A Global version of Grozman's theorem

Olivier Mathieu

Description

A Global version of Grozman's theorem Kenji IOHARA and Olivier MATHIEU April 5, 2012 Abstract Let X be a manifold. The classification of all equivariant bilinear maps between tensor density modules over X has been investigated by P. Ya. Grozman [G1], who has provided a full classification for those which are differential operators. Here we investigate the same question without the hypothesis that the maps are differential operators. In our paper, the ge- ometric context is algebraic geometry and the manifold X is the circle SpecC[z, z?1]. Our main motivation comes from the fact that such a classification is required to complete the proof of the main result of [IM]. Indeed it requires to also include the case of deformations of tensor density modules. Contents 0 Introduction 2 1 The Kaplansky-Santharoubane Theorem 7 2 Germs and bilinear maps 9 3 Degenerate and non-degenerate bilinear maps 13 4 Examples of W-equivariant bilinear maps 16 5 Classification of W-equivariant degenerate bilinear maps 21 1

• operator ?∞


• module over


• since differential operators


• m?n ?


• tensor density


• bilinear maps


• maps between

Sujets

• Mathieu
• Tensor density

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 8
• Langue: English

Extrait

A

Global

version

of Grozman’s theorem

Kenji IOHARA and Olivier MATHIEU

April 5, 2012

Abstract

Let X  Thebe a manifold. classification of all equivariant bilinear maps between tensor density modules over X has been investigated by P. Ya. Grozman [G1], who has provided a full classification for those which are differentialoperators. Here we investigate the same question without the hypothesis that the maps are differential operators. In our paper, the ge- ometric context is algebraic geometry and the manifold X is the circle SpecC[z z −1 ] . Our main motivation comes from the fact that such a classification is required to complete the proof of the main result of [IM]. Indeed it requires to also include the case of deformations of tensor density modules.

Contents

0 Introduction

1 The Kaplansky-Santharoubane Theorem

2 Germs and bilinear maps

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4

5

Degenerate and non-degenerate bilinear maps

Examples of W-equivariant bilinear maps

Classification of W-equivariant degenerate bilinear maps

1

2

7

9

13

16

21

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Bounds for the dimension of the spaces of germs of bilinear maps 25

7 Determination ofG W (M×N P)31 8 On the map B W (M×N P)→ G W (M×N P)37 9 Computation of B W (M×N P)whendimG W (M×N P) = 139 10 Computation of B W (M×N P)whendimG W (M×N P) = 243

11 Conclusion

A Complete expression for D δ 1 ,δ 2 ,γ (x y)

0 Introduction

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The introduction is organized as follows. The first section is devoted to the main definitions and the statement of the Grozman Theorem. In the second section, our result is stated. In the last section, the main ideas of the proof are explained.

0.1 Grozman’s Theorem:

LetXbe a manifold of dimensionn, letW X be the Lie algebra of vector fields overXand letM NandPbe three tensor density modules overX. The precise meaning of tensor density module will be clarified later on and the geometric context (differential geometry, algebraic geometry,. . .) is not yet precised. In a famous paper [G1], P. Ya. Grozman has classified all bilinear differ- ential operatorsπ:M×N→Pwhich areW X  differential-equivariant. Since operators are local [P], it is enough to consider the case of the formal geom- etry, namelyX= SpecC[[z 1  . . .  z n ]]. The most intersting and difficult part of Grozman’s theorem involves the case where dimX= 1, indeed the general case follows from this case. Therefore, we will now assume thatX= SpecC[[z this manifold,]]. For the tensor density modules are the modules Ω δ , where the parameterδruns overC. As aC[[z]]-module, Ω δ a rank one free module whose generatoris

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is denoted by (dz) δ . The structure ofW X -module on Ω δ is described by the following formula:

