Lie algebroid structures on double vector bundles and representation theory of Lie algebroids

LIE ALGEBR OID STR UCTURES ON DOUBLE VECTOR BUNDLES AND REPRESEN T A T ION THEOR Y OF LIE ALGEBROIDS ALFONSO GRACI A-SAZ AND RAJAN AMIT MEHT A Abstract. A V B –algebroid is essent ially deﬁned as a Li e algebroid ob ject in the cat- egory of ve ctor bundles. There i s a one-to-one corresp ondence betw een V B –algebroids and c ertain ﬂat Lie algebroid superconnections, up to a natural notion of equiv alence. In this setting, w e are able to const ruct characte ristic classes, which i n sp ecial cases repro- duce characteristic classes constructed by Crainic and F ernandes. W e give a complete classiﬁcation of regular V B –algebroids, and in the pro cess we obtain anothe r charact er- istic class of Lie algebroids that do es not app ear in the ordinary represen tation theory of Lie algebroids. 1. I ntroduction Double structures, such as double v ector bundles, double Lie group oids, double Lie al- gebroids, and LA –gro up o ids, hav e been ex tensively studied b y Kirill Mack enzie and his collab ora tors [1 1, 12, 13, 14, 15]. In this paper , we study V B –algebr oids, which are essen- tially Lie algebr o id ob jects in the ca tegory o f v ector bundles. The no tion of V B –alg ebroids is e quiv a le n t to that of Mack enzie’s LA –vector bundles [11], whic h are es sentially vector bundle ob jects in the categor y of Lie alg ebroids. Our guiding principle is that V B –algebr oids may be viewed as generalized L ie a lgebroid representations. An obvious drawbac k of the usual notion of Lie algebr o id representations is that there is no na tural “a djoint ” r epresentation; fo r a Lie algebroid A → M , the action of Γ( A ) o n itself via the br ack et is generally not C ∞ ( M )-linear in the ﬁr s t en try . One possible solution to this problem was given b y Evens, Lu, and W einstein [5], in the form of represe ntations “up to homotopy”. Brie ﬂy , a repr esentation up to homotop y is an a ction of Γ( A ) o n a Z 2 -graded complex of vector bundles, where the C ∞ ( M )-linearity co ndition is only required to hold up to an exact term. With this deﬁnition, they were able to constr uct a n adjoint repre s ent ation up to homotop y of A on the “ K –theor etic” formal diﬀerence A ⊖ T M . This representation up to ho mo topy was us e d by Cr ainic and F ernandes [3, 4] to construc t ch ara c teristic class es for a Lie algebroid, the ﬁrst o f which a grees up to a constant with the mo dula r cla ss of [5]. Another notion of an a djoin t represe ntation was given by F ernandes [6], who gener alized Bott’s theory [2] o f secondary , o r “exotic”, character istic class es for reg ula r folia tions. The key element in F ernandes’s construction is the no tio n of a b asic c onne ction , whic h, for a L ie alg ebroid A → M , is a pa ir of A -co nnections on A a nd T M , satis fying cer tain conditions. These conditions imply that, although the individua l co nnections generally hav e nonzer o curv ature, there is a s e nse in which they are ﬂa t on the for mal diﬀerence; therefore, they can b e used to pro duce secondar y characteristic classes. It was shown in [4] that these c haracteris tic cla sses agree up to a consta nt with tho se constructed via the adjoint representation up to homotopy in [3]. Y et a no ther generalized notion of Lie algebro id representation a ppe a rs in V ain trob’s pap er [19] on the sup erg eometric a pproach to Lie algebr oids. There, a mo dule over a Lie alge br oid A is deﬁned as a Q -vector bundle (i.e. vector bundle in the catego ry of Q - manifolds) with base A [1 ]. T o our kno wledge, this idea ha s not prev iously bee n explored in depth. 1 2 ALFONSO GRACIA-SAZ AND RAJAN AMIT MEHT A One can immediately see from the s uper geometric p ersp ective that a V B – algebroid is a sp ecial case of a Lie alg e broid module. Thus, we may in terpret the notion of V B – algebro ids as providing a description of certa in Lie a lgebroid mo dules in the “conv entional” language of br ack ets and anchors. In par ticular, this sp ecia l case includes V aintrob’s a djoint a nd coadjoint mo dules (see Example 3.3). As w e see in Theorem 4.1 1, every V B –alge br oid ma y be nonca nonically “decomp osed” to g ive a ﬂat Lie algebr oid sup erco nnection on a 2-term complex o f vector bundles, a nd conv ersely , one can cons tr uct a decomp osed V B –a lgebroid fr om such a sup erco nnection. W e show in Theorem 4.14 that diﬀerent choices of de c omp o sition co r resp ond to supe rcon- nections that are equiv a lent in a natura l sense; therefor e we ha ve the following key result: Ther e is a one- t o-one c orr esp ondenc e b etwe en isomorphism classes of V B –algebr oids and e quiva lenc e classes of 2 -term ﬂat Lie algebr oi d sup er c onne ct ions. Given a Lie a lgebroid A → M , a decomp osition of the tang ent pro longation V B –a lgebroid T A yields a ﬂat A -sup erco nnection on A [1 ] ⊕ T M , the diagonal co mpo nen ts of whic h for m a basic co nnection in the s ense of F ernandes [6]. In fact, the V B –algebroid T A is a canonical ob ject from which v arious c hoices of decomp ositio n pro du ce al l basic connections. W e int erpret the V B – algebro id T A as playing the role of the adjoin t representation. As in the case of ordina ry re pr esentations, one may obta in characteristic classes fr o m V B –algebro ids; the constructio n of these classes is describ ed in § 5. In the case of T A , one can see that our characteristic cla sses coincide with the Cr ainic-F ernandes classes. Then, we consider V B –alg e broids that are regular in the sense that the cobo undary map in the associa ted 2 -term complex of v ector bundles is of constant r ank. As in the case of representations up to homotopy (in the sense of E vens, Lu, W einstein), there ar e cano nic a l Lie algebroid repres ent ations on the cohomolog y H ( E ) of the complex. How ever, w e ﬁnd that there is an additional piece of data—a c a nonical cla ss [ ω ] in the 2 nd Lie alg ebroid cohomolog y with v alue s in deg ree − 1 mo r phisms of H ( E ). The tw o r epresentations, together with [ ω ], completely classify regular V B –algebr oids. Finally , in the case of the V B –alge broid T A , the class [ ω ] is an inv ariant of the Lie alge- broid A . Mo re sp eciﬁcally , given a reg ula r Lie algebroid A with anchor map ρ , the cohomol- ogy clas s [ ω ] asso cia ted to the V B –algebr oid T A is an element of H 2 ( A ; Hom(coker ρ, k er ρ )). W e see that [ ω ] ma y be interpreted as an obstructio n to the r egularity of the restrictions o f A to leaves of the induced foliation. R emark. After submitting this pap er, we lear ned of the work of Arias Abad and Crainic [1], in which some of the constr uctions in this pap er a re develope d indep endent ly . In particular, they deﬁne the notion of r epr esentation up to homo topy , whic h coincides with our notion of sup e rreprese ntation a s in Deﬁnition 4.7, and whic h is diﬀerent from the notion of repr esen- tation up to homo to p y acco r ding to E vens, Lu, a nd W einstein [5 ]. As we show in Theorem 4.11, a V B –algebro id, a fter choosing a decompo s ition, corresp onds to a s uper representation in t w o degrees. As an analog y , a V B – a lgebroid is to a superr e presentation in t w o degrees what a linea r map is to a matrix, with choice of decompo sition playing the role of c hoice of basis. This is describ ed in further detail for the example T A (whic h b oth here a nd in [1] is int erpreted as the adjoint representation) in § 7. Structure of the paper. • T he ob jects that this pap er deals with are double ve ctor bu nd les equipped with additional structures . W e b eg in in § 2 by r ecalling the deﬁnition of a double vector bundle and describing some of its pro per ties. • I n § 3 w e in tro duce our main ob ject: V B –a lgebroids, and w e present v arious equiv- alent structur es. • I n § 4 w e g ive a one-to- o ne corr esp ondance b et ween isomorphism c lasses o f V B – algebroids and cer tain equiv alence classes of ﬂat superc o nnections, hence interpret- ing V B –algebr oids a s “higher ” Lie algebr oid repr esentations. V B -ALGEBR OIDS AND REPRESENT A TION THEOR Y OF LIE ALGEBROIDS 3 • I n § 5 we deﬁne characteristic classes for every V B – a lgebroid. • I n § 6 we clas s ify a ll regular V B –alg e br oids. • Fina lly , in § 7 w e use the results from § 6 in the ca s e of the “ adjoint representation” to ass o ciate a cohomology class to every regular Lie alg ebroid that has a geometric int erpretatio n in terms o f regula rity ar o und leaves induced by the alg ebroid foliatio n. Ac kno wledgement s. W e were par tially supp orted b y gran ts from Conselho Nacional de Desenv olvimen to Cient ´ ıﬁco e T ecno l´ ogico (CNPq) and the Ja pa nese So ciety for the Pro- motion of Science (J SPS). W e thank the Centre de Recer ca Matem` atica and the Centre Bernoulli fo r their hospitality while this research w as b eing done. W e a lso thank Ec khard Meinrenken. 2. B ac kgr ound: Double vector bundles The main ob j ects that this paper deals with a re double ve ctor bund les (DV Bs) eq uipped with additional structur es. Therefore we sha ll b egin in this s ection by brieﬂy recalling the deﬁnition of a DVB and descr ibing some prop erties o f DVBs that will b e useful la ter. F or details and pr o ofs, see [1 4]. A D VB is essen tially a v ector bundle in the categ ory o f v ector bundles. The notion of a D VB was in tro duced b y Pradines [17] a nd has since b een studied by Mack enzie [14] and Konieczna and Urba ´ nski [9 ]. Mack enzie has also in tro duced higher ob jects ( n -fold v ector bundles [14]) and more g eneral double structures ( LA -group oids and double Lie a lgebroids [11, 12, 13, 15]). Recen tly , Grab owski and Rotkiewicz [7] ha ve studied do uble and n -fold vector bundles from the superg eometric point o f view. Most of the material in this section has a ppe ared in the ab ov e-referenced work o f Mack en- zie. 2.1. Deﬁni tion of D VB. In o rder to deﬁne double v ector bundle, we begin with a com- m utative squar e (2.1) D q D B / / q D A   B q B   A q A / / M , where all four sides are v ector bundles. W e w is h to describe compatibility conditions be- t ween the v a rious v ector bundle structures. W e follow the nota tion of [14]. In particular, the addition ma ps for the t wo vector bundle structures on D ar e + A : D × A D → D a nd + B : D × B D → D . The zero sections are denoted as 0 A : M → A , 0 B : M → B , e 0 A : A → D , and e 0 B : B → D . W e lea ve the pro of of the following prop osition a s an exe rcise. Prop ositio n 2. 1. The fol lowing c onditions ar e e quivalent: (1) q D B and + B ar e ve ctor bun d le morphisms over q A and the addition map + : A × M A → M , r esp e ctively. (2) q D A and + A ar e ve ctor bun d le morphisms o ver q B and the addition map + : B × M B → M , r esp e ctively. (3) F or al l d 1 , d 2 , d 3 , and d 4 in D su ch that ( d 1 , d 2 ) ∈ D × B D , ( d 3 , d 4 ) ∈ D × B D , ( d 1 , d 3 ) ∈ D × A D , and ( d 2 , d 4 ) ∈ D × A D , the following e quations hold: (a) q D A ( d 1 + B d 2 ) = q D A ( d 1 ) + q D A ( d 2 ) , (b) q D B ( d 1 + A d 3 ) = q D B ( d 1 ) + q D B ( d 3 ) , (c) ( d 1 + B d 2 ) + A ( d 3 + B d 4 ) = ( d 1 + A d 3 ) + B ( d 2 + A d 4 ) . Deﬁnition 2.2. A double ve ctor bu n d le (DV B) is a commutativ e s quare (2.1), where all four sides a re v ector bundles, satisfying the c onditions of P rop osition 2.1. 