the following property:For any bounded homomorphism ϕ : A →B of convenient algebras andfor any convenient graded differential algebra (ΩB, dB) for B there exists aunique ϕ-homomorphism Ω(A) →ΩB.Proof.

Put Ω0(A) = A and Ωk(A) := Ω1(A)˜⊗A . .

. ˜⊗AΩ1(A) (k factors).

Then eachΩk(A) is a convenient A-bimodule by 2.7.3, which also defines the multiplicationwith elements of Ω0(A). For k, ℓ> 0 we define the multiplication as the canonicalbilinear mapΩk(A) × Ωℓ(A) →Ωk(A)˜⊗AΩℓ(A) ∼= Ωk+ℓ(A)Thus Ω(A) = Lk Ωk(A) is a convenient graded algebra.

12CAP, KRIEGL, MICHOR, VANˇZURAClaim. There is an isomorphism Ω1(A) ∼= A˜⊗(A/R) of convenient vector spaces.Consider the embedding i : R →A and the projection p : A →A/R, denoted alsoby p(a) =: ¯a.

We consider the following diagram, where the horizontal and thevertical sequences are exact:0xAAµx0 −−−−→A˜⊗RId ˜⊗i−−−−→A˜⊗AId ˜⊗p−−−−→A˜⊗(A/R) −−−−→0xΩ1(A)x0The vertical sequence is splitting: a 7→a ⊗1 is a section for µ and the prolongationof (a, b) 7→a d(b) is a retraction onto Ω1(A) which even factors over Id˜⊗p, sinceby 1.7 the space Ω1(A) is the completion of the kernel of the prolongation of themultiplication map to A ⊗A. So we may invert all arrows of the vertical sequenceand the two sequences are isomorphic as required.Claim.

There is an isomorphism of convenient vector spacesA˜⊗k-timesz}|{A/R˜⊗· · · ˜⊗A/R →Ωk(A)which is induced by the map (a0, ¯a1, . .

., ¯ak) 7→a0da1 ⊗A da2 ⊗A · · · ⊗A dak. Thisis a direct consequence of the last claim and lemma 2.7.We now define d : Ωk(A) →Ωk+1(A) by d(a) = 1 ⊗a −a ⊗1 for a ∈Ω0(A) = Aand for k > 0 as the mapping defined on Ωk(A) ∼= A˜⊗A/R˜⊗.

. .

˜⊗A/R which isassociated to:(a0, ¯a1, . .

., ¯ak) 7→1 ⊗¯a0 ⊗¯a1 ⊗· · · ⊗¯akA × (A/R)k →A˜⊗A/R˜⊗. .

. ˜⊗A/R ∼= Ωk+1(A)Let us show now that d is a graded derivation: We have to show that for ωk ∈Ωk(A)and ωℓ∈Ωℓ(A) we have d(ωkωℓ) = d(ωk)ωℓ+ (−1)kωkd(ωℓ).We proceed byinduction on k. By the claim above it suffices to check the identity for elements of

NON COMMUTATIVE FR ¨OLICHER-NIJENHUIS BRACKET13A × (A/R)i. For k = 0 we have a(b0,¯b1, .

. .

,¯bℓ) = (ab0,¯b1, . .

.,¯bℓ) which is mappedby d to the element (1, ab0,¯b1, . .

.,¯bℓ) which under the isomorphism with Ωℓ(A) goesto d(ab0) ⊗A db1 ⊗A · · · ⊗A dbℓso the result follows from the derivation property ofd : A →Ω1(A).In the general case we first see that using this derivation property again, theproduct of (a0, ¯a1, . .

., ¯ak) and (b0,¯b1, . .

.,¯bℓ) in Ωk+ℓ(A) can be written asa0da1 ⊗A · · · ⊗A dak−1 ⊗A d(akb0) ⊗A db1 ⊗A · · · ⊗A dbℓ−−(a0da1 ⊗A · · · ⊗A dak−1)(akdb0 ⊗A db1 ⊗A · · · ⊗A dbℓ)and from this the result follows easily using the induction hypothesis.So let us turn to the universal property. Let B be a convenient algebra, (ΩB, dB)a convenient differential algebra for B and ϕ : A →B a bounded homomorphismof algebras.

Via ϕ and the multiplication of ΩB all spaces ΩBiare convenient A-bimodules.As dB is a graded derivation the map dB ◦ϕ : A →ΩB1 is a derivation. Thus bythe universal property of Ω1(A) we get a unique bounded bimodule homomorphismϕ1 : Ω1(A) →ΩB1 .

Thus for a ∈A and ω ∈Ω1(A) we have ϕ1(aω) = ϕ(a)ϕ1(ω)and ϕ1(ωa) = ϕ1(ω)ϕ(a).Consider the map f : (Ω1(A))k →ΩBk defined byf(ω1, ω2, . .

., ωk) := ϕ1(ω1)ϕ1(ω2) . .

.ϕ1(ωk) which is obviously bounded and k-linear. Moreover as ϕ1 is a bimodule homomorphism we get f(.

. .

, ωia, ωi+1, . .

.) =f(.

. ., ωi, aωi+1, .

. .

). Thus there is a unique prolongation of f to Ωk(A) which wedefine to be ϕk.

From this definition it is obvious that the maps ϕi form a boundedhomomorphism of graded algebras.The composition:A×A/R × · · · × A/R →A˜⊗A/R . .

. ˜⊗A/R ∼= Ωk(A)ϕk−→ΩBkis given by(a0, ¯a1, .

. ., ¯ak) 7→a0da1 ⊗A da2 ⊗A · · · ⊗A dak 7→ϕ(a0)ϕ1(da1) .

. .ϕ1(dak) = ϕ(a0)dB(ϕ(a1)) .

. .dB(ϕ(ak))and this element is mapped by dB to dB(ϕ(a0))dB(ϕ(a1)) .

. .dB(ϕ(ak)).

This showsthat ϕk+1 ◦d = dB ◦ϕk□2.10. Corollary.

The construction A 7→Ω∗(A) defines a covariant functor fromthe category of convenient algebras with unit to the category of convenient gradeddifferential algebras.So for a bounded algebra homomorphism f : A →B we denote by Ω∗(f) :Ω∗(A) →Ω∗(B) its universal prolongation.

14CAP, KRIEGL, MICHOR, VANˇZURA3. Some related questionsIn the following we treat two questions which arise naturally in the context ofsection 2 but which are not relevant for the developments afterwards.3.1.

The kernel of the multiplication µ : A˜⊗A →A is the very important spaceΩ1(A). What about the analogue with more factors?Proposition.

Let A be a convenient algebra with unit.Then the kernel of then-ary multiplication µn : A ˜⊗n →A is the subspacen−2Xi=0A ˜⊗i ˜⊗Ω1(A)˜⊗A ˜⊗(n−2−i) ⊂A ˜⊗n.Proof. Note that µ2 = µ : A˜⊗A →A.

