Shoikhet's Conjecture and Duflo Isomorphism on (Co)Invariants
aa r X i v : . [ m a t h . QA ] S e p Symmetry, Integrability and Geometry: Methods and Applications SIGMA (2008), 060, 17 pages Shoikhet’s Conjecture and Duf lo Isomorphismon (Co)Invariants ⋆ Damien CALAQUE and Carlo A. ROSSIDepartment of Mathematics, ETH Zurich, 8092 Zurich, Switzerland
E-mail: [email protected], [email protected]
Received May 23, 2008, in final form August 29, 2008; Published online September 03, 2008Original article is available at
Abstract.
In this paper we prove a conjecture of B. Shoikhet. This conjecture statesthat the tangent isomorphism on homology, between the Poisson homology associated toa Poisson structure on R d and the Hochschild homology of its quantized star-product algebra,is an isomorphism of modules over the (isomorphic) respective cohomology algebras. Asa consequence, we obtain a version of the Duflo isomorphism on coinvariants. Key words: deformation quantization; formality theorems; cap-products; Duflo isomorphism
In his seminal paper [7] on the deformation quantization of Poisson manifolds M. Kontsevichproved that the differential graded Lie algebra (shortly, DGLA) of polydifferential operators ona smooth manifold M is formal (i.e. it is quasi-isomorphic, as a DGLA, to its cohomology). Asa consequence one obtains that any Poisson structure π on M can be quantized in the senseof [2], and that the Hochschild cohomology of the deformed quantized algebra is isomorphic tothe Poisson cohomology of ( M, π ). Moreover, it is known that the Hochschild cohomology of anassociative algebra is naturally equipped with an associative cup-product; and Kontsevich provedthat the mentioned isomorphism between Hochschild and Poisson cohomologies is actually mul-tiplicative if M = R d , using a homotopy argument involving the so-called Kontsevich eye . Theproof of this statement, known as the “compatibility with cup-products”, has been clarified in [8],and appeared to have a surprising application to Lie theory [7] (see also [9] and [5]) in providinga new proof, together with a cohomological extension, of the famous Duflo isomorphism [6]. Werecall to the reader that the result of Duflo states that the Poincar´e–Birkhoff–Witt map can bemodified so that it reduces to an isomorphism of algebras on invariants.A homological version of Kontsevich’s formality theorem has been formulated by B. Tsyganin [13] and proved by Shoikhet in [10] (in the case M = R d ) and by Tsygan and Tamarkin in [12],using different approaches. It broadly states that the Hochschild chain complex of the algebraof smooth functions on M is formal as a DG Lie module over the DGLA of polydifferentialoperators. Again, one obtains as a direct consequence of the general formalism on L ∞ -algebras,their modules and L ∞ -morphisms between them, that the Hochschild homology of the deformedquantized algebra is isomorphic to the Poisson homology of ( M, π ). The present paper is mainlyconcerned about the multiplicativity of this isomorphism. Namely, Hochschild (resp. Poisson)homology is naturally a graded module over the Hochschild (resp. Poisson) cohomology algeb-ra, and we prove that the isomorphism induced by the Tsygan–Shoikhet formality intertwines ⋆ D. Calaque and C.A. Rossithese module structures (both cohomology algebras being themselves isomorphic thanks to thecompatibility with cup-products). We call our result the “compatibility with cap-products”.As in the cohomological situation, this compatibility with cap-products has an applicationto Lie theory in providing a version of the Duflo isomorphism for coinvariants.The main goal of this paper is to present a short and comprehensible proof of this conjecture(which is a particular case of a more general result, whose detailed proof is presented in [4]).Here the proof also relies on a homotopy argument, Kontsevich’s eye being replaced by the
I-cube , this time being a manifold with corners of dimension 3.The paper is organized as follows. In Section 2 we state the main result we mentioned inthis introduction. Section 3 is a brief reminder on Kontsevich’s and Shoikhet’s configurationspaces and their compactifications; we also mention how they are related to each other, a factwhich will play a central rˆole in some later computations. In Section 4 we recall the constructionof Kontsevich’s and Shoikhet’s formality L ∞ -quasi-isomorphisms. This is the first time wherethe configuration spaces that were introduced in the previous section appear operatively. Sec-tion 5 is the heart of the paper. It contains the proof of the main result, for which compactifiedconfiguration spaces (and integrals over them) play a crucial rˆole. We finally end the paper withthe proof of a version of the Duflo isomorphism for coinvariants, which we obtain as a conse-quence of our main