4 ALFONSO GRACIA-SAZ AND RAJAN AMIT MEHT A R emark 2.3 . A s mo o th map b etw een vector bundles that r esp ects addition is a vector bundle morphism. F or this reaso n, it is unnece ssary to refer to scalar m ultiplication in condition (3) of Prop ositio n 2.1. Alternatively , Gra bowski and Rotkiewicz [7] hav e giv en an equiv alent deﬁnition of D VBs only in terms of scalar mult iplication; they a lso give an int eresting in terpretation in terms of comm uting Euler v ector ﬁelds. R emark 2.4 . It is sometimes reques ted as part of the deﬁnition of DVB that the double pro jection ( q D A , q D B ) : D → A ⊕ B b e a surjective submer sion. Grabowski a nd Rotkiewicz [7] prov ed that this is a conse quence of the r est of the deﬁnition. 2.2. The core of a D VB. The structure o f a DVB (2.1) obviously includes tw o vector bundles, A and B , ov er M , which are called the side bund les . There is a third v ector bundle C , known as the c or e , deﬁned as the intersection o f the k ernels o f the bundle maps q D A and q D B . Out of the three bundles A , B , and C , the core is specia l in that it naturally e mbeds int o D . In fact, it ﬁts into the sho r t e xact sequence of double vector bundles (2.2) C / /   M   M / / M   / / D / /   B   A / / M / / / / A ⊕ B / /   B   A / / M Given vector bundles A , B , and C , there is a natural double v ector bundle structure o n A ⊕ B ⊕ C with side bundles A and B and c ore C ; this DVB is sa id to b e de c omp ose d . A section (in the categ o ry of double vector bundles) o f (2.2) is equiv alen t to an isomor- phism inducing the identit y map o n A , B , and C , b etw een D and the deco mp os ed D VB A ⊕ B ⊕ C . This isomorphism is called a de c omp osition of D . Gra b owski and Rotkiewicz [7] prov ed tha t decompos itio ns alwa ys exist lo ca lly (ov er op en sets of M ), and a ˇ Cech coho mol- ogy argument shows that decomp ositions exist globally . In fact, the spa ce of decomp ositio ns of D is a nonempty aﬃne s pa ce mo delled on Γ( A ∗ ⊗ B ∗ ⊗ C ). Hence, a section o f (2.2) alwa ys exists, alb eit noncanonically . 2.3. Linear and core sections . Consider a DVB as in (2.1). Ther e ar e tw o sp ecial types of sections o f D o ver B , which we call line ar and c or e 1 sections. As w e will use in v arious pro ofs in Appendix A, statements ab out sec tions of D ov er B can o ften b e reduced to statements ab out linear and core se c tions. Deﬁnition 2.5. A s ection X ∈ Γ( D, B ) is line ar if X is a bundle morphism from B → M to D → A . The space of linear sections is denoted as Γ ℓ ( D , B ). The core s ections arise from s e ctions of the co re bundle C → M , in the following wa y . Let α : M → C be a section o f the core. The comp osition ι ◦ α ◦ q B , where ι is the embedding of C into D , is a map from B to D but is not a right inv erse o f q D B . Instea d, Γ( C ) is em b edded int o Γ( D, B ) by (2.3) α ∈ Γ( C ) 7→ α := ι ◦ α ◦ q B + A e 0 B ∈ Γ( D , B ) . Deﬁnition 2.6. The space Γ C ( D , B ) of c or e se ctions is the image of the map (2.3). In the rest of this pap er we will use the same no tation for α and α if there is no ambiguit y . Let X b e a section of D ov er B . W e s ay that X is q -pr oje ct ible (to X 0 ) if X 0 ∈ Γ( A ) and q D A ◦ X = X 0 ◦ q B . A linear section X is necessarily q -pro jectible to its b ase se ction . All core sections are q -pro jectible to the zero section 0 A . Co n versely , if α is q -pr o jectible to 0 A , then α is a core section if a nd o nly if the map ( α − A e 0 B ) is co nstant on the ﬁbres ov er M . 1 In [16], the term vertic al w as used instead of c or e . How ev er, it i s no w apparent that the present terminology is more appropriate. V B -ALGEBR OIDS AND REPRESENT A TION THEOR Y OF LIE ALGEBROIDS 5 R emark 2.7 . It may b e helpful to see a co o rdinate description of the linear and co re se c tions. Cho ose a decomp osition D ≡ A ⊕ B ⊕ C , and choose lo cal co o rdinates { x i , b i , a i , c i } , where { x i } are co or dinates on M , and { b i } , { a i } , and { c i } are ﬁbre co or dinates on B , A , and C , resp ectively . Let { A i , C i } b e the frame o f sections over B dual to the ﬁbre co o rdinates { a i , c i } . Then X ∈ Γ( D, B ) is linear if and only if it locally takes the form (2.4) X = f i ( x ) A i + g i j ( x ) b j C i , and α ∈ Γ( D, B ) is co r e if and only if it loca lly tak es the form α = f i ( x ) C i . Example 2 .8 . A standar d exa mple of a D VB is (2.5) T E / /   E   T M / / M , where E → M is a vector bundle. The cor e, consisting of vertical v ectors tangent to the zero section of E → M , is naturally iso morphic to E . The linear sections of T E over E are the linear vector ﬁelds, and the cor e sections a r e the ﬁbrewise-constant vertical vector ﬁelds. It is p oss ible to characterize morphisms o f DVBs in terms o f linear and cor e sections. Let D / /   B   A / / M and D ′ / /   B   A ′ / / M be D VBs with cores C and C ′ , r esp ectively . Let F : D → D ′ be a map tha t is linear ov er B , and let F ♯ : Γ( D , B ) → Γ( D ′ , B ) b e the induced map of se ctions. Lemma 2.9. Under the ab ove c onditions, the fo l lowing ar e e quivale nt: (1) F ♯ sends line ar se ctions t o line ar se ctions and c or e se ctions t o c or e se ctions. (2) F is a morphism of ve ctor bun d les fr om D → A t o D ′ → A ′ . In other wor ds, F is line ar with r esp e ct to b oth the horizontal and vertic al ve ctor bun d le str u ctur es, which i s the deﬁnition of a morphism o f double ve ctor bund les. Pr o of. The pro of is a stra ightforw ard exercis e in co or dina tes.  2.4. Ho ri zon tal lifts. Linear sections may b e used to intro duce a concept tha t is equiv alen t to that o f a decompositio n o f a DVB, which will be useful later. It is clear fro m the loca l description of linear sectio ns (2.4) that the space Γ ℓ ( D , B ) is lo cally free as a C ∞ ( M )-mo dule, with rank equal to ra nk( A ) + rank( B ) rank( C ). Therefore, Γ ℓ ( D , B ) is equal to Γ( b A ) for s ome v ector bundle b A → M . There is a s hort exact sequence o f v ector bundles o ver M (2.6) 0 / / B ∗ ⊗ C = Hom( B , C ) i / / b A π / / A / / 0 . Deﬁnition 2.10. A horizontal lift o f A in D is a section h : A → b A of the sho rt exa c t sequence (2.6). Prop ositio n 2.11. Ther e is a one-to-one c orr esp ondenc e b etwe en horizontal lifts and de- c omp ositions D ∼ → A ⊕ B ⊕ C . 6 ALFONSO GRACIA-SAZ AND RAJAN AMIT MEHT A Pr o of. There is a natural horizontal lift in the case of a decomposed double v ector bundle A ⊕ B ⊕ C . Therefor e , there is a map κ from deco mpo sitions D ∼ → A ⊕ B ⊕ C to ho rizontal lifts of D . The spac es of decomp ositio ns and of horizo nt al lifts are b oth a ﬃne spaces mo delled on Γ( A ∗ ⊗ B ∗ ⊗ C ). The map κ is aﬃne, and the asso ciated linea r map is the identit y .  Example 2.1 2 . F or the DVB of Example 2.8, a decompo sition T E ∼ → T M ⊕ E ⊕ E is the same thing as a linear connection on E → M . A horiz ontal lift, in the sense of Deﬁnition 2.10, coincides in this ca se with the usua l notion of a horizontal lift for E → M . 3. Doubles f or Lie algebr oids and vector bund les The main ob ject of s tudy in this pa per consists of a double vector bundle with additiona l structure. Ther e are v ario us equiv alent w ays to descr ibe the a dditional structure, including LA –vector bundles ( § 3.1), V B –alg ebroids ( § 3.2), and Poisson double vector bundles ( § 3 .4). There are a lso in terpretations in terms o f diﬀeren tials ( § 3.5) and super g eometry ( § 3.6). 3.1. LA - v ector bundles. An LA – vector bundle is es sentially a v ector bundle in the ca t- egory of Lie a lgebroids. More precisely , it is a DVB (3.1) D q D E / / q D A   E q E   A q A / / M , where the horizontal sides are Lie algebroids and the structur e maps for the vertical vector bundle str uctures form Lie algebro id morphisms. Spec iﬁc a lly , if q D A is a n a lgebroid mor- phism, then there is an induced Lie a lgebroid structur e on the ﬁbre pr o duct D × A D → E × M E , and w e c a n ask tha t the addition map + A : D × A D → D b e a n alg ebroid morphism. The notion of an LA –vector bundle is due to Mack enzie [11]. R emark 3.1 . Consider a D VB o f the form (3.1). Given a Lie algebroid structure on D → E , there is a t most one Lie algebro id str ucture on A → M such that q D A is an a lgebroid morphism. If such a Lie algebr oid structure ex ists o n A , then we may say that the Lie algebroid s tructure on D → E is q - pr oj e ctible . Thus the deﬁnition of an LA –vector bundle may be res tated in the following w ay: Deﬁnition 3. 2. An LA –ve ctor bund le is a D VB (3.1) equipp ed with a q -pro j ectible Lie algebroid s tructure on D → E suc h that the addition map + A : D × A D → D is an algebro id morphism. Example 3 .3 . (1) The DVB (2 .5 ) is an LA –vector bundle, where T E has the canonical tangen t Lie algebroid structure ov er E . (2) Let A → M b e a Lie algebroid. Then (3.2) T ∗ A / /   A ∗   A / / M and T A / /   T M   A / / M are LA –vector bundles , where the Lie algebroid structure on T ∗ A = T ∗ A ∗ → A ∗ arises from the Poisson structure on A ∗ , and that on T A → T M is the tangent pr olongation of the Lie algebroid structure on A → M . W e remark that the latter is in fact a double Lie algebroid a nd thus ma y b e viewed as LA –vector bundle in t wo diﬀerent w ays. T o avoid co nfusion, w e will alwa ys present LA –vector bundles so that the relev ant Lie alg ebroid structures ar e on the horizo nt al s ides. V B -ALGEBR OIDS AND REPRESENT A TION THEOR Y OF LIE ALGEBROIDS 7 In § 4 , we will see that LA – vector bundles ma y be viewed a s higher r epresentation of Lie alg ebroids. F rom this p oint of view, the LA –vector bundles in (3.2) will play the roles of the co adjoint and the adjoin t representation of A . 3.2. V B -algebroids. There is an alterna tive se t o f compatibilit y conditions for the Lie algebroid and vector bundle str uctur es of (3.1). Recall that the spaces of linear and core sections are deno ted b y Γ ℓ ( D , E ) and Γ C ( D , E ), respectively . Deﬁnition 3.4. A V B –algebr oi d is a D VB a s in (3.1 ), e q uipped with a Lie algebr oid structure on D → E such that the anc hor map ρ D : D → T E is a bundle mo rphism o v er A → T M and wher e the bra ck et [ · , · ] D is suc h that (1) [Γ ℓ ( D , E ) , Γ ℓ ( D , E )] D ⊆ Γ ℓ ( D , E ), (2) [Γ ℓ ( D , E ) , Γ C ( D , E )] D ⊆ Γ C ( D , E ), (3) [Γ C ( D , E ) , Γ C ( D , E )] D = 0 . R emark 3.5 . Since the anchor ma p ρ D is automa tically linear over E , the condition that it be linear ov er A is equiv alent to asking that ρ D be a morphism of D VBs fro m (3.1 ) to (2.5). Given a V B – algebroid (3.1), there is a unique map ρ A : A → T M s uch that the dia gram (3.3) D ρ D / / q D A   T E T q E   A ρ A / / T M commutes, and ρ A is necess arily linear o ver M . F urthermor e, a br ack et [ · , · ] A on Γ( A ) ma y be deﬁned by the proper ty that, if X and Y in Γ ℓ ( D , E ) are q -pro jectible to X 0 and Y 0 , rep ectively , then [ X , Y ] D is q -pro jectible to [ X 0 , Y 0 ] A . The map ρ A and the brack et [ · , · ] A together form a Lie a lgebroid structure on A → M . W e leav e the deta ils a s an ex e r cise for the reader. R emark 3.6 . T o provide some motiv a tion fo r the the brack et conditions in Deﬁnition 3.4, we consider the DVB o f Example 2.8. In this case, the Euler vector ﬁeld ε on E induces a grading o n the space o f vector ﬁelds on E , where X ∈ X ( E ) is homog eneous of deg ree p if [ ε, X ] = pX . Then th e linear v ector ﬁelds (whic h are the elements o f Γ ℓ ( T E , E )) are precisely those of degree 0, and the ﬁbrewise-constant vertical vector ﬁelds (which are the elements of Γ C ( T E , E )) are precisely those of degr e e − 1. In this exa mple, the br ack et conditions simply state that the Lie brack et r e s pe c ts the grading o f v ector ﬁelds. The int erpretatio n of the brack et conditions in terms of a gra ding is c arried out in the general case in § 3 .5. 