A Global version of Grozman's theorem
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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
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A
Global
version
of Grozman’s theorem
Kenji IOHARA and Olivier MATHIEU
April 5, 2012
Abstract
LetX Thebe a manifold. classification of all equivariant bilinear maps between tensor density modules overXhas 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 manifoldXis 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
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2
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9
13
16
21
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Bounds for the dimension of the spaces of germs of bilinear maps 25
7 Determination ofGW(M×N P)31 8 On the map BW(M×N P)→ GW(M×N P)37 9 Computation of BW(M×N P)whendimGW(M×N P) = 139 10 Computation of BW(M×N P)whendimGW(M×N P) = 243
11 Conclusion
A Complete expression for Dδ1,δ2,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, letWXbe 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 areWX differential-equivariant. Since operators are local [P], it is enough to consider the case of the formal geom-etry, namelyX= SpecC[[z1 . . . zn]]. 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 Ωδ, where the parameterδruns overC. As aC[[z]]-module, Ωδa rank one free module whose generatoris
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is denoted by (dz)δ. The structure ofWX-module on Ωδis described by the following formula:
ξ.[f.(dz)δ] = (ξ.f+δfdiv(ξ)).(dz)δ for anyf∈C[[z]] andξ∈WX, where, as usual,ξ.f=gf0, div(ξ) =g0 wheneverξd a non-negative integer, Ωfor someδ =zgdg∈C[[z]]. Whenδis is the space (Ω1X)⊗δ, where Ω1XcaoeKfa¨itsehpsfntreloiaerlhffediX. The space⊕δΩδcan be realized as the space of symbols of twisted pseudo-differential operators on the circle (see e.g. [IM],twistedmeans that complex powers ofddzare allowed) and therefore it carries a structure of Poisson algebra. The Poisson structure (a commutative productPand a Lie bracket B) induces two series ofWX-equivariant bilinear maps, namely the maps Pδ1,δ2: Ωδ1×Ωδ2→Ωδ1+δ2and the mapBδ1,δ2: Ωδ1×Ωδ2→Ωδ1+δ2+1. These operators are explicitly defined by: Pδ1,δ2(f1.(dz)δ1 f2.(dz)δ2) =f1f2(dz)δ1+δ2 Bδ1,δ2(f1.(dz)δ1 f2.(dz)δ2) = (δ2f01f2−δ1f1f20)(dz)δ1+δ2+1 Moreover, the de Rham operator is aWX-equivariant mapd: Ω0→Ω1. So we can obtained additionalWX-equivariant bilinear maps between tensor density module by various compositions ofBδ1,δ2andPδ1,δ2withd. An ex-ample is provided by the mapB1,δ◦(d×id) : Ω0×Ωδ→Ωδ+2. Following Grozman, theclassicalWX-equivariant bilinear maps are (the linear combina-tions of) the mapsBδ1,δ2 Pδ1,δ2, and those obtained by various compositions withd. Grozman discovered one additionalWX-equivariant bilinear map, namely Grozman’s operatorG: Ω−2/3×Ω−2/3→Ω5/3defined by the formula: G(f1.(dz)−2/3 f2.(dz)−2/3) = [2(f0100f2−f0200f1) + 3(f010f02−f01f020)](dz)5/3. With this, one can state Grozman’s result:
Grozman Theorem.Any differentialWX-equivariant bilinear map π: Ωδ1×Ωδ2→Ωγbetween 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 allWX-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=WX. Fix two parametersδ s∈C and setρδ,s(ξ) =ξ+δdivξ+iξαsfor anyξ∈W, whereαs=sz−1dz. By definition, Ωsδis theW-module whose underlying space isC[z z−1] and the action is given byρδ,s. To describe more naturally the actionρδ,s, it is convenient to denote by the symbolzs(dz)δthe generator of this module, and therefore the expressions (zn+s(dz)δ)n∈Zform a basis of Ωsδ follows. It easily that Ωsδand Ωδuare 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 Ωδu, where (δ s) runs overC×C/Z. As before, there areW-equivariant bilinear mapsPuδ11,δ,u22: Ωδu11×Ωuδ22→ Ωδu11++δu22andBδu11,δ,u22: Ωδu11×Ωuδ22→Ωδu11++uδ22+1as well as the de Rham differential, 1 d: Ωu0→Ωu. There is also a mapρ: Ωu1→Ωu0, which is defined as follows. Foru6≡0 modZ, the opeartordis invertible and setρ=d−1. For u≡0 modZ, denote byρ: Ω1u→Ω0uthe composite of the residue map Res : Ω10→Cand the natural mapC→Ω00=C[z z−1 definition,]. By aclassicalbilinear map between tensor density modules over the circle is any linear combination of the operatorsBδu11,δu,22,Puδ11,δu,22and those obtained by composition withdandρ example of a classical operator is. Anρ◦P: . Ωuδ1×Ωu12−δ→Ω0u1+u2 Of course, the Grozman operator provides a family of non-classical op-eratorsGu,v: Ωu−2/3×Ωv−2/3→Ωu5/3+v. Atrivial operatoris a scalar multiple of the bilinear map Ω01×Ω01→Ω00(α β)7→Res(α)Res(β is also). There another non-classicalW-equivariant operator Θ∞: Ω01×Ω01→Ω00which satisfies: dΘ∞(α β) = Res(α)β−Res(β)α for anyα β∈Ω10. 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 ofGu,vor ofΘ∞(modulo a trivial operator).
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In the paper, a more general version, which also involves deformations of tensor densi =zd. For aW-moduleMand ty modules, is proved. SetL0dz s∈C, setMs= Ker(L0−s). LetSbe the class of allWmodulesMwhich satisfies the following condition: there existsu∈C/Zsuch that M=⊕s∈uMsand dimMs= 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 Ω00and Ω10, 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≥xMsand N≥x=⊕<s≥xNs. By definition, agermof bilinear map fromM×NtoPis an element of the spaceG(M×N P lim) :=B(Mx× x→+∞ ≥N≥x P). It turns out thatG(M×N P) is aW by-module. DenoteBW(M×N P) the space ofW-equivariant bilinear mapsπ:M×N→P, byB0W(M×N P) the subspace of allπ∈BW(M×N P) whose germ is zero and byGW(M× N P) the space ofW-equivariant germs of bilinear maps fromM×NtoP. There is a short exact sequence: 0→B0W(M×N P)→BW(M×N P)→ GW(M×N P). The paper contains three parts Part 1 which determinesB0W(M×N P), see Theorem 1, Part 2 which determinesGW(M×N P), see Theorem 2, Part 3 which determines BW(M×N P)→ GW(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 dimGW(M× N P)≤2. So it is enough to determine whenGW(M×N P) is non-zero
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