Skein Modules from Skew Howe Duality and Affine Extensions

We show that we can release the rigidity of the skew Howe duality process for sln knot invariants by rescaling the quantum Weyl group action, and recover skein modules for web-tangles. This skew Howe duality phenomenon can be extended to the affine slm case, corresponding to looking at tangles embed...

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Дата:	2015
Автор:	Queffelec, H.
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Опубліковано:	Інститут математики НАН України 2015
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	Skein Modules from Skew Howe Duality and Affine Extensions / H. Queffelec // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 39 назв. — англ.

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spelling	irk-123456789-1470182019-02-13T01:24:48Z Skein Modules from Skew Howe Duality and Affine Extensions Queffelec, H. We show that we can release the rigidity of the skew Howe duality process for sln knot invariants by rescaling the quantum Weyl group action, and recover skein modules for web-tangles. This skew Howe duality phenomenon can be extended to the affine slm case, corresponding to looking at tangles embedded in a solid torus. We investigate the relations between the invariants constructed by evaluation representations (and affinization of them) and usual skein modules, and give tools for interpretations of annular skein modules as sub-algebras of intertwiners for particular Uq(sln) representations. The categorification proposed in a joint work with A. Lauda and D. Rose also admits a direct extension in the affine case. 2015 Article Skein Modules from Skew Howe Duality and Affine Extensions / H. Queffelec // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 39 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81R50; 17B37; 17B67; 57M25; 57M27 DOI:10.3842/SIGMA.2015.030 http://dspace.nbuv.gov.ua/handle/123456789/147018 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We show that we can release the rigidity of the skew Howe duality process for sln knot invariants by rescaling the quantum Weyl group action, and recover skein modules for web-tangles. This skew Howe duality phenomenon can be extended to the affine slm case, corresponding to looking at tangles embedded in a solid torus. We investigate the relations between the invariants constructed by evaluation representations (and affinization of them) and usual skein modules, and give tools for interpretations of annular skein modules as sub-algebras of intertwiners for particular Uq(sln) representations. The categorification proposed in a joint work with A. Lauda and D. Rose also admits a direct extension in the affine case.
format	Article
author	Queffelec, H.
spellingShingle	Queffelec, H. Skein Modules from Skew Howe Duality and Affine Extensions Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Queffelec, H.
author_sort	Queffelec, H.
title	Skein Modules from Skew Howe Duality and Affine Extensions
title_short	Skein Modules from Skew Howe Duality and Affine Extensions
title_full	Skein Modules from Skew Howe Duality and Affine Extensions
title_fullStr	Skein Modules from Skew Howe Duality and Affine Extensions
title_full_unstemmed	Skein Modules from Skew Howe Duality and Affine Extensions
title_sort	skein modules from skew howe duality and affine extensions
publisher	Інститут математики НАН України
publishDate	2015
url	http://dspace.nbuv.gov.ua/handle/123456789/147018
citation_txt	Skein Modules from Skew Howe Duality and Affine Extensions / H. Queffelec // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 39 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT queffelech skeinmodulesfromskewhowedualityandaffineextensions
first_indexed	2025-07-11T01:09:35Z