3.3. Equi v alence of LA -v ector bundles and V B -al gebroids. LA –vector bundles and V B –algebro ids a re b o th sp eciﬁed by the same type of data—a DV B of the form (3.1), where D → E is equipp ed with a Lie alg ebroid structure satisfying cer tain compatibility conditions. Both se t of compatibilit y co nditions imply that A → M is also a Lie alg e broid. The compatibility conditions for LA –vector bundles require that the v ertical vector bundles resp ect the ho rizontal Lie algebro id, wher eas the compatibility conditions for V B –algebr oids require that the ho rizontal Lie algebroid respe cts the vertical v ector bundles. The following theorem states that the tw o sets o f co mpatibility conditions are equiv alen t. Theorem 3.7. A double ve ctor bund le of t he form (3.1), wher e t he top side is e quipp e d with a Lie algebr oid structur e, satisﬁes the LA –ve ctor bund le c omp atibility c onditions if and only if it satisﬁes t he V B –algebr oid c omp atibility c onditio ns. The pro of of Theo rem 3.7 is given in App endix A. 8 ALFONSO GRACIA-SAZ AND RAJAN AMIT MEHT A R emark 3.8 . Since the notions of LA –vector bundle and V B – algebroid a re equiv alent, w e could at this po in t disco n tinue the use of the term “ V B –alg ebroid” in fav or o f the previously- established ter m “ LA –vector bundle”. Ho wev er, we will see that the constr uctions in § 4 that for m the hear t of this paper directly utilize the conditions in Deﬁnition 3.4; in other words, this pa p er r elies in a n es sential w ay on the V B –alg ebroid p oint of view. F or this reason, w e will con tinue to use the term “ V B –algebr oid”. R emark 3.9 . In the langua ge of catego ry theory , an LA –vector bundle is essentially a vector bundle ob ject in the c ategory o f Lie algebro ids, and a V B – algebro id is ess ent ially a Lie algebro id ob ject in the category of vector bundles. In this sense, Theorem 3.7 is a n analogue of the following c ategory- theoretic r esult: if X a nd Y are alg ebraic catego ries, then an X ob ject in the catego ry of Y is equiv alen t to a Y ob ject in the categ ory of X . See, for insta nc e , [10]. 3.4. Poisson double v ector bundles. Again, co nsider a double vector bundle of the fo rm (3.1), where D → E is a Lie algebroid. If w e dualize D over E , we obtain a new D VB (3.4) D ∗ E / /   E   C ∗ / / M , where the core is A ∗ . F or a disc us sion on the dualiza tion of D VBs, see [1 4]. The algebra of functions o n D ∗ E has a canonical double-grading; we de no te by C ∞ p,q ( D ∗ E ) the spa c e of functions that are homogeneous of degrees p and q ov er E and C ∗ , respec tively . The s pa ce C ∞ 1 , • ( D ∗ E ) of functions that ar e linear over E may b e identiﬁed with Γ( D , E ), and it is clea r from the co or dina te description of Remark 2.7 that, under this identiﬁca- tion, w e hav e Γ ℓ ( D , E ) = C ∞ 1 , 1 ( D ∗ E ) and Γ C ( D , E ) = C ∞ 1 , 0 ( D ∗ E ). Note that the grading on Γ( D , E ) that is induced from the identiﬁcation with C ∞ 1 , • ( D ∗ E ) is not the same as the grading describ ed in Rema rk 3.6, but is shifted by 1 . As usual, the Lie algebroid structure on D → E induces a Poisson s tructure o n D ∗ E that is linear o ver E , in the sense that { C ∞ p, • ( D ∗ E ) , C ∞ p ′ , • ( D ∗ E ) } ⊆ C ∞ p + p ′ − 1 , • ( D ∗ E ). It is then fairly easy to s ee that the compatibility conditio ns o f Deﬁnition 3.4 ar e equiv alent to the condition that { C ∞ • ,q ( D ∗ E ) , C ∞ • ,q ′ ( D ∗ E ) } ⊆ C ∞ • ,q + q ′ − 1 ( D ∗ E ). In other w ords, w e ha ve Theorem 3.1 0. A double ve ctor bund le of the form (3.1), wher e the top side is e quipp e d with a Lie algebr oi d s tructur e, satisﬁes the V B –algebr oi d c omp atibi lity c onditions if and only if the induc e d Poisson structur e on D ∗ E is line ar over C ∗ . Mack enzie [15 ] has deﬁned a Poisson double ve ctor bund le to b e a DVB whose total space is equipp ed with a P oisson structure that is linear ov er both side bundles. Theorem 3.10 states that there is a corr esp ondence b etw een Poisson double vector bundles and V B – algebroid structures. W e note that this res ult was already e s tablished in [15]. An interesting feature of P oisso n double v ector bundles is that the deﬁnition is symmet- ric with resp ect to the r o les of the tw o side bundles. On the other hand, the cor resp on- dence of Theorem 3.10 is not symmetric, whic h implies that there are in fact tw o diﬀeren t V B –algebro id structures asso ciated to each P oisson double v ector bundle. Using b oth cor- resp ondences, we are able to a sso ciate a V B –algebr oid s tructure on D with a V B –a lgebroid structure on ( D ∗ E ) ∗ C ∗ ; the latter is canonically isomorphic to D ∗ A , so w e obtain the following duality result: V B -ALGEBR OIDS AND REPRESENT A TION THEOR Y OF LIE ALGEBROIDS 9 Corollary 3.11. A V B –algebr oi d st ructur e on (3.1) induc es a dual V B –algebr oi d st ructur e on (3.5) D ∗ A / /   C ∗   A / / M . R emark 3.12 . In general, a D VB (3.1) and its t wo neighbors (3.4) and (3.5) ﬁt tog ether to form a tr iple vector bundle (3.6) T ∗ D / /   # # G G G D ∗ E " " E E   D / /   E   D ∗ A # # G G G / / C ∗ " " E E E A / / M If (3.1) is a V B –alg ebroid, then the triple vector bundle (3.6) has the following structures: • T he four hor izontal edges are Lie alg ebroids. • T he r ight and left faces are Poisson double vector bundles. • T he o ther four fac e s ar e V B – algebro ids . All of the above structures and their relations may b e summarized as follows: the c ub e (3.6) is a Lie alg ebroid ob ject in the category of Poisso n double v ector bundles. Example 3.13 . Let E → M b e a v ector bundle. The V B –algebr oid (2.5) is asso ciated to the Poisson double vector bundle T ∗ E / /   E   E ∗ / / M , from whic h we may obtain the dual V B –alge br oid T E ∗ / /   E ∗   T M / / M . Example 3.14 . Let A → M b e a Lie algebroid. The V B –algebr oid T ⋆ A in (3.2) is asso ciated to the Poisson double vector bundle T A ∗ / /   A ∗   T M / / M , and is dual to the V B –alg ebroid T A in (3.2). 3.5. Com patibility in term s of algebroid diﬀ e ren tials. O nce a gain, consider a double vector bundle o f the for m (3.1), where D → E is a Lie algebr o id. W e may iden tify Γ( D ∗ E , E ) with the space of functions on D that ar e linear ov er E . F urthermo re, this identiﬁcation gives Γ( D ∗ E , E ) a g rading according to p olynomia l deg ree ov er A , and this grading may b e extended to ∧ Γ( D ∗ E , E ). W e denote by Ω p,q ( D ) the subspace o f ∧ p Γ( D ∗ E , E ) consisting of those p -forms that ar e of degr ee q over A . Recall that in § 3.4, the space of s ections Γ( D , E ) w as given a g rading, where the linear sections were o f degree 1 and the co r e sections were o f degree 0. The gradings on Γ( D, E ) 10 ALFONSO GRACIA-SAZ AND RAJAN AMIT MEHT A and Ω( D ) agre e up to a shift, in the sense that, for a degree q section X ∈ Γ( D , E ), the op erator ι X on Ω( D ) is of degree q − 1. The Lie algebroid s tr ucture on D → E induces a diﬀerential d D on ∧ Γ( D ∗ E , E ). Theorem 3.1 5. A double ve ctor bund le of the form (3.1), wher e the top side is e quipp e d with a Lie algebr oi d s tructur e, satisﬁes the V B –algebr oi d c omp atibi lity c onditions if and only if d D is of de gr e e 0 with r esp e ct to the “ov er A gr ading”. Pr o of. Applying Lemma 2.9 to the anchor map ρ D , we hav e that the c o mpatibility co nditio n for the anchor is equiv alent to the condition that, for a degree q section X ∈ Γ( D , E ), the degree of ρ D ( X ) is q − 1 as an op er ator on C ∞ ( E ) . Additionally , the compatibility co nditio ns for the bra ck et are equiv a lent to the co ndition that, for sectio ns X and Y o f degrees q a nd q ′ , resp ectively , the degree of ι [ X,Y ] D is q + q ′ − 2 . F or ω ∈ Ω p,q ( D ) and X i ∈ Γ( D, E ) of deg ree q i for i = 0 , . . . , p , the diﬀerential d D is given b y the formula ι X p · · · ι X 0 d D ω = p X i =0 ( − 1) i ρ D ( X i ) ι X p · · · c ι X i · · · ι X 0 ω + X j >i ( − 1) j ι X p · · · ι [ X i ,X j ] D c ι X j · · · c ι X i · · · ι X 0 ω , (3.7) Each term on the right hand side is of degree q + P ( | q i | − 1). The left hand side must be of the sa me degree, whic h implies that d D is of degree 0.  3.6. Sup ergeometric interpretat ion. It was observ ed b y V aintrob [19] that the diﬀer- ent ial p o int of view for a Lie algebroid is more naturally stated in the lang ua ge of s uper ge- ometry in the following way: a Lie a lgebroid structur e on A → M is e q uiv ale nt to a degree 1 homolo gical vector ﬁeld on the graded manifold A [1]. Here, A [1] is the gr aded manifold whose a lgebra of “functions” is ∧ Γ( A ∗ ), and the o per ator d A , as a deriv ation of this algebra , is view ed as a vector ﬁeld on A [1]. The mo diﬁer homolo gic al indicates that d 2 A = 0 . In the case o f a V B –a lgebroid (3.1), we may form the gr aded manifold 2 D [1] E , who s e algebra o f “ functions” C ∞ ( D [1] E ) is Ω • , • ( D ). The o p e rator d D is view ed as a homologica l vector ﬁeld on D [1] E . The algebra C ∞ ( D [1] E ) has a natural double- g rading arising fr om the DVB structure D [1] E / /   E   A [1] / / M , and this double-gra ding coincides with the double-gra ding of Ω( D ) that was in tro duced in § 3.5. In this p oint of view, we may use Theo rem 3.1 5 to e ﬀectively res tate the deﬁnition o f a V B –algebr oid as follows: Theorem 3.1 6. A V B –algebr oid st ructur e on a DVB (2.1) is e quivalent t o a ve ctor ﬁeld d D on D [1 ] E of bide gr e e (1 , 0) such that d 2 D = 0 . 4. (S uper)connections and horizont al lifts One of the main goa ls of this pa p er is to show that isomorphism class es of V B –algebr o ids are in one-to-one corresp ondence with ﬂat Lie algebro id super connections up to a c e rtain notion of equiv alence. With this in mind, we can understa nd a V B – algebro id as a general- ization of a Lie alge br oid r e pr esentation. 2 Since D i s the total space of tw o di ﬀeren t v ecto r bundles, w e use the subscript in [1] E to i ndicate that we are applying the f unctor [1] to the vect or bundle D → E , as opposed to D → A . V B -ALGEBR OIDS AND REPRESENT A TION THEOR Y OF LIE ALGEBROIDS 11 Consider a V B –alg ebroid (3.1) with core C . In § 4.1- § 4.3, we will see tha t there is a natural Lie algebr oid structure o n b A , and that b A p oss e sses natural represe ntations on C and E . In § 4 .4, we us e horizo nt al lifts to o btain A -connections on C and E ; unfortunately , the pro cedure is not canonica l, and the induced co nnections are not ﬂat. Ho wev er, the induced A -connections form par t of a ﬂat A -sup erco nnec tio n on a graded bundle ( § 4.5). Although the ﬂat A -sup erco nnection is noncanonical, diﬀerent choices o f horizontal lifts lead to sup erco nnections that are equiv alent in a w ay that will be descr ibe d in § 4.7 and will be use d in § 6 to classify r egular V B –alge br oids. 4.1. The fat algebroid. Consider a V B –algebr oid (3.1) with co re C . As in § 2.4, let b A denote the vector bundle ov er M whose space of sections is Γ( b A ) = Γ ℓ ( D , E ). The bundle b A has a natura l Lie alg ebroid s tructure with brack et [ · , · ] b A and anchor ρ b A given by [ X , Y ] b A = [ X , Y ] D ρ b A ( X ) = ρ A ( X 0 ) , where X q -pro jects to X 0 . W e refer to b A as the fat algebr oid . The pr o jection map b A → A is a Lie algebro id morphism, the kernel of which may be ident iﬁed with Hom( E , C ). Therefo r e Hom( E , C ) inherits a Lie algebroid structure so that (4.1) 0 / / Hom( E , C ) i / / b A π / / A / / 0 is an exa ct s equence of Lie algebroids over M . Example 4.1 . When D = T A , where A → M is a Lie a lg ebroid, then b A is equal to the ﬁrst jet bundle J 1 A of A . In this case, Crainic and F ernandes [4] have describ ed natural representations o f J 1 A on A and T M . In § 4.3, we will extend this pro cess to all V B – algebroids . 