Affine Braid group, JM elements and knot homology
aa r X i v : . [ m a t h . G T ] J a n AFFINE BRAID GROUP, JM ELEMENTS AND KNOT HOMOLOGY
A. OBLOMKOV AND L. ROZANSKY
Abstract.
In this paper we construct a homomorphism of the affine braid group Br affn in the convolution algebra of the equivariant matrix factorizations on the space X “ b n ˆ GL n ˆ n n considered in the earlier paper of the authors. We explain that the pull-back onthe stable part of the space X intertwines with the natural homomorphism from the affinebraid group Br affn to the finite braid group Br n . This observation allows us derive a relationbetween the knot homology of the closure of β P Br n and the knot homology of the closureof β ¨ δ where δ is a product of the JM elements in Br n Contents
1. Introduction 11.1. Jucys-Murphy elements 21.2. Geometric realization of the affine and finite braid groups 31.3. Geometric trace operator 41.4. Acknowledgements 62. Convolution algebras 62.1. Matrix Factorizations 62.2. Chevalley-Eilenberg complex 72.3. Equivariant matrix factorizations 72.4. Push forwards, quotient by the group action 82.5. Convolutions and reduced spaces 82.6. Convolution on framed spaces 93. Geometric realization of the affine braid group 103.1. Induction functors 103.2. Generators of the finite braid group action 113.3. Generators of the affine braid group action 123.4. Stabilization morphism 12References 131.
Introduction
This paper is an extension of our earlier paper where we constructed a triply-graded knothomology theory [1]. In [1] the homology H p L p β qq of the link L p β q that is a closure of thebraid β P Br n is realized, roughly, as a space of derived global sections of the complex ofequivariant quasi-coherent sheaves S β on the Hilbert scheme of n points on the plane Hilb n .The knot homology of this sort was expected to exist for quite some time [2, 3, 4, 5, 6, 1], The work of A.O. was supported in part by the NSF CAREER grant DMS-1352398.The work of L.R. was supported in part by the Sloan Foundation and the NSF grant DMS-1108727. in particular it was expected that in such theory we would have a natural relation betweenH p L p β qq and H p L p β ˝ T w qq where T w is the full twist braid. This paper shows that thisexpectation is indeed true.Before we proceed to the main statement of the paper, let us recall the main result of [1] .In this paper we use notations V n “ C n , g n “ End p V q , b n , n n are the upper, respectivelystrictly upper, triangular matrices, we also omit the subindex n when the rank is obviousfrom the context.The free nested Hilbert scheme Hilb free ,n is a B -quotient of the sublocus Ą Hilb free ,n Ă b n ˆ n n ˆ V n of the cyclic triples tp X, Y, v q| C x X, Y y v “ V n u . The usual nested Hilbert schemeHilb L ,n is the dg subscheme of Hilb free ,n , it is defined by imposing the equation r X, Y s “ T sc “ C ˚ ˆ C ˚ acts on Hilb free ,n by scaling the matrices. We denote by D perT sc p Hilb free ,n q the derived category of two-periodic complexes of T sc -equivariant quasi-coherent sheaves onHilb free ,n . Let us also denote by B _ the descent of the trivial vector bundle V n on Ą Hilb free ,n tothe quotient Hilb free ,n . Respectively, B stands for the dual of B _ . In [1] we construct for every β P Br n an element S β P D perT sc p Hilb free ,n q such that space of hyper-cohomology of the complex: H k p S β q : “ H p S β b Λ k B q defines an isotopy invariant. Theorem 1.0.1. [1]
For any β P Br n the doubly graded space H k p β q : “ H p k ` writh p β q´ n ´ q{ p S β q is an isotopy invariant of the braid closure L p β q . It is natural to expect that the construction of [1] produces the same triply-graded knothomology as in the original papers [7, 8]. In the subsection 1.3 we remind the construction of S β . Determining the graded dimensions of H k p β q for a given braid is a hard computationalproblem. However, for a special class of braids, including torus braids, the computation isrelatively easy, and we provide the details.1.1. Jucys-Murphy elements.
The braid group Br n is generated by the elements σ i , i “ , . . . , n ´ δ i P Br n : δ i : “ σ i σ i ` . . . σ n ´ . . . σ i ` σ i , i “ , . . . , n ´ Jucys-Murphy (JM) elements .The group of characters of the Borel subgroup B n is generated by the characters χ i : χ p X q “ X ii and we denote by C χ i the corresponding one-dimensional representation. The trivial linebundle C χ i on Ą Hilb free ,n descends to the line bundle L i on the quotient Hilb free ,n . The mainresult of this note is the following Theorem 1.1.1.