last_indexed	2025-07-11T01:09:35Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 030, 36 pages Skein Modules from Skew Howe Duality and Affine Extensions? Hoel QUEFFELEC Mathematical Sciences Institute, The Australian National University, J.D. 27 Union Lane, Acton ACT 2601, Australia E-mail: hoel.queffelec@anu.edu.au URL: http://maths-people.anu.edu.au/~queffelech/ Received July 22, 2014, in final form March 30, 2015; Published online April 15, 2015 http://dx.doi.org/10.3842/SIGMA.2015.030 Abstract. We show that we can release the rigidity of the skew Howe duality process for sln knot invariants by rescaling the quantum Weyl group action, and recover skein modules for web-tangles. This skew Howe duality phenomenon can be extended to the affine slm case, corresponding to looking at tangles embedded in a solid torus. We investigate the relations between the invariants constructed by evaluation representations (and affinization of them) and usual skein modules, and give tools for interpretations of annular skein modules as sub- algebras of intertwiners for particular Uq(sln) representations. The categorification proposed in a joint work with A. Lauda and D. Rose also admits a direct extension in the affine case. Key words: skein modules; quantum groups; annulus; affine quantum groups 2010 Mathematics Subject Classification: 81R50; 17B37; 17B67; 57M25; 57M27 1 Introduction 1.1 Webs and skew-Howe duality Cautis, Kamnitzer and Licata [4, 5] recently introduced a reformulation of the sln Reshetikhin– Turaev invariants for knots and links based on the quantum skew Howe duality. This duality phenomenon involves two commuting actions of Uq(slm) and Uq(sln) on the quantum exterior algebra ∧ q(Cn ⊗Cm), where n corresponds to the sln-invariants we look at, and m governs the braiding of m-fold tensor products of sln-representations. In this framework, braidings arise from the so-called quantum Weyl group action [16, 29] on Uq(slm). This new process is naturally related to the concept of webs, which emerge from the study of sln knot invariants and describe intertwiners of sln-representations (see [25, 26, 33] for detailed studies of the spider categories they form). For each n, sln webs are trivalent oriented graphs with edges labeled with integers in {1, . . . , n}. At each vertex, the sum of the indices of the incoming edges equals the sum of the indices of the outgoing edges: k + l k l k + l k l ?This paper is a contribution to the Special Issue on New Directions in Lie Theory. The full collection is available at http://www.emis.de/journals/SIGMA/LieTheory2014.html mailto:hoel.queffelec@anu.edu.au http://maths-people.anu.edu.au/~queffelech/ http://dx.doi.org/10.3842/SIGMA.2015.030 http://www.emis.de/journals/SIGMA/LieTheory2014.html 2 H. Queffelec Here is an example of a web: k + l kl These diagrams are to be understood up to some local relations (see [6] for example), which are a diagrammatic analogue of relations between morphisms of Uq(sln)-representations that are at the origin of the definition of webs. Note that there are more refined notions of sln-webs, in particular concerning what to do with n-labeled strands. Indeed, a k-labeled strand corresponds to the k-th exterior power ∧k q (Cn) of the standard Uq(sln) representation Cn. The 0-th power is the trivial representation, and it appears natural not to depict it in webs. Similarly, the maximal exterior power ∧n q (Cn) is just the trivial representation, and it is usually forgotten as well (which comes with a correspondence between an edge labeled by k and the same edge with opposite orientation labeled by n − k, see for example [32]). However, it appears that this maximal exterior power plays a non-trivial role in some places, in particular when looking at categorification questions. An heuristic interpretation of this could be the fact that this representation corresponds to the determinant representation, which is indeed a trivial sln- representation, but is not a trivial gln-representation. This non-triviality has been encoded by tags in some places [33, 6], and applications to categorified knot invariants using these tags in the sl2 case can be found in a work by Clark, Morrison and Walker [9]. One can also choose to keep all the n-edges (which we will then depict doubled). Although the difference is at first sight minimal on the level of webs, it seems to play an important role at the categorified level, as suggested by Blanchet’s work [2] and developed in [28]. We sometimes refer to these webs as enhanced. For example, the enhanced web: n n − k − l kl is represented