result; we only observe that we prefer to give a direct computational proof,as opposed to the proof of the same result in [10], where it was proved under the assumptionthat the conjecture (whose proof is the core of this paper) were true. For the manifold V = R d , we consider the differential graded Lie algebras (shortly, DGLA) T • poly ( V ) and D • poly ( V ) of polyvector vector fields on V and of polydifferential operators on V respectively.Further, let γ be a solution of the Maurer–Cartan equation in T • poly ( V ) of the form γ = ~ π, (1)where π is a bivector field and ~ a formal parameter (therefore in particular π is a Poissonstructure). The Formality Theorem of Kontsevich [7] implies that the polydifferential operator B = U ( γ ) = X n ≥ n ! U n ( γ, . . . , γ | {z } n ) U n being the Taylor components of the L ∞ -quasi isomorphism between T • poly ( V ) and D • poly ( V ),satisfies the Maurer–Cartan equation in the DGLA D • poly ( V )[[ ~ ]] of (series of) polydifferentialoperators on V , viewed as a subcomplex of the Hochschild cochain complex of the C [[ ~ ]]-algebra A = C ∞ ( V )[[ ~ ]].We denote by µ the usual multiplication on the algebra A : it may be viewed as an elementof D • poly ( V )[[ ~ ]] of degree 1, and the sum µ + B specifies an associative product ⋆ on A , which isa deformation of the usual product on A , and which moreover satisfies f ⋆ g − g ⋆ f = 2 ~ h π, d f ∧ d g i + O ( ~ ) . In other words, ⋆ is a quantization of the Poisson structure π on V in the sense of [2].The DGLA T • poly ( V ) possesses an associative product ∪ , namely the usual ∧ -product on T • poly ( V ), and a solution γ of the Maurer–Cartan equation defines, by means of the Schouten–Nijenhuis bracket , a differential γ · = [ γ, ] on T • poly ( V ) w.r.t. ∪ . We observe that, if we follow the sign conventions of [1], we have then to modify the Schouten–Nijenhuisbracket as [ α, β ] ′ SN = − [ β, α ] SN , where the Schouten–Nijenhuis bracket on the right-hand side is the usual one. hoikhet’s Conjecture and Duflo Isomorphism on (Co)Invariants 3The (negatively graded) deRham complex Ω −• ( V ) of differential forms on V is naturallya differential graded module (shortly, DGM) over the DGLA T • poly ( V ): the extensions of the Liederivative L by means of the Cartan formula and of the contraction operator ι define respectivelya differential L γ , for γ as in (1), and an action ∩ of T • poly ( V ) on Ω −• ( V ).On the other hand, for γ as above, there is a product ∪ (of degree 1) on the (shifted by 1)Hochschild cochain complex C • ( A, A )[1] of A with values in A ,( ϕ ∪ ψ )( a , . . . , a p + q ) = ϕ ( a , . . . , a p ) ⋆ ψ ( a p +1 , . . . , a p + q ) . Additionally, the Hochschild differential d H on the Hochschild cochain complex of A is modifiedto the Hochschild differential d H ,⋆ w.r.t. ⋆ . All these structures descend to the subcomplex ofpolydifferential operators on V .For the algebra A , we consider the (negatively graded) Hochschild chain complex C −• ( A, A ).For γ as (1), there is an action of C • ( A, A )[1] on C −• ( A, A ) via ϕ ∩ ( a | a | · · · | a n ) = ( a ⋆ ϕ ( a , . . . , a m ) | a m +1 | · · · | a n ) , if m ≤ n ; if m > n , the action is trivial. Furthermore, we also have the differential b ⋆ on C −• ( A, A ), which modifies the usual Hochschild differential b on C −• ( A, A ). The previousformula defines also an action of the DGLA of polydifferential operators on V on C −• ( A, A ).We denote by S n , n ≥
0, the Taylor components of the L ∞ -quasi-isomorphism S from the L ∞ -module C −• ( A, A ) to the DGM Ω • ( V )[[ ~ ]], both over T • poly ( V ), constructed in [10]. TheDGM structure of Ω • ( V )[[ ~ ]] over T • poly ( V ) comes from the Lie derivative of polyvector fields ondifferential forms, while, as shown in [10], composition of the L ∞ -quasi-isomorphism U with theaction L of Hochschild cochains on A on Hochschild chains gives C −• ( A, A ) the structure of an L ∞ -module over T • poly ( V ).For a solution γ of the Maurer–Cartan equation as in (1), we consider the following linearmaps: U γ ( α ) = X n ≥ n ! U n +1 ( α, γ, . . . , γ | {z } n ) , resp. (2) S γ ( c ) = X n ≥ n ! S n ( γ, . . . , γ | {z } n , c ) , (3)for a general polyvector field α on V , resp. Hochschild chain c on A . Theorem 1.
For a solution γ of the Maurer–Cartan equation as in (1) , (2) is a quasi-isomor-phism of complexes U γ : (cid:0) T • poly ( V )[[ ~ ]] , γ · (cid:1) → (cid:0) D • poly ( V )[[ ~ ]] , d H ,⋆ (cid:1) , which additionally preserves the products in the corresponding cohomologies, [ U γ ([ α ] ∪ [ β ])] = [ U γ ([ α ]) ∪ U γ ([ β ])] , square brackets denoting cohomology classes. We refer to [7, 8] for the proof of Theorem 1. The main result of this paper is the proof ofConjecture 3.5.3.1 in [10], which we may state in the following
Theorem 2.
