Deforming the Lie algebra of vector fields on S1 inside the Lie algebra of

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ar X iv :m at h/ 98 12 07 4v 1 [m ath .Q A] 1 1 D ec 19 98 Deforming the Lie algebra of vector fields on S1 inside the Lie algebra of pseudodifferential symbols on S1 V. Ovsienko and C. Roger Dmitri Borisoviqu Fuksu po sluqa xestidestileti Abstract We classify nontrivial deformations of the standard embedding of the Lie algebra Vect(S1) of smooth vector fields on the circle, into the Lie alge- bra ?D(S1) of pseudodifferential symbols on S1. This approach leads to deformations of the central charge induced on Vect(S1) by the canonical cen- tral extension of ?D(S1). As a result we obtain a quantized version of the second Bernoulli polynomial. 1 Introduction The classical deformation theory usually deals with formal deformations of associa- tive (and Lie) algebras. Another part of this theory which studies deformations of Lie algebra homomorphisms, is less known. It should be stressed, however, that, sometimes, this second view point is more interesting and leads to richer results. The Lie algebra of vector fields on the circle Vect(S1) gives an example when such situation occurs. It is well-known that Vect(S1) itself is rigid, but it has many interesting embeddings to other remarkable Lie algebras that can be nontrivially deformed.

semi-classical limits

formal deformations

lie algebra

vect

deformation πt

π˜t contracts

π˜t

poisson algebra

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profil-urra-2012

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Lie algebra Poisson algebra

DeformingtheLiealgebraofvectorﬁeldsonS1insidetheLiealgebraofpseudodiﬀerentialsymbolsonS1V.OvsienkoandC.RogerDmitriBorisoviquFuksuposluqaxestidestiletiAbstractWeclassifynontrivialdeformationsofthestandardembeddingoftheLiealgebraVect(S1)ofsmoothvectorﬁeldsonthecircle,intotheLiealge-braΨD(S1)ofpseudodiﬀerentialsymbolsonS1.ThisapproachleadstodeformationsofthecentralchargeinducedonVect(S1)bythecanonicalcen-tralextensionofΨD(S1).AsaresultweobtainaquantizedversionofthesecondBernoullipolynomial.1IntroductionTheclassicaldeformationtheoryusuallydealswithformaldeformationsofassocia-tive(andLie)algebras.AnotherpartofthistheorywhichstudiesdeformationsofLiealgebrahomomorphisms,islessknown.Itshouldbestressed,however,that,sometimes,thissecondviewpointismoreinterestingandleadstoricherresults.TheLiealgebraofvectorﬁeldsonthecircleVect(S1)givesanexamplewhensuchsituationoccurs.Itiswell-knownthatVect(S1)itselfisrigid,butithasmanyinterestingembeddingstootherremarkableLiealgebrasthatcanbenontriviallydeformed.TheaimofthisarticleistostudytheembeddingsofVect(S1)intotheLiealgebraofpseudodiﬀerentialsymbolsΨD(S1).Wewillclassifythedeformationsofthestandardembeddingwhichpolynomiallydependontheparametersofdeformation.Itturnsoutthatthereexistsathree-parameterfamilyofnontrivialdeformations.Wecomputeauniversalexplicitformuladescribingthisfamily.Thisworkcanbeconsideredasthesecondpartof[12]wheredeformationsofVect(S1)insidethePoissonLiealgebraofLaurentseriesonT∗S1hasbeenclassi-ﬁed.ThelatterLiealgebracanbeconsideredasthesemi-classicallimitofΨD(S1),sothepresentarticleis,insomesense,thequantumversionof[12].

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