4.2. The core-anc hor. T o explicitly desc r ib e the Lie algebro id structure on Hom( E , C ) inherited from (4.1), it is useful to int ro duce an auxilliary map. Since the anc hor ρ D is a morphism of DVBs from D / /   E   A / / M to T E / /   E   T M / / M , it induces a linear map of the core v ector bundles. Deﬁnition 4 .2. The c or e-ancho r ∂ o f a V B –alg ebroid (3.1) is minus the vector bundle morphism induced b y the a nchor map ρ D from the core C of D to the cor e E of T E . The core-anchor ∂ is explicitly given by the equation h ∂ α, e i = − ρ D ( α )( e ) for all α ∈ Γ( C ) and e ∈ Γ( E ∗ ). In other words, s inc e − ρ D ( α ) is a ﬁbr ewise-constant vertical vector ﬁeld on E , it may b e identiﬁed with a section ∂ α o f E . The map ∂ is C ∞ ( M )-linear and therefore is an e lement of Hom( C, E ). The brack et o n Hom( E , C ) is then giv en by (4.2) [ φ, φ ′ ] = φ∂ φ ′ − φ ′ ∂ φ for φ, φ ′ ∈ Hom( E , C ). The a nch or is trivial, so Hom( E , C ) is actually a bundle of Lie algebras . Example 4.3 . In the V B – a lgebroid (3.2), the cor e-anchor ma ps A to T M and is equa l to min us the anc hor of the a lgebroid A → M . 12 ALFONSO GRACIA-SAZ AND RAJAN AMIT MEHT A 4.3. Si de and core represen tations of b A . The fat algebroid has natural representations (i.e. ﬂat co nnections) ψ c and ψ s ∗ on C a nd E ∗ , resp ectively , given by ψ c χ ( α ) := [ χ, α ] D , (4.3) ψ s ∗ χ ( e ) := ρ D ( χ )( e ) , (4.4) for χ ∈ Γ( b A ), α ∈ Γ( C ), and e ∈ Γ( E ∗ ). In (4.4), we view e as a linear function o n E . Since ρ D ( χ ) is a linea r vector ﬁeld on E , it acts on the space of linear functions. As usual, the repr e sentation ψ s ∗ may b e dualized to a representation ψ s on E , given by the equation h ψ s χ ( ε ) , e i := ρ b A ( χ ) h ε, e i − h ε, ψ s ∗ χ ( e ) i W e lea ve the following as an exercise. Prop ositio n 4. 4. The r epr esentations ψ c and ψ s ar e r elate d in the fol lowing ways: (1) ∂ ψ c χ = ψ s χ ∂ (2) φψ s χ − ψ c χ φ = [ φ, χ ] b A for al l χ ∈ Γ( b A ) a nd φ ∈ Hom( E , C ) . The representations o f b A may b e pulled back to obtain representations θ c and θ s of Hom( E , C ) on C and E . Explicitly , these represetnations are given b y θ c φ ( α ) = φ ◦ ∂ ( α ) , (4.5) θ s φ ( ε ) = ∂ ◦ φ ( ε ) , (4.6) for φ ∈ Hom( E , C ), α ∈ Γ( C ), and ε ∈ Γ( E ). W e would like to b e able to push the s ide and c o re represe n tations of b A forward to obtain representations of A ; how ever, this is g enerally not p ossible since the induced r epresentations of Hom( E , C ) in (4.5)-(4.6) ar e nontrivial. If ∂ is o f constant rank, the side and co r e b A - representations do induce A -representations ∇ K on the subbundle K := k er ∂ and ∇ ν on the quotient bundle ν := co ker ∂ . These induced A -representations play an imp ortant r ole in the cla s siﬁcation of r egular V B –alge br oids in § 6. Even if ∂ is not of constant r ank, it is po ssible to noncanonic a lly extend the (pos sibly singular) represe ntations on K and ν to C a nd E , at the co s t of in tro ducing cur v ature. W e discuss this in the following s ection. 4.4. Si de and core A -connections. There does not exist in general a section of the shor t exact sequence (4.1) in the catego r y of Lie algebroids. Nonetheless , sections in the categ ory of v ector bundles do e x ist; in § 2 .4, they w ere ca lled hori zontal lifts . Let us choose a hor iz ontal lift h : A → b A . F or X ∈ Γ( A ), we denote its image in Γ( b A ) by b X = h ( X ). W e may use h to pull bac k the r epresentations ψ c and ψ s to A -connections ∇ c and ∇ s , resp ectively , so that, ∇ c X := ψ c b X , ∇ s X := ψ s b X . (4.7) R emark 4.5 . The connections ∇ s on E and ∇ c on C depend on the c hoice o f horizontal lift. How ev er, they are extensions of the canonica l ﬂat connections ∇ K on the subbundle K of C and ∇ ν on the q uotient bundle ν of E , which w ere in tro duced ab ove. The induced side and co re connections (4.7) will genera lly ha ve nonzer o curv atur e, re- sulting fro m the failure of h to resp ect Lie brack ets. T o this end, we deﬁne Ω ∈ ∧ 2 Γ( A ∗ ) ⊗ Hom( E , C ) as follo ws: Ω X,Y := \ [ X , Y ] − [ b X , b Y ] for X, Y ∈ Γ( A ). W e will la ter req uir e the following iden tit y: V B -ALGEBR OIDS AND REPRESENT A TION THEOR Y OF LIE ALGEBROIDS 13 Lemma 4.6. F or al l X, Y , Z ∈ Γ( A ) , Ω [ X,Y ] ,Z + [Ω X,Y , b Z ] + { cycl. } = 0 . Pr o of. F rom the deﬁnition o f Ω, we have that \ [[ X , Y ] , Z ] = Ω [ X,Y ] , Z + [Ω X,Y , b Z ] + [[ b X , b Y ] , b Z ] . The result follows from the Ja cobi iden tit y .  A dir ect computation reveals tha t the curv atures F c and F s of ∇ c and ∇ s , resp ectively , satisfy the follo wing e q uations for X, Y ∈ Γ( A ): F c X,Y = θ c Ω X,Y = Ω X,Y ◦ ∂ , (4.8) F s X,Y = θ s Ω X,Y = ∂ ◦ Ω X,Y . (4.9) Additionally , the following prop erties are immediate co nsequences of P rop osition 4.4: ∂ ◦ ∇ c X = ∇ s X ◦ ∂ , (4.10) φ ◦ ∇ s X − ∇ c X ◦ φ = [ φ, b X ] , (4.11) for X ∈ Γ( A ) a nd φ ∈ Hom( E , C ). 4.5. The A -Sup erconnection. So far, giv en a V B –algebr oid equipp ed with a horizontal lift, w e ha ve obtained the following data: • a bundle map ∂ : C → E , • (in ge ne r al, nonﬂat) A -co nnections ∇ c and ∇ s on C and E , resp ectively , a nd • a Hom ( E , C )-v a lued A -2-for m Ω. In this section, we will show tha t the ab ove da ta may b e combined to for m a ﬂat A - sup e rconnection. Let us ﬁrst r ecall the deﬁnitions. Let A → M b e a Lie a lgebroid, let Ω( A ) denote the alg ebra o f Lie algebro id forms ∧ Γ( A ∗ ), and let E b e a Z -graded vector bundle over M . The alge bra Ω( A ) a nd the s pace Γ( E ) ar e b oth naturally Z -g raded. W e co nsider the space of E -v alued A -for ms Ω ( A ) ⊗ C ∞ ( M ) Γ( E ) to be Z -gr aded with r esp ect to the tota l g rading. Deﬁnition 4. 7. An A -su p er c onne ction o n E is an o dd op erator D on Ω( A ) ⊗ Γ( E ) such that (4.12) D ( ω s ) = ( d A ω ) s + ( − 1 ) p ω ∧ D ( s ) for all ω ∈ Ω( A ) and s ∈ Γ( E ), where p is the de g ree o f ω . W e say that D is ﬂat if the curvatur e D 2 is zero. R emark 4 .8 . When the graded bundle E is concen trated in degree 0, D eﬁnition 4.7 agrees with the notion of an A -connection in the sense o f F er nandes [6]. On the other hand, when A = T M , the ab ov e notion of an A -superco nnection reduces to that of a sup erconnec tio n in the sens e o f Q uillen [18]. R emark 4.9 . The sup er c onnections of primary in terest in this pap er are of degree 1. F or this reason, in the remainder of this pap er, by “sup erco nnection” we will mean “degr ee 1 sup e rconnection” unless otherwise stated. Let us now return to the s itua tion of a V B – a lgebroid equipp ed with a ho rizontal lift A → b A . Let D c be the degree 1 op erator on Ω( A ) ⊗ Γ( C ) asso ciated to the core connection ∇ c . Similar ly , let D s be the op erator on Ω( A ) ⊗ Γ( E ) as so ciated to ∇ s . W e may extend bo th D c and D s to Ω( A ) ⊗ Γ( C ⊕ E ) b y setting D c ( ω ε ) = D s ( ω α ) = 0 for all ω ∈ Ω( A ), ε ∈ Γ( E ), and α ∈ Γ( C ). 14 ALFONSO GRACIA-SAZ AND RAJAN AMIT MEHT A W e may also view ∂ and Ω as op er ators on Ω( A ) ⊗ Γ( C ⊕ E ), where for ω ∈ Ω p ( A ), α ∈ Γ( C ), and ε ∈ Γ( E ), ∂ ( ωα ) = ( − 1 ) p ω · ∂ ( α ) , ∂ ( ωε ) = 0 , Ω( ω α ) = 0 , Ω( ω ε ) = ( − 1) p ω ∧ Ω( ε ) . Although ∂ and Ω are of degree 0 and 2, resp ectively , as o p e rators on Ω( A ) ⊗ Γ( C ⊕ E ), they may b oth be view ed as degr ee 1 op era tors o n Ω( A ) ⊗ Γ( C [1] ⊕ E ) , where the [1] denotes that se c tions of C ar e consider ed to b e of degre e − 1. Thus D := ∂ + D c + D s + Ω is a degree 1 op erator on Ω( A ) ⊗ Γ( C [1 ] ⊕ E ). Clea rly , D satisﬁe s (4 .12), so D is a degr ee 1 A -sup erconnection on C [1] ⊕ E . Theorem 4.10. The sup erc onne ction D is ﬂat. Pr o of. Let F := D 2 be the curv a tur e of D . Since E nd( C [1] ⊕ E ) is concentrated in degrees − 1, 0, and 1, we may deco mpo se F , which is an End( C [1] ⊕ E )-v a lued A -for m o f total degree 2, as F = F − 1 + F 0 + F 1 , where F i ∈ Ω 2 − i ( A ) ⊗ End i ( C [1] ⊕ E ). Sp e ciﬁcally , we hav e F − 1 = D c ◦ Ω + Ω ◦ D s , F 0 = F c + F s + ∂ ◦ Ω + Ω ◦ ∂ , F 1 = ∂ ◦ D c + D s ◦ ∂ . It is immedia te fro m (4 .8) and (4.9) that F 0 = 0. Simila r ly , F 1 = 0 by (4.1 0). T o s ee that F − 1 = 0 , w e co mpute the follo wing fo r X , Y , Z ∈ Γ ( A ): ι Z ι Y ι X D c ◦ Ω = ∇ c X Ω Y ,Z − Ω [ X,Y ] ,Z + { cycl. } = ∇ c X Ω Y ,Z + [Ω X,Y , b Z ] + { cycl. } = Ω X,Y ∇ s Z + { cycl. } = − ι Z ι Y ι X Ω ◦ D s . Lemma 4.6 w as used in the second line, and (4.11) w as used in the third line. W e conclude that D 2 = 0 , so D is ﬂa t.  4.6. The sup erc onnection in the diﬀeren tial viewp oint. As we saw in § 2.4, a choice of a horizontal lift A → b A is equiv alent to a c hoice of a decomposition D ∼ → A ⊕ E ⊕ C . Given such a c hoice, the space of algebroid cochains may b e deco mp os ed a s (4.13) ∧ Γ( D ∗ E , E ) ∼ → ∧ Γ( A ∗ ) ⊗ C ∞ ( E ) ⊗ ∧ Γ( C ∗ ) . W e restrict our a tten tion to the elemen ts of ∧ Γ( D ∗ E , E ) that are of degree 1 with respect to the “ov er A ” gr ading of § 3.5. Using the decomp osition (4 .13), we may de s crib e the subspace of s uch elements as ∧ Γ( A ∗ ) ⊗  C ∞ ℓ ( E ) ⊕ ∧ 1 Γ( C ∗ )  = Ω( A ) ⊗ (Γ( E ∗ ) ⊕ Γ( C ∗ [ − 1])) . This s ubs pace is inv aria nt under the diﬀeren tial d D , and it is immediate that the restriction of d D to this subspace is a ﬂat A -sup erco nnection on E ∗ ⊕ C ∗ [ − 1]. It ma y b e seen that d D is dua l to the super connection D of § 4.5, in the sense that, for all ω ∈ Ω( A ) ⊗ (Γ( C [1]) ⊕ Γ( E )) and η ∈ Ω( A ) ⊗ (Γ( E ∗ ) ⊕ Γ( C ∗ [ − 1])), hD ω , η i = d A h ω , η i − ( − 1) | ω | h ω , d D η i . F rom this p er s pe c tive, w e s ee that the ﬂatness of D is equiv alen t to the fa c t that d 2 D = 0 . The following theo rem, which ties toge ther the main results of § 4, is an immediate con- sequence of the ab ove discussion. Theorem 4.11. V B -ALGEBR OIDS AND REPRESENT A TION THEOR Y OF LIE ALGEBROIDS 15 (1) Ther e is a one-to-one c orr esp ondenc e b etwe en V B –algebr oid str u ctur es on t he de- c omp ose d DVB A ⊕ E ⊕ C and ﬂat A -sup er c onne ctions on C [1] ⊕ E . (2) L et D b e a DVB such as (3.1) , with side bund les A and E , and with c or e bund le C , wher e A is a Lie algebr oid. After cho osing a horizontal lift Γ ( A ) → Γ( b A ) = Γ ℓ ( D , E ) (or, e quivalently, a de c omp osition D ∼ → A ⊕ E ⊕ C ), ther e is a one-to-one c orr esp ondenc e b etwe en V B –algebr oid structure s on D and ﬂat A -sup er c onne ctions on C [1] ⊕ E . (3) A ﬂ at A -sup er c onne ction on C [1] ⊕ E is e quivalent to an A –c onne ction ∇ c on C , an A –c onne ction ∇ s on E , an op er ator ∂ : C → E , and an op er ator Ω ∈ ∧ 2 Γ( A ∗ ) ⊗ Hom( E , C ) , sa tisfying ∂ ◦ ∇ c X = ∇ s X ◦ ∂ F c X,Y = Ω X,Y ◦ ∂ F s X,Y = ∂ ◦ Ω X,Y D c Ω + Ω D s = 0 (4.14) for al l X , Y ∈ Γ( A ) . Her e, F c and F s ar e the cu rvatur es of ∇ c and ∇ s ; wher e as D c and D s ar e the op er ators on Ω( A ) ⊗ Γ ( C [1] ⊕ E ) asso ciate d to ∇ c and ∇ s . In § 4.7 we explain how the ﬂat A –superco nnection depends on the choice of horizontal lift. Example 4.12 . A V B – algebro id is sa id to b e vac ant if the core is trivial. In this case, there is a unique dec omp o sition D = A ⊕ E , so by Theorem 4.11 there is a one-to-one corres p o ndenc e betw een v aca n t V B –algebr o ids and Lie algebroid representations. Example 4.13 . In the case where M is a p oint, so that A is a Lie alg ebra, it is p er haps surprising that the s itua tion do es not s implify muc h; a fter choos ing a decomp os ition, we still obtain a ﬂat A -superconnectio n on C [1] ⊕ E , where C and E are now vector s paces. In particular, there exis t exa mples that do not co rresp ond to ordinar y Lie algebra repre - sentations. This situation is in contrast to that of r epresentations up to homotopy (in the sense of Evens, Lu, W einstein [5]), which reduce to ordinary representations when A is a Lie algebra. 