For any β P Br n we have H k p S β ¨ δ q “ H k p S β b L δ q , where δ “ ś ni “ δ r i i and L δ “ b ni “ L b r i i . Here and everywhere below we state a GL version of the results of [1]; the paper [1] covers the SL versionof the results, but the proofs of the GL version are essentially identical. FFINE BRAID GROUP, JM ELEMENTS AND KNOT HOMOLOGY 3
The scheme Hilb L ,n is expected to have many features of the usual Hilbert scheme of pointson the plane. However, since the derive structure is non-trivial, the computations on the dgscheme Hilb L ,n are very challenging. In contract, the space Hilb free ,n is smooth manifold andis an iterated tower of projective spaces. In particular, we have the following Proposition 1.1.2.
The line bundle L b ¨ ¨ ¨ b L n ´ is ample on Hilb free ,n . Using the ampleness from the previous conjecture we can use the spectral sequence argu-ment to imply an easy
Corollary 1.1.3.
If the numbers r i are sufficiently large then H k p S δ q “ H p Hilb free ,n , r O Hilb L ,n s vir b Λ k B b L δ q , where r O Hilb L ,n s vir is the notation for the defining complex of the dg scheme Hilb L ,n . Now we explain the method of the proof of the main theorem and describe some otherinteresting algebraic structures that are explored in this paper.1.2.
Geometric realization of the affine and finite braid groups.
The affine braidgroup Br affn is the group of braids whose strands may also wrap around a ‘flag pole’. Thegroup is generated by the standard generators σ i , i “ , . . . , n ´ n that wrapsthe last stand of the braid around the flag pole: σ i “ i ` i ‚ ‚ ‚ ‚ ‚ ‚ ‚‚ ‚ ‚ ‚ ‚ ‚ ‚ ......................................................................................... ..................................................................................................................... ˚ .............. ...................................................... ...................................................... ...................................................... ...................................................... ..................................................................................................................................................................................... and ∆ n “ ‚ ‚ ‚ ‚ ‚ ‚ ‚‚ ‚ ‚ ‚ ‚ ‚ ‚ .................................................................................. .............................................................................................................. ˚ .............. ...................................................... ...................................................... ...................................................... ...................................................... ...................................................... .............................................................................................................................................................................................................................. . The defining relations for this generators are σ n ´ ¨ ∆ n ¨ σ n ´ ¨ ∆ n “ ∆ n ¨ σ n ´ ¨ ∆ n ¨ σ n ´ ,σ i ¨ ∆ n “ ∆ n ¨ σ i , i ă n ´ ,σ i ¨ σ i ` ¨ σ i “ σ i ` ¨ σ i ¨ σ i ` , i “ , . . . , n ´ ,σ i ¨ σ j “ σ j ¨ σ i , | i ´ j | ą . The mutually commuting Bernstein-Lusztig (BL) elements ∆ i P Br affn are defined as fol-lows: ∆ i “ σ i ¨ ¨ ¨ σ n ´ σ n ´ ∆ n σ n ´ σ n ´ ¨ ¨ ¨ σ i “ i ‚ ‚ ‚ ‚ ‚ ‚ ‚‚ ‚ ‚ ‚ ‚ ‚ ‚ .................................................................................. .............................................................................................................. ˚ .............. ............................... ............................... ............................... .............................................. ............... ............... ............... ...................................................... ....................................................................................... ...................... ...................... ...................... ................................................................................................................................................................................................................................................ ........................................................ . A further discussion of their properties can be found in [9] which is the source of our affinebraid pictures.There is a natural homomorphism fgt : Br affn Ñ Br n , geometrically it is defined by remov-ing the flag pole. In particular we have: fgt p ∆ n q “ , fgt p ∆ i q “ δ i , i “ , . . . , n ´ . A. OBLOMKOV AND L. ROZANSKY
The main technical tool in [1] is the realization of Br n inside of the convolution algebraof the category of equivariant matrix factorizations p MF scB n p X p G n q , W q , ¯ ‹q where X p G n q “ b n ˆ G n ˆ n n and W “ Tr p X, g, Y q “ Tr p X Ad g p Y qq . Now we extend this structure:
Theorem 1.2.1.
There is a homomorphism: Φ aff : Br affn Ñ p MF scB n p X p G n q , W q , ¯ ‹q . Note that the paper [10] constructs a homomorphism from the affine braid group to thecategory of matrix factorizations. The construction of [10] relies on the earlier result of Riche[11], the construction in [1] is independent of the results in [11]. It is unclear to us how torelate the results in this paper to the constructions of the paper [10].Given a matrix factorization C in MF scB n p X p G n q , W q and two characters ξ, τ : B Ñ C ˚ wedefine the twisted matrix factorization C x ξ, τ y to be the matrix factorization C b C ξ b C τ . Inthese terms we have Theorem 1.2.2.
