UNIVERSIDADE FEDERAL DO PARANÁ Luiz Henrique Pereira Pêgas THE HOCHSCHILD-KOSTANT-ROSENBERG THEOREM FOR SMOOTH MANIFOLDS. Curitiba, 2010.

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1 UNIVERSIDADE FEDERAL DO PARANÁ Luiz Henrique Pereira Pêgas THE HOCHSCHILD-KOSTANT-ROSENBERG THEOREM FOR SMOOTH MANIFOLDS Curitiba, () 1 / 20

2 Overview: From now on, fix a field K, an associative commutative unital K-algebra A and a m-dimensional smooth manifold M. The following concepts will be treated: Derivations on an K-algebra A; Derivations of high order on A; Multiderivations on A; Iterated derivations on A; Polyderivations on A; The H-K-R theorem for smooth manifolds; () 2 / 20

3 Derivations on an K-algebra A Definition: A K-linear map D : A A is a derivation on the K-algebra A if and only if, it satisfies the Leibniz rule: D(ab) = D(a)b + ad(b) a, b A The space of derivations on A is denoted Der(A). Theorem: The R-vector space of vector fields on a m-dimensional smooth manifold M, X(M), and the R-vector space Der(C (M)) are isomorphics. () 3 / 20

4 Derivations of high order on an K-algebra A Definition: A K-linear map D : A A is a differential operator of order r on the K-algebra A, for r > 0 if and only if, for all g A, the operator D given by D(f ) = D(gf ) gd(f ) f A is of order r 1. D is a differential operator of order 0 if it is a product by an element in A. Definition: A differential operator D : A A of order r on A is called a derivation of order r on A if and only if, it vanishes on K. () 4 / 20

5 A characterization of high order derivations: Theorem: There exists a smooth vector bundle J r (M), whose base space is M and whose space of smooth sections Γ(J r (M)) is isomorphic to the R-vector space of derivations of order r on C (M). Local expression: A derivation D, of order r, on C (M) is written in coordinates as D(f ) = r k=1 1 i 1... i k m k f a i1...i k (x 1,..., x m ) x i, 1... x i k for all f C (M), with a i1...i k smooth. () 5 / 20

6 Construction of the bundle:the idea Definition: For p M, r 1, let I p be the ideal of germs of functions vanishing at p. Define Jp r = (I p /Ip r+1 ) For the bundle structure, take the (adapted) charts induced by M. Note that we can define A differential operator of order r at p is a linear map D p : F p R, such that d g (f ) = D p (gf ) g(p)d p (f ) is of order r 1 at p, for all g F p. It is of order 0 if it is a product by a germ of function. () 6 / 20

7 Notations: C n (A, A) = Hom Vec K(A n, A), n Z, n 0 C (A, A) = n 0 C n (A, A) Definition: We define the product as the K-linear map of degree 0, : C (A, A) C (A, A) C (A, A) such that if f C n (A, A) and g C m (A, A) then (f g)(a 1... a m+n ) = f (a 1... a n )g(a n+1... a m+n ) for all a 1,..., a m+n A. Proposition: (C (A, A), ) is an associative graded K-algebra. () 7 / 20

8 Multiderivations: Definition: The space of multiderivations on an K-algebra A is the subalgebra of (C (A, A), ) generated by Der(A), denoted MDer(A). We denote MDer n (A) = MDer(A) C n (A, A). Theorem: MDer n (C (M)) is isomorphic to Γ((T M) n ), for all n 1. Local expression: An element in D MDer n (C (M)) is written in coordinates as D(f 1... f n ) = m j 1,...,j n=1 D(x j 1... x jn ) f 1 x j 1... f n x jn () 8 / 20

9 Iterated derivations on an K-algebra A Definition: The space of iterated derivations on the K-algebra A is the subalgebra of (C 1 (A, A), ), generated by Der(A), denoted SDer(A). We denote the space of the elements in SDer(A) which can be written as linear combination of at most n iterated composites of elements in Der(A) by SDer n (A). Theorem: If D SDer n (A), then D is a derivation of order n on A. () 9 / 20

10 Theorem: If D is a derivation of order r, with r > 0, on C (M), then D SDer r (C (M)). We have There is a bijection between derivations of order r on C (M), and SDer r (C (M)), for all r > 0. () 10 / 20

11 Polyderivations: Definition: The space of polyderivations on an K-algebra A is the subalgebra of (C (A, A), ) generated by SDer(A), denoted D poly (A). We denote D n,r poly (A) the space of polyderivations of degree n and order r, i.e., elements generated by SDer r (A) belonging to C n (A, A). Local expression: If D D n,r poly (C (M)), then D is written in coordinates as D(f 1... f n ) = a Ik1...I kn x I k 1 I k1,...,i kn I k1 f 1... Ikn f n x I kn () 11 / 20

