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IMRN International Mathematics Research Notices 2002, No. 16 Multi-Parameter Deformations of the Module of Symbols of Differential Operators B. Agrebaoui, F. Ammar, P. Lecomte, and V. Ovsienko 1 Introduction In this paper, we discuss the following general principle. Given a Lie algebra g and a g- module V , one can canonically associate to (g, V) a commutative associative algebra. The generators of this commutative algebra are the nontrivial cohomology classes in H 1 (g;End(V)), while the relations between the generators are encoded by elements of H 2 (g;End(V)). More precisely, the relations correspond to the obstructions for integra- bility of infinitesimal deformations of V . The classical deformation theory of Lie algebras and modules over Lie algebras traditionally deals with one-parameter deformations (cf. [13, 14, 21, 24]). It is, however, natural to consider, as in other deformation theories, “multi-parameter” deformations, that is, deformations of Lie algebras over commutative algebras. This viewpoint has been adopted in [9], and the existence of the so-called miniversal deformation has been proven. A construction of miniversal deformations of Lie algebras was given in [10].
Voir
- been calculated
- lie algebra
- module over
- structure has
- any deformation
- multi-parameter deformations
- commutative algebras
- algebra intrinsically associated
- lie algebras
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IMRNInternationalMathematicsResearchNotices2002,No.16Multi-ParameterDeformationsoftheModuleofSymbolsofDifferentialOperatorsB.Agrebaoui,F.Ammar,P.Lecomte,andV.Ovsienko1IntroductionInthispaper,wediscussthefollowinggeneralprinciple.GivenaLiealgebragandag-moduleV,onecancanonicallyassociateto(g,V)acommutativeassociativealgebra.ThegeneratorsofthiscommutativealgebraarethenontrivialcohomologyclassesinH1(g;End(V),whiletherelationsbetweenthegeneratorsareencodedbyelementsofH2(g;End(V).Moreprecisely,therelationscorrespondtotheobstructionsforintegra-bilityofinfinitesimaldeformationsofV.TheclassicaldeformationtheoryofLiealgebrasandmodulesoverLiealgebrastraditionallydealswithone-parameterdeformations(cf.[13,14,21,24]).Itis,however,naturaltoconsider,asinotherdeformationtheories,“multi-parameter”deformations,thatis,deformationsofLiealgebrasovercommutativealgebras.Thisviewpointhasbeenadoptedin[9],andtheexistenceoftheso-calledminiversaldeformationhasbeenproven.AconstructionofminiversaldeformationsofLiealgebraswasgivenin[10].Similarmethodswasappliedin[22,23]todeformationsofhomomorphismsofsomeinfinite-dimensionalLiealgebras.ThecanonicalnotionofminiversaldeformationleadstoanaturalcommutativealgebraintrinsicallyassociatedwithaLiealgebra(orwithamoduleoveraLiealgebra).Thisinterestingalgebraiccharacteristicdeservesafurtherinvestigation.Inthispaperweconsiderthespace,D(Rn),oflineardifferentialoperatorsonRnviewedasamoduleovertheLiealgebra,Vect(Rn),ofsmoothvectorfieldsonRn.Thismodulestructurehasbeenrecentlystudiedin[3,4,5,6,7,12,16,17,20](seealsoReceived30January2001.Revisionreceived23November2001.
848B.Agrebaouietal.thereferencestherein).Themoduleofdifferentialoperatorscanbeviewedasadeforma-tionofthecorrespondingmoduleofsymbols;thegeneralframeworkofthedeformationtheory(see,e.g.,[10,11,13,14,21,24]),therefore,relatesthisstudytothecohomologyoftheLiealgebraofvectorfields(cf.[7,17]).Themainpurposeofthispaperistointroducethecommutativealgebraassoci-atedtotheVect(Rn)-moduleofsymbolsofdifferentialoperatorsonRn.Geometricallyspeaking,symbolsaresymmetriccontravarianttensorfieldsonRn,or,inotherwords,polynomialfunctionsonT∗Rn.Wewilldescribetheminiversaldeformationofthismodule.LetFλbethespaceoftensordensitiesofdegreeλ∈RonRn.Thetwo-parameterfamilyofVect(Rn)-modules,Dλ,µ,oflineardifferentialoperatorsfromFλtoFµwillprovideuswithanimportantclassofexamplesofnontrivialdeformationsofthemoduleofsymbols.ThefirstcohomologyspaceoftheLiealgebraofvectorfields,classifyingtheinfinitesimaldeformationsofthemoduleofsymbolshasbeencalculated,foranarbi-trarysmoothmanifold,in[17](seealso[3]forthedetailsintheone-dimensionalcase).Ofcourse,notforeveryinfinitesimaldeformationthereexistsaformaldeformationcon-tainingthelatterasaninfinitesimalpart.TheobstructionsarecharacterizedintermsofNijenhuis-Richardsonproductsofnontrivialfirstcohomologyclasses.Themainprob-lemconsideredinthispaperistodeterminetheintegrabilitycondition,thatis,anec-essaryandsufficientconditionforaninfinitesimaldeformationthatguaranteesexis-tenceofaformaldeformation.WeprovidesuchaconditioninthecaseofRn,where.2≥n2ThegeneralframeworkWestartwiththenotionof(multi-parameter)deformationsoveracommutativealgebra.Ourapproachwillbesimilartothoseof[22,23];itcorrespondstothenotionofminiversaldeformations[10]inaspecialcasewhenonecanchooseabasisofthefirstcohomologyspace.2.1DeformationsovercommutativealgebrasConsideraLiealgebragoverC(orR)and(V,ρ)ag-module,whereVisavectorspaceandρisahomomorphismρ:g→−dnE(V.).2()1