For a solution γ of the Maurer–Cartan equation as in (1) , (3) is a quasi-isomor-phism of complexes S γ : ( C −• ( A, A ) , b ⋆ ) → (Ω • ( V )[[ ~ ]] , L γ ) and additionally preserves the action of T • poly ( V ) in the corresponding cohomologies, [[ α ] ∩ S γ ([ c ])] = [ S γ ( U γ ([ α ]) ∩ [ c ])] , with the previous notation for cohomology classes. D. Calaque and C.A. Rossi
We briefly discuss in this Section configuration spaces of i ) points in the complex upper-halfplane H and on the real axis R , and ii ) points in the interior of the punctured unit disk D andon the boundary S , and their compactifications `a la Fulton–MacPherson. For a pair of non-negative integers ( n, m ), the (open) configuration space C + n,m is defined as C + n,m = { ( p , . . . , p n , q , . . . , q m ) ∈ H n × R m : p i = p j , i = j, q < · · · < q m } /G , where G is the semidirect product R + ⋉R , acting via rescalings and translations. If 2 n + m − ≥ C + n,m is a smooth real manifold of dimension 2 n + m −
2. We may consider more generalconfiguration spaces C + A,B , where A is any finite set and B is any ordered finite set.The configuration space C n is defined as C n = { ( p , . . . , p n ) ∈ C n : p i = p j , i = j } /G , where G is the semidirect product R + ⋉ C , acting via rescalings and complex translations. If2 n − ≥ C n is a smooth real manifold of dimension 2 n −
3. Again, we may consider moregeneral configuration spaces C A , for any finite set A .Both configuration spaces C + A,B and C A are orientable, see e.g. [1].Configuration spaces C + A,B and C A admit compactifications `a la Fulton–MacPherson, denotedby C + A,B and C A respectively: they are smooth manifolds with corners and we refer to [7, 8, 3]and [4] for their explicit constructions. As in Subsection 3.1, for a pair of non-negative integers ( n, m ), m ≥
1, the (open) configurationspace D + n,m is defined as D + n,m = ( ( p , . . . , p n , q , . . . , q m ) ∈ ( D × ) n × ( S ) m : ( p i = p j , i = j,q < · · · < q m < q , ) /S , where D × denotes the punctured unit disk, and where we introduced a cyclic order on S ; thegroup S acts by rotations. If 2 n + m − ≥ D + n,m is a smooth real manifold of dimension2 n + m −
1. As before, we may consider configuration spaces D + A,B , where A is any finite setand B is any cyclically ordered finite set. When | B | = 1, we omit the superscript +.For a positive integer n , we consider the configuration space D n = (cid:8) ( p , . . . , p n ) ∈ ( C × ) n : p i = p j , i = j (cid:9) / R + , where R + acts by rescaling. It is obviously a smooth real manifold of dimension 2 n −
1, when2 n − ≥
0. We may consider configuration spaces D A , with A any finite set.Finally, D + A,B and D A are orientable, by the same arguments as in [1].Configuration spaces D + A,B and D A , admit compactifications `a la Fulton–MacPherson, de-noted by D + A,B and D A respectively, which are smooth manifolds with corners.Being D + A,B a stratified space, its boundary strata of codimension 1 are given in the followinglist:hoikhet’s Conjecture and Duflo Isomorphism on (Co)Invariants 5 i ) There is a subset A of A , obeying 1 ≤ | A | ≤ | A | , such that ∂ A , D + A,B ∼ = D A × D + A \ A ,B . (4)Clearly, 2 | A | − ≥ | A | − | A | ) + | B | − ≥
0. Intuitively, this corresponds to thesituation, where points in D × labelled by A collapse together to the origin. ii ) There is a subset A of A , obeying 2 ≤ | A | ≤ | A | , such that ∂ A D + A,B ∼ = C A × D + A \ A ⊔{•} ,B . (5)We must impose 2 | A | − ≥ | A | − | A | + 1) + | B | − ≥
0. Intuitively, thiscorresponds to the situation, where points in D × labelled by A collapse together toa single point in D × labelled by • . iii ) Finally, there is a subset A of A and an ordered subset B of successive elements of B ,obeying 0 ≤ | A | ≤ | A | and 2 ≤ | B | ≤ | B | , such that ∂ A ,B D + A,B ∼ = C + A ,B × D + A \ A ,B \ B ⊔{•} . (6)We impose 2 | A | + | B | − ≥ | A | − | A | ) + ( | B | − | B | + 1) − ≥
0. Intuitively, thiscorresponds to the situation, where points in D × labelled by A and points in S labelledby B collapse together to a single point in S labelled by • . We may use the action of S to construct a section of D + A,B , namely we fix one point ◦ in S to 1. This section is diffeomorphic, by means of the M¨obius transformation ψ : H ⊔ R −→ D ⊔ S \{ } ; z z − i z + i , where D is the unit disk, to a smooth section of C + A ⊔{•} ,B \{◦} , given by fixing one point • in thecomplex upper half-plane H to i by means of the action of G .Then, the compactified configuration space D + A,B can be identified with C + A ⊔{•} ,B \{◦} , and weobserve that the cyclic order on the points in S translates naturally into an order on the pointson the real axis R .We further consider the manifold D A , and notice the identification D A ∼ = C A ⊔{•} : to be moreprecise, by means of complex translation, we may put one point • in C n +1 at the origin, andusing rescalings, one can put the remaining points in the punctured unit disk with boundary.Analogously as before, the compactification D A of D A can be identified with C A ⊔{•} .We consequently identify the codimension 1 boundary strata of D + A,B with those of C + A ⊔{•} ,B \{◦} (higher codimension can be worked out along the same lines very easily): i ) A boundary stratum as in (4) corresponds to the situation, where points labelled by A ⊔{•} collapse together to a single point in H , which takes the rˆole of the marked point • . ii ) A boundary stratum as in (5) corresponds to the situation, where points labelled by A collapse together to a single point in H , which will not be the new marked point • . iii ) A boundary stratum as in (6), where ◦ / ∈ B , corresponds to the situation, where pointslabelled by A ⊔ B collapse to a single point in R , which will not be the new markedpoint ◦ . iii ) Finally, a boundary stratum as in (6), where ◦ ∈ B , corresponds to the situation, wherepoints labelled by the set A \ A ⊔ {•} ⊔ B \ B collapse to a single point in R , which willbe the new marked point ◦ . D. Calaque and C.A. Rossi Here is a short review of the formulæ we will need to construct the aforementioned L ∞ -quasi-isomorphisms U and S . L ∞ -quasi-isomorphism U For the sake of simplicity, we denote by [ n ], for a positive integer n , the set { , . . . , n } . For anypair of non-negative integers ( n, m ), such that 2 n + m − ≥
0, an admissible graph
Γ of type( n, m ) is by definition a directed graph with labels obeying the following requirements: i ) The set of vertices V Γ is given by [ n ] ⊔ [ m ]; vertices labelled by integers in [ n ], resp. [ m ],are called vertices of the first, resp. second type; further, the labelling of vertices of thefirst type specifies an order on them. The set of vertices factorizes into V Γ = V ⊔ V ,where V , resp. V , is the set of vertices of the first type, resp. second type.ii) Every edge in E Γ starts at some vertex of the first type; there is at most one edge betweenany two distinct vertices of Γ; no edge starts and ends at the same vertex.For a given vertex v of Γ, we denote by star( v ) the subset of E Γ of edges starting at v : then,we assume that, for any vertex of the first type v of Γ, the elements of star( v ) are labelled as( e v , . . . , e | star( v ) | v ). By definition, the valence of a vertex v is the cardinality of the star of v . Theset of admissible graphs of type ( n, m ) is denoted by G n,m .We also need the following lemma, borrowed from [7], to which we also refer for a more carefulexplanation of the origins of the form ω . Lemma 1.