in the tagged version of webs as n − k − l kl The only difference between the two pictures above lies in the way to deal with the strands decorated with the maximum exterior power of the fundamental representation: while we keep them completely in the first case, they only appear locally in the second case. The Jones polynomial and its sln analogues naturally take place in the spider categories. Their reformulation in terms of quantum skew Howe duality proved to be a very powerful tool for understanding these categories, and led Cautis, Kamnitzer and Morrison [6] to solve conjectures on generators and relations for categories of representations Repq(sln). Furthermore, this process admits a very natural categorification [4], linking [28] topological categorifications based on skein theory [1, 18, 19, 20] and categorified quantum groups [21, 22, 23, 24]. Skein Modules from Skew Howe Duality and Affine Extensions 3 However, the skew Howe duality process is quite rigid, allowing to deal only with ladder webs, which are a particular class of webs with only upward oriented edges. This is a generalization to the web case of the notion of upward-oriented tangles, with the additional requirement that webs are presented in a rigid structure where strands are either vertical (the uprights of the ladder) or elementary horizontal pieces (the rungs of the ladder). For example, a ladder version of the previous web would be n l k n − k − l Furthermore, the relation established by Cautis–Kamnitzer–Licata between the braiding (or R-matrix) and the quantum Weyl group action does not allow to ignore crossings involving 0- labeled strands. For example, the definition of the braidings as it appears in [29] gives for such crossings a smoothing map Ψ as follows ΨT ′i  k 0  = (−1)kqk k 0 while we would like this crossing to be smoothed without creating any coefficient, since we do not want to consider 0-labeled strands in the skein context. Similarly, definitions provided in [4] wouldn’t produce any coefficients, but the use of tags in some of the Reidemeister-like web moves produces difficulties. In this paper, we find an appropriate rescaling of the Weyl group action that removes the rigidity in the diagrammatic formulation of link (and knotted web) invariants: the goal is to find a skew-Howe based process from which we can extract smoothing rules that yield a skein module for web-tangles. The next paragraph gives more details about these ideas. 1.2 Obtaining a skein module We give in this paper a detailed explanation of the skew-Howe duality process, focusing on obtaining sln skein modules from this rather rigid context, for any value of n. One of the problems that usually appears when looking at a local crossing in a skein context is that it can be understood in different ways. For example, k l can be translated as a positive (k, l) crossing, or (if we look at it from left to right) as a nega- tive (l, k) crossing with the l strand in reverse direction. These two crossings would give rise to different smoothings in their ladder transcriptions. The refinement we introduce in this paper is based on both a convenient rescaling of Lusztig’s definition of the braidings [29] with a glm-information, and keeping the whole enhanced informa- tion of webs, following ideas of Blanchet [2]. It is interesting to note that the original construction of Murakami–Ohtsuki–Yamada [34] was actually also using n-labeled strands and is consistent with this presentation. In the sl2 case, this leads to a rather unusual presentation of the skein 4 H. Queffelec module, since 2-labeled crossings produce when smoothed some non-trivial coefficients, so that the smoothing map Ψ would behave as follows Ψ ( ) = q−2 , Ψ ( ) = q2 . In the sl3 case, we similarly keep 1-, 2- and 3-labeled strands, which produce again different coefficients in the smoothings. The main result is then that there exists a version of the skew-Howe duality process from which the definition of the braiding can be used locally to define an invariant of framed web- tangles. A good understanding of the behavior of the braidings back in the representation- theory world is of great help in the proof of the invariance under Kauffman’s web-moves and considerably simplify them, and also clarifies the categorification of these results. 