We prove the assertion by induction on n.Consider the following commutative diagram:0u0uA ˜⊗(n−1) ˜⊗Ω1(A)uw Ω1(A)wu00w n−2Xi=0A ˜⊗i ˜⊗Ω1(A)˜⊗A ˜⊗(n−2−i)!˜⊗AwuA ˜⊗(n+1)uA ˜⊗(n−1) ˜⊗µwµn ˜⊗ANNNNNNPµn+1A˜⊗Awuµ00wn−2Xi=0A ˜⊗i ˜⊗Ω1(A)˜⊗A ˜⊗(n−2−i)uw A ˜⊗nwµnuAwu0000The right hand column is the defining sequence for Ω1(A) and it is splitting. Themiddle column being the right hand one tensored with A ˜⊗(n−1) from the left isthen again splitting and thus exact.

The bottom row is exact by the inductionhypothesis and is also splitting since µn admits many obvious sections. The middlerow is the bottom one tensored with A from the right and it is again splitting andthus exact.

The left hand side vertical arrow is multiplication from the right. Thetop horizontal arrow is total multiplication onto the left of Ω1(A).Let us now take an element x ∈A ˜⊗(n+1) which is in the kernel of µn+1.

Then asimple diagram chasing shows that x is in the sum of the two subspaces of A ˜⊗(n+1)which are above and to the left. The converse is trivial, so the result follows.□

NON COMMUTATIVE FR ¨OLICHER-NIJENHUIS BRACKET153.2. We have seen in 2.6 that Ω1(A) is the representing object for the functorDer(A,) on the category of A-bimodules.

Which functor is represented by Ωn(A)?Recall that Ωn(A) = Ω1(A)˜⊗A . .

. ˜⊗AΩ1(A) (n times).

We consider the n-linearmappingdn : An →(A/R)n →Ωn(A),dn(a1, . .

.an) := da1 ⊗A · · · ⊗A dan.We view it as a Hochschild cochain which is bounded as a multilinear mappingand normalized, i. e. it factors to (A/R)n. It is well known that the normalizedHochschild complex leads to the usual Hochschild cohomology, see [Cartan, Eilen-berg, 1956, p. 176].Lemma. The mapping dn is a normalized and bounded Hochschild cocycle withvalues in the A-bimodule Ωn(A).Proof.

By definition of the right A-module structure on Ωn(A) we havedn(a1, . .

., an)an+1 = (da1 ⊗A · · · ⊗A dan)an+1= da1 ⊗A · · · ⊗A d(anan+1) −(da1 ⊗A · · · ⊗A dan−1)an ⊗A dan+1= dn(a1, . .

., anan+1) −dn(a1, . .

., an−1an, an+1)+ (da1 ⊗A · · · ⊗A dan−2)an−1 ⊗A (dan ⊗A dan+1)= . .

.=nXi=1(−1)n−idn(a1, . .

. , aiai+1, .

. ., an+1) + (−1)na1dn(a2, .

. ., an+1),and thus as required0 = a1dn(a2, .

. ., an+1) +nXi=1(−1)idn(a1, .

. ., aiai+1, .

. ., an+1)+ (−1)n+1dn(a1, .

. .

, an)an+1=: (δdn)(a1, . .

., an+1),where δ denotes the usual Hochschild coboundary operator.□3.3. Proposition.

Let M be an A-bimodule. Then the mapping(dn)∗: HomAA(Ωn(A), M) →¯Zn(A, M)is an isomorphism onto the space of all normalized and bounded Hochschild cocycleswith values in M.Proof.

Clearly for any bimodule homomorphism Φ : Ωn(A) →M the n-linearmapping Φ ◦dn : ¯An →M is a normalized and bounded Hochschild cocycle. Let us

16CAP, KRIEGL, MICHOR, VANˇZURAassume conversely that c : An →M is a normalized bounded cocycle. In the proofof 2.9 we got a natural isomorphism of convenient vector spacesA˜⊗k-timesz}|{A/R˜⊗· · · ˜⊗A/R →Ωk(A)which is given by a0⊗¯a1⊗· · ·⊗¯ak 7→a0da1⊗Ada2⊗A· · ·⊗Adak.

Using this we defineΦc : Ωn(A) →M by Φc(a0da1 . .

. dan) := a0c(a1, .

. ., an).

Then clearly Φ ◦dn = c.Obviously Φc is a homomorphism of left A-modules and from the definition of theright A-module structure on Ωn(A) we see that δc = 0 translates into Φc being aright module homomorphism, by a computation which is completely analogous tothe one in the proof of 3.2. Obviously both constructions are bounded.□3.4.

Is it possible to recognize the Hochschild coboundaries in the description¯Zn(A, M) ∼= HomAA(Ωn(A), M)?In order to answer this question we consider the canonical normalized mapping,where a 7→¯a is the quotient mapping A →A/R:ϕ : An−1 →A˜⊗n−1z}|{(A/R)˜⊗. .

. ˜⊗(A/R) ˜⊗Aϕ(a1, .

. ., an−1) := 1 ⊗¯a1 ⊗· · · ⊗¯an−1 ⊗1Then ∂ϕ ∈¯Zn(A; A˜⊗(A/R) ˜⊗(n−1) ˜⊗A) is given by∂ϕ(a1, .

. ., an) = a1ϕ(a2, .

. ., an) +n−1Xi=1(−1)iϕ(a1, .

. ., aiai+1, .

. ., an)+ (−1)nϕ(a1, .

. ., an−1)an= a1 ⊗¯a2 ⊗· · · ⊗¯an ⊗1+n−1Xi=1(−1)i1 ⊗¯a1 ⊗· · · ⊗aiai+1 ⊗· · · ⊗¯an ⊗1+ (−1)n1 ⊗¯a1 ⊗· · · ⊗¯an−1 ⊗an.By proposition 3.3 there exists a unique bimodule homomorphism I : Ωn(A) →A˜⊗(A/R) ˜⊗(n−1) ˜⊗A such that ∂ϕ = I ◦dn.A short computation (again essentially the same as in the proof of lemma 3.2)shows that this bimodule homomorphism I coincides with the following compositionof canonical mappings:Ωn(A) = Ω1(A)˜⊗A .

. .

˜⊗AΩ1(A)i⊗···⊗i−−−−→i⊗···⊗i−−−−→(A˜⊗A)˜⊗A . .

. ˜⊗A(A˜⊗A) ∼=n+1z}|{A˜⊗.

. .

˜⊗A →A˜⊗n−1z}|{(A/R)˜⊗. .

. ˜⊗(A/R) ˜⊗A,where i is the injection Ω1(A) = ker µ →A˜⊗A.

NON COMMUTATIVE FR ¨OLICHER-NIJENHUIS BRACKET173.5. Proposition.

Let Φ : Ωn(A) →M be a bimodule homomorphism. Then thecorresponding normalized Hochschild cocycle Φ◦dn is a coboundary if and only if Φfactors over I to a bimodule homomorphism ˜Φ : A˜⊗(A/R) ˜⊗(n−1) ˜⊗A →M, so thatΦ = ˜Φ ◦I.In more details: for any bimodule homomorphism Ψ : A˜⊗(A/R) ˜⊗(n−1) ˜⊗A →Mwe have Ψ ◦I ◦dn = ∂ψ where the normalized bounded cochain ψ : An−1 →M isgiven byψ(a1, .