There exists a smooth -form ω on C , , with the following properties: i ) The restriction of ω to the boundary stratum C = S equals the deRham differential of theangle function measured in counterclockwise direction from the positive imaginary axis. ii ) The restriction of ω to C , , where the first point in the complex upper half-plane goes tothe real axis, vanishes. For any pair of non-negative integers ( n, m ), such that 2 n + m − ≥
0, there are naturalsmooth projections from C + n,m onto C , (provided n ≥
2) or onto C , (if n, m ≥ n, m ) is associated its Kontsevich’s weight W Γ via W Γ = Z C + n,m ^ e ∈ E Γ ω e = Z C + n,m ω Γ , (7)where, for an edge e of Γ, ω e denotes the pull-back of ω to C + n,m via the projection π e from C + n,m onto C , , onto the pair of points labelled by the endpoints of e . The labelling of Γ specifies anorder of the forms ω e in the above product.To an admissible graph Γ of type ( n, m ), to n polyvector fields γ , . . . , γ n and to m functions f , . . . , f m on V , such that | star( k ) | = | γ k | + 1, k = 1 , . . . , n , we associate a function U Γ ( γ , . . . , γ n )( f , . . . , f m )by the following rule: to a vertex v of the first type, resp. second type, we associate the polyvectorfield γ v , resp. the function f v . For a function I from E Γ to [ d ], we associate to the vertex v ofthe first type, resp. second type, the function ϕ Iv = γ I ( e v ) ,...,I ( e | star( v ) | v ) v , resp. ϕ Iv = f v . hoikhet’s Conjecture and Duflo Isomorphism on (Co)Invariants 7with the same notations as before. The function I labels edges of Γ by (standard) coordinatesof V . Then, we have the following assignment, for an admissible graph Γ of type ( n, m ): U Γ ( γ , . . . , γ n )( f , . . . , f m ) = X I : E Γ → [ d ] Y v ∈ V Γ Y e ∈ E Γ : e =( ∗ ,v ) ∂ I ( e ) ϕ Iv . (8)It is clear that U Γ , with Γ as above, maps n polyvector fields { γ , . . . , γ n } to a polydifferentialoperator of (shifted) degree m − n -th Taylor component U n of Kontsevich’s L ∞ -quasi-isomorphism U by combining (7) and (8), namely U n = X m ≥ X Γ ∈G n,m W Γ U Γ . (9) Theorem 3 (Kontsevich) . The Taylor components (9) combine to an L ∞ -quasi-isomorphism U : T • poly ( V ) → D • poly ( V ) , of L ∞ -algebras, whose first order Taylor component reduces to the Hochschild–Kostant–Rosen-berg quasi-isomorphism in cohomology. For a proof of Theorem 3, we refer to [7]. L ∞ -quasi-isomorphism S An admissible graph of type ( n, m ), for any two non-negative integers such that 2 n + m − ≥ n, m ) is denoted by G n,m, .A (partial) counterpart of Lemma 1 in this framework is the following lemma.We define a smooth 1-form on the configuration space C , via ϕ D ( p, q, r ) = ϕ ( q, r ) − ϕ ( q, p ) , for any three pairwise distinct points p , q , r in H ⊔ R : we then set ω D = d ϕ D . Lemma 2.