4.7. Dep endence o f D on the hori zon tal l i ft. As we saw in Theore m 4.1 1, a V B – algebroid str ucture on a D VB (3.1) is, a fter c hoos ing a hor izontal lift, eq uiv ale n t to a ﬂat A -sup erconnection on C [1] ⊕ E . It is then r e asonable to wonder how ﬂat A -sup erconnectio ns behave under a change of horizontal lift. In this section, we will obtain a simple de- scription that may be interpreted as a na tural notion of equiv alence b etw een tw o ﬂat A - sup e rconnections. The se t of horizontal lifts is an aﬃne spa ce mo delle d on the v ector spac e Γ( A ∗ ⊗ E ∗ ⊗ C ). More sp eciﬁca lly , consider tw o hor izontal lifts h, ˚ h : Γ( A ) → Γ( b A ). F or X ∈ Γ( A ) we denote b X := h ( X ) a nd ˚ b X := ˚ h ( X ). Let σ X ∈ Hom( E , C ) b e deﬁned as (4.15) σ X := ˚ b X − b X . Equation (4.15) deﬁnes a unique σ ∈ Γ( A ∗ ⊗ E ∗ ⊗ C ) = Ω 1 ( A ) ⊗ Hom( E , C ). W e may extend σ to a n o p er ator of to ta l deg ree 0 o n Ω( A ) ⊗ Γ( E ⊕ C [1]), where for ω ∈ Ω p ( A ), α ∈ Γ( C ), and ε ∈ Γ( E ), σ ( ω α ) = 0 , σ ( ω ε ) = ( − 1) p ω ∧ σ ( ε ) . Theorem 4.14 . L et X → b X and X → ˚ b X b e two horizontal lifts r elate d by σ ∈ Γ( A ∗ ⊗ E ∗ ⊗ C ) via (4.15) . L et D and ˚ D b e the c orr esp onding su p er c onne ctions. Then (4.16) ˚ D = D + [ σ, D ] + 1 2 [ σ , [ σ, D ]] . 16 ALFONSO GRACIA-SAZ AND RAJAN AMIT MEHT A In addition, [ σ , [ σ, [ σ, D ] ]] = 0 . If we denote ad( P 1 ) P 2 := [ P 1 , P 2 ] for op er ators P 1 and P 2 on Ω( A ) ⊗ Γ( E ⊕ C [1]) , then (4.16) c an b e r ewritten as ˚ D = ∞ X n =0 1 n ! (ad( σ )) n D = exp(ad( σ )) D = u ◦ D ◦ u − 1 . In the last e quation, u is the automorphism in Ω( A ) ⊗ Γ( E ⊕ C [1]) deﬁne d by u = 1 + σ . Pr o of. Let us write ea ch sup erconnection as sum of co nnections and oper ators, a s in § 4.5: D = D c + D s + ∂ + Ω , ˚ D = ˚ D c + ˚ D s + ˚ ∂ + ˚ Ω . A direct calculation from (4.15) gives us ι X ˚ D c = ∇ c ˚ b X = ∇ c b X + ∇ c σ X = ι X D c + σ X ◦ ∂ , (4.17) ι X ˚ D s = ∇ s ˚ b X = ∇ s b X + ∇ s σ X = ι X D s + ∂ ◦ σ X , (4.18) ˚ ∂ = ∂ , (4.19) ˚ Ω X,Y = ˚ \ [ X , Y ] − [ ˚ b X , ˚ b Y ] = \ [ X , Y ] + σ [ X,Y ] − [ b X + σ X , b Y + σ Y ] . (4.20) According to (4.11), w e have [ σ X , b Y ] = σ X ◦ ∇ s Y − ∇ c Y ◦ σ X , [ b X , σ Y ] = ∇ c X ◦ σ Y − σ Y ◦ ∇ s X , and according to (4.2), we have [ σ X , σ Y ] = σ X ◦ ∂ ◦ σ Y − σ Y ◦ ∂ ◦ σ X , so that ˚ Ω X,Y = Ω X,Y + σ [ X,Y ] − σ X ∇ s Y + σ Y ∇ s X − ∇ c X σ Y + ∇ c Y σ X − σ X ∂ σ Y − σ Y ∂ σ X . (4.21) Then w e can r ewrite (4.17), (4.18), (4.19), and (4.2 1) as ˚ D c = D c + σ ∂ , ˚ D s = D s − ∂ σ, ˚ ∂ = ∂ , ˚ Ω = Ω − D c σ + σ D s − σ ∂ σ. (4.22) On the o ther hand, we can write the left-hand side of (4 .16) in terms of D c , D s , ∂ and Ω a s follows: D = D s + D c + ∂ + Ω , [ σ , D ] = σD s + σ ∂ − D c σ − ∂ σ, [ σ , [ σ , D ]] = − 2 σ ∂ σ , [ σ , [ σ , [ σ , D ]]] = 0 . (4.23) Finally , comparing (4.22) and (4.23) completes the proo f.  V B -ALGEBR OIDS AND REPRESENT A TION THEOR Y OF LIE ALGEBROIDS 17 5. Characteristic classes Given a Lie alg ebroid A → M equipped with a repr esentation (i.e. a ﬂat A -connection) on a vector bundle E → M , Crainic [3] has constr ucted Chern-Simons- type secondar y characteristic class es in H 2 k − 1 ( A ). In this section we extend his co nstruction to ﬂat A - sup e rconnections on graded vector bundles. I n the cas e of ﬂat A -sup erconnectio ns aris ing from V B –a lg ebroids, w e will see that the a sso ciated characteristic clas s es do not dep end on the choice of horizontal lift; in other words, this constr uc tio n gives us V B –algebr oid inv ariants. Let A → M b e a Lie algebr oid, and let E = L E i → M b e a Z -graded vector bundle 3 equipp e d with a ﬂat A -s uper connection D . In other words, D is a degree 1 o per ator on Ω( A ) ⊗ Γ( E ) satisfying (4 .12) and such that D 2 = 0. Before we can deﬁne characteristic classes asso ciated to D , we will require a few piec es of bac kground. First, there is a na tural pairing (5.1) Ω( A ) ⊗ Γ( E ) × Ω ( A ) ⊗ Γ( E ∗ ) → Ω( A ) given by h ω a, η ς i = ( − 1) | a || η | ω ∧ η h a, ς i for a ll ω , η ∈ Ω( A ), a ∈ Γ( E ), and ς ∈ Γ( E ∗ ). The adjoint connection D † is an A -sup erc onnection on E ∗ deﬁned b y the equation (5.2) d A h a, ς i = hD a, ς i + ( − 1) | a | h a, D † ς i . It is immedia te fro m the deﬁnition that D 2 = 0 implies that ( D † ) 2 = 0 . Second, a choice o f metric on E i for all i gives an isomo rphism g : E ∼ → E ∗ , which preser ves parity but fails to be deg ree-pres e r ving; rather, it identiﬁes the degree i co mpo ne nt of E with the degree − i comp onent o f E ∗ . Nonetheles s, suc h a choice allows us to transfer D † to a ﬂat A -sup erconnection g D o n E . The supe r connection g D of co urse depends on g , and since g is not degree-pre s erving, g D is not homogeneous of degr ee 1 . T o emphasize this fact, we will refer to g D a s a “no nhomogeneous s uper connection”. Third, let I b e the unit interv a l, and consider the pro duct Lie algebroid A × T I → M × I . If the ca nonical co ordinates on T [1] I ar e { t , ˙ t } , then any Lie alg ebroid p -form B ∈ Ω p ( A × T I ) may be uniquely ex pressed as B p ( t )+ ˙ tB p − 1 ( t ), where B p and B p − 1 are t -dep endent elements of Ω p ( A ) and Ω p − 1 ( A ), resp ectively . F urthermore, in terms of the co or dinates o n T [1] I , the Lie algebroid diﬀer ent ial is d A × T I = d A + ˙ t ∂ ∂ t . T ogether, D and g D determine an A × T I -(nonhomogeneous) super connection T D , g D on p ∗ E , where p is the pr o jection map from M × I to M , suc h that (5.3) T D , g D ( a ) = t D ( a ) + (1 − t )( g D ( a )) , where a ∈ Γ( E ) is viewed as a t -independent elemen t of Γ( p ∗ E ). Equation (5.3 ), tog ether with the Leibniz rule (4.12), completely determines T D , g D as an op er ator on Ω( A × T I ) ⊗ Γ( E ). F or positive integers k , the k -th Cher n-Simons forms are then (5.4) cs g k ( D ) := Z d t d ˙ t str  ( T D , g D ) 2 k  . The in tegral in (5.4 ) is a Bere zin integral. F or the purp ose of clar ity , we will sp e ll o ut what (5 .4) means in mor e detail. Since ( T D , g D ) 2 k is an even oper ator on Ω( A × T I ) ⊗ Γ( E ), its sup er trace is an even (in general nonhomogeneo us) element o f Ω( A × T I ). If w e ex pr ess str  ( T D , g D ) 2 k  in the form B even ( t ) + ˙ tB od d ( t ), then cs g k ( D ) = R 1 0 B od d ( t ) dt . Therefor e cs g k ( D ) ∈ Ω od d ( A ). R emark 5.1 . The integral in (5.4) may b e explicitly c o mputed. T he r esult is that, up to a constant, cs g k ( D ) is given by str  D ( g DD ) k − 1 − ( g DD ) k − 1 ( g D )  . 3 W e assume that the tot al r ank of E is ﬁnite, so as to ensure that the supertrace in (5.4) i s we ll-deﬁned. 18 ALFONSO GRACIA-SAZ AND RAJAN AMIT MEHT A Since the pro ofs of the following s tatement s ar e similar to those of Crainic and F erna ndes [4], w e postp one them to Appendix B. Prop ositio n 5. 2. F or al l k , cs g k ( D ) is close d. Lemma 5.3. If k is even, t hen cs g k ( D ) = 0 . Prop ositio n 5. 4. The c oho molo gy class of cs g k ( D ) is an element of H 2 k − 1 ( A ) . In other wor ds, the c omp onents of [cs g k ( D )] in al l de gr e es o ther t han 2 k − 1 vanish. Prop ositio n 5. 5. The c ohomolo gy class o f cs g k ( D ) do es not dep end on g . In s ummary , we have well-deﬁned Chern- Simons clas ses [cs k ( D )] ∈ H 2 k − 1 ( A ) asso ciated to an y ﬂat A -sup erco nnection D . Let us no w return to V B – a lgebroids. W e hav e s een in § 4.5 that, giv en a V B – algebro id (3.1), a choice of a horizontal lift A → b A lea ds to a ﬂat A - s uper connection on C [1] ⊕ E . Therefore, the above pro cedure applies, and w e may obtain Cher n-Simons classes. Theorem 5.6. The Chern-Simons classes [cs k ( D )] do not dep end on the choic e of horizontal lift. Ther efor e the Chern-Simons classes arisi ng fr om ﬂat A -sup er c onne ctions on C [1] ⊕ E ar e V B –algebr oid inva riants. 6. Cl assifica tion of regular V B –algebroids Let D b e a DVB such as (3.1), with s ide bundles A and E , and with cor e bundle C , where A is a Lie alg e broid. In this s e c tion we classify the V B –a lgebroid structures on D that a re regular in a sens e tha t will be deﬁned b elow. As we saw in Theo rem 4.11, given a ho rizontal lift A → b A , a V B –algebr o id structure on D is equiv alent to c ho osing A –connections ∇ c and ∇ s on C and E , r esp ectively , an op erator ∂ : C → E , a nd a n op erato r Ω ∈ Ω 2 ( A ) ⊗ Γ(Hom( E , C )), satisfying (4.14). As we saw in § 4.7, only ∂ is intrinsically deﬁned, whereas ∇ c , ∇ s , a nd Ω dep end on the choice of horizontal lift acco r ding to (4.22). As a consequence, the set of is omorphism cla sses of V B –a lgebroid structures on D is in one-to -one corres po ndence with the set of tuples ( ∇ s , ∇ c , ∂ , Ω) satisfying (4.14), mo dulo the action of Γ( A ∗ ⊗ E ∗ ⊗ C ) describ ed by (4.22). Nevertheless, as was explained in § 4.4, when ∂ is o f constant rank, ∇ c and ∇ s induce the following t w o A –connectio ns that depend only on the total V B –algebr oid structure, and not on the choice o f horizontal lift: • a ﬂat A –connection ∇ K on the s ubbundle K := ker ∂ ⊆ C , and • a ﬂat A –connection ∇ ν on the q uotient bundle ν := cok er ∂ = E / im ∂ . The elements ( A, E , C , ∂ , ∇ K , ∇ ν ) are all inv aria nt under isomorphis ms of V B –algebro ids. W e will see that any suc h 6-tuple can alwa ys b e “ex tended” to a V B –algebro id, and w e will classify the ex tens io ns up to isomorphism. Deﬁnition 6.1. A V B – algebro id is ca lled r e gular when the core-a nchor ∂ : C → E has constant rank. Note that in a regular V B –algebroid, the L ie alg e broids D → E and A → M do not have to b e regula r (i.e. the anc hor maps do no t ha ve to hav e constant ra nk). F or example, a V B – algebroid where ∂ is an isomorphism is clear ly reg ular; how ev er, in such a V B – algebro id the anchors ρ D and ρ A need not b e of constant rank. This fact will b e more clear ly illustra ted in § 6 .1. There are tw o sp ecial t yp es of reg ular V B –a lgebroids. W e will describe them now, and then we will show that any regula r V B –a lgebroid can be uniquely decomposed as a direct sum o f these tw o sp ecial t yp es of V B – a lgebroids. T his will allow us to g ive a complete description of a ll regular V B – algebro ids up to isomorphism. V B -ALGEBR OIDS AND REPRESENT A TION THEOR Y OF LIE ALGEBROIDS 19 6.1. V B –algebroids of t ype 1 . Deﬁnition 6.2. W e say that a V B –algebr oid is of typ e 1 when the core- anchor ∂ is an isomorphism of v ector bundles. There is one ca nonical example (which turns out to b e the only o ne). Let A → M b e a Lie algebroid and E → M be a vector bundle. Cons ide r the pullback of T E b y the anchor ρ A of A in the following dia gram: (6.1) ρ ∗ A ( T E ) / /   T E   A ρ A / / T M Then there is a natural pullbac k L ie alg e broid structure (see [8]) on ρ ∗ A ( T E ) → E such that (6.2) ρ ∗ A ( T E ) / /   E   A / / M is a V B –algebroid of type 1. The core of (6.2) may b e canonically iden tiﬁed with E , and the core-anchor map 4 is − id E . Let us try to construct the most g e ne r al V B –alge br oid of t ype 1. Let us ﬁx the sides A and E . W e ma y assume that C = E and ∂ = − 1. Now we need to deﬁne ∇ s , ∇ c , and Ω satisfying (4.14). In this case, the equations become: • ∇ s = ∇ c , • a nd -Ω is the cur v a ture of ∇ s . Hence, putting a V B – algebro id structure on A ⊕ E ⊕ E is the same thing as deﬁning an A –connection on E . If w e wan t to clas sify them up to isomorphism we nee d to include the action of Γ( A ∗ ⊗ E ∗ ⊗ E ) = Ω 1 ( A ) ⊗ End( E ) by (4.22). Given any tw o A –c o nnections ∇ and ˚ ∇ on E there exist a unique σ ∈ Ω 1 ( A ) ⊗ End( E ) such that ˚ ∇ = ∇ + σ . In other words: Prop ositio n 6.3. Given side bund les A and E , t her e exists a unique V B –algebr oid of typ e 1 up to isomorphi sm, namely ρ ∗ A ( T E ) . 