. .

, an−1) = Ψ(1 ⊗¯a1 ⊗· · · ⊗¯an−1 ⊗1).Proof. Let Φ ◦dn be a coboundary.

Then there is an (n −1)-linear mapping c :An−1 →M such that ∂c = Φ ◦dn. This mapping c induces a unique bimodulehomomorphism˜Φ : A˜⊗(A/R) ˜⊗(n−1) ˜⊗A →M,˜Φ(a0 ⊗¯a1, .

. ., ¯an, an+1) = a0 · c(a1, .

. ., an) · an+1.and we have ˜Φ ◦I ◦dn = ˜Φ ◦∂ϕ, and moreover(˜Φ ◦∂ϕ)(a1, .

. ., an) = ˜Φ(a1 ⊗¯a2 ⊗· · · ⊗¯an ⊗1)+n−1Xi=1(−1)i ˜Φ(1 ⊗¯a1 ⊗· · · ⊗aiai+1 ⊗· · · ⊗¯an ⊗1)+ (−1)n ˜Φ(1 ⊗¯a1 ⊗· · · ⊗¯an−1 ⊗an)= ∂c(a1, .

. ., an).So we get Φ ◦dn = ∂c = ˜Φ ◦I ◦dn and the result follows from 3.3.The second assertion of the proposition follows also from the last computa-tion.□3.6.

Corollary. For a convenient algebra A and a convenient bimodule M over Awe haveHn(A, M) ∼=HomAA(Ωn(A), M)I∗(HomAA(A˜⊗¯A ˜⊗(n−1) ˜⊗A, M)).□4.

The calculus of Fr¨olicher and Nijenhuis4.1. In this section let A be a convenient algebra with unit and let Ω(A) = Ω∗(A)be the universal graded differential algebra for A.

The space Derk Ω(A) consistsof all bounded (graded) derivations of degree k, i.e. all bounded linear mappingsD : Ω(A) →Ω(A) with D(Ωℓ(A)) ⊂Ωk+ℓ(A) and D(ϕψ) = D(ϕ)ψ + (−1)kℓϕD(ψ)for ϕ ∈Ωℓ(A).

Obviously Derk Ω(A) is a closed linear subspace of L(Ω(A), Ω(A))and thus a convenient vector space.

18CAP, KRIEGL, MICHOR, VANˇZURALemma. The space Der Ω(A) = Lk Derk Ω(A) is a convenient graded Lie algebrawith the graded commutator [D1, D2] := D1 ◦D2 −(−1)k1k2D2 ◦D1 as bracket.

Thismeans that the bracket is graded anticommutative, [D1, D2] = −(−1)k1k2[D2, D1],and satisfies the graded Jacobi identity[D1, [D2, D3]] = [[D1, D2], D3] + (−1)k1k2[D2, [D1, D3]](so that ad(D1) = [D1,] is itself a derivation).Proof. Plug in the definition of the graded commutator and compute.

The bound-edness of the bracket follows from 1.11.□4.2. Fields.

Recall from 2.6 that d∗: HomAA(Ω1(A), A) →Der(A; A) is an isomor-phism, which we will also denote by L. We denote the space HomAA(Ω1(A), A) byX(A) and call it the space of fields for the algebra A. Then L : X(A) →Der(A; A)is an isomorphism of convenient vector spaces.

The space of derivations Der(A; A)is a convenient Lie algebra with the commutator [,] as bracket, and so we havean induced Lie bracket on X(A) = HomAA(Ω1(A), A) which is given by L([X, Y ])a =[LX, LY ]a = LXLY a −LY LXa. It will be referred to as the Lie bracket of fields.4.3.

Lemma. Each field X ∈X(A) = HomAA(Ω1(A), A) is by definition a boundedA-bimodule homomorphism Ω1(A) →A.

It prolongs uniquely to a graded derivationj(X) = jX : Ω(A) →Ω(A) of degree −1 byjX(a) = 0for a ∈A = Ω0(A),jX(ω) = X(ω)for ω ∈Ω1(A)jX(ω1 ⊗A · · · ⊗A ωk) ==k−1Xi=1(−1)i−1ω1⊗A · · · ⊗A ωi−1 ⊗A X(ωi)ωi+1 ⊗A · · · ⊗A ωk+(−1)k−1ω1⊗A · · · ⊗A ωk−1X(ωk)for ωi ∈Ω1(A). The derivation jX is called the contraction operator of the field X.Proof.

This is an easy computation□With some abuse of notation we write also ω(X) = X(ω) = jX(ω) for ω ∈Ω1(A)and X ∈X(A) = HomAA(Ω1(A), A).4.4. A derivation D ∈Derk Ω(A) is called algebraic if D | Ω0(A) = 0.ThenD(aω) = aD(ω) and D(ωa) = D(ω)a for a ∈A, so D restricts to a boundedbimodule homomorphism, an element of HomAA(Ωl(A), Ωl+k(A)).Since we haveΩl(A) = Ω1(A)˜⊗A .

. .

˜⊗AΩ1(A) and since for a product of one forms we have

NON COMMUTATIVE FR ¨OLICHER-NIJENHUIS BRACKET19D(ω1 ⊗A · · · ⊗A ωl) = Pli=1(−1)ikω1 ⊗A · · · ⊗A D(ωi) ⊗A · · · ⊗A ωl, the deriva-tion D is uniquely determined by its restrictionK := D|Ω1(A) ∈HomAA(Ω1(A), Ωk+1(A));we write D = j(K) = jK to express this dependence. Note the defining equationjK(ω) = K(ω) for ω ∈Ω1(A).

Since it will be very important in the sequel we willuse the notationΩ1k = Ω1k(A) : = HomAA(Ω1(A), Ωk(A))Ω1∗= Ω1∗(A) =∞Mk=0Ω1k(A).Elements of the space Ω1k will be called field valued k-forms, those of Ω1∗will becalled just field valued forms.4.5. In 4.3 we have already met some algebraic graded derivations: for a fieldX ∈X(A) the derivation jX is of degree −1.

The basic derivation d is of degree 1.Note also that LX := d jX + jX d translates to LX = [jX, d] and that this extendsLX from a derivation A to a derivation of degree 0 of Ω∗(A).4.6 Theorem. (1) For K ∈Ω1k+1(A) and ωi ∈Ω1(A) the formulajK(ω0 ⊗A · · · ⊗A ωℓ) =ℓXi=0(−1)ikω0 ⊗A · · · ⊗A K(ωi) ⊗A · · · ⊗A ωkdefines an algebraic graded derivation iK ∈Derk Ω(A) and any algebraic derivationis of this form.

(2) The mapj : Ω1k+1 = HomAA(Ω1(A), Ωk+1(A)) →DeralgkΩ(A)where DeralgkΩ(A) denotes the closed linear subspace of Derk Ω(A) consisting of allalgebraic derivations is an isomorphism of convenient vector spaces. (3) By j([K, L]∆) := [jK, jL] we get a bracket [,]∆on the space Ω1∗−1 whichdefines a convenient graded Lie algebra structure with the grading as indicated, andfor K ∈Ω1k+1,and L ∈Ω1ℓ+1 we have[K, L]∆= jK ◦L −(−1)kℓjL ◦K.