ξ.[f.(dz) δ ] = (ξ.f+δfdiv(ξ)).(dz) δ for anyf∈C[[z]] andξ∈W X , where, as usual,ξ.f=gf 0 , div(ξ) =g 0 wheneverξ d  a non-negative integer, Ωfor some δ = z g d g∈C[[z]]. Whenδis is the space (Ω 1 X ) ⊗δ , where Ω 1 X caoeKfa¨itsehpsfntreloiaerlhffediX. The space⊕ δ Ω δ can be realized as the space of symbols of twisted pseudo- differential operators on the circle (see e.g. [IM],twistedmeans that complex powers of d dz are allowed) and therefore it carries a structure of Poisson algebra. The Poisson structure (a commutative productPand a Lie bracket B) induces two series ofW X -equivariant bilinear maps, namely the maps P δ 1 ,δ 2 : Ω δ 1 ×Ω δ 2 →Ω δ 1 +δ 2 and the mapB δ 1 ,δ 2 : Ω δ 1 ×Ω δ 2 →Ω δ 1 +δ 2 +1 . These operators are explicitly defined by: P δ 1 ,δ 2 (f 1 .(dz) δ 1  f 2 .(dz) δ 2 ) =f 1 f 2 (dz) δ 1 +δ 2 B δ 1 ,δ 2 (f 1 .(dz) δ 1  f 2 .(dz) δ 2 ) = (δ 2 f 0 1 f 2 −δ 1 f 1 f 2 0 )(dz) δ 1 +δ 2 +1 Moreover, the de Rham operator is aW X -equivariant mapd: Ω 0 →Ω 1 . So we can obtained additionalW X -equivariant bilinear maps between tensor density module by various compositions ofB δ 1 ,δ 2 andP δ 1 ,δ 2 withd. An ex- ample is provided by the mapB 1,δ ◦(d×id) : Ω 0 ×Ω δ →Ω δ+2 . Following Grozman, theclassicalW X -equivariant bilinear maps are (the linear combina- tions of) the mapsB δ 1 ,δ 2  P δ 1 ,δ 2 , and those obtained by various compositions withd. Grozman discovered one additionalW X -equivariant bilinear map, namely Grozman’s operatorG: Ω −2/3 ×Ω −2/3 →Ω 5/3 defined by the formula: G(f 1 .(dz) −2/3  f 2 .(dz) −2/3 ) = [2(f 0 1 00 f 2 −f 0 2 00 f 1 ) + 3(f 0 1 0 f 0 2 −f 0 1 f 0 2 0 )](dz) 5/3 . With this, one can state Grozman’s result:

Grozman Theorem.Any differentialW X -equivariant bilinear map π: Ω δ 1 ×Ω δ 2 →Ω γ between tensor density modules is either classical, or it is a scalar multiple of the Grozman operator.

0.2 The result of the present paper:

In this paper, a similar question is investigated, namely the determination of allW X -equivariant bilinear mapsπ:M×N→Pbetween tensor density

