Symmetrization maps and differential operators. 1. Symmetrization maps

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1 July 26, 2011 Symmetrizatio maps ad differetial operators Paul Garrett garrett/ The symmetrizatio map s : Sg Ug is a liear surjectio from the symmetric algebra Sg to the uiversal evelopig algebra Ug of a algebra g, completely characterized by beig the idetity map o g ad o the scalars. This map is peculiar, effectively attemptig to parametrize a o-commutative algebra by a commutative oe. It is liear, but caot quite be a rig homomorphism. Nevertheless, begiig with Harish-Chadra s work, the symmetrizatio map plays a importat role. The itet is to have a commutative algebra be mapped as surjectively as possible to a o-commutative algebra by a liear map as much a algebra homomorphism as possible. These are coflictig requiremets. Give the techicality of this map, coordiate-free characterizatio is all the more importat. Throughout, k is a field of characteristic 0. All algebras are k-algebras, i particular requirig that k is i the ceter. Uless specifically desigated as algebras, all algebras are associative. 1. Symmetrizatio maps [1.1] What should a symmetrizatio map be? Of course, commutative algebras caot liearly surject to o-commutative algebras without losig the algebra homomorphism property, leavig the mystery of what structure might remai. For a commutative algebra S ad a arbitrary associative algebra A, a requiremet that a k-liear map f : S A be a algebra homomorphism sharply restricts the image fs: it must lie iside a commutative sub-algebra of A. O the other had, for A = Ug the uiversal evelopig algebra of a algebra g, the o-commutativity is ot severe, sice Ug is commutative modulo lower-degree terms, as we will see i the proof of surjectivity below. I other words, the associated graded algebra of the filtratio by degree o Ug is commutative, so is the uiversal commutative algebra Sg o the vector space g. There is the atural algebra homomorphism F : U g Sg, which reasoably-eough has a large kerel, geerated by commutators xy yx for x, y g. So, agai, it is ureasoable to hope for a two-sided iverse to F, but it is plausible to ask for a merely-liear right iverse s : Sg Ug. Some further algebraic structure must be required, or such a map is certaily ot uique, ad, cocommitatly, probably ot useful. The desired sort of liear map s : S A from a commutative algebra S to a ot-ecessarily-commutative algebra A ought to be as much a algebra homomorphism as possible, meaig that wheever sx ad sy commute, we should have sxy = sxsy. However, the oly systematic thig that ca be said is that sx commutes with itself, so the oly uiversally safe coditio to impose is sx = sx This may seem very weak, but the multiomial theorem effectively exploits this, over a field of characteristic 0. For example, s2xy = s x+y 2 x 2 y 2 = s x+y 2 sx 2 sx 2 = sx+y 2 sx 2 sy 2 = sx sy+sy sx I fact, by slightly more elaborate idetities below the coditio sx = sx for a collectio {x} of geerators ca be used to completely determie the liear map s. 1

2 Paul Garrett: Symmetrizatio maps ad differetial operators July 26, 2011 Thus, a symmetrizatio map s : Sg Ug is required to be the idetity o g, to be liear, ad to have the property sx = sx for all x g. Proof is required that such a map exists, is uique, ad gives a liear isomorphism. [1.2] Uiversal algebras We will prove that ay symmetrizatio map s : Sg Ug satisfies sx 1... x = 1 x 1... x for x 1,..., x g where S is the permutatio group o {1, 2,..., }. I fact, this idetity has othig to do with algebras g, isofar as it holds for a over-lyig symmetrizatio map t : S from the symmetric algebra to the uiversal associate algebra for ay vector space. The characterizatio of is that it has the followig uiversal property: there is a liear map such that, for every liear map A to a associative algebra A, there is a uique algebra map A through which the origial A factors. The diagram is A That is, the fuctor takig to is a left adjoit to the forgetful fuctor F that seds a associative algebra to the uderlyig vector space: for every associative algebra A, Hom algebras, A Hom vectorspaces, F A The costructio, as proof of existece, of is by tesors: = where is the uiversal object for -multi-liear maps from... : there is a fixed -multi-liear... such that every -multi-liear... W factors through a uique liear map W. The diagram is... W The multiplicatio o is give summad-wise m by the iocuous =0 m+ u 1... u m v 1... v u 1... u m v 1... v Similarly, S is the uiversal commutative algebra over a vector space : there is a liear map S such that, for every liear map A to a commutative algebra A, there is a uique algebra map A through which the origial A factors. The diagram is S A 2

