The Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Noncommutative Contact Algebras Hideki Omori Yoshiaki Maeda Naoya Miyazaki Akira Yoshioka Vienna, Preprint ESI 395 (1996) October 21, 1996 Supported by Federal Ministry of Science and Research, Austria Available via

2 NONCOMMUTATIVE CONTACT ALGEBRAS HIDEKI OMORI Faculty of Science and Technology, Science University of Tokyo, Noda, 278, Japan YOSHIAKI MAEDA Faculty of Science and Technology, Keio University, Yokohama, 223, Japan NAOYA MIYAZAKI Faculty of Science and Technology, Science University of Tokyo, Noda, 278, Japan AND AKIRA YOSHIOKA Faculty of Technology, Science University of Tokyo, Tokyo, 162, Japan 1. Introduction In [5], Gerstenhaber proposed a denition of the deformation of an algebra which associates to an associative algebra A an associative structure on the space A[[h]] of formal power series in h with coecients in A. Here h is a central element in A[[h]]. Stimulated by this work, several authors have studied such deformations from a geometrical point of view. In particular, Bayen et al. [1] proposed a notion of deformation quantization, which is a deformation of Poisson algebras. In this context, the question naturally arises of how to generalize these deformations for various geometric structures. In this paper, we will introduce the notion of a noncommutative contact algebra. On a contact manifold M, the Lagrange bracket f ; g L denes a Lie bracket on the space C 1 (M) of all smooth functions on M, under which

3 2 HIDEKI OMORI ET AL. C 1 (M) forms a Lie algebra. For a deformation of contact structures, one might attempt rst to deform the Lie algebra (C 1 (M); f ; g L ). However, there exists no non-trivial deformation of the Lie algebra (C 1 (M); f ; g L ) in general [4]. In our process of \noncommutatizing" contact algebras, the deformation parameter will no longer be a central element. A distinguishing feature of contact algebras, as opposed to Poisson algebras, is the existence of the characteristic vector eld. We will noncommutatize the characteristic vector eld together with the algebra C 1 (M). Thus, the algebra we present here is a slight generalization of deformation quantization. Our main purpose in this paper is also to show the existence of the noncommutative contact algebra on an arbitrary contact manifold. 2. Contact algebra Let M be a contact manifold of dimension (2n+1) and! its contact 1-form; (d!) n ^! 6= 0. We denote by C 1 (M) the algebra of all smooth functions on M. There exist local coordinates (x 1 ; ; x n ; y 1 ; ; y n ; z) such that! can be written locally as! = dz nx j=1 (x j dy j? y j dx j ): (1) The contact 1-form! gives a Jacobi structure on M (cf. [6]): a pair (; D) consisting of a vector eld D 2?(T M) and a skew-symmetric bivector eld on T M satisfying the following; (i) [D; ] = 0, (ii) [; ] = 2D ^, (iii) rank = 2n, where [ ; ] stands for the Schouten bracket. In terms of the local coordinates where the 1-form! has the form (1), we have = nx x j z ^ 1 2 For f; g 2 C 1 (M), set nx j=1 (x x j + y j@ y j ) and D z: ff; gg = (df; dg) : C 1 (M) C 1 (M)! C 1 (M): (2) The bracket f ; g is skew-symmetric and satises ff; ghg = ff; ggh + gff; hg; Dff; gg = fdf; gg + ff; Dgg: (3) We call the triple (C 1 (M); f ; g; D) the contact algebra on M.

4 NONCOMMUTATIVE CONTACT ALGEBRAS 3 We remark that Lichnerowicz [6] introduced a Lie algebra structure on C 1 (M) by setting ff; gg L = ff; gg + fd(g)? D(f)g: (4) The bracket f ; g L given by (4) is called the Lagrange bracket on M. 3. Noncommutative contact manifold We rst generalize the notion of deformation quantization (cf. [9]): let A be an associative, complete topological algebra. We denote by ab the product of a; b 2 A. We set [a; b] = a b? b a and k =. {z } k Denition 1 A is called a formal -regulated algebra if there exists an element 2 A with = and a closed linear subspace B such that (A.1) [; A] A. (A.2) [A; A] A. (A.3) A = Q 1 k=0 k B (topological direct sum). (A.4) The mappings : A! A, : A! A dened by a! a, a! a respectively are linear isomorphisms. is called the regulator. By (A.2), the product denes a commutative associative product on the factor space A=A identied with B. We denote the commutative product on B by a b. Formally setting [?1 ; a] =??1 [; a]?1 ; (5) we see by (A.1) and (A.4) that [?1 ; a] is well-dened. Since [?1 ; A] A, ad(?1 ) is a derivation of (A; ), which also induces a derivation of (B; ). According to the decomposition (A.3), we can write a b = a b + 1 (a; b) + + k k (a; b) + ; ad(?1 )a = 0 (a) + 1 (a) + + k k (a) + ; (6) uniquely for any a; b 2 B. Remark 1 Since ad(?1 ) is a derivation of (A; ), 0 is a derivation of (B; ). Note that the commutator bracket [a; b] is a biderivation of A A to A. Therefore, the skew part? of 1 1 is a skew-biderivation of B B to B. Denition 2 Let M be a contact manifold and (C 1 (M); f ; g; D) the contact algebra on M. A formal -regulated algebra A = B[[]] of formal power series in with coecients in B is called the noncommutative contact algebra on M if B = C 1 (M), 1 = f ; g, and 0 = D in (6).