For any i “ , . . . , n we have Φ aff p ∆ i q “ Φ aff p qx χ i , y . Results of this paper are based on a realization that the ordinary braid group Br n actsnaturally on the framed version X ,fr p G n q of space X p G n q : X ,fr p G n q “ tp X, g, Y, v q P X p G n q ˆ V n | , C x X, Ad g p Y qy v “ V n , g ´ p v q P V u where V is the subset of V consisting of vectors with non-zero last coordinate. Thereis a natural map fgt : X ,fr p G n q Ñ X p G n q , and a pull-back along fgt provides a naturalanalog of homomorphism Φ aff which we restrict on the finite part of the braid group Br n “ C x σ , . . . , σ n ´ y : Φ fr : Br n Ñ MF scB p X ,fr , W q . Theorem 1.2.3.
There is convolution algebra structure ¯ ‹ on MF scB n p X ,fr p G n q , W q and thepull-back map fgt ˚ : MF scB n p X ,fr p G n q , W q Ñ MF scB n p X p G n q , W q is a homomorphism of the convolution algebras. The convolution algebra structures are compatible with the forgetful homomorphism fgt : Theorem 1.2.4.
We have fgt ˚ ˝ Φ aff “ Φ fr ˝ fgt . Geometric trace operator.
The variety Ą Hilb free ,n embeds inside X p G n q via the map j e : p X, Y, v q Ñ p
X, e, Y, v q . The diagonal copy B “ B ∆ ã Ñ B respects the embedding j e andsince j ˚ e p W q “
0, we obtain a functor: j ˚ e : MF scB n p X p G n q , W q Ñ MF scB ∆ p Ą Hilb free ,n , q . Respectively, we get a geometric version of ”closure of the braid” map: L : MF scB n p X p G n q , W q Ñ D perT sc p Hilb free ,n q . FFINE BRAID GROUP, JM ELEMENTS AND KNOT HOMOLOGY 5
The main result of [1] could be restated in more geometric terms via the geometric trace map: T r : Br n Ñ D perT sc p Hilb free ,n q , T r p β q : “ ‘ k L p Φ fr p β qq b Λ ‚ B . Theorem 1.3.1. [1]
The composition H ˝ T r : Br n Ñ D perT sc p pt q categorifies the Jones-Oceanutrace and thus defines a triply graded homology of links. Theorem 1.0.1 now follows from the theorems in this section. Indeed, let ∆ “ ś i ∆ k i i and δ “ ś i δ k i i then we have L ˝ Φ fr p β ¨ δ q “ L ˝ Φ fr ˝ fgt p β ¨ ∆ q “ L ˝ fgt ˚ ˝ Φ aff p β ¨ ∆ q “ L p fgt ˚ ˝ Φ aff p β qq b L δ . To summarize, we constructed the following commutative diagram:(1.1) Br affn Br n L p MF scB n p X p G n q , W q , ¯ ‹q p MF scB n p X ,fr p G n q , W q , ¯ ‹q Vect ´ gr fgt Φ aff cl Φ fr H p q fgt ˚ H ˝ T r Here L is the set of (isotopy classes of) oriented links in a 3-sphere, cl is the closure of abraid and H p q is the triply graded link homology defined in[1].The left commutative diagram has two important generalizations. The first generalizationuses the concatenation homomorphism cnt : Br affn ˆ Br affm Ñ Br affn ` m which is geometricallyan insertion of the affine braid element on m strands in place of the flag pole of the n -strandbraid: Br affn ˆ Br affm Br affn ` m p MF scB n p X p G n q , W q , ¯ ‹q ˆ p MF scB m p X p G m q , W q , ¯ ‹q p MF scB n ` m p X p G n ` m q , W q , ¯ ‹q , cnt Φ aff ˆ Φ aff Φ aff ind n here ind n is the induction functor described in the section 3.1. The second generalizationuses the concatenation map cnt : Br affn ˆ Br m Ñ Br n ` m which is an insertion of an ordinarybraid on m strands in place of the flag pole of the affine braid: Br affn ˆ Br m Br n ` m p MF scB n p X p G n q , W q , ¯ ‹q ˆ p MF scB m p X ,fr p G m q , W q , ¯ ‹q p MF scB n ` m p X ,fr p G n ` m q , W q , ¯ ‹q , cnt Φ aff ˆ Φ fr Φ fr ind n A. OBLOMKOV AND L. ROZANSKY here ind n is the functor from the section 3.1. In particular, the left square of our maindiagram 1.1 is the last diagram with m “ X “ b ˆ G n ˆ n and its bigger version which we call ‘non-reduced space’. Weneed this section for the proofs of our main result but this section also could be useful for thereader who is interested in the results of [1] but not interested in the details of the proofs.In the section 3 we explain the construction of the homomorphism from [1] and explain howit extends to the case of the affine braid groups. We also prove our main result about theforgetful pull-back functor.1.4. Acknowledgements.
