Universal Vassiliev Invariants

A universal Vassiliev invariant of links onΣis a mapV: L^pΣqÑC^pΣq±8_m0C^pmqpΣqof filtered vector spaces such that

VGrλId_CpΣq. (4.2)

We refer to (4.2) as the defining equation of a universal Vassiliev invariant; it implies the following corollary to Proposition 4.2.

Proposition 4.3. If V is a universal Vassiliev invariant for Σ, then λ:C^{pΣq Ñ} LGr^pΣq is an isomorphism of graded Poisson algebras with inverseλ¹VGr.

Assume for the remainder of this chapter that^BΣ H. The construction in [AMR2] of a universal Vassiliev invariant of links on Σbuilds on the construction in [B] of a universal Vassiliev invariant of non-associative tangles in the standard square S II with top I^t1^uand bottomI^t0^u; by this we mean a family of filtered, linear maps parametrized by all boundary markings onS:

V:L^pS;^Bpq,^Bpq

qÑC^pS;^B ,^Bq

and satisfying the obvious analogue of (4.2). It will be useful to have refined versions of the chord tangle spaces. To be specific, let T ^„ S be a tangle (chord tangle without chords), and defineC^pS;T^q„C^pS;^B T,^BT^qto be the (homogeneous) subspace generated by chord tangles with skeletonT. Aperturbation of the skeletonis any element^pPm^qPC^pS;T^q such that P0 T. Also, we allow the second factor ofSto shrink and stretch so that the spaces of tangles in this square are closed under composition. In particular, this means thatC^pS;^ÒÒqis an algebra with composition as multiplication, the unit being the trivial tangle ^{Ò Ò}. To define V we fix, once and for all, two parameters, theassociator Φ P

C^pS;^ÒÒÒqand theR-matrix R ^PC^pS;^ÒÒqsubject to various conditions (cf. [B]); we mention a couple of them. BothΦandRare perturbations of their skeletons; this implies that they are invertible elements in the algebras they belong to. Also,Rsatisfies the identity

RR¹ higher degree terms. (4.3)

One constructs an element C ^P C^pS;^Òqin terms ofΦ; it is a perturbation of its skeleton.

Bending Cappropriately, it may be regarded as a member of either of the spacesC^pS; x

q

andC^pS;x^{q; we put} V

R, V

R¹ , (4.4a)

V

Φ, V

Φ¹, (4.4b)

V C, V C. (4.4c)

Any non-associative tangle inSmay be obtained from the above six elementary ones by ca-bling, extension and composition, so requiring thatVis compatible with these operations on non-associative tangles and chord tangles, of course, (over)determinesV. Only the care-ful choice of^pΦ,R^qensures that this procedure leads to well-defined mapsV:L^pS;^Bpq,^Bpq

q

4.3 Universal Vassiliev Invariants 39

Figure 4.2: A partition ofΣ_0,3.

· · · ·

Figure 4.3: The hexagon tangle.

Ñ C^pS;^B ,^Bq. That V is filtered and satisfies the defining equation of a universal Vas-siliev invariant follows from (4.3) and the fact thatΦ,Rand Care perturbations of their skeletons. Note that ifT^„Sis a non-associative tangle, then

V^pT^qPC^pS;π^pT^qq. (4.5)

The above construction yields, of course, a universal Vassiliev invariant of non-associative tangles in any embedded squareS^„Σ; we use this below.

Now assume thatΣis not itself a square. The universal Vassiliev invariant of links on Σdepends not only on^pΦ,R^qbut also on apartitionofΣ. A partitionPis determined by a finite collection of embedded intervals^pIk,^BIk^q„pΣ,^BΣqchosen such that cuttingΣalong these intervals results in a decomposition

ΣpYiSi^qYpYjHj^q

consisting of squaresS_i and hexagons H_j. The sides of these polygons are alternately an interval I_k and a piece of^BΣ. A possible partition of the three-holed sphereΣ_0,3 is illus-trated in Figure 4.2. For technical reasons we assume that no two hexagons are adjacent;

in particular, the decomposition contains at least one square. (By the Euler characteristic the number of hexagons is constant, but we will not use that). Also part of the structure is a choice of top and bottom on all polygons; for a square this means that one of the two embedded intervals bordering it is the top of that square and the other one is the bottom, whereas for a hexagon either the top or the bottom consists of two of theIkbordering it and the opposite side is the remaining one of these intervals. The choice of tops and bottoms must be consistent, i.e., result in an unambiguous direction ‘up’ onΣ.