3 Paul Garrett: Symmetrizatio maps ad differetial operators July 26, 2011 That is, the fuctor takig to S is a left adjoit to the forgetful fuctor F that seds a commutative algebra to the uderlyig vector space: for every associative algebra A, Hom commutative S, A Hom vectorspaces, F A [1.3] Uiversal symmetrizatio map To avoid presumig well-defiedess, ad to avoid coordiatedepedecy issues, we first defie a uiversal symmetrizatio map. Say that a liear map f : S A of S to a associative algebra A is a symmetrizatio map if it is liear ad if fx = fx for all x. The uiversal symmetrizatio map j : S Q is a symmetrizatio map to a associative algebra Q such that, give aother symmetrizatio map S A, there is a uique algebra homomorphism Q A through which S A factors. That is, we have a diagram sym Q alg S A The usual categorical argumet gives uiqueess up to uique isomorphism, assumig existece. Sice the idetity map S S is a symmetrizatio map, ad is ijective o, ecessarily the copy of iside S ijects to Q. sym Existece of the uiversal symmetrizatio is straighforward, as follows. For a vector space, let i : S S be the atural ijective liear map. Let I be the two-sided ideal i S geerated by all images i sx sx for x, ad let Q = / S I be the quotiet, with j : S Q the atural liear map. The uiversal properties of S yield the desired uiversal properties of j ad Q. [1.4] The uiversal formula We ca deduce formulas i Q for jx 1... x i terms of the jx l. For example, from jx + y 2 = jx + y 2 for x, y g we obtai ad the deduce jx 2 + 2jxy + jy 2 = jx + jy 2 = jx 2 + jxjy + jyjx + jy 2 jxy = 1 2 jxjy + jyjx Amog may possible approaches to obtai the geeral expressio, we ca cosider scalars t 1,..., t ad x 1,..., x i g, ad expad jt 1 x t x two differet ways, with the expoet the same as the umber of summads. Without writig out either expressio etirely, over a field of characteristic 0 equality of the two sides for all scalars t i implies equality of the two sides as polyomials i idetermiates t i with values i Q. Equatig the coefficiets of the middle term t 1... t gives jx 1... x = jx 1... jx which gives the uiversal formula for the uiversal symmetrizatio map: jx 1... x = 1 jx 1... jx 3

4 Paul Garrett: Symmetrizatio maps ad differetial operators July 26, 2011 The formula shows that the map Q iduced from S, composed with the quotiet S Q is surjective, sice the formula exhibits every elemet of Q as a liear combiatio of moomials i elemets jx with x. Agai, sice the idetity S S is ijective o, ijects to Q. Thus, sice geerates ad ay algebra homomorphism image thereof, the image j geerates Q. [1.5] Caoical symmetrizatio map to There is a caoical symmetrizatio-like map S. This also depeds upo the uderlyig field beig of characteristic 0. From their characterizatios as uiversal algebra ad uiversal commutative algebra for, there is a caoical surjectio S with kerel the two-sided ideal geerated by commutators xy yx with x, y i. This quotiet respects the gradig by degree, ad is the direct sum of the caoical maps q : Sym For each, we will costruct a liear sectio s : Sym, that is, a liear map such that q s is the idetity map o Sym. Each elemet of the permutatio group S o thigs gives a multiliear map :... }{{} by x 1... x = x 1... x ad thus gives a uique map of to itself. The symmetric th power Sym is the S co-fixed vectors i, that is, the largest S -quotiet of o which S acts trivially. O the other had, sice the characteristic is 0, there is a averagig map α of to the S fixed vectors i, by α β = 1 β for β This map is visibly the idetity map o fixed vectors. Thus, there is a direct sum decompositio = ker α fixed vectors where kerα cotais o o-zero fixed vectors. We claim that the S -fixed vectors i map isomorphically to the S -cofixed vectors Sym. O oe had, the map of to its ow fixed vectors by α must factor through q, so the fixed vectors iject to the co-fixed vectors. To prove surjectivity, we make the subordiate claim that the kerel K of the quotiet map is geerated by elemets m m for m ad S. Certaily this maps to 0 uder ay S -homomorphism to a S -module o which S acts trivially. O the other had, S acts trivially o the quotiet of by K, sice m + K = m + m m + K = m + K This is the sub-claim. The q α m = q 1 m = 1 q m = 1 q m = q m 4