5 4 HIDEKI OMORI ET AL. Example 1 A typical example of a noncommutative contact algebra is the noncommutative 3-sphere given in [9], which is obtained by quantizing the Hopf bration : S 3! S 2. Example 2 Another example of a noncommutative contact algebra is an algebra 0 (T N) of symbols of order 0 on the cotangent bundle T N over a compact manifold N modulo the algebra?1 (T N) of symbols of order?1 (cf. [10]). This gives a noncommutative contact algebra dened on the unit cosphere bundle S N. Theorem 1 For the contact algebra (C 1 (M); f ; g; D) on a contact manifold M, there exists a noncommutative contact algebra C 1 (M)[[]]. Theorem 1 shows that every contact algebra extends to a formal - regulated algebra, i.e. every contact algebra is deformation quantizable. 4. Proof of Theorem 1 Let M be a contact manifold. Consider the direct product manifold R + M, with R + the set of positive real numbers. Dene a closed 2-form = d(!) on R + M, where is the coordinate function on R +. It is easy to see that is a symplectic 2-form on R + M. Thus, it denes a Poisson bracket f ; g s on C 1 (R + M). We see that ff; gg s =?1 ff; gg (7) for f; g 2 C 1 (M), where f ; g is given by (2). Let B = C 1 (M)[[?1 ]] be the set of all formal power series ~ f in?1 with coecients in C 1 (M): ~f(; p) = f 0 (p) +?1 f 1 (p) + +?k f k (p) + : (8) It is obvious that the bracket f ; g s in (7) gives a Poisson bracket on B such that fb; Bg s?1 B: (9) We now consider a deformation quantization of B: an associative algebra B[[h]] with a star product such that h is in the center and for ~ f; ~g 2 B ~f ~g = ~ 0 ( ~ f; ~g) + h~ 1 ( ~ f; ~g) + + h k ~ k ( ~ f; ~g) + ; (10) where ~ 0 ( ~ f; ~g) = ~ f ~g; ~ 1 ( ~ f; ~g) = 1 2 f ~ f; ~gg s : (11) We dene the following 1-parameter conformal symplectomorphism on (R + M; ); t (; x) = (e t ; x); t 2 R: (12) Setting t (h) = e t h, we extend t on B to B[[h]]:

6 NONCOMMUTATIVE CONTACT ALGEBRAS 5 Lemma 1 Let B[[h]] be a deformation quantization of (B; ; f ; g s ): Then, the map t is an automorphism of (B[[h]]; ) if and only if ~ k satises ~ k (?l f;?m g) 2?(k+l+m) C 1 (M) (13) for all f; g 2 C 1 (M) and for all k; l; m 0. We note the following: Proposition 1 Let B[[h]] be a deformation quantization of (B; ; f ; g s ): If t is an automorphism of (B[[h]]; ), then the subalgebra of all t -invariant elements in B[[h]] is a noncommutative contact algebra on M. Proof. Set = h?1. Denote by ~ B[[h]] the space of t -invariant elements of B[[h]]. We have ~ B[[h]] = C 1 (M)[[]] For f; g 2 C 1 (M), we have f g = f g + 2 ff; gg (mod 2 ); [?1 ; f] = 0 (f) (mod ); (14) which gives Proposition 1. To obtain Theorem 1, it suces to show the following: Lemma 2 There exists a deformation quantization B[[h]] of (B; ; f ; g s ) with the property (13). Proof. Since is a symplectic form on R + M, the Poisson algebra (C 1 (R + M); ; f ; g s ) is deformation quantizable ([2], [3], [8]). We recall the stepwise construction of the bilinear map ~ k in (9). As in [8], given f~ j g k?1, ~ j=0 k is determined by the following equations: ~? k (f; gh)? g~? k (f; h)? ~? k (f; g)h = hhf; gi? ; hi + k + hhf; hi? ; gi + k + hhg; hi+ ; fi? k ~ + k (f; gh)? ~+ k (h; gf)? ~+ k (f; g)h + ~+ k (h; g)f; = hhf; gi + ; hi + k? hhh; gi+ ; fi + k + hhh; fi? ; gi? k ; where ~ i (f; g) = 1 2 f~ i(f; g) ~ i (g; f)g, and hhf; gi ; hi m = X +j=m; i;j1 ~ i (~ j (f; g); h) (m 2): (15) On each coordinate neighborhood, we can construct ~ k inductively as a bidierential operator satisfying (14) and (16) (cf. [8]). A partition of unity argument yields Lemma 2. Since there always exists a contact one-form on every orientable, compact 3-manifold (cf. [7]), we have

7 6 HIDEKI OMORI ET AL. Corollary 1 Let M be a orientable compact 3-manifold. Then there exists a noncommutative contact algebra on M. References 1. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Deformation theory and quantization, Ann. Phys. 111 (1978), M. De Wilde, P. Lecomte, Existence of star-products and formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds, Lett. Math. Phys. 7 (1983), B. Fedosov, A simple geometric construction of deformation quantization, J. Di. Geom., 40 (1994), M. Flato, A. Lichnerowicz and D. Sternheimer, Deformations 1-dierentiables des algebres de Lie attachees a une variete symplectique ou de contact, Compositio Math. 31 (1975) M. Gerstenhaber, On the deformation of rings and algebras, Ann. Math. 79 (1964), A. Lichnerowicz, Les varietes de Jacobi et leurs algebres de Lie associees, J. Math. Pures et Appl. 57 (1978), J. Martinet, Formes de contact sur les varietes de dimension 3, Proc. Liverpool Singularities Sympos. II, Lecture Notes in Math., vol. 209, Springer-Verlag, Berlin and New York, 1971, H. Omori, Y. Maeda, A. Yoshioka, Deformation quantization of Poisson algebras, Contemp. Math. 179 (1994), H. Omori, Y. Maeda, N. Miyazaki and A. Yoshioka, Noncommutative 3-sphere, to appear. 10. H. Omori, Y. Maeda, A. Yoshioka and O. Kobayashi, On regular Frechet Lie groups VI, Tokyo J. Math. 6 (1983),