LetLbe a link onΣ. By isotopy we assume thatLis in general position with respect to the embedded intervals and that each intersectionL^XHj looks like Figure 4.3. (possibly turned upside down). Now choose parenthesizations of all the setsL^XIk subject to the condition that the parenthesization of the top (bottom) endpoints of a hexagon is the union of the parenthesizations of the bottom (top) endpoints. In this way all intersectionsL^XS_i are non-associative tangles, and we can define

VP^pL^q

¹

i

V^pL^XSi^q

¹

j

π^pL^XHj^qPC^pΣq where the product is composition of chord tangles.

Theorem 4.4 (Andersen, Mattes & Reshetikhin). The map VP:L^pΣqÑ C^pΣqis a universal Vassiliev invariant.

Notice that for any linkLonΣ

VP^pL^qPC^pΣ;π^pL^qq (4.6)

40 Chapter 4Quantization of the Algebra of Chord Diagrams

Sl S Sr

Figure 4.4: A squareS0containing another squareS.

because of (4.5). By definition of the universal Vassiliev invariant of non-associative tangles in a square it is also clear thatVPis compatible with cabling of links and chord diagrams.

It is natural to ask howVPdepends onP. On one hand, we can refinePby bisecting one of the squaresSi with an extra embedded interval running parallel to the top and bottom ofSi. This modification leavesVPunchanged since the universal Vassiliev invariant forSis compatible with composition. On the other hand, we can consider the action ofΓ pΣqon the setP^pΣqof all partitions ofΣ; given a mapg^PΓ pΣq, the imageg^pP^qof the embedded intervals constitutingPis another partition. Evidently

V_gpP^qpg^pL^qqg^pVP^pL^qq, L^PL^pΣq. (4.7) We now introduce a useful computational tool, namely universal Vassiliev invariants of non-associative tangles inΣS. They also depend on a partitionPofΣ, now required to becompatiblewithSin the sense that there exists a squareS0inPcontainingSas depicted in Figure 4.4. For technical reasons we also assume thatS0is not adjacent to a hexagon; this is no restriction since we can refineP. We begin with the easy case whenS S0(so that Sl Sr ^Hq. For a tangleT^PL^pΣS0;^BpqpΣS0^q,^Bpq

pΣS0^qqwe proceed as for links and define

VP^pT^q

¹

i^0

V^pT^XSi^q

¹

j

π^pT^XHj^qPC^pΣS0;^{B p}ΣS0^q,^BpΣS0^qq

remembering, of course, that the parentheses on T^XI^pΣS0^q are already fixed to be

B pq

pΣS0^q. Adhering to the rule for parentheses in top and bottom intervals of hexagons is no problem since these polygons do not neighbourS0.

The general case builds on the first one. LetT ^P L^pΣS;^BpqpΣS^q,^Bpq

pΣS^qq. Put Tl T^XSl, and choose parentheses onTl^XI ^pSl^qand Tl^XI^pSl^qto obtain a boundary marking^pBpqSl,^BpqSl^qonSl; in this wayTl^PL^pSl;^BpqSl,^BpqSl^q. Similarly for the right hand square. Define boundary markings onS0by

B

pqS0^ppB

pqSl^B

pqS^qBpqSr^q, ^BpqS0^ppB

pq

Sl^B

pq

S^qBpqSr^q

so that T^XpΣS0^qcan be regarded as an element inL^pΣS0;^BpqpΣS0^q,^Bpq

pΣS0^qq. Put

VP^pT^qVP^pT^XpΣS0^qqV^pT_l^qV^pTr^qPC^pΣS;^{B p}ΣS^q,^BpΣS^qq.

That these maps are well-defined universal Vassiliev invariants of tangles inΣSis proved much like Theorem 4.4 (cf. [AMR2]). Compatibility with cabling is immediate from the construction as in the case of links on Σ. For (compatible) non-associative tangles TS ^P

L^pS;^Bpq,^Bpq

qandT^ΣS ^PL^pΣS;^Bpq,^Bpqq, the compositionT^ΣSTSis a link onΣ, and the three kinds of universal Vassiliev invariants fit together in

VP^pT^ΣSTS^qVP^pT^ΣS^qV^pTS^q

4.4-Products and Standard Situations 41

as one readily deduces from the definitions. This formula is ubiquitous in calculations in the sequel.