[,]∆is called the algebraic bracket or also the abstract De Wilde, Lecomtebracket see [DeWilde, Lecomte, 1988].

20CAP, KRIEGL, MICHOR, VANˇZURAProof. The first assertion is clear from the definition.Clearly the map D 7→D|Ω1(A) is bounded.

To show that j is bounded recallthat Derd Ω(A) is a closed subspace of L(Ω(A), Ω(A)) ∼=Qk L(Ωk(A), Ω(A)). By2.7.2 it suffices to show that j is bounded as a map to LA(Ω1(A), .

. ., Ω1(A); Ω(A))and by the linear uniform boundedness principle 1.9.2 it is enough to show that forall ωi ∈Ω1(A) the map K 7→jK(ω1 ⊗A · · · ⊗A ωk) is bounded.

But this is clear by(1).For the third assertion it suffices to evaluate [jK, jL] at some ω ∈Ω1(A).□4.7. The exterior derivative d is an element of Der1 Ω(A).

In view of the formulaLX = [jX, d] = jX d + d jX for fields X, we define for K ∈Ω1k the Lie derivationLK = L(K) ∈Derk Ω(A) by LK := [jK, d].Then the mapping L : Ω1∗→Der Ω(A) is obviously bounded and it is injectiveby the universal property of Ω1(A), since LKa = jKda = K(da) for a ∈A.Theorem. For any graded derivation D ∈Derk Ω(A) there are unique homomor-phisms K ∈Ω1k and L ∈Ω1k+1 such thatD = LK + jL.We have L = 0 if and only if [D, d] = 0.

D is algebraic if and only if K = 0.Proof. D|A : a 7→Da is a derivation A →Ωd(A), so by 2.5 it is of the formD|A = K ◦d for a unique K ∈Ω1k.The defining equation for K is Da = jKda = LKa for a ∈A.

Thus D −LK isan algebraic derivation, so D −LK = jL by 4.4 for unique L ∈Ω1k+1.Since we have [d, d] = 2d2 = 0, by the graded Jacobi identity we obtain 0 =[jK, [d, d]] = [[jK, d], d] + (−1)k−1[d, [jK, d]] = 2[LK, d]. The mapping L 7→[jL, d] =LL is injective, so the last assertion follows.□4.8.

The Fr¨olicher-Nijenhuis bracket. Note that j(IdΩ1(A))ω = kω for ω ∈Ωk(A).

Therefore we have L(IdΩ1(A))ω = j(IdΩ1(A))dω −d j(IdΩ1(A))ω = (k +1)dω −kdω = dω. Thus L(IdΩ1(A)) = d.4.9.

Let K ∈Ω1k and L ∈Ω1ℓ. Then obviously [[LK, LL], d] = 0, so we have[L(K), L(L)] = L([K, L])for a uniquely defined [K, L] ∈Ω1k+ℓ.

This vector valued form [K, L] is called theabstract Fr¨olicher-Nijenhuis bracket of K and L.Theorem. The space Ω1∗= Lk Ω1k with its usual grading and the Fr¨olicher-Nijen-huis bracket is a convenient graded Lie algebra.

IdΩ1(A) ∈Ω11 is in the center, i.e. [K, IdΩ1(A)] = 0 for all K.

NON COMMUTATIVE FR ¨OLICHER-NIJENHUIS BRACKET21L : (Ω1∗, [,]) →Der Ω(A) is a bounded injective homomorphism of gradedLie algebras. For fields in HomAA(Ω1(A), A), i. e. bounded derivations of A, theFr¨olicher-Nijenhuis bracket coincides with the bracket defined in 4.2.Proof.

Boundedness of the bracket follows from the fact that the map LK 7→Kis bounded as it is just the composition of the restriction to A with the boundedinverse to d∗constructed in 2.6.For X, Y ∈HomAA(Ω1(A), A) we have j([X, Y ])da = L([X, Y ])a = [LX, LY ]a.The rest is clear.□4.10. Lemma.

For homomorphisms K ∈Ω1k and L ∈Ω1ℓ+1 we have[LK, jL] = j([K, L]) −(−1)kℓL(jL ◦K), or[jL, LK] = L(jL ◦K) −(−1)k j([L, K]).Proof. For a ∈A we have [jL, LK]a = jL jK da −0 = jL(K(da)) = (jL ◦K)(da) =L(jL ◦K)a.

So [jL, LK] −L(jL ◦K) is an algebraic derivation. [[jL, LK], d] = [jL, [LK, d]] −(−1)kℓ[LK, [jL, d]] == 0 −(−1)kℓL([K, L]) = (−1)k[j([L, K]), d]).Since [, d] kills the ‘L’s’ and is injective on the ‘j’s’, the algebraic part of [jL, LK]is (−1)k j([L, K]).□4.11.

Theorem. For homomorphisms Ki ∈Ω1ki and Li ∈Ω1ki+1 we have[LK1 + jL1, LK2 + jL2] =(1)= L[K1, K2] + jL1 ◦K2 −(−1)k1k2jL2 ◦K1+ i[L1, L2]∆+ [K1, L2] −(−1)k1k2[K2, L1].Each summand of this formula looks like a semidirect product of graded Lie algebras,but the mappingsj : Ω1∗−1 →EndK(Ω1∗, [,])ad : Ω1∗→EndK(Ω1∗−1, [,]∆),adK L = [K, L],do not take values in the subspaces of graded derivations.We have instead forhomomorphisms K ∈Ω1k and L ∈Ω1ℓ+1 the following relations:jL ◦[K1, K2] = [jL ◦K1, K2] + (−1)k1ℓ[K1, jL ◦K2](2)−(−1)k1ℓj(adK1 L) ◦K2 −(−1)(k1+ℓ)k2j(adK2 L) ◦K1

22CAP, KRIEGL, MICHOR, VANˇZURAadK[L1, L2]∆= [adK L1, L2]∆+ (−1)kk1[L1, adK L2]∆−(3)−(−1)kk1 ad(j(L1) ◦K)L2 −(−1)(k+k1)k2 ad(j(L2) ◦K)L1The algebraic meaning of the relations of this theorem and its consequences ingroup theory have been investigated in [Michor, 1990]. The corresponding productof groups is well known to algebraists under the name ‘Zappa-Szep’-product.Proof.

Equation (1) is an immediate consequence of 4.10. Equations (2) and (3)follow from (1) by writing out the graded Jacobi identity, or as follows: ConsiderL(jL ◦[K1, K2]) and use 4.10 repeatedly to obtain L of the right hand side of (2).Then consider j([K, [L1, L2]∆]) and use again 4.10 several times to obtain i of theright hand side of (3).□4.12.

Naturality of the Fr¨olicher-Nijenhuis bracket. Let f : A →B be abounded algebra homomorphism.