The -form ω D extends to a smooth -form on C , , with the following properties: i ) its restriction to C , , when q approaches the real axis, vanishes; ii ) its restriction to C , × C , , when p and q collapse together to the real axis, equals − π ∗ ω ; iii ) its restriction to C × C , , when p and q collapse together in the upper half-plane, equals π ∗ ω − π ∗ ω ; iv ) its restriction to C , × C , (resp. C × C , ), when p and r collapse together to the real axis(resp. in the upper half-plane), vanishes; v ) its restriction to C , × C , , when q and r collapse together to the real axis, equals π ∗ ω ; vi ) its restriction to C × C , , when q and r collapse together in the upper half-plane, equals π ∗ ω − π ∗ ω . As above, we define a Shoikhet’s weight associated to a graph without loop Γ with m + n + 1vertices labelled by V (Γ) := { , . . . , n, , . . . , m } . To any edge e = ( i, j ) ∈ E (Γ), we associatea smooth 1-form ω D,e on D + n,m by the following rules: D. Calaque and C.A. Rossi • if neither i nor j lies in { , } , then ω D,e := π ∗ (0 ,i,j ) ω D , where π (0 ,i,j ) : D + n,m ∼ = C + n +1 ,m − −→ C , , (cid:2) ( z , . . . , z n , z , . . . , z m ) (cid:3) (cid:2) ( z , z i , z j ) (cid:3) ; • if i = 0 and j = 1, then ω D,e := π ∗ ( i,j ) ω , where π ( i,j ) : D + n,m ∼ = C + n +1 ,m − −→ C , , (cid:2) ( z , . . . , z n , z , . . . , z m ) (cid:3) (cid:2) ( z i , z j ) (cid:3) ; • if j = 1 and i = 0, then ω D,e := p ∗ ( i,j ) ω , wherep ( i,j ) : D + n,m −→ D , ∼ = C , , (cid:2) ( z , . . . , z n , z , . . . , z m ) (cid:3) (cid:2) ( z i , z j ) (cid:3) ; • if i = 1 or j = 0 or ( i, j ) = (0 , ω D,e = 0.Then, as above, ω D, Γ := ^ e ∈E (Γ) ω D,e defines a differential form on D + n,m . Definition 1.
The Shoikhet weight W D, Γ of the directed graph Γ is W D, Γ := Z D + n,m ω D, Γ . We consider an admissible graph in G n,m, , such that | star(0) | = l . To n polyvector fields { γ , . . . , γ n } on V , such that | star( k ) | = | γ k | + 1, k = 1 , . . . , n , and to a Hochschild chain c = ( a | a | · · · | a m − ) of degree − m + 1, we associate an l -form on V (whose actual degree is − l ,following the grading introduced in [10]) via h α, S Γ ( γ , . . . , γ n , c ) i = U Γ ( α, γ , . . . , γ n )( a , . . . , a n ) . Finally, the n -th Taylor component S n of the L ∞ -quasi-isomorphism S from the L ∞ -module C −• ( A, A ) to the L ∞ -module Ω −• ( V ) over T • poly ( V ) is given by S n = X m ≥ X Γ ∈G n,m, W Γ ,D S Γ . (10) Theorem 4 (Shoikhet) . The Taylor components (10) combine to an L ∞ -quasi-isomorphism S : C −• ( A, A ) → Ω −• ( V ) , of L ∞ -modules over T • poly ( V ) , whose -th order Taylor component reduces to the Hochschild–Kostant–Rosenberg quasi-isomorphism in homology. We refer to [10] for the proof of Theorem 4.
In this Section we sketch the proof of Theorem 2, Section 2, whose strategy owes to the homotopyargument used in [7, 8]; for a more detailed version of the proof of Theorem 2 in an even moregeneral case, we refer to [4].We must consider separately the case, where S γ acts on Hochschild chains i ) of degree m = 0and ii ) of degree − m ≤ −
1: as we will soon notice, the geometric aspects of the two cases arequite different.hoikhet’s Conjecture and Duflo Isomorphism on (Co)Invariants 9 D , and Hochschild chains of degree 0 By the definition of the action ∩ , if c has degree 0, the only Hochschild cochains acting on c non-trivially must be functions, in which case the action is simply multiplication on the rightw.r.t. the product ⋆ .We consider the curve ℓ on the configuration space D , , with initial point on α , and finalpoint b , which corresponds to the following embedding of the open unit interval into D , : (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) s α b the curve al l Figure 1.
The curve ℓ in D , . Since D , ∼ = C , D , coincides with Kontsevich’s eye: its “pupil” α represents the boundarystratum D × D , of codimension 1, while the point b represents the boundary stratum C , ×C +0 , × D , of codimension 2, graphically α b Figure 2.
The boundary strata α and b of D , . The subset Y + n,m of D + n,m , for n ≥
1, consisting of those configurations, whose projectiononto D , belongs to the curve ℓ , is a smooth submanifold with corners of D + n,m of codimension 1.Pictorially, a typical configuration in Y n, looks like as follows: Figure 3.
A typical configuration in Y n, . The dashed line represents the curve, along which the first point in D × (labelled by ◦ ) moves,going from the origin to the unit circle.We are interested in the boundary strata of Y n, of codimension 1, which correspond to thechosen point on the pupil α and to the point b of D , , namely i ) configurations in D n, , whose projection onto D , is the initial point of the curve ℓ (thecorresponding strata are denoted collectively by Y n, ); ii ) configurations in D n, , whose projection onto D , is the final point of the curve ℓ (thecorresponding strata are denoted collectively by Y n, );0 D. Calaque and C.A. RossiLet γ be a solution of the Maurer–Cartan equation as in (1), α a polyvector field on V ofdegree − c = a a Hochschild chain of degree 0 for the algebra A . Proposition 1.
For γ , α and c as above, the following identities hold true: α ∩ S γ ( c ) = X n ≥ n ! X Γ ∈G n +1 , , W D, Γ S Γ ( α, γ, . . . , γ | {z } n , c ) , (11) S γ ( U γ ( α ) ∩ c ) = X n ≥ n ! X Γ ∈G n +1 , , W D, Γ S Γ ( α, γ, . . . , γ | {z } n , c ) , (12) where the weights W iD, Γ , i = 0 , , are defined via W i Γ = Z Y in, ω D, Γ . Proof .