4.4

-Products and Standard Situations

For partitionsPofΣwe consider the completed map VPVP:L^pΣqÑC^pΣq.

By Proposition 4.3 and Remark 3.3, Theorem 3.11 applies to S C^pΣq,F L^pΣq and VVP:

Theorem 4.5 (Andersen, Mattes & Reshetikhin). For any partition P ofΣthere is a-product

Pwith coefficients

cr^pD,E^qpV_P^pV_P^1pD^qV_P^1pE^{qqqpm1 m2 rq} for chord diagrams D and E with m1and m2chords, respectively.

Different partitions may yield different-products as we shall see, but at least we have Theorem 4.6. If P1and P2are two partitions ofΣ, then the endomorphismτofC^pΣqrrh^ss deter-mined by

τ^pD^q

¸

r

pV_P2V_P1^1pD^{qqpm rq}h^r, D^PC^pmqpΣq is an equivalence fromP₁toP₂.

Proof. Theorem 3.12 applies since^pVP₁^qGrλ^pVP₂^qGr. ^l Remark 4.7. From formula (4.6) follows immediately that the AMR-products and the equivalences between them preserve the skeletons of the chord diagrams. Therefore the above two theorems also hold for C^pΣ;G^qif we simply carry along the representations associated to core components in the definitions.

Remark 4.8. With a little more effort one can show thatPpreserves more than skeletons;

ifDandEare chord diagrams thencr^pD,E^qis a linear combination of chord diagrams each of which is obtained fromDEby addingrchords appropriately. Formally, this is proved by generalizing the results about non-associative tangles in the complement of an embedded squareS^„Σto the case of two disjoint embedded squares assumed (by isotopy) to contain the ‘non-trivial’ parts ofD, respectivelyE.

Proposition 4.9. The mapP^pΣqÑpC^pΣqqgiven by P^ÞÑPisΓ pΣq-equivariant.

Proof. LetPbe a partition ofΣ, and letg ^PΓ pΣq. For chord diagramsDandEwe have by (4.7) and the analogous identity forV_P¹

gVP^pV_P^1pD^qV_P^1pE^qqV_gpP^qpg^pV_P^1pD^qV_P^1pE^qqq

V_gpP^qpgV_P^1pD^qgV_P^1pE^qq

V_gpP^qpV_g¹

pP^qpg^pD^qqV_g¹

pP^qpg^pE^qqq.

It follows from Theorem 4.5 thatg^pDPE^qg^pD^q_gpP^qg^pE^q; this is exactly the statement

gPg^pP^q, cf. Proposition 3.14. ^l

42 Chapter 4Quantization of the Algebra of Chord Diagrams

4.4.1 Standard Situations

We shall encounter a couple of standard situations in computations involvingP,P^PP^pΣq and the equivalences between these-products. The common set-up of the standard situ-ations is as follows: There is an embedded square^pS;I ,I^{q „} Σwith boundary mark-ing ^pBpq,^Bpq

q, and we have compatible elements L ^P C^pmqpS;^B ,^Bqand T ^P C^pm1qpΣ S;^B,^{B q}. We writeDTL^PC^{pm1 mqpΣq}.

In the first standard situationEis a chord diagram onΣwithm2chords, and we want to calculateDPEwherePis some partition ofΣ. By a homotopy we may assume firstly thatSis contained in the interior of a squareS0fromPand secondly thatEis represented by an element E ^P C^pΣS^qso thatE E^{H P} C^pΣq. We refinePwith two embedded intervals as illustrated in Figure 4.5 in order to makePcompatible withS. Having settled these technical issues, we derive

VP V_P^1pD^qV_P^1pE^qVP V_P^1pTL^qV_P^1pE^Hq

VP^prV_P^1pT^qV^1pL^qsrV_P^1pE^qHsq

VP^rpV_P^1pT^qV_P^1pE^qqV^1pL^qs

VP V_P^1pT^qV_P^1pE^qL so that

cr^pD,E^q VP V_P^1pD^qV_P^1pE^{q pm1 m m2 rq}

VP V_P^1pT^qV_P^1pE^qL^{pm1 m m2 rq}

VP V_P^1pT^qV_P^1pE^{q pm1 m2 rq}L.