We would like to thank Roman Bezrukavnikov, Eugene Gorsky,Andrei Negut¸, Jake Rasmussen for useful discussions. L.R. is especially thankful to DmitryArinkin for illuminating discussions. A.O. Is especially thankful to Andrei Negut¸ for illu-minating discussions. Both authors are very thankful to an anonymous referee who mademany very valuable suggestions that helped to improve the text. Work of A.O. was partiallysupported by NSF CAREER grant DMS-1352398. The work of L.R. is supported by the NSFgrant DMS-1108727. 2.
Convolution algebras
In this section we define convolution algebras on the categories of matrix factorizations onseveral auxiliary spaces. First we discuss the spaces and maps between them. The main spaceused for our constructions of the convolution algebras is the space X ℓ p G n q “ g ˆ p G n ˆ n n q ℓ . It has a natural G n ˆ B ℓn -action p b , . . . , b ℓ q ¨ p X, g , Y , . . . , g ℓ , Y ℓ q “ p X, g ¨ b ´ , Ad b p Y q , . . . , g ℓ ¨ b ´ ℓ , Ad b ℓ p Y ℓ qq ,h ¨ p X, g , Y , . . . , g ℓ , Y ℓ q “ p Ad h p X q , h ¨ g , Y , . . . , h ¨ g ℓ , Y ℓ q The space is X is particularly important. The central object of our study is the matrixfactorizations on this space with the potential: W p X, g , Y , g , Y q “ Tr p X p Ad g p Y q ´ Ad g p Y qqq . Below we briefly discuss the categories of matrix factorizations and their equivariant ana-logues.2.1.
Matrix Factorizations.
Matrix factorizations were introduced by Eisenbud [12] andlater the subject was further developed by Orlov [13], one can also consult [14] for an overview.Below we present only the basic definitions and do not present any proofs.Let us remind that for an affine variety Z and a function F P C r Z s there exists a triangu-lated category MF p Z , F q . The objects of the category are pairs F “ p M ‘ M , D q , D : M i Ñ M i ` , D “ F, where M i are free C r Z s -modules of finite rank and D is a homomorphism of C r Z s -modules. FFINE BRAID GROUP, JM ELEMENTS AND KNOT HOMOLOGY 7
Given F “ p M, D q and G “ p N, D q the linear space of morphisms Hom p F , G q consists ofhomomorphisms of C r Z s -modules φ “ φ ‘ φ , φ i P Hom p M i , N i q such that φ ˝ D “ D ˝ φ .Two morphisms φ, ρ P Hom p F , G q are homotopic if there is homomorphism of C r Z s -modules h “ h ‘ h , h i P Hom p M i , N i ` q such that φ ´ ρ “ D ˝ h ´ h ˝ D .In the paper [1] we introduced a notion of equivariant matrix factorizations which weexplain below. First let us remind the construction of the Chevalley-Eilenberg complex.2.2. Chevalley-Eilenberg complex.
Suppose that h is a Lie algebra. Chevalley-Eilenbergcomplex CE h is the complex p V ‚ p h q , d q with V p p h q “ U p h qb C Λ p h and differential d ce “ d ` d where: d p u b x ^ ¨ ¨ ¨ ^ x p q “ p ÿ i “ p´ q i ` ux i b x ^ ¨ ¨ ¨ ^ ˆ x i ^ ¨ ¨ ¨ ^ x p ,d p u b x ^ ¨ ¨ ¨ ^ x p q “ ÿ i ă j p´ q i ` j u b r x i , x j s ^ x ^ ¨ ¨ ¨ ^ ˆ x i ^ ¨ ¨ ¨ ^ ˆ x j ^ ¨ ¨ ¨ ^ x p , Let us denote by ∆ the standard map h Ñ h b h defined by x ÞÑ x b ` b x . Suppose V and W are modules over the Lie algebra h then we use notation V b ∆ W for the h -modulewhich is isomorphic to V b W as a vector space, the h -module structure being defined by ∆.Respectively, for a given h -equivariant matrix factorization F “ p M, D q we denote by CE h b ∆ F the h -equivariant matrix factorization p CE h b ∆ F , D ` d ce q . The h -equivariant structure onCE h b ∆ F originates from the left action of U p h q that commutes with right action on U p h q usedin the construction of CE h .A slight modification of the standard fact that CE h is the resolution of the trivial moduleimplies that CE h b ∆ M is a free resolution of the h -module M .2.3. Equivariant matrix factorizations.