The proof of (11) and (12) relies mainly on the evaluation of the weights W iD, Γ , i = 0 , G n +1 , , , the weight W D, Γ vanishes, if 1 has at least oneincoming edge: namely, if 1 has one arrow coming from 0, then the integral vanishes, since theangle form is the derivative of a (locally) constant function. Further, Kontsevich’s VanishingLemma [7, Lemma 6.4] applies to the remaining cases, whence only the case matters, where 0and 1 collapse together, and then Lemma 2 does the job. Otherwise, the identity W D, Γ = W D, Γ (13)holds true, where Γ is the admissible graph in G n, , obtained from Γ by collapsing the vertices 0and 1. Here is a graphical representation of a general component Z of Y n +1 , Figure 4.
A typical configuration in Y n, . Second, for an admissible graph Γ in G n +1 , , , the weight W D, Γ , restricted to a component Z of Y n +1 , of the form Z ∼ = C A , × C + A , × D A , , ≤ | A | , vanishes, unless there are no edges outgoing from A or from A , in which cases the weightfactorizes as W D, Γ | Z = W Γ W Γ W D, Γ , (14)and Γ is the admissible subgraph of Γ, whose vertices of the first type are labelled by A ,Γ is the graph, whose vertices are labelled by A ⊔ { , } , and obtained by collapsing Γ to thevertex 2, and Γ is the graph, whose vertices are labelled by A ⊔ { } , obtained by collapsing Γ to the vertex 1.hoikhet’s Conjecture and Duflo Isomorphism on (Co)Invariants 11Graphically, a typical configuration in the component Z looks like as A A A 0
Figure 5.
A typical configuration in Y n, . First of all, if there is an edge e.g. from A to A , the corresponding contribution vanishesby means of Lemma 2, i ); the same argument implies the claim in all other cases. We observethat this also implies that Γ i , i = 1 , ,
3, is admissible.Further, the first two factors in the factorization of the weight W D, Γ reduce to usual Kont-sevich’s weights, again in virtue of Lemma 2, v ). (cid:4) The curve ℓ in D , “interpolates” between Y n +1 , and Y n +1 , : we may evaluate weights ofadmissible graphs in G n +1 , , on the remaining boundary strata of codimension 1 of Y n +1 , , and,by means of Stokes’ Theorem, this implies that (11) and (12) coincide at the level of cohomology;for a complete discussion of the corresponding homotopy formula, we refer to [4]. D +1 , and Hochschild chains of higher degree We prove now Theorem 2 in the case, where (3) is applied to Hochschild chains c of higher(negative) degree.The open unit square in C can be embedded in the open configuration space D +1 , via ( s, t ) (cid:2)(cid:0) s, , e π i t (cid:1)(cid:3) , where the square brackets denote equivalence classes w.r.t. the action of S ; wemay take a possible closure σ of it in the compactification D +1 , as follows: α βξ a bde f hj o pq boundary of the surface interior of the surface an imbedding of the plane square, σ σ in the open I−cube0 1 s e t i π Figure 6.
The boundary of σ in D +1 , . Topologically, D +1 , is a “cube with two eyes”, or I-cube: we will be mostly interested, in theforthcoming discussion, in the boundary stratum ξ of codimension 1, which is D × D +0 , , andin the boundary strata o , q of codimension 2, which are C , × C +0 , × D +0 , and C , × C +0 , × D , respectively; graphically2 D. Calaque and C.A. Rossi ξ
01 2 o q Figure 7.
The boundary strata ξ , o and q of D +1 , . We observe that the straight line on the boundary stratum ξ corresponds to the edge { s = 0 } ,the boundary stratum o corresponds to the edge { s = 1 } and the boundary stratum q correspondsto (a way of imbedding) the point (1 , m, n ), with n ≥ m ≥
2, the subset Y + n,m of those confi-gurations in D + n,m , whose projection onto D +1 , belongs to σ , is a smooth, orientable submanifoldwith corners of D + n,m of codimension 1. Graphically, Figure 8.
A typical configuration in Y + n,m . We will need the boundary strata of Y + n,m of codimension 1, corresponding to configurationsin D + n,m , whose projection onto the I-cube is in the boundary of σ (collectively denoted by Y + ,∂σn,m ).More precisely, Y + ,∂σn,m factorizes into eight different types, denoted by Y + ,xn,m , and we will consideronly x to be the straight line on ξ or o and q .We consider a solution γ of the Maurer–Cartan equation as in (1), a polyvector field α on V ,and a Hochschild chain c = ( a | · · · | a m ) in A , m ≥ Proposition 2.
For γ , α and c of degree − m ≤ − as above, the following identities hold true: α ∩ S γ ( c ) = X n ≥ n ! X Γ ∈G n +1 ,m +1 , W ξD, Γ S Γ ( α, γ, . . . , γ | {z } n , c ) , (15) and S γ ( U γ ( α ) ∩ c ) = P n ≥ n ! P Γ ∈G n +1 ,m +1 , W oD, Γ S Γ ( α, γ, . . . , γ | {z } n , c ) , | α | = − , P n ≥ n ! P Γ ∈G n +1 ,m +1 , W qD, Γ S Γ ( α, γ, . . . , γ | {z } n , c ) , | α | ≥ , (16) where the weights W xD, Γ , x = ξ, o, q , are defined via W xD, Γ = Z Y + ,xn +1 ,m ω D, Γ . Proof .