6.2. V B –algebroids of t ype 0 . Deﬁnition 6.4. W e say that a V B –a lgebroid is of typ e 0 when the co r e anc hor is zer o. Fix the sides A and E , and the core C . Let us try to cons tr uct the most gene r al V B – algebroid with ∂ = 0. W e need to deﬁne ∇ s , ∇ c , and Ω sa tisfying (4.14). In this cas e, the equations become: • ∇ s is a ﬂat A –co nnection on E , • ∇ c is a ﬂat A –co nnection on C , • D c ◦ Ω + Ω ◦ D s = 0 (6.3) T o classify these V B –algebr oids up to isomorphis m we need to include the action of Γ( A ∗ ⊗ E ∗ ⊗ C ) by (4.22). In this case , if σ ∈ Γ( A ∗ ⊗ E ∗ ⊗ C ) acts on ( ∇ s , ∇ c , Ω), the connnections ∇ s and ∇ c remain in v a riant, wherea s Ω bec o mes: (6.4) ˚ Ω = Ω + σ D s − D c σ Equations (6.3) and (6.4) can be interpreted in terms of cohomo lo gy . Namely , the ﬂat A -connections on C and E induce a ﬂat A -co nnec tio n on Hom( E , C ), whos e cov ariant deriv ativ e D is giv en by the equation (6.5) D α := αD s + ( − 1 ) p D c α 4 Note that a minus sign alr eady app ears in the deﬁnition of the core-anc hor (Deﬁnition 4. 2). 20 ALFONSO GRACIA-SAZ AND RAJAN AMIT MEHT A for α ∈ Ω p ( A ) ⊗ Γ(Hom( E , C )). Then (6.3) says that D Ω = 0 , whereas (6.4) says that ˚ Ω = Ω + D σ . Hence, the coho mology class [Ω] ∈ H 2 ( A ; Ho m( E , C )) is well-deﬁned and inv ariant up to iso morphism of V B –algebr oids. This gives us the following re s ult: Prop ositio n 6. 5. T yp e 0 V B –algebr oids with s ides A and E , and c or e C ar e classiﬁe d up to isomorphism by triples ( ∇ s , ∇ c , [Ω]) , wher e • ∇ s is a ﬂat A –c onne ction on E , • ∇ c is a ﬂat A –c onne ction on C , • [Ω ] is a c ohomolo gy class in H 2 ( A ; Ho m( E , C )) . 6.3. The general case. Given tw o V B –algebr oids D 1 / /   E 1   A / / M D 2 / /   E 2   A / / M ov er the same Lie algebroid A , we ca n obtain the dir e ct sum V B –a lgebroid D 1 ⊕ A D 2 / /   E 1 ⊕ M E 2   A / / M . Note that the co r e of D 1 ⊕ A D 2 is the direct sum o f the cores o f D 1 and D 2 . Theorem 6.6. Given a r e gular V B –algebr oid D , ther e exist un ique (u p to isomorphism) V B –algebr oids D 0 of t yp e 0, and D 1 of typ e 1, such that D is isomorp hic to D 0 ⊕ A D 1 . Pr o of. • Existence. Let D b e a r e gular V B –algebr o id as in (3.1). Then the core-anchor ∂ : C → E induces the following vector bundles: K := k er ∂ ⊆ C , F := im ∂ ⊆ E , and ν := co ker ∂ = E /F . They ﬁt in to the s hort exact sequences: K / / C / / F , F / / E / / ν . (6.6) Let us cho ose splittings of the sequences (6.6), which would g ive isomorphisms C ≈ K ⊕ F E ≈ ν ⊕ F (6.7) Next we mak e a c hoice o f ho rizontal lift X ∈ Γ( A ) → b X ∈ Γ( b A ). As we saw in Theorem 4.11, the V B – algebro id s tructure in the DVB D is determined b y the tuple ( ∇ s , ∇ c , ∂ , Ω). W e wr ite a “blo ck-matrix dec o mpo sition” o f eac h one of these opera to rs with resp ect to the direct sums in (6.7): (6.8) ∇ s =  ∇ ν 0 Λ ∇ F  , ∇ c =  ∇ K Γ 0 ∇ F  , ∂ =  0 0 0 − 1  , Ω =  α ⋆ ⋆ ⋆  . In (6.8), ⋆ means a n unsp eciﬁed op erator . The zeros in ∇ s and ∇ c are a consequence of the ﬁrst equation in (4.1 4). The b ottom– right blo cks of ∇ s and ∇ c (whic h we denote ∇ F ) are the same, also becaus e o f the ﬁrs t equa tion in (4.14). The comp onents α , Λ, a nd Γ are describ ed as follo ws: α ∈ Λ 2 Γ( A ) ⊗ Γ( ν ) → Γ( K ) Λ ∈ Γ( A ) ⊗ Γ( ν ) → Γ( F ) Γ ∈ Γ( A ) ⊗ Γ( F ) → Γ( K ) V B -ALGEBR OIDS AND REPRESENT A TION THEOR Y OF LIE ALGEBROIDS 21 The o pe r ators α , Λ, and Γ dep end on the choice of splittings of (6.6), as well as on the choice of horizontal lift. The A -connection ∇ F depe nds o n the c hoice o f horizontal lift. If the op era tors in (6.8) w ere blo ck-diagonal, then we could break them apar t to form t wo separate V B -algebr oid structures, one with s ide bundle K and cor e ν , and the other with F a s b oth the side and core. Luckily , it is p ossible to mak e all the o pe r ators in (6.8) blo ck-diagonal via a c hange of horizontal lift, as follows. As we expla ine d in § 4.7, a change of horizontal lift co rresp onds to an element σ ∈ Γ( A ∗ ⊗ E ∗ ⊗ C ). If σ is written in blo ck matrix form as (6.9) σ =  σ 11 σ 12 σ 21 σ 22  , then, according to (4.17) and (4.18), the side and co re connections for the new horizontal lift will be ˚ ∇ s = ∇ s + ∂ σ =  ∇ ν 0 Λ ∇ F  +  0 0 − σ 21 − σ 22  , ˚ ∇ c = ∇ c + σ ∂ =  ∇ K Γ 0 ∇ F  +  0 − σ 12 0 − σ 22  . Therefore, if we c ho ose (6.10) σ =  0 Γ Λ 0  , it will make the new connections ˚ ∇ s and ˚ ∇ c blo ck–diagonal. Consequently , the second and third eq uations in (4.14) imply that ˚ Ω will also be blo ck–diagonal. In particula r, ˚ Ω will necessarily take the form (6.11) ˚ Ω =  ω 0 0 − R F  , where R F is the curv ature of ∇ F . Using (6.10) in (4.2 1), we can relate the upp er-left block ω of ˚ Ω to the upper- left block α o f Ω in the following wa y: (6.12) ω X,Y = α X,Y − Γ X ◦ Λ Y + Γ Y ◦ Λ X . Since ˚ ∇ s , ˚ ∇ c , and ˚ Ω are blo ck dia gonal, their diagonal blo cks give us the data fo r tw o V B -algebro ids: a t yp e 0 V B -a lgebroid D 0 , with side bundle ν and cor e bundle K , and a t yp e 1 V B - a lgebroid D 1 , with F as b oth side and core. • Uniqueness. Based o n the cla ssiﬁcation of V B -alg ebroids of type 0 ( § 6.2) and type 1 ( § 6.1), we may characterize the V B -alg ebroids D 0 and D 1 up to is o morphism as follows: • D 1 is determined up to is omorphism b y its side bundle F , • D 0 is determined up to isomorphism b y its side bundle ν and core bundle K , the ﬂat A –connections ∇ ν and ∇ K , and the cohomology class [ ω ] ∈ H 2 ( A ; Ho m( ν, K )), given by (6.12). W e ha ve alr eady seen that the bundles F , ν , K , and the ﬂat A –connections ∇ ν and ∇ K are canonical. T o c omplete the pr o of we need to show that the cohomology clas s of ω does not depend on the choice of splittings of (6.6), nor on the choice o f horizontal lift. First, the cohomo logy class do es no t depend on the choice of hor izontal lift, thanks to our a nalysis o f t ype 0 and type 1 V B –a lgebroids. If w e ﬁx the c hoice o f complemen ts but change to a diﬀerent horizontal lift that s till makes the op erators in (6.11) blo ck-diagona l, this corres p o nds to c ho osing arbitrary blocks in the main diagonal of (6 .9). No tice that the cohomolog y class o f ω do es no t change, and in fact all the repr esentativ es of the cohomology class of ω may b e obtained in this wa y . 22 ALFONSO GRACIA-SAZ AND RAJAN AMIT MEHT A Second, supp ose that w e ha v e chosen splittings o f the sequences (6.6 ) and a hor izontal lift such that the op erator s are a lready blo ck–diagonal like in (6.11). Then a change of splitting of the se c ond sequence in (6.6) may b e expres sed in blo ck form by a matrix  1 0 g 1  for some linear map g : ν → F . Under the c hange of splitting, ∇ c and ∂ will hav e the same matrix forms, whereas the new blo ck ma tr ix forms for ∇ s and Ω will b e ∇ s =  1 0 − g 1   ∇ ν 0 0 ∇ F   1 0 g 1  =  ∇ ν 0 − g ∇ ν + ∇ F g 1  and Ω will b e: Ω =  ω 0 0 − R F   1 0 g 1  =  ω 0 ω − R F g − R F  It is clear tha t ω , as deﬁned b y (6.1 2), stays the same. A similar calculation shows that ω does not depend on the choice of splitting of the ﬁrst s e quence in (6 .6). Notice that as a consequence of the ab ove a nalysis, if w e were to start with an ar bitrary choice of splittings o f (6.6) and an arbitrar y choice of horizontal lift, then ω may c hange under a change of splitting, but only by an exact term. This can alternatively be shown by a direct (and leng th y) calculation.  Corollary 6.7 (Classiﬁcatio n of regula r V B –alg ebroids) . A r e gular V B –algebr oi d is de- scrib e d, up to isomorphism, by a u nique tuple ( M , A, E , C , ∂ , ∇ K , ∇ ν , [ ω ]) , wher e • M is a manif old, • A → M is a Lie algebr oid , • E → M and C → M ar e ve ctor bu nd les, • ∂ : C → E is a morphism o f ve ctor bund les, • ∇ K is a ﬂat A –c onne ction on K := k er ∂ , • ∇ ν is a ﬂat A –c onne ction on ν := cok er ∂ , • [ ω ] is a c ohomolo gy class in H 2 ( A ; Ho m( ν, K )) . 7. E xample: T A Let A → M b e a Lie algebro id. If A is a regula r Lie algebroid, i.e. if the a nchor ma p ρ A : A → T M is of co nstant r ank, then the V B –a lgebroid T A in (3.2) is regula r . Then, by Coro llary 6.7, there is an asso ciated cohomology class [ ω ] ∈ H 2 ( A ; Ho m( ν, K )), where K a nd ν ar e the kernel and co kernel of ρ A , resp ectively . Since the cons tr uction of the V B –algebro id T A from A is functoria l, the class [ ω ] is a c haracter is tic class of A . In this section, w e will giv e a g eometric interpretation of [ ω ] in this case. As was noted in E xample 4.1, the fat alg ebroid b A in this case is s imply the ﬁrst jet bundle J 1 A of A . There is a natur al ma p j : Γ( A ) → Γ( J 1 A ), whic h ho wev er is not C ∞ ( M )-linear; instead, it satisﬁes the pr o p e rty j ( f X ) = f j ( X ) + d f · X for f ∈ C ∞ ( M ) and X ∈ Γ( A ). H ere, d f · X ∈ Hom( T M , A ) is viewed as a jet alo ng t he zero section o f A . If we c ho ose a linear connection e ∇ : X ( M ) × Γ( A ) → Γ( A ), w e ma y obtain a ho r izontal lift X ∈ A 7→ b X ∈ J 1 A , wher e b X := j ( X ) − e ∇ X . The resulting side and core connections are described a s follows: ∇ c X Y = [ X, Y ] A + e ∇ ρ A ( Y ) X , (7.1) ∇ s X φ = [ ρ A ( X ) , φ ] + ρ A  e ∇ φ X  , (7.2) for X, Y ∈ Γ( A ) a nd φ ∈ Γ( T M ). V B -ALGEBR OIDS AND REPRESENT A TION THEOR Y OF LIE ALGEBROIDS 23 Additionally , one can der ive the fo llowing expression for Ω ∈ Ω 2 ( A ) ⊗ Γ(Hom( T M , A )): (7.3) Ω X,Y φ = [ e ∇ φ X , Y ] + [ X , e ∇ φ Y ] − e ∇ φ [ X , Y ] − e ∇ ∇ s X φ Y + e ∇ ∇ s Y φ X . 7.1. The case ρ = 0 . It is p er haps instructive to b egin with the case where the anchor map ρ A is trivia l (or in other w ords, where A is simply a bundle of Lie algebras ). Since for the V B –algebr oid T A in (3.2) w e have ∂ = − ρ A , the cas e ρ A = 0 co r resp onds to the ca se where T A is a V B – a lgebroid of type 0 (see § 6.2). The v anis hing of the cohomology c lass [Ω] is equiv ale nt to the existence of a connection e ∇ for whic h Ω, describ ed by (7.3), v anishes. Since ∇ s bec omes trivial when ρ A = 0, w e immediately see that Ω φ measures the failure of e ∇ φ to b e a deriv a tion of the Lie brack et. Therefo re, Ω = 0 precisely when, for a ny φ ∈ X ( M ), parallel transp or t along φ induces Lie alg ebra isomorphisms of the ﬁbres o f A . In fa c t, it can b e shown that if Ω = 0, one ca n us e pa rallel transpo rt to lo ca lly trivia lize A as a Lie alge bra bundle. Conv ersely , g iven a loc al trivialization of A , o ne can deﬁne parallel transp ort in a wa y that resp ects the Lie bra ck ets on the ﬁbres o f A . Th us we have the following result: Prop ositio n 7. 