Two forms K ∈Ω1k(A) = HomAA(Ω1(A), Ωk(A))and K′ ∈Ω1k(B) = HomBB(Ω1(B), Ωk(B)) are called f-related or f-dependent, if wehave(1)K′ ◦Ω1(f) = Ωk(f) ◦K : Ω1(A) →Ωk(B),where Ω∗(f) is described in 2.10.Theorem. (2) If K and K′ as above are f-related then jK′ ◦Ω(f) = Ω(f) ◦jK : Ω(A) →Ω(B).

(3) If jK′ ◦Ω(f)|d(A) = Ω(f) ◦jK|d(A), then K and K′ are f-related, whered(A) ⊂Ω1(A) denotes the space of exact 1-forms. (4) If Kj and K′j are f-related for j = 1, 2, then jK1 ◦K2 and jK′1 ◦K′2 aref-related, and also [K1, K2]∆and [K′1, K′2]∆are f-related.

(5) If K and K′ are f-related then LK′ ◦Ω(f) = Ω(f) ◦LK : Ω(A) →Ω(B). (6) If LK′ ◦Ω(f) | Ω0(A) = Ω(f) ◦LK | Ω0(A), then K and K′ are f-related.

(7) If Kj and K′j are f-related for j = 1, 2, then their Fr¨olicher-Nijenhuis brack-ets [K1, K2] and [K′1, K′2] are also f-related.Proof. (2).

Since both sides are graded derivations over Ω(f) it suffices to checkthis for a 1-form ω ∈Ω1(A). By 4.6 and 2.10 we have Ωk(f)jK(ω) = Ωk(f)K(ω) =K′(Ω1(f)ω) = jK′Ω1(f)(ω).

(3) follows from the universal property of Ω1(A) because K′ ◦Ω1(f) ◦d andΩk(f) ◦K ◦d are both derivations from A into Ωk(B) which is an A-bimodule viaf and the multiplication in Ω(B). (4) is obvious; the result for the bracket then follows from 4.6.3.

(5) The algebra homomorphism Ω(f) intertwines the operators jK and jK′ by(2), and Ω(f) commutes with the exterior derivative d. Thus Ω(f) intertwines thecommutators [jK, d] = LK and [jK′, d] = LK′.

NON COMMUTATIVE FR ¨OLICHER-NIJENHUIS BRACKET23(6) For an element g ∈Ω0(A) we have LK Ω(f) g = jK d Ω(f) g = jK Ω(f) dgand Ω(f) LK′ g = Ω(f) jK′ dg. By (3) the result follows.

(7) The algebra homomorphism Ω(f) intertwines LKj and LK′j, so also theirgraded commutators which equal L([K1, K2]) and L([K′1, K′2]), respectively. Nowuse (6) .□5.

Distributions and integrability5.1. Distributions.

By a distribution in a convenient algebra A we mean a c∞-closed sub-A-bimodule D of Ω1(A).The distribution D is called globally integrable if there exists a c∞-closed subal-gebra B of A such that D is the c∞-closure in Ω1(A) of the subspace generated byA(d(B)) and d(B)A.The distribution D is called splitting if there exists a bounded projection P ∈Ω11(A) = HomAA(Ω1(A), Ω1(A)) onto D, i.e. P ◦P = P and D = P(Ω1(A)).

Thenthere is a complementary submodule ker P ⊂Ω1(A).The distribution D is called involutive if the c∞-closed ideal (D)Ω∗(A) generatedby D in the graded algebra Ω∗(A) is stable under d, i.e. if d(D) ⊂(D)Ω∗(A).5.2.Comments.

One should think of this as follows: In differential geometry,where we have A = C∞(M, R) for a manifold M, a distribution is usually given asa sub vector bundle E of the tangent bundle TM. Then D is the A-bimodule ofthose 1-forms which annihilate the subbundle E of TM.

Global integrability thenmeans that it is integrable and that the space of functions which are constant alongthe leaves of the foliation generates those forms. This is a strong condition: Thereare foliations where this space of functions consists only of the constants, and thiscan be embedded into any manifold.

So in C∞(M, R) there are always involutivedistributions which are not globally integrable. To prove some Frobenius theorema notion of local integrability would be necessary.5.3 Curvature and cocurvature.

Let P ∈Ω11(A) = HomAA(Ω1(A), Ω1(A)) be aprojection, then the image P(Ω1(A)) is a splitting distribution, called the verticaldistribution of P and the complement ker P is also a splitting distribution, called thehorizontal one. ¯P := IdΩ1(A) −P is a projection onto the horizontal distribution.We consider now the Fr¨olicher-Nijenhuis bracket [P, P] of P and defineR = RP = [P, P] ◦Pthe curvature,¯R = ¯RP = [P, P] ◦¯Pthe cocurvature.The curvature and the cocurvature are elements of Ω12(A) = HomAA(Ω1(A), Ω2(A)).The curvature kills elements of the horizontal distribution, so it is vertical.

Thecocurvature kills elements of the vertical distribution.Since the identity Id ∈Ω11(A) lies in the center of the Fr¨olicher-Nijenhuis algebrawe get [ ¯P, ¯P] = [Id −P, Id −P] = [P, P] and hence ¯RP = R ¯P . We shall also

24CAP, KRIEGL, MICHOR, VANˇZURAneed the homomorphisms of graded algebras Ω(P), Ω( ¯P) : Ω(A) →Ω(A) withΩ0(P) = Ω0( ¯P) = IdA which are induced by the bimodule homomorphisms P, ¯P :Ω1(A) →Ω1(A).5.4. Lemma.

In the setting of 5.3 the following assertions hold:1. For ω ∈Ω1(A) we haveRP (ω) = [P, P](P(ω)) = −2(Ω( ¯P) ◦d ◦P)(ω)¯RP (ω) = [P, P]( ¯P(ω)) = −2(Ω(P) ◦d ◦¯P)(ω).2.

For the c∞-closed ideals generated by the distributions ker P and P(Ω1(A))we have (ker P)Ω∗(A) = ker Ω(P) and (P(Ω1(A)))Ω∗(A) = ker Ω( ¯P).3. The curvature R = [P, P] ◦P is zero if and only if the horizontal distributionis involutive.

The cocurvature ¯R = [P, P]◦(Id−P) is zero if and only if the verticaldistribution P(Ω1(A)) is involutive.Proof. (1) It suffices to show the first equation.