1.3 Affine extensions The skew-Howe duality process is based on two commuting actions of Uq(sln) and Uq(slm) on the module ∧N q (Cn⊗Cm). Uq(sln) corresponds to the quantum invariant we are looking at, and we want to keep it unchanged, but Uq(slm) appears more as a parameter, and we may want to consider extensions of it. A first step is to replace Uq(slm) by its affine version Uq(ŝlm). Classical representation-theoretic tools tell us that we can extend the action of Uq(slm) to a Uq(ŝlm) one, keeping by construction the commutation property with the Uq(sln) action. These extensions can be achieved by the process of evaluation representations [8]. This natu- rally provides knotted-web invariants for the cylinder, and the only question is to relate these invariants to the usual skein module associated to the thickened surface. We show that the evaluation representations with a particular choice of the parameter give the skein module of the filled cylinder, that can be refined by passing to the affinization of the representations. These constructions therefore provide a very natural extension of Jones’ construction in the case of web-tangles drawn on the cylinder. We also investigate better descriptions of the annular skein module in terms of Uq(slm), and we take a first step towards a realization as a sub-algebra of intertwiners for an expli- cit Uq(sln) representation, which would give to it the same kind of representation-theory flavored interpretation as we have in the linear case. Many proofs use the fact that relations for Uq(slm) and Uq(ŝlm) locally have the same form. Thus, just as at the uncategorified level, the categorification of the skew Howe process provided in [28] admits a direct extension to the affine case. Note that a recent paper by Mackaay and Thiel [31] presents a categorification of affine q-Schur algebras. Although their paper does not directly deal with annular knots, it would be interesting to understand its implications in terms of categorified invariants of annular web- tangles and the links with categorified quantum skew-Howe duality. 2 Skew Howe duality and skein modules 2.1 Skew Howe duality 2.1.1 Context We first give a short description of the skew Howe duality phenomenon for usual sln as explained in [5] and [6]. Skein Modules from Skew Howe Duality and Affine Extensions 5 We look at the quantum group Uq(slm) as the C[q, q−1]-algebra generated by the Chevalley elements Ei, Fi, K ±1 i , for 1 ≤ i ≤ n− 1, subject to the relations: KiK −1 i = K−1 i Ki = 1, KiKj = KjKi, KiEjK −1 i = qaijEj , KiFjK −1 i = q−aijFj , EiFj − FjEi = δij Ki −K−1 i q − q−1 , E2 i Ej − ( q + q−1 ) EiEjEi + EjE 2 i = 0 if j = i± 1, F 2 i Fj − ( q + q−1 ) FiFjFi + FjF 2 i = 0 if j = i± 1, EiEj = EjEi, FiFj = FjFi if \|i− j\| > 1. The idempotented version of Uq(slm) will be denoted U̇q(slm). Generators are 1λ, Ei1λ and Fi1λ, for all weights λ. The unit is then replaced by a collection of orthogonal idempotents 1λ indexed by the weight lattice of slm, 1λ1λ′ = δλλ′1λ, such that if λ = (λ1, λ2, . . . , λm−1), then Ki1λ = 1λKi = qλi1λ, Ei1λ = 1λ+αiEi, Fi1λ = 1λ−αiFi, where λ+ αi =  (λ1 + 2, λ2 − 1, λ3, . . . , λm−2, λm−1) if i = 1, (λ1, λ2, . . . , λm−3, λm−2 − 1, λm−1 + 2) if i = m− 1, (λ1, . . . , λi−1 − 1, λi + 2, λi+1 − 1, . . . , λm−1) otherwise. Uq(slm) can be endowed with the structure of a Hopf algebra, with coproduct ∆: Uq(slm) 7→ Uq(slm)⊗ Uq(slm) given on Chevalley generators by ∆(Ei) = 1⊗ Ei + Ei ⊗Ki, ∆(Fi) = K−1 i ⊗ Fi + Fi ⊗ 1, ∆(K±1 i ) = K±1 i ⊗K ±1 i . Define ∧ q(Cr) as the algebra generated by r variables∧ q (Cr) = C [ q, q−1 ] 〈X1, . . . , Xr〉/ ( X2 i , XiXj + q−1XjXi for i < j ) . This algebra can be given a Uq(slr) action, extending the natural representation1. More precisely: EiXi = Xi+1, EiXj = 0 if j 6= i, FiXi+1 = Xi, FiXj = 0 if j 6= i+ 1, KiXi = q−1Xi, KiXi+1 = qXi+1, KiXj = Xj otherwise. We now consider ∧ q(Cn ⊗ Cm), where, following [6], the generating variables can be deno- ted zij with 1 ≤ i ≤ n, 1 ≤ j ≤ m, subject to skew-commutation relations. There are two isomorphisms:∧ q (Cn)⊗m ← ∧ q (Cn ⊗ Cm)→ ∧ q (Cm)⊗n. We can thus endow this module with actions of Uq(sln) and Uq(slm), which Cautis, Kamnitzer and Licata have proved to commute, calling this quantum skew Howe duality. Furthermore, 1Actually, we choose here a non-standard form (dual) for the natural representation in order to obtain the same conventions as in [28]. 