12 The Hochschild differential Definition: The Hochschild differential on C (A, A) is the linear map δ H : C (A, A) C +1 (A, A) such that, if f C n (A, A), then (δ H f )(a 1... a n+1 ) = a 1 f (a 2... a n+1 )+ n + ( 1) i f (a 1... a i a i+1... a n+1 )+ Proposition: i=1 + ( 1) n+1 f (a 1... a n )a n+1 The Hochschild differential is a degree 1 derivation on (C (A, A), ). () 12 / 20

13 The Hochschild complex of polyderivations Theorem: Any multiderivation is a Hochschild cocycle. Theorem: (D poly (A), δ H ) is a filtered subcomplex of (C (A, A), δ H ). Proposition (Gutt-Rawnsley): If D D n,r poly (C (M)) is a Hochschild cocycle, then there exists a cochain E D n 1,r+1 poly (C (M)) and η MDer n (C (M)) alternating, such that D = δ H (E) + η () 13 / 20

14 The Hochschild-Kostant-Rosenberg theorem for smooth manifolds. Lets denote the space of multivector fields on M by Ω (M) and D = C (M) D poly (C (M)). Theorem (Cahen-De Wilde-Gutt): The complexes (D, δ H ) and (Ω (M), d), where d : Ω (M) Ω +1 (M) is the null differential, are quasi-isomorphics. Define J 0 = Id and J n : D n poly (C (M)) D n poly (C (M)), for n 1, by J n (D) = Alt(D), D D n poly (C (M)). Note that: 1 Ω n (M) Alt(MDer n (C (M))); 2 For any cochain E, Alt(δ H E) = 0. Then J n δ H = d J n 1 ; 3 H n (Ω n (M)) Ω n (M). () 14 / 20

15 Sketch of the proof: J n : H n (D) H n (Ω n (M)) is an isomorphism. J n is injective: If θ is the equivalence class of an n-cocycle D = δ H E + η such that J n(θ) = 0, then 0 = J n(θ) = [J n (D)] = [J n (δ H E + η)] = [η] = η. J n is surjective: To any η Ω n (M) we can uniquely associate an element η Alt(MDer n (C (M))), which is a cocycle in D n poly (C (M)). Just note that J n([ η]) = [J n ( η)] = η, uniquely associated to η. () 15 / 20

16 Acknowledgements Universidade Federal do Paraná, Departamento de Matemática - for the opportunity and good environment to work; CAPES - for financial support; Dr. Eduardo Hoefel, my advisor - for kindness and patience during this work; My wife - for all... () 16 / 20

17 Bibliography Cahen, M., De Wilde, M., Gutt, S.: Local cohomology of the algebra of C functions on a connected manifold, Lett. in Math. Phys., 4: , Cannas da Silva, A., Weinstein, A. Geometric Models for Noncommutative Algebras, University of California, Berkeley, Doubek, M., Markl, M., Zima, P.: Deformation theory (lecture notes), Archivum Mathematicum, 43(5): , 2007, arxiv: v3[math.ag]. Forger, M., Paufler, C., Römer, H.: The Poisson Bracket for Poisson Forms in Multisymplectic Field Theory, Rev. Math. Phys., 15(7): , Gelfand, S. I., Manin, Yu. I. Methods of Homological Algebra, Springer, New York, Gerstenhaber, M.: The cohomology structure of an associative ring, Ann. of Math., 78(2): ,1963. () 17 / 20

18 Bibliography Gutt, S., Rawnsley, J.: Equivalence of star products on a symplectic manifold: an introduction to Deligne s Čech cohomology classes, J. Geom. Phys., 29: , Kontsevich, M.: Deformation quantization of Poisson manifolds, Lett. in Math. Phys., 66: , Kostrikin, A. I., Shafarevich I. R. Basic Notions of Algebra, Springer-Verlag, New York, Kostrikin, A. I., Shafarevich I. R. Algebra V: homological algebra, Springer-Verlag, New York, Marsden, J. E., Ratiu, T. Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer-Verlag, Warner, F. W. Foundations of Differentiable Manifolds and Lie Groups, Springer-Verlag, New York, () 18 / 20

19 Auxiliary formulae: The Hochschild differential of an iterated derivation: δ H (X i 1... X i j )(a b) = = a(x i 1... X i j )(b) (X i 1... X i j )(ab) + (X i 1... X i j )(a)b = j 1 = (X i )(a)(x i Î k I k )(b) k=1 I k () 19 / 20

20 Auxiliary formulae: For D = X 1... X n : a (b) = (X 1... X n )(ab) a(x 1... X n )(b) = n = (X 1... X n )(a) b + (X 1... ˆX i... X n )(a)x i (b) i<j n i=1 (X 1... ˆX i... ˆX j... X n )(a)(x i X j )(b) I k XÎk (a)x Ik (b) n X i (a)(x 1... ˆX i... X n )(b) i=1 By induction, D is a differential operator of order n. () 20 / 20