Let us assume that there is an action of theLie algebra h on Z and F is a h -invariant function. Then we can construct the followingtriangulated category MF h p Z , W q .The objects of the category are triples: F “ p M, D, Bq , p M, D q P MF p Z , W q where M “ M ‘ M and M i “ C r Z s b V i , V i P Mod h , B P ‘ i ą j Hom C r Z s p Λ i h b M, Λ j h b M q and D is an odd endomorphism D P Hom C r Z s p M, M q such that D “ F, D tot “ F, D tot “ D ` d ce ` B , where the total differential D tot is an endomorphism of CE h b ∆ M , that commutes with the U p h q -action.Note that we do not impose the equivariance condition on the differential D in our defi-nition of matrix factorizations. On the other hand, if F “ p M, D q P MF p Z , F q is a matrixfactorization with D that commutes with h -action on M then p M, D, q P MF h p Z , F q .There is a natural forgetful functor MF h p Z , F q Ñ MF p Z , F q that forgets about the correc-tion differentials: F “ p M, D,
Bq ÞÑ F : “ p M, D q . Given two h -equivariant matrix factorizations F “ p M, D, Bq and ˜ F “ p ˜ M , ˜ D, ˜ Bq thespace of morphisms Hom p F , ˜ F q consists of homotopy equivalence classes of elements Ψ P A. OBLOMKOV AND L. ROZANSKY
Hom C r Z s p CE h b ∆ M, CE h b ∆ ˜ M q such that Ψ ˝ D tot “ ˜ D tot ˝ Ψ and Ψ commutes with U p h q -actionon CE h b ∆ M . Two maps Ψ , Ψ P Hom p F , ˜ F q are homotopy equivalent if there is h P Hom C r Z s p CE h b ∆ M, CE h b ∆ ˜ M q such that Ψ ´ Ψ “ ˜ D tot ˝ h ´ h ˝ D tot and h commutes with U p h q -action on CE h b ∆ M .Given two h -equivariant matrix factorizations F “ p M, D,
Bq P MF h p Z , F q and ˜ F “p ˜ M , ˜ D, ˜ Bq P MF h p Z , ˜ F q we define F b ˜ F P MF h p Z , F ` ˜ F q as the equivariant matrix fac-torization p M b ˜ M , D ` ˜ D, B ` ˜ Bq .2.4. Push forwards, quotient by the group action.
The technical part of [1] is theconstruction of push-forwards of equivariant matrix factorizations. Here we state the mainresults, the details may be found in section 3 of [1]. We need push forwards along projectionsand embeddings. We also use the functor of taking quotient by group action for our definitionof the convolution algebra.The projection case is more elementary. Suppose Z “ X ˆ Y , both Z and X have h -actionand the projection π : Z Ñ X is h -equivariant. Then for any h invariant element w P C r X s h there is a functor π ˚ : MF h p Z , π ˚ p w qq Ñ MF h p X , w q which simply forgets the action of C r Y s .We define an embedding-related push-forward in the case when the subvariety Z j ã ÝÑ Z isthe common zero of an ideal I “ p f , . . . , f n q such that the functions f i P C r Z s form a regularsequence. We assume that the Lie algebra h acts on Z and I is h -invariant. Then there existsan h -equivariant Koszul complex K p I q “ p Λ ‚ C n b C r Z s , d K q over C r Z s which has non-trivialhomology only in degree zero. Then in section 3 of [1] we define the push-forward functor j ˚ : MF h p Z , W | Z q ÝÑ MF h p Z , W q , for any h -invariant element W P C r Z s h .Finally, let us discuss the quotient map. The complex CE h is a resolution of the trivial h -module by free modules. Thus the correct derived version of taking h -invariant part of thematrix factorization F “ p M, D,
Bq P MF h p Z , W q , W P C r Z s h isCE h p F q : “ p CE h p M q , D ` d ce ` Bq P MF p Z { H, W q , where Z { H : “ Spec p C r Z s h q and use the general definition of h -module V :CE h p V q : “ Hom h p CE h , CE h b ∆ V q . Convolutions and reduced spaces.