6 H. Queffelec Uq(sln) and Uq(slm) form a Howe pair, which is a key argument in [6].2 The actions of the two quantum groups can be deduced, for Uq(slm) for example, from the expressions on the variables zij : Ejzij = zi,j+1, Ekzij = 0 if k 6= j, Fjzi,j+1 = zi,j , Fkzij = 0 if k 6= j + 1, Kjzij = q−1zij , Kjzi,j+1 = qzi,j+1, Kkzij = zij otherwise. We can assign degree one to each generating variable. Given an integer N , the subspace of degree N decomposes as an sln-representation as follows:∧N q (Cn ⊗ Cm) = ⊕ a1+···+am=N ∧a1 q (Cn)⊗ · · · ⊗ ∧am q (Cn). Each direct summand is an m-fold tensor product of minuscule sln-representations, but is not stable under the action of Uq(slm). However, it appears (tracking it from the explicit definition of the actions) that each subspace ∧a1 q (Cn)⊗ · · · ⊗ ∧am q (Cn) is a Uq(slm) weight space of weight (a2−a1, a3−a2, . . . , am−am−1) In particular, if m = 2, the subspaces are of the form ∧k q (Cn)⊗∧l q(Cn), which are both Uq(sln)-modules and Uq(sl2) weight spaces of weight l − k. The action of Uq(slm) can be explicitly tracked (see Table (2.2) for the case where m = n = N = 2), and we see that Ei : ∧a1 q (Cn) ⊗ · · · ⊗ ∧ai q (Cn) ⊗ ∧ai+1 q (Cn) ⊗ · · · ⊗ ∧am q (Cn) 7→ ∧a1 q (Cn) ⊗ · · · ⊗∧ai−1 q (Cn)⊗ ∧ai+1+1 q (Cn)⊗ · · · ⊗ ∧am q (Cn). This Uq(slm) action can be depicted by some particular diagrams called ladders. To a direct summand ∧a1 q (Cn) ⊗ · · · ⊗ ∧am q (Cn) we assign a sequence (a1, . . . , am) depicted as weighted upward strands (a1, . . . , am) 7→ a1 · · · ai ai+1 · · · am Strands labeled by zero will be erased, and we will sometimes depict the n-labeled strands doubled. We represent the action of Ei and Fi as follows Ei 7→ a1 · · · ai ai+1 ai − 1 ai+1 + 1 · · · am and Fi 7→ a1 · · · ai ai+1 ai + 1 ai+1 − 1 · · · am (2.1) where we only depicted the strands 1, i, i + 1 and m: straight strands with indices aj , j 6= 1, i, i+ 1, m have to be added in place of the dots. The diagrams are to be read from bottom to top. We can define the notion of ladder as any morphism obtained by composition of identities and elementary morphisms given by the images of Ei and Fi. The above diagrams have an interpretation in terms of webs. Recall that sln webs are trivalent oriented graphs with edges indexed by integers 1, . . . , n, so that at each vertex, the 2Note that proving the commutation is an easy computation, while it is much harder to prove that both algebras are each other commutant. Skein Modules from Skew Howe Duality and Affine Extensions 7 sum of outgoing labels equals the sum of ingoing labels (in the literature, the n-strands are usually erased or only kept as tags on the other strands, which we will not do here). These graphs are considered modulo some local relations (see for example [6], or Definition 2.4, for a more precise description). The webs may be understood as sln analogues of the skein module in the sl2 case (see also [34] or [32] for more details). An interesting fact is that all of the web relations in the spider category can be recovered from the relations in Uq(slm) via its action on webs3. We refer for this and for a complete description of the spider category to [6]. We give below an example of the translation process, which gives a ladder whose closure is the web depicted in the introduction (with k = l = 1) with n = 3, m = 3 and N = 3: E1F2E2F11(3,−3) 7→ 3 1 2 11 In the case where m = n = N = 2, the Uq(slm) action can be explicitly given: summand generator image under E image under F∧2 q ⊗ ∧0 q z11 ⊗ z21 z11 ⊗ z22 + q−1z12 ⊗ z21 0 ∧1 q ⊗ ∧1 q z11 ⊗ z12 0 0 z21 ⊗ z22 0 0 z11 ⊗ z22 qz12 ⊗ z22 qz11 ⊗ z21 z12 ⊗ z21 z12 ⊗ z22 z11 ⊗ z21∧0 q ⊗ ∧2 q z12 ⊗ z22 0 q−1z12 ⊗ z21 + z11 ⊗ z22 (2.2) The previous table corresponds in the diagrammatic world to the next situation: ∧2 q ⊗ ∧0 q ∧1 q ⊗ ∧1 q ∧0 q ⊗ ∧2 q // // oooo where in the above pictures, 0-strands are depicted dotted and 2-strands are doubled. 