Of course, we can reverse the roles ofDandEand get a parallel result. A first application of this standard situation yields

Theorem 4.10 (Andersen, Mattes & Reshetikhin). A subspaceI ^„ C^pΣq spanned by local relations is a-ideal with respect toPfor any partition P ofΣ.

Proof. Previously (cf. 2.4.1) we noted that I is a Poisson ideal so the statement of the theorem makes sense. To prove it we must verify condition (3.13). Consider a generator D^PI. There exists an embedded square^pS;I ,I^q„Σwith boundary marking^pB ,^Bq such that D T^°_iλiLi; hereLi ^P C^pS,^B ,^Bqand λi ^P C are the chord tangles and the scalars defining the relevant local relation, andT ^PC^pm1qpΣS;^B,^{B q}is an arbitrary

S I+(S) I−(S)

I−(S0) I+(S0)

Figure 4.5: Two intervals inS0refiningP.

4.4-Products and Standard Situations 43

chord tangle. LettingE be a chord diagram with m2 chords and choosing an arbitrary parenthesization on^pB ,^Bq, the standard situation yields

cr^pD,E^q

¸

i

λ_icr^pTL_i,E^q

¸

i

λ_i VP V_P^1pT^qV_P^1pE^{qpm1 m2 rq}L_i

VP V_P^1pT^qV_P^1pE^{q pm1 m2 rq}

¸

i

λiLi ^PI

as desired. In the same way,cr^pE,D^qPI. ^l

In the second standard situation we aim to computeτ^pD^qPC^pΣqrrh^sswhereτis the canon-ical equivalence fromP₁toP₂for partitionsP1andP2ofΣ, cf. Theorem 4.6. By homotopy and the refinement procedure illustrated in Figure 4.5, we may assume thatSis compatible with bothP1andP2. We get

V_P2V_P¹

1

pD^qV_P2V_P¹

1

pTL^qV_P2^pV_P¹

1

pT^qV^1pL^qqpV_P2V_P¹

1

pT^qqL so that

τr^pD^qpV_P2V_P1^1pD^{qqpm1 m rqp}V_P2V_P1^1pT^qL^{qpm1 m rqp}V_P2V_P1^1pT^{qqpm1 rq}L.

Not surprisingly the first application of the second standard situation is the following re-sult.

Theorem 4.11. LetI ^„C^pΣqbe a subspace spanned by local relations, and let P1and P2be two partitions ofΣ. The canonical equivalenceτ:C^pΣqrrh^ssÑC^pΣqrrh^ssfromP₁ toP₂ descends to C^pΣq{Ito yield an equivalence between the induced-products.

Proof. This is analogous to the proof of Theorem 4.10; in the notation of that proof we deduce

τr^pD^q

¸

i

λiτr^pTLi^q

¸

i

λi^pV_P2V_P1^1pT^{qqpm1 rq}Li^pV_P2V_P1^1pT^{qqpm1 rq}

¸

i

λiLi ^PI

as required, cf. (3.16). ^l

Applying the preceding two theorems to the loop relation (2.21), we obtain

Theorem 4.12. The AMR-products onC^pΣqand the canonical equivalences between them de-scend to the Poisson loop algebraZ_s,f^pΣqvia the resolving map Rs,f:C^pΣqÑZ_s,f^pΣq.

Chapter 5

Quantization of the Moduli Space

In this chapter we prove, under the assumption^BΣH, that the-productsP, P^PP^pΣq onC^pΣqand the canonical equivalences between them descend toO^pM^pΣ;G^qqifGis one of the groups GLn^pC^qand SLn^pC^q. These results are achieved by presentingO^pM^pΣ;G^qq as an explicit quotient of C^pΣq; the description we give is also valid in the case whereΣ is closed. Obtaining it relies on Sikora’s work [S]; in the general linear case we adapt the methods of his paper to derive a parallel version of its main result, and in the special linear case we simply translate the main result into our context.

We round off the chapter with Andersen’s explicit formula forP in the abelian case GGL1^pC^q[A]; it is a corollary thatPis independent ofPandΓ pΣq-invariant. We also provide counterexamples illustrating that this corollary fails in general.

5.1 The General Linear Case

We consider the groupGGLn^pC^qequipped with the orthogonal structure B^pX,Y^qTr^pXY^q, X,Y^Pgl_n^pC^q.