5 Paul Garrett: Symmetrizatio maps ad differetial operators July 26, 2011 That is, every elemet q m i the quotiet Sym is hit by a fixed vector, amog them the fixed vector α m. This proves that the fixed vectors surject to the co-fixed vectors, ad, thus, map isomorphically. Let σ : Sym be the iverse of the isomorphism of fixed to co-fixed vectors, ad let σ : S be the direct sum of these isomorphisms. We claim that σ is a symmetrizatio map, meaig that σx = σx for x. Note that for x the elemet x S is q x... x. Coveietly, x... x is already a fixed vector, ad the averagig map is the idetity o it. Thus, σx = α q x... x = x... x = σx... σx = σx Thus, ideed, σ : S is a symmetrizatio map. [1.6] The uiversal symmetrizatio idetified Now we will see that the caoical symmetrizatio σ : S is uiversal. Amog all the other symmetrizatio-like maps from S, we have this caoical σ to. Thus, this σ must factor through a uique algebra homomorphism σ : Q from the uiversal symmetrizatio j : S Q. Ad Q is a caoical image f of. This fits ito a diagram S S j ic σ f Q σ Thus, j : S f ad σ : S are both symmetrizatio maps, ad σ factors through j. As observed above, j is ijective o the copy of i S, ad j geerates Q as a algebra. Likewise, σ is ijective o, ad obviously σ geerates as a algebra. Therefore, the composite Q must be the idetity o. That is, Q =, ad the caoical symmetrizatio σ : S is the uiversal oe. I particular, σ is the uique symmetrizatio map S with the property of beig the idetity o the copies of. [1.7] Symmetrizatio Sg Ug Now retur to algebras = g. Note that ay associative algebra A has a atural algebra structure give by [x, y] = xy yx. The the uiversal evelopig algebra Ug of g is a associative algebra characterized as follows. There is there is a algebra map i : g Ug such that, for every algebra map g A to a associative algebra A, there is a uique associative algebra map Ug A through which the origial g A factors. The diagram is Ug assoc g A That is, the fuctor takig g to Ug is a left adjoit to the forgetful fuctor F that seds a associative algebra to the uderlyig vector space: for every associative algebra A, Hom assoc Ug, A Hom g, F A 5

6 Paul Garrett: Symmetrizatio maps ad differetial operators July 26, 2011 The Poicaré-Birkhoff-Witt theorem proves that i ijects g to U g. This categorical characterizatio shows that i : g Ug is uique up to uique isomorphism, if it exists. To prove existece, costruct Ug as the quotiet of g by the two-sided ideal geerated by all xy yx [x, y] with x, y g, deducig the desired properties of Ug from those of g. We wat a symmetrizatio map s : Sg Ug with s the idetity o the copies of g. Sice Ug is geerated by the image of g, there is at most oe such symmetrizatio, give by the uiversal formula above, if a symmetrizatio map exists. For existece, let s : Sg Ug be the compositio s Sg σ g quot Ug As a corollary, this symmetrizatio s : Sg Ug is the uiversal symmetrizatio map for algebras g, i the followig sese. For a symmetrizatio map f : Sg A to a associative algebra A which is a algebra map o g, there is a uique associative algebra homomorphism Ug A through which f factors by s : Sg Ug. That is, Ug sym g liear Sg assoc sym A [1.7.1] Remark: This defiitio produces the same outcome as attemptig to defie a symmetrizatio map Sg Ug by takig the idetity map o g ad extedig it by the uiversal formula from above. However, doig this directly faces two possible difficulties. First, it is ot clear a priori that there exists ay s : Sg Ug with sx = sx for all x g, ad s is the idetity o g, because this might impose mutually coflictig coditios. Secod, we might attempt to avoid potetial coflicts by forgettig about the requiremets sx = sx, istead merely choosig a basis {x i } for g, takig the correspodig basis for Sg cosistig of elemets x i1... x i where i 1 i 2... i ad defiig a liear map s : Sg Ug o that basis by the i fact, uiversal formula sx i1... x i = 1 x i1... x i Ug where i 1 i 2... i Amog other flaws, it is ot clear that this is idepedet of the basis for g, ad it is surely a fool s errad to try to prove it by choosig two bases for g ad comparig. [1.8] Surjectivity of Sg Ug It is mildly surprisig that the symmetrizatio map s : Sg Ug is surjective, so is a liear isomorphism. The surjectivity is ot merely a curiosity. Let g = i 0 i g < g = i 0 i< g 6

7 Paul Garrett: Symmetrizatio maps ad differetial operators July 26, 2011 ad S g, S g, U g, U < g the correspodig images. I all cases, the parameter is the degree. We prove that S g surjects to U g, by iductio o degree. At degree 0 the scalars, by covetio, the symmetrizatio map is the idetity. At degree 1, the symmetrizatio map is the idetity, beig the idetity o g by defiitio. Cosider a moomial u = x 1... x U g Every permutatio S is a product = σ 1... σ l of adjacet traspositios σ i,i+1, ad σ i,i+1 u = x σi,i x σi,i+1 = x 1... x i 1 x i+1 x i x i+2 x = x 1... x i 1 [x i + 1, x i ]x i+2 x + x 1... x i 1 x i x i+1 x i+2 x U < g + u That is, by iductio o the umber of adjacet traspositios eeded to express a permutatio, u u U < g I particular, u 1 u U < g ad the sum is i the image of the symmetrizatio map, by the uiversal formula. This proves the surjectivity. Ijectivity follows from the Poicaré-Birkhoff-Witt theorem. Thus, the symmetrizatio map is a liear isomorphism Sg Ug. 7