2.1.2 Quantum Weyl group action The action of the Weyl group Sm of slm on the weights q-deforms to give rise to a braid group action on representations of Uq(slm). This phenomenon is referred to as the quantum Weyl group action (see [5, 16, 29]). Generators of the braid group action are elements of the completion Ũq(slm) of Uq(slm). This ring is defined (see [16] for example) as a quotient of the ring of series ∞∑ k=1 Xk of elements of Uq(slm), acting on each irreducible representation V (λ) of highest weight λ by zero but for 3This statement holds in the case where n-strands are only kept as tags. We will therefore add some relations on the n-strands later. 8 H. Queffelec finitely many terms Xk. We then consider the quotient of this ring by the two-sided ideal of elements acting by zero on all V (λ). Following [29], to si the elementary transposition corresponding to the root αi, we associate the map T ′′i ∈ Ũq(slm): T ′′i 1λ := ∑ a−b+c=−λi (−1)bq−ac+bE (a) i F (b) i E (c) i 1λ. With this definition, T ′′i gives an endomorphism of any finite-dimensional representation. Note that if v is a weight vector of weight λ, Ti(v) is a weight vector of weight si(λ). Taking m = 2 for simplicity, we have T ′′± ∈ Ũq(sl2), acting on ∧N q (Cn ⊗C2). This stabilizes the whole representation, and gives a morphism of Uq(sln) representations, from ∧k q (Cn) ⊗∧l q(Cn) to ∧l q(Cn) ⊗ ∧k q (Cn). It is shown in [5] that this Uq(sln) endomorphism recovers the braiding. This is the starting point of a reinterpretation of Reshetikhin–Turaev invariants in terms of skew-Howe duality [4], which admits natural categorifications [4, 28]. The name quantum Weyl group is used by different authors with slightly different sig- nifications. The first one, where we use the notation T ′′i , consists in considering morphisms of representations, acting on the category of finite-dimensional modules. We can also use it to build automorphisms of the quantum group itself, by conjugation. Following [16], we denote the latter CT ′′i : X 7→ T ′′i XT ′′ i −1. We will use both versions in this paper. We will need some results concerning the behavior of these elements for later use. For w = si1 · · · sin element of the Weyl group written in reduced form, where si are simple reflections, we define T ′′w = T ′′i1 · · ·T ′′ ik . Proposition 2.1 ([8, Theorem 8.1.2], [29, Section 37.1.3], [16]). CT ′′i (Ei1λ) = −q−λiFi1si(λ), CT ′′i (1λFi) = −qλi1si(λ)Ei. For w ∈W such that w(αi) = αj, CT ′′w(Ei) = Ej. Other intertwiners, defined in [29], may also be of interest: T ′i1λ := ∑ a−b+c=λi (−1)bq−ac+bF (a) i E (b) i F (c) i 1λ. We have an analogue of Proposition 2.1: Proposition 2.2. CT ′i (1λEi) = −q−λi1si(λ)Fi, CT ′i (Fi1λ) = −qλiEi1si(λ). For w ∈W such that w(αi) = αj, CT ′w(Ei) = Ej. The relation between the actions of both definitions is given by: Proposition 2.3 ([29, Sections 5.2.3, 37.1.2]). T ′′i and T ′i , where the bar corresponds to chan- ging q to q−1, are inverse of each other. T ′′i 1λ and (−1)λiqλiT ′i1λ act the same way on any integrable module. Skein Modules from Skew Howe Duality and Affine Extensions 9 2.1.3 Skew Howe duality and quantum invariants for knots As we have seen, the skew-Howe duality process gives us different pieces of the Jones (or Reshetikhin–Turaev) invariants: • minuscule representations ∧k q (Cn) of Uq(sln). This means that we are looking at knot invariants where we decorate the strands with minuscule representations. In particular, this does not deal with the colored Jones polynomial or its sln generalizations, where the strands of the knot can be decorated with any finite-dimensional representations. Paths using Jones-Wenzl projectors, and their categorifications in the categorified case, can be given to relate the general invariants to the ones we study here [10, 12, 35, 37]. • elementary morphisms between tensor products of these representations, given as images of Ei and Fi ∈ Uq(slm). These morphisms involve minuscule representations, but do not directly deal with duals, which in the language of knots means that we are looking at upward tangles (or their generalization for webs). The bridge with general knots or links is established in our case in [6] (see also [32]). • braiding between minuscule representations, understood in terms of the quantum Weyl group action of Uq(slm). Again, this is given in the framework of ladders, and relaxing this structure will be one of the goals of the next section. 