The proof follows along the same lines of the proof of Proposition 1, with some duechanges; once again, we refer to [4] for the complete proofs, while here we make some necessarycomments on the degrees, which hold true in this particular situation.hoikhet’s Conjecture and Duflo Isomorphism on (Co)Invariants 13We observe that, for any admissible graph Γ in G n +1 ,m, , the weight W ξD, Γ vanishes, if thevertex 1 has at least one incoming edge, by the very same arguments sketched in the proof ofProposition 1. Otherwise, the identity W ξD, Γ = W D, Γ holds true, where Γ in G n,m, is obtained from Γ by collapsing the vertices 0 and 1: thisgeneralizes (13) in the proof of Proposition 1, and the proof uses almost the same arguments.On the other hand, a general component Z of the boundary strata of Y + ,on,m , resp. Y + ,qn,m hasthe explicit form Z ∼ = C + A , × C + A ,B × D + A ,B , resp. (17) Z ∼ = C + A ,B × C + A ,B × D + A ,B , (18)where A i , i = 1 , ,
3, are disjoint subsets of [ n ], with 1 ≤ | A | ≤ n , 0 ≤ | A | ≤ n , 0 ≤ | A | ≤ n ,with n = | A | + | A | + | A | , and B i , i = 1 , ,
3, are disjoint ordered subsets of [ m ] of successiveelements, such that 1 ≤ | B | ≤ m , 2 ≤ | B | ≤ m , 1 ≤ | B | ≤ m , and m = | B | + | B | + | B | .Pictorially, A A A A21 o qBB B B B
13 223
Figure 9.
Typical components of Y + ,on,m and Y + ,qn,m . We consider a component Z of Y on +1 ,m as in (17), resp. of Y qn +1 ,m as in (18), and for anadmissible graph Γ as before, we denote by Γ the admissible subgraph of Γ, whose vertices arelabelled by A ⊔ B , by Γ the graph, whose vertices are labelled by A ⊔ B , and obtained bycollapsing Γ to a single vertex, and by Γ the graph, whose vertices are labelled by A ⊔ B ,obtained by collapsing Γ to a single vertex.The weight W oD, Γ , restricted to a component Z of Y + ,on +1 ,m as above vanishes, unless there areno edges going from A to A or A , or from A to A , | B | = 2, and α has degree −
1. In fact,the weight W oD, Γ factorizes as W oD, Γ | Z = W Γ W Γ W D, Γ , with B = ∅ . In particular, the third weight on the right-hand side is non-trivial only if 2 | A |− | A |− | α | +1(since all vertices of the first type in A are 2-valent except the first vertex): therefore, theintegral is non-trivial only if | α | = − W qD, Γ , restricted to a component Z of Y + ,qn +1 ,m as above vanishes, unless there areno edges going from A to A or A , or from A to A , | B | = 2, and | B | = | α | + 1 (whichimplies that α has degree bigger or equal than 0). In such cases, the weight W oD, Γ factorizes as W oD, Γ | Z = W Γ W Γ W D, Γ , ≤ | B | . Dimensional reasons as in the case of a component Z of Y + ,qn +1 ,m force the degreeof α to be bigger or equal than 0: namely, the degree of the third integrand is 2( | A |− | α | +1,and the integral is non-trivial only if it equals 2 | A | + | B | −
2. Since 1 ≤ | B | , the non-trivialitycondition forces | α | = | B | −
1, whence the claim.This result obviously generalizes (14) in the proof of Proposition 1, and its proof is the sameas the proof of (14). (cid:4)
The surface σ “interpolates” between the boundary strata Y + ,ξn +1 ,m , Y + ,on +1 ,m and Y + ,qn +1 ,m : the“interpolation” in this situation is of course more complicated than the one in Subsection 5.1,since we have to keep track of the boundary of σ . In fact, the weighted sums as in (15) and (16),where we integrate over boundary strata of Y + ,xn +1 ,m , x = h, j, p , vanish; the weighted sums overthe remaining boundary strata of Y + n +1 ,m can be also explicitly evaluated, and, by means ofStokes’ Theorem, they produce a homotopy formula, proving that the left-hand sides of (15)and (16) coincide at the level of cohomology. Let g be a finite dimensional Lie algebra over C .First of all, the (modified) Duflo element [6] is defined via J := det (cid:16) e ad / − e − ad / ad (cid:17) ∈ b S( g ∗ ) g . We have a morphism of g -modules D := sym ◦ ( J / · ) : S( g ) −→ U( g ) , where U( g ) is the Universal Enveloping Algebra of g , and sym denotes the usual symmetrizationmap from the symmetric algebra S( g ) to U( g ).The following result generalizes to coinvariants the well-known Duflo isomorphism [6]. Theorem 5.