For a Borel group B , we treat B -modules as T -equivariant n “ Lie pr B, B sq -modules. For a space Z with B -action and for W P C r Z s B wedefine MF B p Z , W q as the full subcategory of MF n p Z , W q whose objects are matrix factoriza-tions p M, D, Bq , where M is a B -module and the differentials D and B are T -invariant. Thecategory MF B ℓ p Z , W q has a similar definition.The categories that we use in [1] are subcategories MF scB ℓ p X ℓ , F q Ă MF B ℓ p X ℓ , F q thatconsist of the matrix factorizations which are equivariant with respect to the action of T sc and G -invariant.The space X has natural projections π ij on X onto the corresponding factors. Since π ˚ p W q ` π ˚ p W q “ π ˚ p W q , there is a well-defined binary operation on matrix factorizations FFINE BRAID GROUP, JM ELEMENTS AND KNOT HOMOLOGY 9 MF scB p X , W q :(2.1) F ‹ G : “ π ˚ p CE h p q p π ˚ p F q b π ˚ p G qqq T p q . This operation defines an associative product and we call the corresponding algebra theconvolution algebra . For computational reasons we also introduce a smaller ‘reduced’ space X ℓ : “ b ˆ G ℓ ´ ˆ n with the B ℓ -action: p b , . . . , b ℓ q ¨ p X, g , . . . , g ℓ ´ , Y q “ p Ad b p X q , b g b ´ , b g b ´ , . . . , Ad b ℓ p Y qq . In particular the space X has the following B -invariant potential: W p X, g, Y q “ Tr p X Ad g p Y qq . The proposition 5.1 from [1] provides a functor:Φ : MF scB p X , W q Ñ MF scB p X , W q which is an embedding of the categories. Without the B -equivariant structure the functoris an ordinary Kn¨orrer functor [15], the equivariant version of the Kn¨orrer functor is definedas composition of the equivariant pull-back and push-forward (see section 5 of [1]):Φ : “ j x ˚ ˝ π ˚ y , where π y : r X Ñ X , r X : “ b ˆ G ˆ n ˆ G ˆ n is the projection π y p X, g , Y , g , Y q “p X, g ´ g , Y q and j x is the natural embedding of r X into X .Let us also introduce a convolution algebra structure on the category of matrix factoriza-tions MF scB p X , W q . There are the following maps ¯ π ij : X Ñ X :¯ π p X, g , g , Y q “ p X, g , Ad g p Y q `` q , ¯ π p X, g , g , Y q “ p X, g g , Y q , ¯ π p X, g , g , Y q “ p Ad ´ g p X q ` , g , Y q . Here and everywhere below X ` and X `` stand for the upper and strictly-upper triangularparts of X . The map ¯ π ˆ ¯ π is B -equivariant but not B -equivariant. However in section5.4 of [1] we show that for any F , G P MF scB p X , W q there is a natural element(2.2) p ¯ π b B ¯ π q ˚ p F b G q P MF scB p X , ¯ π ˚ p W qq , such that we can define the binary operation on MF scB p X , W q : F ¯ ‹ G : “ ¯ π ˚ p CE n p q pp ¯ π b B ¯ π q ˚ p F b G qq T p q q and Φ intertwines the convolution structures:Φ p F q ‹ Φ p G q “ Φ p F ¯ ‹ G q . Convolution on framed spaces.
As we mentioned in the introduction, it is naturalto consider the framed version of our basic spaces. The framed version of the non-reducedspace is an open subset X ℓ,fr Ă X ℓ ˆ V defined by the stability condition: C x Ad ´ g i p X q , Y i y g ´ i p u q “ V, g ´ i p u q P V i “ , . . . , ℓ ´ , where V Ă V is a subset of vectors with a non-zero last coordinate. Similarly, we define theframed reduced space X ,fr Ă X ˆ V with the stability condition(2.3) C x X, Ad g p Y qy u “ V, g ´ p u q P V . Let us also define X ,fr to be the intersection ¯ π ´ p X ,fr q X ¯ π ´ p X ,fr q where ¯ π ij are themaps X ˆ V Ñ X ˆ V which are just extensions of the previously discussed maps by the identity map on V . Similarly we have the natural maps π ij : X ,fr Ñ X ,fr and both reducedand non-reduced spaces have natural convolution algebra structure defined by the formulas(2.1) and (2.2)We denote by fgt the maps X ,fr Ñ X , X ,fr Ñ X that forget the framing. Lemma 12.3of [1] says that the corresponding pull-back morphism is an homomorphism of the convolutionalgebras: fgt ˚ p F ‹ G q “ fgt ˚ p F q ‹ fgt ˚ p G q . Finally, let us mention that we can restrict the Kn¨orrer functor Φ on the open set X ,fr toobtain the functor Φ : MF scB p X ,fr , W q Ñ MF scB p X ,fr , W q . This functor intertwines the convolution algebra structures on the reduced and non-reducedframed spaces. 3.
