The Lie bialgebra of loops on surfaces of Goldman and Turaev

Transcription

1 The Lie bilgebr of loops on surfces of Goldmn nd Turev Consider ny surfce F.Letˆπ be the set of free homotopy clsses of loops on F.LetZ be the vector spce with bsis ˆπ \{null} where null is the homotopy clss of the null loop. We consider the null loop to be 0 in Z. For ny two loops α, β in generl position on F,letα β denote the set of (simple) intersections of α nd β. We cn give Z Lie lgebr structure by tking two loops, nd summing over their intersections of the loop obtined by removing the intersection nd smoothing. Precisely, we get the formul [, b ] = ɛ(q; α, β) α q β q (0.1) q α β where ɛ(q; α, β) =±1 s determined by the figure q q α β β α ε = 1 ε = 1 nd α q β q is the product of the loops α, β bsed t q. Tht is, α q β q = γ where γ is the smoothed version q α β γ The coproduct is given by summing over self-intersections nd removing the intersection point, nd smoothing to obtin two pths. Precisely, let α denote the set of self-intersections of α (ssuming α is in generl position). The coproduct is then given by summing over selfintersections nd removing the intersection point, nd smoothing to obtin two pths. Then δ( α ) = q α α q 1 α q ( b := b b ) (0.) where α 1 q,α q re the two pieces of α determined by cutting α t q s follows:

2 q α α 1 q α q

3 Anlogue: the necklce Lie lgebr of Ginzburg, Bocklndt, nd Le Bruyn This is essentilly replcing the surfce with the double of quiver. Tke quiver Q ( collection of vertices nd directed edges between them, where multiple edges with the sme endpoints re llowed) nd consider, insted of ˆπ, the set of loops in the double quiver Q = Q Q (here Q replces every edge e Q with the reverse edge e, reversing orienttion). As vector spce, our Lie lgebr L hssbsisthissetofloops(withoutpreferred choice of initil rrow). More precisely, let Q be quiver, which we consider the set of rrows, nd let I be the set of vertices. Let Q be the double quiver (for ech e Q, e is the reversed edge). Let E Q be the vector spce with bsis Q. Let B = C I be the semisimple ring with the set I s idempotents. We give E Q B-bimodule structure s follows: for ech rrow e let e s be the strting vertex, nd e t be the terminl vertex. Then we set e s e = e = ee t for ech e Q, while ve =0=ew for ny v e s,w e t.letp be the pth lgebr in Q, i.e. P = T B E Q. Then the Lie lgebr L := P/[P, P]where[P, P] is the commuttor in the ssocitive lgebr P. Note tht ny pth which is not loop is killed in this quotient, nd loops lose their choice of initil rrow. Now we define the Lie brcket structure s follows. Insted of intersections of two pths, we consider pirs of edges e in one of the loops nd e in the other, nd consider tht n intersection. Then we join the two pths together s before by creting γ which smooths thetwo,inthiscsebyremovingthee nd e nd joining the edges. If we consider the e nd e beds tht we cut the two necklces t, nd then join the corresponding ends to form new necklce, we get the picture which gives this Lie lgebr its nme: α e e* β summnd of [ α, β]

4 Precisely, we hve [ 1 k,b 1 b b l ]= [ i, j ]( i ) t i+1 i 1 b j+1 b j 1, (0.3) i,j 1 j =( i ), i Q [ i, j ]= 1 i =( j ), j Q (0.4) 0 otherwise The coproduct is given s follows. Agin, insted of self-intersections, we consider pirs of rrows e, e in pth, nd we then smooth by removing these two rrows, creting two pths which re wedged together: e Precisely, we hve e* α summnd of δ (α) δ( 1 n )= i<j [ i, j ]( j ) t j+1 i 1 ( i ) t i+1 j 1 (0.5)

5 Turev s quntiztion of the Lie bilgebr of loops The quntiztion of Z is bsed on skein lgebr which is closely tied to the Jones polynomil. Tke three-mnifold M. Define Conwy triple to be triple of links (L +,L,L 0 ) in M where ll three links re identicl outside of bll, nd inside the bll the three look like L+ L L 0 Then we let A be the C[h]-module generted by the set of isotopy clsses of links L modulo isotopy clsses of unlinked nullhomotopic loops, nd modulo the reltions L + L h ɛ L 0 (0.6) for ny Conwy triple (L +,L,L 0 ), with ɛ set s follows: { 0 #(L 0 )=#(L + ) 1=#(L ) 1, ɛ = 1 #(L 0 )=#(L + )+1=#(L )+1. This gives skein C[h]-module. (0.7) Now, to define bilgebr, we need to restrict to the cse M = F [0, 1] where F is surfce. Then we get LL = {(, t) F [0, 1] t 1 nd (, t 1) L, or t 1 nd (, t) L }, (0.8) tht is, we stck L on top of L. To define the coproduct, we need to define colorings. Given link L, we mke link digrm of link isotopic to L with only simple crossings. Then, we color L with two colors, 1 nd, such tht ny segment between crossings hs single color, nd t ech crossing q we hve d c c d q q s(q) = 1 b b s(q) = 1