The map D restricts to an isomorphism of algebras S( g ) g ˜ −→ U( g ) g = Z (cid:0) U( g ) (cid:1) on invariants, and induces an isomorphism of S( g ) g -modules S( g ) g ˜ −→ U( g ) g = A (cid:0) U( g ) (cid:1) oncoinvariants. Here Z ( B ) denotes the center of an algebra B , and A ( B ) = B/ [ B, B ] its abelianization.We sketch here a proof of Theorem 5 in the spirit of the approach of [7, 9] to the originalDuflo isomorphism.We consider the Kirillov–Kostant–Souriau Poisson bivector π on g ∗ and associated product ⋆ .Since the product ⋆ obeys x ⋆ y − y ⋆ x = [ x, y ] g , viewing x, y ∈ g as linear functions on g ∗ , thereis an algebra morphism I : U( g ) −→ (cid:0) S( g ) , ⋆ (cid:1) ; x x. The map I − ◦ U γ induces an algebra isomorphism by means of Theorem 1, Section 2,S( g ) g = Z (cid:0) S( g ) , { , } (cid:1) −→ Z (cid:0) U( g ) (cid:1) = U( g ) g while, dually, S γ ◦I induces an isomorphism of S( g ) g -modules by means of Theorem 2, Section 2,U( g ) g = A (cid:0) U( g ) (cid:1) −→ A (cid:0) S( g ) , { , } (cid:1) = S( g ) g . We may set ~ = 1, as the Poisson structure is linear. hoikhet’s Conjecture and Duflo Isomorphism on (Co)Invariants 15In [11], the restriction of U γ to S( g ) has been shown to be the identity, which, coupled withKontsevich’s discussion [7, Section 8], implies that I − ◦ U γ = I − = D : S( g ) → U( g ).Shoikhet’s proof of the fact that the restriction of U γ to functions is the identity can besummarized as follows: for any function f ∈ S( g ), U γ ( f ) is expressed only via so-called innerwheels, whose weights vanish by the main result of [11]. Pictorially, an inner wheel looks like asfollows: i Figure 10.
An inner wheel.
Theorem 5 follows then from the following
Proposition 3.
The restriction of S γ on S( g ) coincides with the identity map. Proof .
By the arguments of [10, Paragraph 3.6.1], the only admissible graphs contributingnon-trivially to S γ are of the form a a i Figure 11.
A typical admissible graph in S γ on functions and its counterpart in the upper half-plane. Further, we use a correspondence between weights of admissible graphs in G n, , and admis-sible graphs in G n +1 , , by means of the M¨obius transformation ψ : H ⊔ R −→ D ⊔ S \{ } ; z z − iz + i , which induces (see Subsection 3.3) isomorphisms C n +1 , ˜ −→ D n, , to prove that S γ = id.First of all, given on D , the 1-form ω D as in Lemma 2, then the 1-form ω := ψ ∗ ω D on C , is a difference of usual Kontsevich’s angle forms as in Lemma 1. Then the weight W D, Γ of anadmissible graph Γ in G n, , is pulled-back to a weight W Γ ′ , Γ ′ being admissible in G n +1 , : thevertex 0 is mapped to a vertex of the first type, while the only vertex of the second type ismapped to ∞ in the complex upper half-plane. The factors of ω D, Γ are pulled-back to i ) usual6 D. Calaque and C.A. Rossiforms ω e , whenever e is an edge from some vertex (of the first and of the second type) to 1, andthe “new” e is now an edge from the (inverse image of the) starting point to i, and ii ) differencesbetween ω e and ω e (i) , if e is an edge between two vertices (of the first and second type, neitherof which is 0) and the new edge e connects the (inverse images of the) endpoints, while e (i) is anedge, whose starting point is the starting point of e and whose endpoint is i. We have used herethe arguments exposed in Appendix 1 of [4], to which we refer for a more detailed discussion, aswell as for the properties of ω D . We also used the fact that the form ω vanishes when its finalpoint goes to infinity, which finally implies the above correspondence.We observe that dashed arrows denote forms ω D,e on D n, in the left-most graph of Fig. 11,while we have used plain, resp. dashed, arrows to denote forms ω e on C n +1 , , resp. differences ofsuch forms, in the right-most graph.Finally, we use the following graphical calculus for replacing dashed edges by plain ones: = −i ii Figure 12.
Replacing dashed edges by plain ones in an inner wheel.
The second graph on the right-hand side has a double edge, whence its weight vanishes. Thus,the Shoikhet weight of the left-most graph in Fig. 11 equals the usual Kontsevich weight of theright-most one (i.e. with all edges turned into black ones), which is zero by [11]. (cid:4)
We finally observe that Proposition 3 has been proved in Subsubsection 3.6.2 of [10] underthe assumption of the validity of Conjecture 3.5.3.1, which is implied by Theorem 2: our proof,on the contrary, is purely based on the main result of [11] and on the properties of the forms ω D . In this paper, we have proved Shoikhet’s conjecture using configuration space integrals: it isworthwhile noticing that we did not exploit all boundary strata of the I-cube, which replaces inour homotopy argument Kontsevich’s eye (namely, we made use only of the boundary stratum ξ of codimension 1 and of the boundary strata o and q of codimension 2). In fact, Shoikhet’sconjecture can be viewed as a special case of a more general result, which involves a Maurer–Cartan element of a more general shape, i.e. a sum of polyvector fields of different degrees, towhich, via Kontsevich’s formality, corresponds also a sum of polydifferential operators, also ofdifferent degrees: accordingly, all boundary strata of the surface σ in the I-cube contribute tothe proof of this more general result [4]. Acknowledgement
We thank Giovanni Felder for useful discussions and comments. The research of D.C. (on leaveof absence from Universit´e Lyon 1) is fully supported by the European Union thanks to a MarieCurie Intra-European Fellowship (contract number MEIF-CT-2007-042212).hoikhet’s Conjecture and Duflo Isomorphism on (Co)Invariants 17
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