For ω ∈Ω1(A) we have:[P, P](ω) = [ ¯P, ¯P](ω) = j([ ¯P, ¯P])(ω)= [L ¯P , j ¯P ](ω) + L(j ¯P ¯P)(ω)by 4.10= L ¯P j ¯P (ω) −j ¯P L ¯P (ω) + L ¯P (ω)since j ¯P ¯P = ¯P 2 = ¯P= 2(j ¯P d ¯P(ω) −d ¯P(ω)) −j ¯P j ¯P d(ω) + j ¯P d(ω).For ω, ϕ ∈Ω1(A) we havej ¯P j ¯P (ω ⊗A ϕ) = j ¯P ( ¯P(ω) ⊗A ϕ + ω ⊗A ¯P(ϕ))= ¯P(ω) ⊗A ϕ + 2 ¯P(ω) ⊗A ¯P(ϕ) + ω ⊗A ¯P(ϕ)= (2Ω( ¯P) + j ¯P )(ω ⊗A ϕ),thusj ¯P j ¯P |Ω2(A) = (2Ω( ¯P) + j ¯P )|Ω2(A).So we have[P, P](ω) = 2(j ¯P d ¯P(ω) −d ¯P(ω) −Ω( ¯P)(d(ω)))RP (ω) = [P, P](P(ω)) = −2Ω( ¯P)dP(ω)as required. (2) The kernel of the bounded algebra homomorphism Ω(P) is a c∞-closed idealand contains ker P. On the other hand any ω ∈Ω1(A) ⊗A · · · ⊗A Ω1(A) ∩ker Ω(P)(non-completed tensor product) may be written as a finite sum ω = Pi ω1,i⊗A· · ·⊗Aωk,i with the property that Pi P(ω1,i)⊗A· · ·⊗AP(ωk,i) = 0.

Since P + ¯P = IdΩ1(A)we have ωj,i = P(ωj,i)+ ¯P(ωj,i) for all j and i. Thus each summand of ω splits into

NON COMMUTATIVE FR ¨OLICHER-NIJENHUIS BRACKET25a sum of products of P(ωj,i) and ¯P(ωj,i) and the sum of those products containingonly P(ωj,i) vanishes. So at least one ¯P(ωj,i) appears in each summand and thewhole sum is in the ideal generated by ker Ω1(P) = ¯P(Ω1(A)).By 1.7 Ωk(A)∩ker(Ω(P)) is the completion of Ω1(A)⊗A · · ·⊗A Ω1(A)∩ker Ω(P)so it must be the c∞-closure in Ωk(A) of this space and hence must also be containedin the c∞-closed ideal.The second assertion follows by symmetry.

(3) We have to prove only the first assertion. The distribution ker ¯P is involutiveif and only if for all ω ∈Ω1(A) we have dPω ∈(ker P)Ω∗(A) = ker Ω(P).

By (2)this is equivalent to R(ω) = −2Ω( ¯P)(dP(ω)) = 0 for all ω ∈Ω1(A).□5.5. Lemma (Bianchi identity).

If P ∈Ω11(A) is a projection with curvature Rand cocurvature ¯R, then we have[P, R + ¯R] = 02[R, P] = jR ¯R + j ¯RR.Proof. We have [P, P] = R + ¯R by 5.3 and [P, [P, P]] = 0 by the graded Jacobiidentity.

So the first formula follows. We have R = [P, P] ◦P = j[P,P ] ◦P.

By4.11.2 we get j[P,P ] ◦[P, P] = 2[j[P,P ] ◦P, P] −0 = 2[R, P]. Therefore 2[R, P] =j[P,P ] ◦[P, P] = j(R + ¯R) ◦(R + ¯R) = jR ◦¯R + j ¯R ◦R since R has vertical valuesand kills vertical vectors, so jR ◦R = 0; likewise for ¯R.□6.

Bundles and connectionsLet G be a Lie group in the usual sense.We want to carry over to non-commutative /n differential geometry the concepts of principal bundles, charac-teristic classes, and Chern-Weil homomorphism. The last two concepts still makedifficulties, since we do not know how to express local triviality and only some ofthe usual properties hold in the general setup we use.6.1.

Definition. By a bundle in non-commutative differential geometry we meana convenient algebra A together with a closed subalgebra B ֒→A.The bundle is said to have a finite dimensional Lie group G as structure group ifwe have an injective homomorphism λ : G →Aut(A), such that λ : G →L(A, A)is smooth and B = AG, the subalgebra of all elements fixed by the G-action.We remark that for the notion of a principal bundle one should add requirementslike quantum transitiveness on the fiber, compare with [Narnhofer, Thirring, Wick-licky, 1988], but this is still not enough to get the Chern-Weil homomorphism, seealso 6.9.If p : P →M is a smooth principal bundle in the usual sense, we put A =C∞(P, R) and B = C∞(M, R), which is embedded into A via p∗.

Then clearly allrequirements are satisfied.

26CAP, KRIEGL, MICHOR, VANˇZURA6.2. Lemma.

For each g ∈G the algebra automorphism λg : A →A extends toan automorphism of the algebra of differential forms as follows:A −−−−→Ω(A)λgyλgyA −−−−→Ω(A).Proof. This follows from the universal property 2.9.□6.3.

Horizontal forms. Recall, that on a classical bundle the horizontal formsare exactly those which annihilate vertical vectors.

Guided by this we define thespace of horizontal 1-forms Ωhor1(A) as the closed A-bimodule generated by Ω1(B)in Ω1(A), in the bornological topology. Likewise we define the algebra Ωhor(A) ofall horizontal forms as the closed subalgebra of Ω(A) generated by A + Ω(B).So Ωhor1(A) is the closed linear subspace generated by all elements of the forma(db)a′ for a, a′ ∈A and b ∈B.Since in Ω1(A) ⊂A˜⊗A we have a(db)a′ =a(1 ⊗b −b ⊗1)a′ = a ⊗ba′ −ab ⊗a′, we get A˜⊗A/Ωhor1 (A) = A˜⊗BA where A isviewed as a B-bimodule.

The situation is explained in the following diagram000yyy0 −−−−→Ωhor1(A)Ωhor1(A) −−−−→0yyy0 −−−−→Ω1(A)−−−−→A˜⊗Aµ−−−−→A −−−−→0yyy0 −−−−→Ω1(A)/Ωhor1 (A) −−−−→A˜⊗BAµ−−−−→A −−−−→0yyy000which has exact columns and also rows since the middle row is splitting.6.4.Principal connections. We have a good description of horizontal forms,whereas vertical vector fields do not exist in sufficient supply, thus we describeconnections in the form of horizontal projections.So a connection on a bundleB ֒→A is an element χ ∈Ω11(A) = HomAA(Ω1(A), Ω1(A)) which satisfies χ ◦χ = χ(equivalently jχ◦χ = χ), such that the image of χ is Ωhor1(A), the space of horizontal1-forms of the bundle.

NON COMMUTATIVE FR ¨OLICHER-NIJENHUIS BRACKET27Note that a connection χ : Ω1(A) →Ωhor1 (A) has a unique extension as anA-bimodule homomorphismΩk(A) = Ω1(A) ⊗A · · · ⊗A Ω1(A)Ω(χ)−−−→Ωhor1(A) ⊗A · · · ⊗A Ωhor1 (A)ω1 ⊗· · · ⊗ωk 7→χ(ω1) ⊗· · · ⊗χ(ωk).A connection χ on a bundle with structure group G is called a principal connec-tion if it is G-equivariant: χ ◦λg = λg ◦χ for all g ∈G.For a usual principal bundle this corresponds to the projection of forms ontohorizontal forms, which describe the vertical distribution. This explains our choiceof names here and in 5.3.PROBLEM: What means ‘locally trivial’ for a bundle?