That is, we fixt^Pgl_n^pC^qbgl_n^pC^qto be the Ad-invariant symmetric tensor corresponding to the pairingB. Colouring all core components of chord diagrams with the defining rep-resentationιId_GLnpC^qof GLn^pC^qyields a Poisson homomorphismC^pΣqÑC^pΣ; GLn^pC^qq. We write

Ψ: C^pΣqÑC^pΣ; GLn^pC^qqÝΨÑt O^pM^pΣ; GLn^pC^qqq for the composite Poisson homomorphism, cf. Theorem 2.22.

Theorem 5.1. Assume that^BΣH. For any partition P ofΣthe-productPonC^pΣqdescends viaΨto a-product onO^pM^pΣ; GLn^pC^qqq.

Theorem 5.2. Assume that^BΣ  H, and let P1 and P2be two partitions ofΣ. The canonical equivalence fromP₁toP₂onC^pΣqdescends viaΨtoO^pM^pΣ; GLn^pC^qqqto yield an equivalence between the induced-products.

Remark 5.3. Theorem 5.1 was also stated in [AMR2]. The proof appearing below roughly follows the outline of the justification supplied in that paper. The primary deviation is that we shall not claim that the kernel ofΨis generated by local relations.

44

5.1 The General Linear Case 45

5.1.1 The Relevant Loop Relation

LetE_i,j ^P gl_n^pC^qbe the matrix whose sole non-zero entry is a 1 in the^pi,j^qth entry, and define

B_i,jEi,j Ej,i, B_i,jEi,jEj,i; 1^¤i j^¤n.

These matrices along withEi,i,i1, . . . ,nare readily seen to constitute an orthogonal basis forgl_n^pC^q. Recalling Remark 2.8, we normalize suitably and perform a simple calculation to obtain

pι^bι^qpt^qt

¸

i,j

Ei,j^bEj,i^PEnd^pC^nqbEnd^pC^nq

which under the isomorphism End^pC^nqbEnd^pC^nq End^pC^nbC^nqcorresponds to the transposition of the factors. HenceΨsatisfies the relation (cf. (2.24b))

(5.1)

that is, the loop relation (2.21) with parameters^ps,f^qp1, 0^q. Thus we derive a triangle of Poisson homomorphisms

C^pΣq

R_1,0 Ψ

Z1,0^pΣq

Ψ O^pM^pΣ; GLn^pC^qqq

The Poisson structure on the loop algebra will not occupy us in the study of KerΨ, so we agree to writeZ^pΣqZ_1,0^pΣq.

5.1.2 The Universal GLn-Representation

Our strategy is to introduce a commutative complex algebraRn^pΣq R^pΣ; GLn^pC^qq en-dowed with a GLn^pC^q-action such thatΨfactors through the algebra of fixed points:

Rn^pΣq

GL_n^pC^q

Z^{pΣq Ψ} O^pM^pΣ; GLn^pC^qqq

A universal property definesRn^pΣq; it admits a representationρ^Σ ρΣ,GL_n^pC^q: π1^pΣqÑ

GLn^pRn^pΣqq (the universal GLn-representation of π1^pΣq) such that for any representation ρ: π1^pΣq Ñ GLn^pA^q, Abeing a commutative complex algebra, there exists a unique ho-momorphismhρ:Rn^pΣqÑAfitting into the diagram

GLn^pRn^pΣqq

GL_n^ph_ρ^q

π1^pΣq

ρΣ

ρ GLn^pA^q

(5.2)

46 Chapter 5Quantization of the Moduli Space

Here is an explicit construction ofRn^pΣq. Let^xgλ,λ^P Λ| rµ,µ ^P M^ybe a presentationP ofπ1^pΣqsatisfying that all relationsrµare written as products of generatorsgλ. LetQn^pΛq

be the polynomial algebraC^rx^λ_i,j,d_λ^swhereλ^PΛandi,j1, . . . ,n. Define matricesA_λ

px_i,j^{λq P} Mn^pQn^pΛqq,λ ^P Λ. HereMn denotes the functor assigning the complex algebra Mn^pR^qofnnmatrices to a commutative complex algebraR; note that Ris included in Mn^pR^qas the central subalgebra of scalar matrices. LetI^pP^q„Qn^pΛqbe the ideal generated byd_λdetA_λ1 and all entries inA_λ1A_λk1 for each relationrµg_λ1g_λk. Set

Rn^pΣ;P^qR^pΣ; GLn^pC^q,P^qQn^pΛq L

I^pP^q; q:Qn^pΛqÑRn^pΣ;P^q.