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modules, without the hypothesis thatπis a differential operator. Since differential operators are local, we will establish a global (=non-local) version of Grozman Theorem. For this purpose, we will make new hypotheses. From now on, the context is the algebraic geometry, and the manifoldXof investigation is thecircle, namelyC ∗ = SpecC[z z −1 ]. SetW=W X . Fix two parametersδ s∈C and setρ δ,s (ξ) =ξ+δdivξ+i ξ α s for anyξ∈W, whereα s =sz −1 dz. By definition, Ω sδ is theW-module whose underlying space isC[z z −1 ] and the action is given byρ δ,s . To describe more naturally the actionρ δ,s , it is convenient to denote by the symbolz s (dz) δ the generator of this module, and therefore the expressions (z n+s (dz) δ ) n∈Z form a basis of Ω sδ  follows. It easily that Ω sδ and Ω δ u are equal ifs−uis an integer. we will Therefore, consider the parametersas an element ofC/Z. We will not provide a rigorous and general definition of thetensor density modules [M2]). Just(see e.g. say that thetensor density modulesconsidered here are theW-modules Ω δ u , where (δ s) runs overC×C/Z. As before, there areW-equivariant bilinear mapsP uδ 1 1 ,δ ,u 2 2 : Ω δ u 1 1 ×Ω u δ 2 2 → Ω δ u 1 1 + + δ u 2 2 andB δ u 1 1 ,δ ,u 2 2 : Ω δ u 1 1 ×Ω u δ 2 2 →Ω δ u 1 1 + +u δ 2 2 +1 as well as the de Rham differential, 1 d: Ω u 0 →Ω u . There is also a mapρ: Ω u 1 →Ω u 0 , which is defined as follows. Foru6≡0 modZ, the opeartordis invertible and setρ=d −1 . For u≡0 modZ, denote byρ: Ω 1 u →Ω 0 u the composite of the residue map Res : Ω 1 0 →Cand the natural mapC→Ω 0 0 =C[z z −1  definition,]. By aclassicalbilinear map between tensor density modules over the circle is any linear combination of the operatorsB δ u 1 1 ,δ u, 2 2 ,P uδ 1 1 ,δ u, 2 2 and those obtained by composition withdandρ example of a classical operator is. Anρ◦P: . Ω u δ 1 ×Ω u 1 2 −δ →Ω 0 u 1 +u 2 Of course, the Grozman operator provides a family of non-classical op- eratorsG u,v : Ω u −2/3 ×Ω v −2/3 →Ω u 5/3 +v . Atrivial operatoris a scalar multiple of the bilinear map Ω 01 ×Ω 01 →Ω 00 (α β)7→Res(α)Res(β is also). There another non-classicalW-equivariant operator Θ ∞ : Ω 01 ×Ω 01 →Ω 0 0 which satisfies: dΘ ∞ (α β) = Res(α)β−Res(β)α for anyα β∈Ω 1 0 . Indeed Θ ∞ is unique modulo a trivial operator. Our result is the following:

Theorem:(restricted version) LetX Anybe the circle.W-equivariant bilinear map between tensor density module is either classical, or it is a scalar multiple ofG u,v or ofΘ ∞ (modulo a trivial operator).

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In the paper, a more general version, which also involves deformations of tensor densi =z d . For aW-moduleMand ty modules, is proved. SetL 0 dz s∈C, setM s = Ker(L 0 −s). LetSbe the class of allWmodulesMwhich satisfies the following condition: there existsu∈C/Zsuch that M=⊕ s∈u M s and dimM s = 1 for alls∈u. TheZ-cosetuis called thesupport ofM, and it is denoted by SuppM. It turns out that all modules of the classShave been classified by Ka- plansky and Santharoubane [KS])] and, except deformations of Ω 00 and Ω 1 0 , all modules of the classSare tensor density modules. full result is the Our classification of allW-equivariant bilinear maps between modules of the class S.

0.3 About the proofs:

In order to describe the proof and the organization of the paper, it is necessary to introduce the notion of germs of bilinear maps. For any three vector spacesM NandP, denote byB(M×N P) the space of bilinear mapsπ:M×N→P. Assume now thatM Nand PareW-modules of the classS. Forx∈R, setM ≥x =⊕ <s≥x M s and N ≥x =⊕ <s≥x N s . By definition, agermof bilinear map fromM×NtoPis an element of the spaceG(M×N P lim) :=B(M x × x→+ ∞ ≥ N ≥x  P). It turns out thatG(M×N P) is aW by-module. DenoteB W (M×N P) the space ofW-equivariant bilinear mapsπ:M×N→P, byB 0 W (M×N P) the subspace of allπ∈B W (M×N P) whose germ is zero and byG W (M× N P) the space ofW-equivariant germs of bilinear maps fromM×NtoP. There is a short exact sequence: 0→B 0 W (M×N P)→B W (M×N P)→ G W (M×N P). The paper contains three parts Part 1 which determinesB 0 W (M×N P), see Theorem 1, Part 2 which determinesG W (M×N P), see Theorem 2, Part 3 which determines B W (M×N P)→ G W (M×N P), see Theorem 3. Part 1 is discussed in Section 5. The map Θ ∞ is an example of a degen- erate map. Part 2 is the main difficulty of the paper. One checks that dimG W (M× N P)≤2. So it is enough to determine whenG W (M×N P) is non-zero

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