Does it imply the exis-tence of connections?6.5. Curvature.

Let χ be a connection on a non-commutative bundle B ֒→A.The curvature R = R(χ) of the connection is given byR = [χ, χ] ∈Ω12(A) = HomAA(Ω1(A), Ω2(A)),the abstract Fr¨olicher-Nijenhuis bracket of χ with itself.6.6. Lemma.

The curvature of a connection satisfiesR ∈HomAAΩ1(A)/Ωhor1 (A), Ωhor2 (A).If the connection is principal then also R is G-equivariant.Proof. By definition Ωhor1(A) = χ(Ω1(A)) is globally integrable, thus Rχ = [χ, χ] ◦χ = 0 and we haveR : = [χ, χ] = ¯Rχ = [χ, χ] ◦(Id −χ)by 5.3= −2Ω(χ) ◦d ◦(Id −χ)by 5.4.1.The last expression implies the first assertion.

If χ is a principal connection it isG-equivariant and by 4.12 also R = [χ, χ] is G-equivariant.□6.7. Steps towards the Chern-Weil homomorphism.

Let B ⊂A be a non-commutative bundle with structure group G. Let g denote the Lie algebra of G.We differentiate the action λ : G →Aut(A) and get bounded linear mappingsgTeλ−−−−→Der(A; A)xd∗gλ′−−−−→HomAA(Ω1(A), A).Using this we define a mappingα : Ω1(A)/Ωhor1 (A) →A ⊗g∗(IdA ⊗evX)α(ω) := λ′(X)(ω) for X ∈g, ω ∈Ω1(A).

28CAP, KRIEGL, MICHOR, VANˇZURA6.8. Lemma.

This mapping α is well defined, an A-bimodule homomorphism, andis G-equivariant for the action λg ⊗Ad(g−1)∗on the right hand side.Proof. For X ∈g, a, a′ ∈A, and ω ∈Ω1(A) we have(A ⊗evX)α(aωa′) = λ′(X)(aωa′)= aλ′(X)(ω)a′since λ′(X) ∈HomAA(Ω1(A), A)= (A ⊗evX)(aα(ω)a′),so α is a bimodule homomorphism.

For b ∈B we have(A ⊗evX)α(a(db)a′) = aλ′(X)(db)a′= a(Teλ.X)(b)a′ = 0since λg(b) = b.So α annihilates horizontal forms and is thus well defined. In order to prove that αis G-equivariant we begin with the following computation, where g ∈G:λg(Teλ.X)(a) = λg ddt|0λexp tX(a)= ddt|0λgλexp tX(a)since λg is linear and bounded= ddt|0λg exp(tX) g−1(λg(a))= Teλ(Ad(g)X)(λg(a)).By the universality of d we have Ω1(λg) ◦d = d ◦λg and thus we getλg(λ′(X)(a da′)) = λg(aλ′(X)(da′)) = λg(a)λg(λ′(X)da′)= λg(a) λg((Teλ.X)(a′))= λg(a) (Teλ.

Ad(g)X)(λg(a′))= λg(a) λ′(Ad(g)X))(dλg(a′))= λ′(Ad(g)X))(λg(a da′)).So finally we have(A ⊗evX)α(Ω1(λg)ω) = λ′(X)(Ω1(λg)ω)= λg(λ′(Ad(g−1)X)ω)= (λg ⊗evAd(g−1)X)α(ω)= (A ⊗evX)(λg ⊗Ad(g−1))α(ω),so α ◦Ω1(λg) = (λg ⊗Ad(g−1)) ◦α as required.□

NON COMMUTATIVE FR ¨OLICHER-NIJENHUIS BRACKET296.9. Remarks.

We stop our development here and add just some remarks aboutthe Chern-Weil homomorphism. To continue from this point one should add require-ments to the bundle A which imply that α is invertible (the inverse then describesthe fundamental vector field mapping) and that the extension of the inverse toinvariant polynomials on g factors to the ¯Ω(B).A good model for the Chern-Weil homomorphism is described in the paper[Lecomte, 1985] where the following construction is given:Let P →M be a smooth principal fiber bundle with structure group G. Thenwe have the following exact sequence of vector bundles over M:0 →P[g, Ad] →TP/GT p−−→TM →0.The smooth sections of these bundles give rise to the following exact sequence ofLie algebras:0 →Xvert(P)G →Xproj(P)G →X(M) →0,namely first all vertical G-equivariant vector fields (the Lie algebra of the gaugegroup), second the all projectable G-equivariant vector fields on P (the infinitesimalprincipal bundle automorphisms), third all vector fields on the base.

The ‘dual’ ofthis sequence of Lie algebras is0 ←(Ω∗(A)/Ωhor(A))G ←Ω∗(A)G ←Ω∗(B) ←0,where A = C∞(P, R) and B = C∞(M, R). For general algebras this sequence is notexact.

For any short exact sequence of Lie algebras [Lecomte, 1985] has describeda generalization of the Chern-Weil homomorphism in purely algebraic terms, usingChevalley cohomology of the Lie algebras in question. This should be the startingpoint of the Chern-Weil homomorphism in non-commutative differential geometry.7.

Polyderivations and the Schouten-Nijenhuis bracketIn this section we describe the analogue of the Schouten-Nijenhuis bracket inthe setting of non-commutative differential geometry. It turns out that one has torequire skew symmetry in the construction in order to get a meaningful theory.

Inthe end we obtain the Poisson structures for convenient algebras. The results inthis section are also a generalization for non-commutative algebras of the results in[Krasil’shchik, 1988], which were the original motivation for the developments here,but our approach is different: we first show that the Nijenhuis-Richardson bracket(c.f.

[Nijenhuis, Richardson, 1967] and [Lecomte, Michor, Schicketanz]) passes to theconvenient setting and then by restricting it to a suitable space of polyderivations(the non-commutative analog of multi vector fields) we derive a generalization ofthe Schouten-Nijenhuis bracket.

30CAP, KRIEGL, MICHOR, VANˇZURA7.1. It has been noticed in [De Wilde, Lecomte, 1985] that for any smooth manifoldM the Schouten-Nijenhuis bracket on the space C∞(ΛTM) of all multivector fieldsimbeds as a graded sub Lie algebra into the space Λ∗+1(C∞(M, R); C∞(M, R)) withthe Nijenhuis-Richardson bracket (see 7.2 for a description of this space).

Lecomtetold us, that a very elegant proof of this fact can be given in the following way:The space C∞(M, R) of smooth functions is the degree −1 part of the Schouten-Nijenhuis algebra.By the universal property of the Nijenhuis-Richardson alge-bra (Λ∗+1(C∞(M, R); C∞(M, R)), [,]∧) described in [Lecomte, Michor, Schick-etanz] the identity on C∞(M, R) prolongs to a unique homomorphism Φ of gradedLie algebras from the Schouten-Nijenhuis algebra into the Nijenhuis-Richardsonalgebra, and a simple computation described in [Lecomte, Melotte, Roger, 1989]shows that Φ(T) = d∗(T) = T ◦(d × . .