We now prove thatRn^pΣ;P^qsatisfies the universal property, implying in particular that dif-ferent presentations ofπ1^pΣqyield canonically isomorphic algebras (all denoted byRn^pΣq).

The formulas

det^pMn^pq^qpA_λ^qqq^pdetA_λ^q, q^pd_λ^qq^pdetA_λ^q1

prove thatMn^pq^qpAλ^qPGLn^pRn^pΣ;P^qq, and it is clear that we have a representation ρ^ΣρΣ,GL_n^pC^q,P:π1^pΣqÑGLn^pRn^pΣqq; ρ^Σpgλ^qMn^pq^qpAλ^q. (5.3) IfAis a commutative complex algebra admitting a representationρ:π1^pΣqÑGLn^pA^q, we definehρ:Qn^pΛqÑAby

hρ^px^λ_i,j^qρ^pgλ^qi,j, hρ^pdλ^qdetρ^pgλ^q

1. (5.4)

SinceMn^phρ^qpAλ^qρ^pgλ^q, it follows thatI^pP^q„Kerhρ; the induced maphρ:Rn^pΣqÑA is obviously the unique homomorphism making the triangle (5.2) commutative.

The action of GLn^pC^qonRn^pΣqis the prime application of the universal property. Ele-mentsA^PGLn^pC^qgive rise to representations

A¹ρ^ΣA:π1^pΣqÑGLn^pRn^pΣqq

and the corresponding endomorphisms A:Rn^pΣqÑRn^pΣqdefine a GLn^pC^q-action; this is a direct consequence of the uniqueness of (5.2). By (5.4) and (5.3) we have

Aq^px^λ_i,j^qpA¹Mn^pq^qpAλ^qA^qi,j, Aq^pdλ^qdet^pMn^pq^qpAλ^qq

1

q^pdλ^q. (5.5) We extend the action of GLn^pC^qtoMn^pRn^pΣqqby

AMA^pAMi,j^qA¹, M^PMn^pRn^pΣqq, A^PGLn^pC^q. (5.6) Consequently,

Lemma 5.4. The inclusionRn^pΣq„Mn^pRn^pΣqqis equivariant.

We lift the GLn^pC^q-actions toQn^pΛqand Mn^pQn^pΛqq. By definition we can regardQn^pΛq

as the algebra of polynomial functions from^pMn^pC^qC^qΛtoC, and henceMn^pQn^pΛqqas the algebra of polynomial functions from^pMn^pC^qC^qΛ toMn^pC^q. Since GLn^pC^qacts on Mn^pC^qby conjugation and trivially onC, it acts on the product^pMn^pC^qC^qΛ. The induced actions on the function sets Map^ppMn^pC^qC^qΛ,C^qand Map^ppMn^pC^qC^qΛ,Mn^pC^qq pre-serve the property of being polynomial, thereby defining the desired actions denoted also by.

5.1 The General Linear Case 47

Lemma 5.5. We have dλ, detAλ^PQn^pΛq

GL_n^pC^qand Aλ^PMn^pQn^pΛqq

GL_n^pC^q.

Proof. This is immediate when regarding dλ, detAλ and Aλ as functions on ^pMn^pC^q

C^qΛ. ^l

The analogues of (5.6) and Lemma 5.4 hold:

Lemma 5.6. For any F^PMn^pQn^pΛqqand A^PGLn^pC^qwe have AFA^pAFi,j^qA¹ Proof. Thinking ofFandFi,jas functions we derive

pAF^qppMλ,sλ^qλ^qAF^pA^1pMλ,sλ^qλ^qA¹

A^pFi,j^pA^1pMλ,sλ^qλ^qqA¹

A^ppAFi,j^qppMλ,sλ^qλ^qqA¹

where^pMλ,sλ^qλ^PpMn^pC^qC^qΛ. ^l

Corollary 5.7. The inclusion Qn^pΛq„Mn^pQn^pΛqqis equivariant.

All four GLn^pC^q-actions are related by

Proposition 5.8. The maps in the commutative square Mn^pQn^pΛqq

Mn^pq^q

Tr

Mn^pRn^pΣqq

Tr

Qn^pΛq

q Rn^pΣq

are equivariant.

