IMRN International Mathematics Research Notices No

B. Agrebaoui, F. Ammar, P. Lecomte, And V. Ovsienko

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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].

• been calculated


• lie algebra


• module over


• structure has


• any deformation


• multi-parameter deformations


• commutative algebras


• algebra intrinsically associated


• lie algebras

Sujets

• Lie algebra

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 60
• Langue: English

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IMRN InternationalMathematicsResearchNotices 2002,No.16 Multi-ParameterDeformationsoftheModule ofSymbolsofDifferentialOperators B.Agrebaoui,F.Ammar,P.Lecomte,andV.Ovsienko 1Introduction Inthispaper,wediscussthefollowinggeneralprinciple.GivenaLiealgebra g anda g - module V ,onecancanonicallyassociateto (g,V) acommutativeassociativealgebra. Thegeneratorsofthiscommutativealgebraarethenontrivialcohomologyclassesin H 1 (g; End (V) ,whiletherelationsbetweenthegeneratorsareencodedbyelementsof H 2 (g; End (V) .Moreprecisely,therelationscorrespondtotheobstructionsforintegra- bilityofinfinitesimaldeformationsof V . TheclassicaldeformationtheoryofLiealgebrasandmodulesoverLiealgebras traditionallydealswithone-parameterdeformations(cf.[13,14,21,24]).Itis,however, naturaltoconsider,asinotherdeformationtheories,“multi-parameter”deformations, thatis,deformationsofLiealgebrasovercommutativealgebras.Thisviewpointhas beenadoptedin[9],andtheexistenceoftheso-calledminiversaldeformationhasbeen proven.AconstructionofminiversaldeformationsofLiealgebraswasgivenin[10]. Similarmethodswasappliedin[22,23]todeformationsofhomomorphismsofsome infinite-dimensionalLiealgebras. Thecanonicalnotionofminiversaldeformationleadstoanaturalcommutative algebraintrinsicallyassociatedwithaLiealgebra(orwithamoduleoveraLiealgebra). Thisinterestingalgebraiccharacteristicdeservesafurtherinvestigation. Inthispaperweconsiderthespace, D(R n ) ,oflineardifferentialoperatorson R n viewedasamoduleovertheLiealgebra,Vect (R n ) ,ofsmoothvectorfieldson R n . Thismodulestructurehasbeenrecentlystudiedin[3,4,5,6,7,12,16,17,20](seealso Received30January2001.Revisionreceived23November2001.

848B.Agrebaouietal. thereferencestherein).Themoduleofdifferentialoperatorscanbeviewedasadeforma- tionofthecorrespondingmoduleofsymbols;thegeneralframeworkofthedeformation theory(see,e.g.,[10,11,13,14,21,24]),therefore,relatesthisstudytothecohomology oftheLiealgebraofvectorfields(cf.[7,17]). Themainpurposeofthispaperistointroducethecommutativealgebraassoci- atedtotheVect (R n ) -moduleofsymbolsofdifferentialoperatorson R n .Geometrically speaking,symbolsaresymmetriccontravarianttensorfieldson R n ,or,inotherwords, polynomialfunctionson T ∗ R n .Wewilldescribetheminiversaldeformationofthis module. Let F λ bethespaceoftensordensitiesofdegree λ∈R on R n .Thetwo-parameter familyofVect (R n ) -modules, D λ,µ ,oflineardifferentialoperatorsfrom F λ to F µ will provideuswithanimportantclassofexamplesofnontrivialdeformationsofthemodule ofsymbols. ThefirstcohomologyspaceoftheLiealgebraofvectorfields,classifyingthe infinitesimaldeformationsofthemoduleofsymbolshasbeencalculated,foranarbi- trarysmoothmanifold,in[17](seealso[3]forthedetailsintheone-dimensionalcase). Ofcourse,notforeveryinfinitesimaldeformationthereexistsaformaldeformationcon- tainingthelatterasaninfinitesimalpart.Theobstructionsarecharacterizedintermsof Nijenhuis-Richardsonproductsofnontrivialfirstcohomologyclasses.Themainprob- lemconsideredinthispaperistodeterminetheintegrabilitycondition,thatis,anec- essaryandsufficientconditionforaninfinitesimaldeformationthatguaranteesexis- tenceofaformaldeformation.Weprovidesuchaconditioninthecaseof R n ,where . 2≥n 2Thegeneralframework Westartwiththenotionof(multi-parameter)deformationsoveracommutative algebra.Ourapproachwillbesimilartothoseof[22,23];itcorrespondstothenotionof miniversaldeformations[10]inaspecialcasewhenonecanchooseabasisofthefirst cohomologyspace. 2.1Deformationsovercommutativealgebras ConsideraLiealgebra g over C (or R )and (V,ρ) a g -module,where V isavectorspace and ρ isahomomorphism ρ:g→− dnE (V.) .2()1