. × d), where d is the exterior differential.This shows that the Schouten-Nijenhuis bracket which we will construct belowboils down to the usual one in the commutative case A = C∞(M, R).7.2 The Nijenhuis-Richardson bracket in the convenient setting.

Let Vbe a convenient vector space.We consider the space Λk(V ) of all bounded k-linear skew symmetric functionals V × . .

. × V →R, where Λ0(V ) = R. ThenΛ(V ) = Lk≥0 Λk(V ) is a graded commutative convenient algebra with the usualwedge product(1)(ϕ ∧ψ)(v1, .

. .

, vk+ℓ) ==1k!ℓ!Xσsign σ ϕ(vσ1, . .

. , vσk)ψ(vσ(k+1), .

. .

, vσ(k+ℓ)),where the sum is over all permutation of k + ℓsymbols.Now let W be another convenient vector space. We need the space Λk(V ; W) ofall bounded k-linear mappings V ×.

. .×V →W.

Then Λ(V ; W) = Lk≥0 Λk(V, W)is a graded convenient vector space and a graded convenient module over the gradedcommutative algebra Λ(V ) with the wedge product (1) from above. If A is a con-venient algebra then Λ(V ; A) is an associative graded convenient algebra with the(formally) same wedge product.Now for K ∈Λk+1(V ; V ) and Φ ∈Λp(V ; W) we define(2)(iKΦ)(v1, .

. .

, vk+p) ==1(k+1)! (p−1)!Xσsign σ Φ(K(vσ1, .

. .

, vσ(k+1)), vσ(k+2), . .

. , vσ(k+p)).Then the following results hold; for proofs see [Nijenhuis, Richardson, 1967], [Mi-chor, 1987], and [Lecomte, Michor, Schicketanz] for multigraded versions; the ex-tension to the convenient setting does not offer any difficulties.

(3) For K ∈Λk+1(V ; V ), ϕ ∈Λp(V ), and Φ ∈Λ(V ; W) we have iK(ϕ ∧Φ) =iKϕ ∧Φ + (−1)kpϕ ∧iKΦ so iK is a graded derivation of degree k of theΛ(V )-module Λ(V ; W) and any derivation is of that form.

NON COMMUTATIVE FR ¨OLICHER-NIJENHUIS BRACKET31(4) The space of graded derivations of the graded Λ(V )-module Λ(V ; W) is agraded Lie algebra with bracket the graded commutator [D1, D2] = D1D2 −(−1)d1d2D2D1, see 3.1. (5) For K ∈Λk+1(V ) and L ∈Λℓ+1(V ) we have [iK, iL] = i([K, L]∧) where[K, L]∧= iKL −(−1)kℓiLK.

So by (4) we get a graded Lie algebra(Λ∗+1(V ; V ), [,]∧), called the Nijenhuis-Richardson algebra. (6) If µ ∈Λ2(V ; V ), i. e. µ : V × V →V is bounded skew symmetric bilinear,then [µ, µ]∧= 2iµµ = 0 if and only if (V, µ) is a convenient Lie algebra.7.3.

Polyderivations. Let A be a convenient algebra and let Lk(A) ⊂Λk+1(A; A)be the space of all maps K such that for any a1, .

. .ak ∈A the linear map a 7→K(a, a1, .

. ., ak) is a derivation of A.

Obviously this is a closed linear subspace andthus each Lk(A) is a convenient vector space. We call L(A) := Lk≥0 Lk(A) thespace of all skew symmetric polyderivations of A.

Obviously Lk(A) is not an Asubmodule of Λk+1(A; A) in general.7.4. Theorem.

Let A be a convenient algebra. Then (L(A), [,]∧) is a gradedLie subalgebra of the Nijenhuis-Richardson algebra (Λ∗+1(A; A), [,]∧).So (L(A), [,]∧) is a convenient graded Lie algebra called the Schouten-Ni-jenhuis algebra of A.Proof.

It suffices to show that for Ki ∈Lki(A) the bracket [K1, K2]∧again lies inL(A). This means that we have to show that for arbitrary elements a, b ∈A wehave:iab[K1, K2]∧= (ia[K1, K2]∧)b + a(ib[K1, K2]∧)From 7.2.

(5) we see that for a ∈A = Λ0(A; A) and K ∈Λk+1(A) we have(1)iaiK −(−1)kiKia = i([a, K]∧) = i(iaK).If furthermore L ∈Lℓwe obviously have from the polyderivation property of L:i(K ∧a)L = iKL ∧a + K ∧iaL,(2)i(a ∧K)L = a ∧iKL + (−1)(k+1)ℓiaL ∧K. (3)Using this we may compute as follows, where we delete ∧if one of the factors is in

32CAP, KRIEGL, MICHOR, VANˇZURAthe algebra A:iab[K1, K2]∧= iab(i(K1)K2) −(−1)k1k2iab(i(K2)K1) ==i(iabK1)K2 + (−1)k1i(K1)(iabK2)−−(−1)k1k2i(iabK2)K1 −(−1)(k1+1)k2i(K2)(iabK1) ==i(iaK1b)K2) + i(a(ibK1)K2)++ (−1)k1i(K1)(iaK2b) + (−1)k1i(K1)(aibK2)−−(−1)k1k2i(iaK2b)K1 −(−1)k1k2i(aibK2)K1−−(−1)(k1+1)k2i(K2)(iaK1b) −(−1)(k1+1)k2i(K2)(aibK1) ==(i(iaK1)K2)b + (iaK1) ∧(ibK2)++ a(i(ibK1)K2) + (−1)k1k2(iaK2) ∧(ibK1)++ (−1)k1(i(K1)(iaK2))b + (−1)k1a(i(K1)(ibK2))−−(−1)k1k2(i(iaK2)K1)b −(−1)k1k2(iaK2) ∧(ibK1)−−(−1)k1k2a(i(ibK2)K1) −(iaK1) ∧(ibK2)−−(−1)(k1+1)k2(i(K2)(iaK1))b −(−1)(k1+1)k2a(i(K2)(ibK1)) ==(ia(i(K1)K2))b −(−1)k1k2(ia(i(K2)K1))b++ a(ib(i(K1)K2)) −(−1)k1k2a(ib(i(K2)K1)) ==(ia[K1, K2]∧)b + a(ib[K1, K2]∧)□7.5.Definition. Let A be an algebra.A 2-derivation µ ∈L1(A) is called aPoisson structure on A if [µ, µ]∧= 0.7.6.

Theorem. Let µ be a Poisson structure for the algebra A.

Then µ : A×A →Ais a Lie algebra structure. Furthermore we haveµ(ab, c) = aµ(b, c) + µ(a, c)b,µ(a, bc) = bµ(a, c) + µ(a, b)c.The mapping ˇµ : A →Der(A), a 7→µ(a,) is a homomorphism of Lie algebras(A, µ) →(Der(A), [,]), where the second bracket is the Lie bracket (commuta-tor), see 4.2.This is the non-commutative generalization of the Poisson bracket of differentialgeometry.Proof.

7.2. (6) implies that µ is a Lie algebra structure.

The other assertion is justthe property of a polyderivation.□

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