Proof. The traces are invariant under conjugation with complex matrices and hence equiv-ariant by (5.6) and Lemma 5.6, respectively. The equivariance ofqneed only be verified on the generatorsx^λ_i,j,d_λ. FixA^PGLn^pC^q. For any^pM_λ,s_λ^q_λ^PpMn^pC^qC^qΛwe have

pAx_i,j^λ0qppMλ,sλ^qλ^qx^λ_i,j^0pA^1pMλ,sλ^qλ^q

x^λ_i,j^0ppA¹MλA,sλ^qλ^q

pA¹Mλ0A^qi,j

pA¹Aλ₀^ppMλ,sλ^qλ^qA^qi,j

ppA¹Aλ₀A^qppMλ,sλ^qλ^qqi,j

so that

q^pAx^λ_i,j^0qq^ppA¹A_λ0A^qi,j^qpA¹Mn^pq^qpA_λ0^qA^qi,j Aq^px^λ_i,j^0q

48 Chapter 5Quantization of the Moduli Space

by (5.5). The elements dλ are invariant by Lemma 5.5, so (5.5) also takes care of those.

RegardingMn^pq^q, we derive forM^PMn^pQn^pΛqq

Mn^pq^qpAM^qMn^pq^qpA^pAMi,j^qA^1q

A^pq^pAMi,j^qqA¹

A^pA^pq^pMi,j^qqqA¹

AMn^pq^qpM^q

by Lemma 5.6, the equivariance ofqand formula (5.6). ^l

Proposition 5.9. The image of the universalGLn-representationρ^Σ: π1^pΣq Ñ Mn^pRn^pΣqqis invariant under the action ofGLn^pC^q.

Proof. It suffices to consider a generator gλ. The matrix Aλ ^P Mn^pQn^pΛqq is invariant by Lemma 5.5; the result now follows from the previous proposition since Mn^pq^qpAλ^q

ρ^Σpgλ^qis then invariant, too. ^l

Remark 5.10. Before continuing the investigation of Rn^pΣq, we explore its relationship withM^pΣ; GLn^pC^qq. We think of GLn^pC^qas an affine subset ofMn^pC^qCC^{n2 1}, namely

GLn^pC^qtpA,d^qPMn^pC^qC^|ddetA1^u.

Recalling the construction of the combinatorial complex KP (cf. 2.1), we infer (at least in the case |^Λ|  ⁸) that the vanishing set of the idealI^pP^{q „} Qn^pΛq O^ppMn^pC^qC^qΛq is exactlyA^pKP; GLn^pC^qq„GLn^pC^qEpKPqGLn^pC^{qΛ „p}Mn^pC^qC^qΛ. By Hilbert’s Null-stellensatz, restriction of functions provides an isomorphism

Qn^pΛq La

I^pP^qÑO^pA^pKP; GLn^pC^qqq.

The former space is, of course, nothing butRn^pΣ;P^q{?0, so we have a commutative trian-gle

Qn^pΛq

q Rn^pΣ;P^q

p

O^pA^pKP; GLn^pC^qqq

where Kerp^?0. It also follows thatpis a GLn^pC^q-equivariant surjection since the other two maps in the diagram enjoy this property.

5.1.3 Diagrams and Relative Diagrams

Recall that a diagram onΣis simply the homotopy class of a map from a finite collection of oriented circles toΣ, or, in other words, a set of conjugacy classes in π1^pΣq. We need a relative version of this concept; a relative diagram Dis the unit interval I union a finite collection of oriented circles, and arelative diagram onΣis a map f: D^Ñ ΣIsuch that f^pi^qpx0,i^q,i0, 1, regarded up to homotopy rel^BI. Post-composing with the projection p: ΣI ^Ñ Σ(a homotopy equivalence), one sees that such an object is nothing but an

5.1 The General Linear Case 49

γ₁ γ2

γ3

Figure 5.1: A decorated diagram.

γ1

γ2

Figure 5.2: A decorated relative diagram.

element ofπ1^pΣqtogether with a finite set of conjugacy classes in this group. LetZ^pΣ,x0^q

denote the complex vector space freely generated by relative diagrams onΣ; it is equipped with a natural algebra structure: For relative diagramsfi:Di^ÑΣI,i1, 2 we define

DD1^YD2

L

tB I1^BI2^u (5.7)

and f f1f2: D^ÑΣIby f^pd^q

#

px, 1^{2t^q, d^PD1^{^}f1^pd^qpx,t^q

px, 1^{2t 1^{2^q, d^PD2^{^}f2^pd^qpx,t^q The unit for the multiplication is

e:I^ÑΣI, e^pt^qpx0,t^q.