1 . L et A → M b e a Lie algebr oid with ρ A = 0 , and c onside r the typ e 0 V B –algebr oid T A in (3.2) . The c oh omolo gy class [Ω] ∈ H 2 ( A ; Ho m( T M , A )) vanishes if and only if the bu nd le of Lie algebr as A is lo c al ly trivializable as a Lie algebr a bund le. 7.2. The general case. Now we will consider the gener al cas e of a regular Lie alge br oid A → M . L e t K ⊆ A b e the kernel of ρ A , and let F ⊆ T M b e the image of ρ A . The v a nis hing of the cohomology class [ ω ] is equiv alent to the existence of a connection e ∇ and splittings A ∼ = K ⊕ F and T M ∼ = ν ⊕ F such that ω , deﬁned in (6.12), v anishes . First, if we choos e a splitting of the shor t exact sequence of vector bundles K → A → F , then we obtain an F -connec tio n ∇ K on K and a K -v alued 2 -form B ∈ Ω 2 ( F ) ⊗ Γ( K ), deﬁned b y the proper ties (7.4) [ φ, k ] A = ∇ K φ k , [ φ , φ ′ ] A = B ( φ, φ ′ ) + [ φ, φ ′ ] T M for φ, φ ′ ∈ Γ( F ) and k ∈ Γ( K ). Note that this ∇ K is not the sa me as the one in (6.8). Cho ose a n extension of ∇ K to a T M -connectio n e ∇ K on K . Second, c ho ose a splitting of the seq uence F → T M → ν . This induces a ν - c onnection ∇ F on F , wher e ∇ F ψ φ is the co mp onent o f [ ψ , φ ] in F , for ψ ∈ Γ( ν ) a nd φ ∈ Γ( F ). Note that this ∇ F is no t the same as the one in (6.8). Cho ose an extention of ∇ F to a T M - connection e ∇ F on F . Third, we may deﬁne a T M -c onnection e ∇ on A as follows: (7.5) e ∇ ψ X = e ∇ K ψ X K + e ∇ F ψ X F + B ( ψ F , X F ) for ψ ∈ X ( M ) and X ∈ Γ( A ) . Here, X K and X F are the compo nent s of X in K and F , resp ectively , and ψ F is the co mpo nent of ψ in F . W e hav e constructed the connection e ∇ in (7.5) such that it has the following prop erties: • I f Y K = 0, then ∇ c X Y is in F . W e see this by substituting (7.5) into (7.1) . • I f ψ ∈ Γ( ν ), then ∇ s X ψ is also in Γ( ν ). W e see this by substituting (7.5) in to (7.2). In other words, if the co r e and side connections are expr e ssed in blo ck form as in (6.8), then they will both b e blo ck-diagonal, and as a cons e q uence, Ω will also b e block-diagonal. Therefore, to compute ω , we simply need to restric t Ω to Γ( ν ) , and the result lies in K . 24 ALFONSO GRACIA-SAZ AND RAJAN AMIT MEHT A Using (7 .4) and (7.5) in (7.3), we obtain from a long but dir ect computation the follo wing equation for X, Y ∈ Γ( A ) and ψ ∈ Γ( ν ): Ω X,Y ψ =[ e ∇ K ψ X K , Y K ] K + [ X K , e ∇ K ψ Y K ] K − e ∇ K ψ [ X K , Y K ] K + e R K X F ,ψ Y K − e R K Y F ,ψ X K + e ∇ K ψ B ( X F , Y F ) − B ( e ∇ F ψ X F , Y F ) − B ( X F , e ∇ F ψ Y F ) . (7.6) Here, e R K is the curv a ture of e ∇ K . Prop ositio n 7.2. ω X,Y ψ vanishes for al l X , Y ∈ Γ( A ) and ψ ∈ Γ( ν ) if and only if the fol lowing statements ar e tru e: (1) e ∇ K is a deri vation of the br acket on K , (2) e R K φ,ψ vanishes for all φ ∈ Γ( F ) a nd ψ ∈ Γ( ν ) , and (3) B ( e ∇ F ψ φ, φ ′ ) + B ( φ, e ∇ F ψ φ ′ ) − e ∇ K ψ B ( φ, φ ′ ) vanishes for al l φ, φ ′ ∈ Γ( F ) and ψ ∈ Γ( ν ) . Pr o of. First, notice that the res trictions w e ha ve imp osed in the choice of T M –co nnection e ∇ on A in the above construction are equiv alen t to asking that the oper ators in (6.11) are blo ck-diagonal. As was men tioned in the uniqueness part of the pro of of Theorem 6.6, with this restriction to the choices w e still get all the forms in the coho mology class [ ω ]. Hence, [ ω ] = 0 if and o nly if there is a choice of complemen ts and linear connection a s the ones ab ov e for which ω = 0. Second, b y alternatively setting X F , Y F = 0, X K , Y F = 0, and X K , Y K = 0 in (7.6), we obtain the required result.  Given a lea f L of the foliation F , the structure of the r estricted Lie algebroid A | L is completely determined by the data  [ · , · ] K , ∇ K , B  ov er L . Thus, we may interpret the three conditions in P rop osition 7.2 as sa ying that, if ψ resp ects the foliation, then para lle l transp ort along ψ gives isomor phisms of the res trictions of A to the leav es. In other w ords, [ ω ] is the obstruction to lo ca l trivializability of A , in the following se ns e: Theorem 7. 3. Th e c oho molo gy class [ ω ] ∈ H 2 ( A ; Ho m( ν, K )) vanishes if and only if, ar oun d any le af L , ther e lo c al ly exists a tubu lar neighb orho o d e L and an identiﬁc at ion e L ≡ L × U su ch t hat the Lie algebr oid A | e L is isomorphic to the cr oss pr o duct of A | L and the trivial Lie alge br oid over U . Clearly , the v anishing of [ ω ] imp oses a strong reg ularity condition on the Lie algebroid structure of A . In general, we may view [ ω ] as a measure of ho w the Lie a lgebroid structure on A | L depe nds o n the choice of L . Appendix A. Proof of T heorem 3.7 Let us ﬁrst concen trate on the a sp ects of the compatibility conditions that r elate to the anchor map ρ D . F or V B –a lgebroids, the requirement is that ρ D be a bundle morphism as in (3 .3). F or LA –vector bundles, w e r equire that the diagram (3.3), as well as the diagr a m (A.1) D (2) ρ (2) D / / + A   T E (2) T (+)   D ρ D / / T E , commute. Here, D (2) := D × A D and T E (2) := T E × T M T E . It is immediately clear that the V B –alge broid and LA –vector bundle compatibilit y conditions for ρ D are equiv alen t to each other. In what follows, we will assume that they are s atisﬁed. V B -ALGEBR OIDS AND REPRESENT A TION THEOR Y OF LIE ALGEBROIDS 25 W e now turn to the asp ects o f the compatibility co nditions that in v olve the brack ets. F or V B –algebro ids, these are conditions (1 )- (3 ) in Deﬁnition 3.4. F or LA –vector bundles, we require that the Lie algebr oid structure on D → E b e q –pr o jectible (as deﬁned in Remar k 3.1), and tha t the map + A : D × A D → D b e a n alg e broid morphism. W e note that Deﬁnition 3.4 re fer s only to brack ets of linear and core sections. In order to prove the equiv a lence of the V B –a lg ebroid and LA –vector bundle co mpa tibilit y conditions, we will rewrite the la tter in ter ms of line a r and core s ections. First, let us consider the condition tha t the Lie algebroid structure on D → E is to be q -pro jectible. Lemma A.1. The algebr oid structu r e on D → E is q -pr oje ctible if and only if, for al l X , Y ∈ Γ ℓ ( D , E ) and α, β ∈ Γ C ( D , E ) , (1) [ X , Y ] D is q -pr oje ctible, (2) [ X , α ] D is q -pr oje ctible to 0 A , (3) [ α, β ] D is q -pr oje ctible to 0 A . Pr o of. As in Remark 2 .7, let us pick a decomp osition D ∼ → A ⊕ E ⊕ C a nd choose lo ca l co ordinates { x i , a i , e i , c i } on D , where { x i } are co o rdinates on M , and { a i } , { e i } , and { c i } are ﬁbre co ordinates on A , E , and C , r e sp e ctively . Let { A i , C i } be the fra me of sections dual to the ﬁbre co o rdinates { a i , c i } . In Remark 2.7 we describ ed the form of line a r and core sections in these co o rdinates. W e now notice that a section X ∈ Γ( D , E ) is q –pro jectible to a sectio n X 0 = f i ( x ) A i ∈ Γ( A, M ) if and only if it is of the form X = f i ( x ) A i + g i ( x, e ) C i This co ordinate descriptio n shows that the spa ce o f q –pro jectible sections of D → E is exactly (A.2) Γ l ( D , E ) + C ∞ ( E ) ⊗ Γ c ( D , E ) In ter ms of the br ack ets, q -pr o jectibilit y is equiv alen t to the following t w o prop erties: • I f X and Y in Γ( D , E ) ar e q -pro jectible, then [ X, Y ] D is q - pro jectible. • I f α ∈ Γ( D , E ) is q -pro jectible to 0 A and X ∈ Γ( D , E ) is q -pr o jectible, then [ X, α ] D is q -pro jectible to 0 A . F rom (A.2) we can see tha t these t w o proper ties a re sa tis ﬁe d if and only if they ar e s atisﬁed for linear and core se c tio ns. Conditions (1)–(3) in the statement o f this lemma are exactly these t w o prop erties restricted to line a r and co re sections.  Second, we w ant to tra nsform the condition that + A be a Lie algebroid morphism into a condition inv olving only linear and core sections . In order to do so, we need so me deﬁnitions. Let e X b e a sectio n of D (2) ov er E (2) := E × M E . W e say that e X is + -pr oj e ctible to X ∈ Γ( D, E ) if + A ◦ e X = X ◦ +, i.e. if it is a “q– pro jectible” section of the D VB D (2) / / + A   E (2) +   D / / E . If tw o s ections X , X ′ ∈ Γ( D , E ) are b oth q -pro jectible to the s ame X 0 ∈ Γ( A ), then we ma y form the pro duct X × X ′ ∈ Γ( D (2) , E (2) ). In particular, any q -pro jectible section X ∈ Γ( D , E ) induces the lift X (2) := X × X . In addition, giv en a n y sec tio n α ∈ Γ( D , E ) that is q –pro jectible to 0 A , we can deﬁne sections α + := 1 2 ( α × e 0 E + e 0 E × α ) and α − := 1 2 ( α × e 0 E − e 0 E × α ) Let us intro duce the follo wing notation, just for the next lemma: 26 ALFONSO GRACIA-SAZ AND RAJAN AMIT MEHT A • Γ (2) l denotes the set o f lifts of se c tions X ∈ Γ l ( D , E ) to se ctions X (2) ∈ Γ( D (2) , E (2) ), • Γ + c denotes the set of lifts o f sections α ∈ Γ c ( D , E ) to sections α + ∈ Γ( D (2) , E (2) ), • Γ − c denotes the set of lifts o f sections α ∈ Γ c ( D , E ) to sections α − ∈ Γ( D (2) , E (2) ), • C ∞ ( E ) (2) denotes the pullbac k of C ∞ ( E ) to functions on E (2) via + : E (2) → E . Now we ar e rea dy for: Lemma A.2. Supp ose that the Lie a lgebr oi d structur e on D → E is q -pr oje ctible, so ther e is an induc e d Lie algebr oid structu re on D (2) → E (2) . The addition map + A is a Lie algebr oid morphism if a nd only if, for al l X, Y ∈ Γ ℓ ( D , B ) and α, β ∈ Γ C ( D , B ) , (1) ([ X , Y ] D ) (2) is + - pr oje ctible t o [ X , Y ] D , (2) ([ X , α ] D ) + is + - pr oje ctible t o [ X , α ] D , (3) [ α, β ] D = 0. Pr o of. The condition that + A is a Lie alg ebroid morphism is equiv alent to the statemen t that “If e X , e Y ∈ Γ( D (2) , E (2) ) ar e +–pr o jectible to X , Y ∈ Γ( D , E ), resp ectively , then [ e X , e Y ] D (2) is +-pro jectible to [ X , Y ] D . ” Call this prope r ty P . Let us choose the same lo cal co ordinates a s in the pro of o f Lemma A.1. The induced co ordinates o n D (2) are { x i , e i 1 , e i 2 , a i , c i 1 , c i 2 } . W e als o int ro duce co or dinates e i ± := 1 2 ( e i 1 ± e i 2 ) and c i ± := 1 2 ( c i 1 ± c i 2 ). Then { A (2) i , C + i , C − i } is the frame of sectio ns of D (2) ov er E (2) dual to the ﬁbre co o rdinates { a i , c i + , c i − } . W e notice that a sectio n e X ∈ Γ( D (2) , E (2) ) is +– pro jectible to X = f i ( x, e ) A i + g i ( x, e ) C i ∈ Γ( D , E ) if and o nly if it is of the form e X = f i ( x, 2 e + ) A (2) i + g i ( x, 2 e + ) C + i + h i ( x, e + , e − ) C − i . Next w e notice that generic s e c tions X (2) ∈ Γ (2) l , α + ∈ Γ + c , and α − ∈ Γ − c hav e, resp ectively , the form: X (2) = f i ( x ) A (2) + g i j ( x ) e j + C + i , α + = h i ( x ) C + i , α − = h i ( x ) C − i . These co ordinate des criptions show tha t the spa c e of sections of D (2) → E (2) that ar e +–pro jectible is exactly C ∞ ( E ) (2) ⊗ Γ (2) l + C ∞ ( E ) (2) ⊗ Γ + c + C ∞ ( E (2) ) ⊗ Γ − c . F rom here, a dire ct co mputation shows that prop erty P is true in general if and only if it is true for e X a nd e Y in Γ 2 l ∪ Γ + c ∪ Γ − c . (This requires using the co mpatibility conditions for the anchor.) W e are left with six pa rticular cases of the s tatement of prop erty P . Finally , the bra cket o n D (2) satisﬁes the prop erty that, if ( X , X ′ ) and a nd ( Y , Y ′ ) are compatible pairs of sections of D → E , then [ X × X ′ , Y × Y ′ ] D (2) = [ X, Y ] D × [ X ′ , Y ′ ] D . W e systematically apply this fact to prop erty P in the six particular cas e s w e hav e, and we obtain conditions (1)–(3) in the statement of this lemma, hence completing its pro of.  