Geometric realization of the affine braid group
Induction functors.
The standard parabolic subgroup P k has Lie algebra generatedby b and E i ` ,i , i ‰ k . Let us define space X p P k q : “ b ˆ P k ˆ n and let us also use notation X p G n q for X . There is a natural embedding ¯ i k : X p P k q Ñ X and a natural projection¯ p k : X p P k q Ñ X p G k q ˆ X p G n ´ k q . The embedding ¯ i k satisfies the conditions for existenceof the push-forward and we can define the induction functor:ind k : “ ¯ i k ˚ ˝ ¯ p ˚ k : MF scB k p X p G k q , W q ˆ MF scB n ´ k p X p G n ´ k q , W q Ñ MF scB n p X p G n q , W q Similarly we define the space X ,fr p P k q Ă b ˆ P k ˆ n ˆ V as an open subset defined bythe stability condition (2.3). The last space has a natural projection map ¯ p k : X ,fr p P k q Ñ X p G k q ˆ X ,fr p G n ´ k q and the embedding ¯ i k : X ,fr p P k q Ñ X ,fr p G n q and we can define theinduction functor:ind k : “ ¯ i k ˚ ˝ ¯ p ˚ k : MF scB k p X p G k q , W q ˆ MF scB n ´ k p X ,fr p G n ´ k q , W q Ñ MF scB n p X ,fr p G n q , W q It is shown in section 6 (proposition 6.2) of [1] that the functor ind k is the homomorphismof the convolution algebras:ind k p F b F q ¯ ‹ ind k p G b G q “ ind k p F ¯ ‹ G b F ¯ ‹ G q . To define the non-reduced version of the induction functors one needs to introduce the space X ˝ p G n q “ g ˆ G n ˆ n ˆ n which is a slice to the G n -action on the space X . In particular, thepotential W on this slice becomes: W p X, g, Y , Y q “ Tr p X p Y ´ Ad g p Y qqq . Similarly to the case of the reduced space, one can define the space X ˝ p P k q : “ g ˆ P k ˆ n ˆ n and the corresponding maps i k : X ˝ p P k q Ñ X ˝ p G n q , p k : X ˝ p P k q Ñ X ˝ p G k q ˆ X ˝ p G n ´ k q .Thus we get a version of the induction functor for non-reduced spaces:ind k : “ i k ˚ ˝ p ˚ k : MF scB k p X p G k q , W q ˆ MF scB n ´ k p X p G n ´ k q , W q Ñ MF scB n p X p G n q , W q It is shown in proposition 6.1 of [1] that the Kn¨orrer functor is compatible with the inductionfunctor: ind k ˝ p Φ k ˆ Φ n ´ k q “ Φ n ˝ ind k . FFINE BRAID GROUP, JM ELEMENTS AND KNOT HOMOLOGY 11
Generators of the finite braid group action.
Let us define B -equivariant em-bedding i : X p B n q Ñ X , X p B q : “ b ˆ B ˆ n . The pull-back of W along the map i vanishes and the embedding i satisfies the conditions for existence of the push-forward i ˚ : MF scB p X p B n q , q Ñ MF scB p X p G n q , W q . We denote by C r X p B n qs P MF scB p X p B n q , q the matrix factorization with zero differential that is homologically non-trivial only in evenhomological degree. As it is shown in proposition 7.1 of [1] the push-forward¯ n : “ i ˚ p C r X p B n qsq is the unit in the convolution algebra. Similarly, n : “ Φ p ¯ n q is also a unit in non-reducedcase.Let us first discuss the case of the braids on two strands. The key to construction of thebraid group action in [1] is the following factorization in the case n “ W p X, g, Y q “ y p g x ` g x q g { det , where det “ det p g q and g “ „ g g g g , X “ „ x x x , Y “ „ y Thus we can define the following strongly equivariant Koszul matrix factorization:¯ C ` : “ p C r X s b Λ x θ y , D, , q P MF scB p X , W q ,D “ g y det θ ` r g p x ´ x q ` g x s BB θ , where Λ x θ y is the exterior algebra with one generator.This matrix factorization corresponds to the positive elementary braid on two strands.Using the induction functor we can extend the previous definition on the case of the arbi-trary number of strands. For that we introduce an insertion functor:Ind k,k ` : MF scB p X p G q , W q Ñ MF scB n p X p G n q , W q Ind k,k ` p F q : “ ind k ` p ind k ´ p ¯ k ´ ˆ F q ˆ ¯ n ´ k ´ q , and similarly we define non-reduced insertion functorInd k,k ` : MF scB p X p G q , W q Ñ MF scB n p X p G n q , W q . Thus we define the generators of the braid group as follows:¯ C p k q` : “ Ind k,k ` p ¯ C ` q , C p k q` : “ Ind k,k ` p C ` q . The section 11 of [1] is devoted to the proof of the braid relations between these elements:¯ C p k ` q` ¯ ‹ ¯ C p k q` ¯ ‹ ¯ C p k ` q` “ ¯ C p k q` ¯ ‹ ¯ C p k ` q` ¯ ‹ ¯ C p k q` , C p k ` q` ‹ C p k q` ‹ C p k ` q` “ C p k q` ‹ C p k ` q` ‹ C p k q` . Let us now discuss the inversion of the elementary braid. In view of inductive definition ofthe braid group action, it is sufficient to understand the inversion in the case n “ C ´ : “ C ` x´ χ , χ y P MF scB p X p G q , W q , and the definition of ¯ C ´ is similar. As we will see below, the definition of C ´ is actuallysymmetric with respect to the left-right twisting: C ´ “ C ` x χ , ´ χ y . Theorem 3.2.1.
We have: (3.1) C ` ‹ C ´ “ . Proof of this relation in the case of SL spaces in given in the section 9 of [1]. The sameproof works for GL -case considered in this paper.3.3. Generators of the affine braid group action.
The new generators that we wouldneed to construct the action of the affine braid group are of the form n x α, β y . The proposition9.1 of [1] states that only the sum of the weights α ` β matters. More precisely, we have thefollowing homotopy n x α, β y „ n x α ` γ, β ´ γ y . Also note that the element n x ř ni “ χ i , y is a central element of the convolution algebraand elements n x χ i , y , i “ , . . . , n generate a commutative subalgebra of the convolutionalgebra. In particular, in the case n “ C ` x´ χ , χ y “ x χ ` χ , y ‹ C ` x´ χ , χ y ‹ x´ χ ´ χ , y “ C ` x χ , ´ χ y Theorem 3.3.1.
The assignment σ ˘ i ÞÑ C p i q˘ , ∆ i ÞÑ n x χ i , y extends to the algebra homomorphism Φ aff : Br affn Ñ MF scB p X , W q . Proof.
Since the elements n x χ i , y mutually commute, it is enough to check the equation(3.2) C p i q` ‹ n x χ i ` , y ‹ C p i q` “ n x χ i , y . Let us first discuss the case n “
2. In this case the only relation that we need to show is C ` ‹ x χ , y ‹ C ` “ x χ , y . This relation follows from the previous theorem. Denote ζ “ χ ` χ , then(3.3) C ` ‹ x χ , y ‹ C ` “ C ` ‹ x´ χ ` ζ, y ‹ C ` “ C ` ‹ x´ χ , y ‹ C ` ‹ x χ , y‹ x ζ ´ χ , y “ C ` ‹ x , ´ χ y ‹ C ` ‹ x χ , y ‹ x χ , y “ C ` ‹ C ´ ‹ x χ , y “ x χ , y . The case of general n follows from the case n “ i ` ,i to the equation (3.3) we get therequired equation (3.2). (cid:3) Stabilization morphism.
To complete our proof of the theorem 1.2.4 we need to provethe following
Proposition 3.4.1.
We have the following formulas for the action of the forgetful functor: fgt ˚ : n x χ n , y ÞÑ n , ¯ n x χ n , y ÞÑ ¯ n . Proof.
Let us show the first equation since the second one is analogous. Indeed, the space X ,fr has coordinates p X, g , Y , g , Y , v q and the stability condition implies that g ´ p v q is thevector that has non-zero last component. Hence, the function S “ p g ´ p v qq n is an invertiblefunction on X ,fr and the multiplication by S yields a invertible homomorphism of the matrixfactorizations on X ,fr that identifies fgt ˚ p n x χ n , yq “ x χ n , y with n . (cid:3) FFINE BRAID GROUP, JM ELEMENTS AND KNOT HOMOLOGY 13
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A. Oblomkov, Department of Mathematics and Statistics, University of Massachusetts atAmherst, Lederle Graduate Research Tower, 710 N. Pleasant Street, Amherst, MA 01003 USA
E-mail address : [email protected] L. Rozansky, Department of Mathematics, University of North Carolina at Chapel Hill, CB
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