It is convenient to represent (relative) diagrams onΣ bydecorated (relative) diagrams. By this we mean (relative) diagrams along each component of which one or more elements ofπ1^pΣqare written. We give a couple of examples of how decorated (relative) diagrams represent (relative) diagrams onΣ. The decorated diagram in Figure 5.1 determines the diagram onΣgiven by a mapS^1ÑΣrepresenting the conjugacy class ofγ1γ2γ3^Pπ1^pΣq. Similarly, the relative diagram onΣrepresented by the decorated relative diagram in Figure 5.2 is defined by a map f: I ^Ñ ΣI such that the loop pf is in the homotopy class γ1γ2 ^P π1^pΣq. How to interpret general decorated (relative) diagrams is obvious from these examples. Decoration of a component with 1 ^P π1^pΣqis sometimes suppressed in the notation. If the component in question is the interval of a relative diagram, it may be omitted entirely; the potential confusion with a (relative) decorated diagram is non-serious as we shall later.

Remark 5.11. It is obvious that two decorated (relative) diagrams represent the same (rel-ative) diagram onΣif and only if there exists a bijection between the circles of the two dia-grams such that the products along corresponding circles are conjugate elements ofπ1^pΣq, and, in the relative case, the products along the two intervals are equal.

Multiplication of (relative) diagrams onΣ lifts to the setting of decorated (relative) dia-grams in the obvious way; take the union of all components carrying along the decoration, and glue the intervals in the relative case (cf. (5.7)).

Parts (local and non-local on Σ) of decorated (relative) diagrams play an important role in the sequel. The ubiquitous example is the braid (over- and undercrossings being ignored)Bσcorresponding to a permutationσ^PSm; we depict it as (notice the notation for bundles of strands):

Bσ σ

· · ·

· · ·

σ

50 Chapter 5Quantization of the Moduli Space

For example, ifσ^p1, 2, 3^qPS3we have Bσ

We define idealsIn^pΣqI^pΣ; GLn^pC^qq„Z^pΣqandIn^pΣ;x0^q„Z^pΣ,x0^qto be generated by the following three kinds of expressions:

1 n (5.8a)

¸

σ^PS_{n 1}

ǫ^pσ^q σ (5.8b)

¸

σ^PS_n

ǫ^pσ^q γ σ

¸

τ^PS_n

ǫ^pτ^q γ⁻¹ τ ^pn!^q2 (5.8c)

The first two relations are local onΣ, whereas the third one is not. We write Dn^pΣqZ^pΣqLIn^pΣq, Dn^pΣ,x0^qZ^pΣ,x0^q

L

In^pΣ;x0^q

for the quotient algebras. Certain elementary decorated (relative) diagrams deserve special attention:

Lγ γ , ELγ γ 1 , Eγ γ

For easy reference we record the following simple fact about them.

Proposition 5.12. Dn^pΣqis generated by Lγ,γ ^P π1^pΣq, andDn^pΣ,x0^qis generated by E_g1 λ , λ^PΛand ELγ,γ^Pπ1^pΣq.

The algebrasDn^pΣqandDn^pΣ,x0^qare related by a pair of maps. In one directionι: Z^pΣqÑ Z^pΣ,x0^qis given on decorated diagrams by simply adding an interval decorated by 1 ^P π1^pΣq. This is well-defined on the level of diagrams onΣand clearly induces an algebra homomorphismι: Dn^pΣq Ñ Dn^pΣ,x0^q; its image is central, so we may viewDn^pΣ,x0^qas an algebra overDn^pΣq. On the other hand, closing up the interval of a decorated relative diagram to a circle and thereby obtaining a decorated diagram results in a mapZ^pΣ,x0^qÑ

Z^pΣq; it descends to a linear map Tr : Dn^pΣ,x0^qÑDn^pΣq. By relation (5.8a) we have TrιnId : Dn^pΣqÑDn^pΣq.

In particular,ιembedsDn^pΣqas a subalgebra ofDn^pΣ,x0^q; this justifies the aforementioned convention of occasionally omitting a trivially decorated interval of a decorated relative diagram, and we often suppressιin the notation.

In document Afhandling (Sider 44-68)