6 where either = c, b = d (ech strnd hs solid color), or else = d>b= c. In the ltter cse the intersection is clled color-cutting intersection. Let f be the number of colorcutting intersections, nd let f be the number of color-cutting intersections of negtive sign s(q). Let L i be the subset of L of color i for i {1, }; L i is lso link. Here we smooth the color-cutting intersections by chnging from n L + /L type crossing to n L 0 -type one. Finlly, let N =#(L) #(L 1 ) #(L ). Now, the coproduct is defined by (L) = colorings ( 1) f h f+n L 1 L. (0.9) The cossocitivity property follows redily from showing tht (1 ) = ( 1) =, where is defined similrly to but with three colors. Well-definition follows from cseby-cse nlysis t n intersection. The bilgebr condition follows by seeing tht colorings of the product of two links must be the product of colorings on ech one (becuse of the coloring rule). I do not know whether there is n ntipode or whether the PBW property is stisfied for this lgebr (I hve not given it gret del of thought nd I didn t see it in Turev s pper).

7 The quntiztion of the necklce Lie lgebr We define Hopf lgebr emulting Turev s construction, using our dictionry between loops on surfces nd loops in quivers, between simple crossings nd pirs of n rrow in Q nd its reverse in Q. Following is digrm of quntum element: Quntum object X: * v b Clssicl Projection: X expressed lgebriclly: Incidence reltions: u * * b* b v b* (,3)(b,4)(b*,1)(*,) & (,6)(*,5) & v u= v= =b w=b s t s w t We will mod out by reltions of the following form, where ɛ {0, 1}: * = * + h ε Precisely, consider the spce of rrows with heights, AH := Q N. LetE Q,h = C AH be the vector spce with bsis AH. This cn lso be viewed s B-module by e s (e, h) =(e, h) = (e, h)e t,ndv(e, h) =0=(e, h)w for ny v e s,w e t.letlh:=t B E Q,h /[TE Q,h,TE Q,h ] to be the spce of cyclic words in AH which form pths once heights re forgotten. There is cnonicl projection LH L given by forgetting the heights. Let SLH[h] be the symmetric lgebr in LH[h] (this is not over B). A monomil here hs the form ( 1,1,h 1,1 ) ( 1,l1,h 1,l1 )&(,1,h,1 ) (,l,h,l )& &( k,1,h k,1 ) ( k,lk,h k,lk )&v 1 & &v m, (0.10) with i,j Q, v i I B. Letà be the subquotient obtined by only considering monomils where every rrow is t distinct height (ll h i,j re distinct), nd so tht we don t cre wht the ctul heights re, only the order in which they pper. (So we think of this s creting link by putting ech rrow t discrete height, connected together by verticl segments).

8 We mod out by the reltions X X i,j,i,j hδ i,i X i,j,i,j, where i i,h i,j <h i,j, nd (i,j )withh i,j <h i,j <h i,j (0.11) wherewethinkofx s the opposite crossing, nd X s the smoothing nlgous to the smoothing we did in Turev s bilgebr. Precisely, when i i, X i,j,i,j is the sme s X but with the heights h i,j nd h i,j interchnged, nd X i,j,i,j replces the components ( i,1,h i,1 ) ( i,li,h i,li )nd( i,1,h i,1) ( i,l i,h i,l i ) with the single component [ i,j, i,j ]( i,j) t ( i,j+1,h i,j+1 ) ( i,j 1,h i,j 1 )( i,j +1,h i,j +1) ( i,j 1,h i,j 1). (0.1) Similrly, X i,j,i,j isthesmesx but with the heights h i,j nd h i,j interchnged, nd X i,j,i,j is given by replcing the component ( i,1,h i,1 ) ( i,li,h i,li ) with the two components [ i,j, i,j ]( i,j ) t ( i,j +1,h i,j +1) ( i,j 1,h i,j 1 )&( i,j ) t ( i,j+1,h i,j+1 ) ( i,j 1,h i,j 1). (0.13) Our quntiztion A of L is given, s module, by quotienting à by the reltions (0.11). The product is gin given by putting one loop on top of the other. Tht is, XX hs the heights of X followed by the heights of X, which re ll greter thn the heights of X but both re in the sme order. The identity is the empty loop, i.e. the one coming from 1 SLH. The coproduct is given gin by colorings: we sum over ll wys of picking rrow nd reversed-rrow pirs to be color-cutting intersections, nd we smooth ech color-cutting intersection by removing the reversed rrow pirs, while multiplying in n pproprite sign nd powers of h. Then the color-1 prt is tensored by the color- prt with the forementioned coefficient. Following is n exmple: 1 Let P := {(i, j) 1 i k, 1 j l i } be our set of pirs, which hs ddition defined by (i, j)+j := (i, j + j ):=(i, j ), where 1 j l i nd j + j j (mod l i ). (0.14) We lso use the nottion (i,j) := i,j nd h (i,j) := h i,j.now,wechoosesetofpirsi P together with self-piring φ : I I tht is involutive nd hs no fixed points, nd which stisfies the condition: For ny (i, j) I where φ(i, j) =(i,j ), we hve l i,l j > 0nd i,j = i,j. Also, let V = {1,...,m} correspond to the vertex idempotents in (0.10). Then, coloring of X with (I,φ)-cutting pirs is mpping c : P V {1, } stisfying the conditions:

9 (1) For ech (i, j) P \ I, wehvec(i, j) =c(i, j + 1); () for ech (i, j) I, wehve c(i, j) =c(φ(i, j)+1) c(i, j +1) = c(φ(i, j)), nd we hve c(i, j) >c(φ(i, j)) iff h i,j >h φ(i,j). Given coloring (I,φ,c), we define mp f : P P by f(i, j) =(i, j)+1if(i, j) / I nd f(i, j) =φ(i, j) + 1 otherwise. Note tht in the ltter cse, i,j = φ(i,j)+1.notelsothtf is invertible by f 1 (i, j) =(i, j) 1if(i, j) 1 / I nd f 1 (i, j) =φ((i, j) 1) otherwise. Then we cn prtition P into orbits under f, P = P 1 P q. Also, note tht ech orbit P i is monochrome: c(p i )={t} for some 1 t n. For ech orbit P i we define corresponding element Y i of A s follows: Suppose P i = {x 1,...,x p } P for f(x i )=x i+1 nd f(x p )=x 1. Then, for ech x i we let y i =( xi,h xi )ifx i / I nd y i =( xi ) s otherwise. Then we set Y i = y 1 y p LH A. Let us suppose tht the Y i re rrnged so tht Y 1,...,Y r hve color 1 nd Y r+1,...,y q hve color. Also suppose tht we order V so tht v 1,...,v u hve color 1 nd v u+1,...,v m hve color. To define the sign, prtition I into I Q = {(i, j) I i,j Q} nd I Q = {(i, j) I i,j Q },sothtφ(i Q )=I Q nd vice-vers. For ech (i, j) I Q set s i,j =1ifh i,j <h φ(i,j) nd s i,j = 1 otherwise. Now, define the sign s(i,φ,c) s follows: Reclling k from (0.10), we set n 1 (X) = n colorings(i,φ,c) s(i,φ,c)= (i,j) I Q s i,j. (0.15) N = k q, (0.16) s(i,φ,c)h #(I)/4+N / X 1 I,φ,c Xn I,φ,c, (0.17) extending it C[h]-linerly to ll of A. Herenotetht#(I)/4 is the sme s hlf the number of color-cutting intersections, nd N = k q = k u (r u) is the nlogue to the previous N. As before, cossocitivity is proved by defining colorings on three colors:

10 well-definition is proved by nlysis t n intersection, nd the bilgebr condition is proved by showing tht colorings of XX come from products of colorings on X with colorings on X. By discreteness, it is esy to define n ntipode (becuse color-cutting intersections in coproducts decrese the number of rrows). We lso hve the PBW property: Theorem 0.1. A is isomorphic s C[h]-module to UL = SL, by descending the forgetful mp à SL. I proved PBW using detiled combintoril rgument bsed on Bergmn s Dimond Lemm.

11 References [Ber78] George M. Bergmn, The dimond lemm for ring theory, Adv. in Mth. 9 (1978), no., [BLB0] Rf Bocklndt nd Lieven Le Bruyn, Necklce lie lgebrs nd noncommuttive symplectic geometry, Mth.Z.40 (00), no. 1, [Gin01] Victor Ginzburg, Non-commuttive symplectic geometry, quiver vrieties, nd operds, Mth. Res. Lett. 8 (001), no. 3, [Kon93] Mxim Kontsevich, Forml (non)commuttive symplectic geometry, TheGelfnd Mthemticl Seminrs, , Birkhäuser Boston, Boston, MA, 1993, pp [Sch04] [Tur91] Trvis Schedler, A Hopf lgebr quntizing necklce Lie lgebr cnoniclly ssocited to quiver, mth.qa/ (004). Vldimir G. Turev, Skein quntiztion of poisson lgebrs of loops on surfces, Ann. Sci. École Norm. Sup. (4) 4 (1991), no. 6,