The compatibility conditions inv olving the bra ck ets in the deﬁnition of LA –vector bundle hav e been rewritten in terms of linear and core sections a s conditions (1 )-(3) in Lemmas A.1 and A.2 . The c ompatibility conditions inv olving the brac kets in th e deﬁnition of V B – algebroid w ere conditions (1)-(3) in Deﬁnition 3.4. Lemma A.3 b elow sho ws that the t w o sets of conditio ns are equiv alent, b y means of a characteriza tion of linear and core sections in ter ms of q - a nd + -pro jectibilit y , hence completing the pro of of Theorem 3.7. Lemma A.3. V B -ALGEBR OIDS AND REPRESENT A TION THEOR Y OF LIE ALGEBROIDS 27 (1) A se ction X ∈ Γ( D, E ) that is q - pr oje ctible to X 0 ∈ Γ( A ) is line ar if and only if X (2) is + -pr oje ctible to X . (2) A se ction α ∈ Γ( D , E ) that is q -pr oje ctible t o 0 A is a c or e se ction if and o nly if α + is + - pr oje ctible t o α . Pr o of. The pro of is a computatio n in co o rdinates, or a direct chec k of the deﬁnitions.  Appendix B. P r oofs fr om § 5 W e begin with the following le mmas, which co nsist of straightf orward extensions of well- known results in standard Chern-W eil theory . F or all the lemmas, we supp ose that B → M is a Lie alg ebroid and E → M is a gra ded vector bundle. Lemma B.1. F or any ( n onhomo gene ous) B -sup er c onn e ction O on E and any End( E ) - value d B - form θ , the op er ator [ O , θ ] on Ω( B ) ⊗ Γ( E ) is Ω( B ) -line ar (and t her efor e may b e viewe d as an End( E ) -value d B -form), a nd str([ O , θ ]) = d B str( θ ) . Pr o of. Lo cally , c ho ose a homogeneous frame { a i } for E , and express O as d B + η , where η is an End( E )-v a lued B -for m. The result follows from the fac t that str([ η , θ ]) = 0.  Lemma B.2. Supp ose t hat E is e quipp e d with a metric g : E ∼ → E ∗ , as in § 5. F or any (nonhomo gene ous) B -sup er c onne ction O on E , (1) d B str( O 2 k ) = 0 for al l k . (2) str ( O 2 k ) = ( − 1) k str(( g O ) 2 k ) for al l k . Pr o of. F or the ﬁrst statement, we note that O 2 k is a n End( E )-v a lued B -form, so b y Lemma B.1 w e ha ve that d B str( O 2 k ) = str([ O , O 2 k ]) = 0. F or the s e cond statemen t, it follows from (5.2) that, for any a ∈ Γ( E ) and ς ∈ Γ( E ∗ ), hO 2 k a, ς i = −hO 2 k − 2 a, ( O † ) 2 ς i = ( − 1) k h a, ( O † ) 2 k ς i .  Next, we present so me lemmas rega rding sup erc onnections that are built out of pairs and triplets of sup erc o nnections. Lemma B.3. L et O 1 and O 2 b e (nonhomo gene ous) B -sup er c onne ctions on E . As in (5.3 ) , let T O 1 , O 2 b e the ( B × T I ) -sup er c onne ction such that T O 1 , O 2 ( a ) = t O 1 ( a ) + (1 − t ) O 2 ( a ) , wher e a ∈ Γ( E ) is viewe d as a t - indep endent se ction of the pul lb ac k of E to M × I . Then Z d t d ˙ t ˙ t ∂ ∂ t ( T O 1 , O 2 ) 2 k = O 2 k 1 − O 2 k 2 . Pr o of. Using the Leibniz rule a nd the fact that the diﬀ erential for B × T I is d B + ˙ t ∂ ∂ t , we compute ( T O 1 , O 2 ) 2 = t 2 O 2 1 + (1 − t ) 2 O 2 2 + t (1 − t )[ O 1 , O 2 ] + ˙ t ( O 1 − O 2 ) . By the F undamental Theo rem of Calculus, we ha ve Z d t ∂ ∂ t ( T O 1 , O 2 ) 2 k = O 2 k 1 − O 2 k 2 + O ( ˙ t ) . Finally , we see that Z d ˙ t ˙ t  O 2 k 1 − O 2 k 2 + O ( ˙ t )  = O 2 k 1 − O 2 k 2 .  28 ALFONSO GRACIA-SAZ AND RAJAN AMIT MEHT A Lemma B.4. L et O 1 , O 2 , and O 3 b e (nonhomo gene ous) B -sup er c onn e ctions on E . let T O 1 , O 2 , O 3 b e the ( B × T I × T I ′ ) -sup er c onne ction such that T O 1 , O 2 , O 3 ( a ) = st O 1 ( a ) + (1 − s ) t O 2 ( a ) + (1 − t ) O 3 ( a ) , wher e t and s ar e c o or dina tes on I and I ′ , r esp e ctively, and a ∈ Γ( E ) is viewe d as an s - and t -indep endent se ct ion of the pul lb ack of E to M × I × I ′ . Then Z d t d ˙ t ˙ t ∂ ∂ t ( T O 1 , O 2 , O 3 ) 2 k = T 2 k O 1 , O 2 − O 2 k 3 and Z d s d ˙ s ˙ s ∂ ∂ s ( T O 1 , O 2 , O 3 ) 2 k = T 2 k O 1 , O 3 − T 2 k O 2 , O 3 . W e omit the pr o of of Lemma B.4, since it is similar to that of Lemma B.3. Pr o of of Pr op osition 5.2. Le t us set B = A × T I a nd O = T D , g D in part (1) of Lemma B.2. Since d A × T I = d A + ˙ t ∂ ∂ t , w e ha ve that d A Z d t d ˙ t str  ( T D , g D ) 2 k  = Z d t d ˙ t ˙ t ∂ ∂ t str  ( T D , g D ) 2 k  , which by Lemma B.3 is str( D 2 k ) − str(( g D ) 2 k ). Since b oth D and g D are ﬂat, we co nclude that d A cs g k ( D ) = 0.  Pr o of of L emma 5.3. It is cle a r from the deﬁnitions that g T D , g D = T g D , D , so by part (2) of Lemma B.2 we have that str  ( T D , g D ) 2 k  = ( − 1) k str  ( T g D , D ) 2 k  . On the o ther hand, the substitution u = 1 − t yields the equation Z d t d ˙ t str  ( T D , g D ) 2 k  = − Z d t d ˙ t str  ( T g D , D ) 2 k  , so we conclude that cs g k ( D ) = ( − 1 ) k − 1 cs g k ( D ), and therefore if k is even we hav e cs g k = 0.  Pr o of of Pr op osition 5.4. B y Lemma 5.3, we may restrict our s elves to the case where k is o dd. Let O b e a degree 1 sup er connection such that g O = O . Such an O may b e constr uc ted as a “blo ck-diagonal” A -superc onnection, where the blo cks are self-adjoint A -connections on E i for each i . Since T D , O is ho mogeneous of degree 1 , it is manifestly the case that I := R d t d ˙ t str  ( T D , O ) 2 k  is an elemen t of Ω 2 k − 1 ( A ). T o complete this pro of w e will show that 2 I and cs g k ( D ) diﬀer by an exa ct ter m. Since g T D , O = T g D , O , w e ha ve b y part (2) of L e mma B .2 that Z d t d ˙ t s tr  ( T D , O ) 2 k  = − Z d t d ˙ t str  ( T g D , O ) 2 k  . Then 2 Z d t d ˙ t str  ( T D , O ) 2 k  = Z d t d ˙ t str  ( T D , O ) 2 k  − Z d t d ˙ t str  ( T g D , O ) 2 k  , which by Lemma B.4 is (B.1) Z d t d ˙ t d s d ˙ s ˙ s ∂ ∂ s str  ( T D , g D , O ) 2 k  . Since d A × T I × T I ′ = d A + ˙ t ∂ ∂ t + ˙ s ∂ ∂ s , w e ha ve by part (1) of Lemma B.2 that (B.1) equals (B.2) − d A Z d t d ˙ t d s d ˙ s str  ( T D , g D , O ) 2 k  − Z d t d ˙ t d s d ˙ s ˙ t ∂ ∂ t str  ( T D , g D , O ) 2 k  . V B -ALGEBR OIDS AND REPRESENT A TION THEOR Y OF LIE ALGEBROIDS 29 Ignoring the exact ter m in (B.2 ), w e see by Lemma B .4 that the se c ond term is Z d s d ˙ s str  ( T D , g D ) 2 k  − Z d s d ˙ s str  O 2 k  . The latter term v anishes since the integrand do e s not dep end on ˙ s , and the ﬁrst ter m is cs g k ( D ). W e conclude that cs g k ( D ) and the (2 k − 1 )-form 2 R d t d ˙ t str  ( T D , O ) 2 k  diﬀer by a n exact term.  Pr o of of Pr op osition 5.5. Le t g and g ′ be metrics E ∼ → E ∗ , as in § 5. By an argument simila r to that in the pro o f o f P r op osition 5.4, w e may see that the equation (B.3) cs g k ( D ) − c s g ′ k ( D ) = Z d s d ˙ s str  ( T g ′ D , g D ) 2 k  holds up to an exact term. Thus, we need to show that the r ight hand side of (B.3) is exact. First, let γ b e a smoo th path of metrics such that γ (0) = g and γ (1) = g ′ , and for all r ∈ [0 , 1] le t θ r be the degr ee 1 E nd( E )-v alued A -for m deﬁned as θ r := γ ( r ) D − g D . Since γ ( r ) D is ﬂat for all r , we have that (B.4) 0 =  γ ( r ) D  2 = ( g D + θ r ) 2 = [ g D , θ r ] + θ 2 r . F or an s -independent section a , we hav e T γ ( r ) D , g D ( a ) = sθ r ( a ) + g D ( a ) , so, using the Leibniz r ule fo r superc o nnections, w e can co mpute  T γ ( r ) D , g D  2 = s 2 θ 2 r + s [ g D , θ r ] + ˙ sθ r = ( s 2 − s ) θ 2 r + ˙ sθ r . (B.5) In the las t step o f (B.5) we have used (B.4). Th us w e see that, up to a constant factor , Z d s d ˙ s str  ( T γ ( r ) D , g D ) 2 k  = str  θ 2 k − 1 r  . W e co nclude that the right hand side of (B.3 ), which we ar e trying to pr ov e is exact, equals R d r ∂ ∂ r str( θ 2 k − 1 r ) up to a constant factor . Second, let u r ∈ End( E ) b e deﬁned b y the proper t y h s, s ′ i γ ( r ) = h u r ( s ) , s ′ i g . It ma y b e direc tly check ed that γ ( r ) D = u − 1 r ◦ g D ◦ u r . W e then s ee that ∂ θ r ∂ r = ∂ ∂ r h γ ( r ) D i = ∂ u − 1 r ∂ r ◦ g D ◦ u r + u − 1 r ◦ g D ◦ ∂ u r ∂ r = ∂ u − 1 r ∂ r u r ◦ γ ( r ) D + γ ( r ) D ◦ u − 1 r ∂ u r ∂ r =  γ ( r ) D , u − 1 r ∂ u r ∂ r  . (B.6) In the las t line o f (B.6), we have used the iden tit y ∂ u − 1 r ∂ r u r + u − 1 r ∂ u r ∂ r = 0 . Using the pr op erty  γ ( r ) D , θ 2 r  = 0, which follows from (B.4 ), we deduce that ∂ θ r ∂ r θ 2 k − 2 r =  γ ( r ) D , u − 1 r ∂ u r ∂ r θ 2 k − 2 r  . 30 ALFONSO GRACIA-SAZ AND RAJAN AMIT MEHT A Finally , w e see tha t ∂ ∂ r str( θ 2 k − 1 r ) = (2 k − 1) str  ∂ θ r ∂ r θ 2 k − 2 r  = (2 k − 1) d A str  u − 1 r ∂ u r ∂ r θ 2 k − 2 r  , where in the last line we ha ve used Lemma B.1 . Thus w e conc lude that R d r ∂ ∂ r str( θ 2 k − 1 r ) is exact, which is wha t w e w anted to prov e.  Pr o of of The or em 5.6. Let D and ˚ D b e the sup erc o nnections ar ising from tw o horizo n tal lifts. As we saw in the pr o of of Pr op osition 5.4, the cohomo logy class of cs g k ( D ) equals that of 2 R d t d ˙ t str  ( T D , O ) 2 k  , where O is s e lf-adjoint. Therefore, up to a n exact term, 1 2  cs g k ( D ) − cs g k ( ˚ D )  is (B.7) Z d t d ˙ t str  ( T D , O ) 2 k  − Z d t d ˙ t s tr  ( T ˚ D , O ) 2 k  . Again using an a rgument from the pro of o f Propo sition 5.4, w e have that, up to an exact term, (B.7) is (B.8) Z d s d ˙ s str  ( T D , ˚ D ) 2 k  . Let σ be deﬁned as in ( 4.15), and let us deﬁned a path u r of automorphisms of Ω( A ) ⊗ Γ( C [1] ⊕ E ) b y u r = 1 + r σ . Then, b y setting D r := u − 1 r ◦ D ◦ u r , Theor e m 4.14 shows that D 0 = D and D 1 = ˚ D . T hus, b y the same a r gument as in the pro of of Prop os ition 5.5, w e see that (B.8) is exa ct.  References [1] C. A r ias Abad and M. Crainic. Represen tations up to homotop y of Lie algebroids, 2009. ar Xiv: 0901.0319. [2] R. Bott. Lectures on characte ristic classes and foliations. In L e ctur es on algebr aic and diﬀer ential top olo gy (Se c ond L atin A meric an Scho ol in Math., Mexico City, 1971) , pages 1–94. Lecture Notes in Math., V ol. 279. 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Ehresmann doubles and Drinfel’d doubles for Li e algebroids and Li e bialgebroids, 2006. ar X iv: mat h/0611799. [16] R. A. Mehta. Sup er gr oup oids, double struct ur es, and e quivariant c ohomolo gy . PhD thesis, U ni v ersity of Cali fornia, Berkeley , 2006. arXiv: math.DG/0605356. [17] J. Pradines. Repr ´ esen tat ion des jets non holonomes par des morphismes v ectoriels doubles soud ´ es. C. R. A c ad. Sci. Paris S´ er. A , 278:1523–152 6, 1974. [18] D. Quillen. Sup erconnections and the Chern c haracter. T op olo gy , 24(1):89– 95, 1985. [19] A. Y. V a ˘ ınt rob. Lie algebroids and homological v ector ﬁelds. Usp ekhi Mat. Nauk , 52( 2(314)):161 –162, 1997. Dep ar tment of Mat hema tics, University of Toronto, 4 0 Saint George Street, Room 6290, Toronto, Ont ario, Canada M5S 2E4 E-mail addr ess : alfonso@math.t oronto.edu Dep ar tment of Mat hema tics, W ashington University in Sain t Louis, One Brookings Drive, Saint Louis, Missouri, USA 63130 E-mail addr ess : raj@math.wustl .edu

Lie algebroid structures on double vector bundles and representation theory of Lie algebroids

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