2.2 Skein modules 2.2.1 Braidings for skein modules Let us now turn toward knots, or ladder analogues of them. The previous diagrammatic process gives us an algebraic interpretation of ladder webs, as well as a definition of the braiding for the tensor product of two minuscule representations. This braiding corresponds in the diagrammatic world to a crossing between two adjacent strands in a ladder, the explicit formulas for T ′i or T ′′i giving a way to smooth it and replace it by a sum of ladders without crossing. We start by defining more precisely the notion of skein module, before relating it to the previous analysis. By skein module, we usually refer here both to the module itself and to the Kauffman bracket defining a map from web-tangles to the module. The definition below is adapted from [6]. Definition 2.4. Let nWeb, the sln web skein module, be the Z[q, q−1]-module generated by webs (planar oriented trivalent graphs with preserved flow), possibly with boundary, up to isotopy and the following web relations: k + l k l k + l = [ k + l l ] q k + l , k l k + l k = [ n− k l ] q k (2.3) k l m k + l +m = k l m k + l +m , k l s r = [ r + s r ] q k l r + s (2.4) k l s r = ∑ t [ k − l + r − s t ] q k ls − t r − t (2.5) 10 H. Queffelec All equations come with the ones obtained by mirror image and arrow reversion. Recall that [k] = qk−q−k q−q−1 , [k]! = [k][k − 1] · · · [1] and [ p k ] q = [p]! [k]![p−k]! . The skein module described above can be given the structure of a category, with objects given by oriented points with labels on a horizontal line (the boundary of strands), and morphisms the webs joining the dots on two such parallel lines. A subcategory is of particular interest for the skew-Howe interpretation, and turns out to essentially represent all the information we need. Assuming a value of N is fixed, let us call Φ: U̇q(slm) 7→ nWeb the map described in equation (2.1). Definition 2.5. Define nWeb+ m to be the image category Φ(U̇q(slm)), with objects, sequences (a1, . . . , am) (0 ≤ ai ≤ n) labeling points on a horizontal line, together with a zero object, and morphisms, sln webs between such sequences (in the sense of Definition 2.4), that are composition of the images of Ei and Fi, as depicted in equation (2.1). We sometimes refer to such webs as upward webs, or ladders. These ladder webs have their boundary split in two parts, with all strands oriented inside for the bottom part, and outside for the upper part. Although general webs are more general than this particular situation, it is shown in [6] that they can be related to the particular class of webs obtained from ladders using the tool of pivotal categories. The use of tags makes the situation somewhat simpler (but harder to fit in a skein module formulation!), but the next relations (and the ones obtained by symmetry on the next ones) are particular realizations of the ones given in [6] in the case where we keep the n-labeled strands (and are special cases of Definition 2.4): n = 1, = , k n n− k k = k In the above pictures, the n-th strands are depicted doubled to emphasize their particular role. The skein module nWeb is a natural target for maps from (equivalence classes of) diagrams of knotted webs. We call knotted webs, or web-tangles, the natural generalization of knots and tangles to webs. For example, knotted webs are isotopy classes of embeddings into R3 of closed webs, and produce diagrams of knotted webs as generic projections onto a plane. Web-tangles are the natural generalization allowing boundaries. Recall from [17] (see also [3, Theorem 2]) the relations that generalize Reidemeister moves (in a framed version, where a numbered circle on a strand stands for twists): any two diagrams representing the same web-tangle are related by a sequence of moves of the following kind: ' , ∼ , (2.6) ∼ , ∼ (2.7) Skein Modules from Skew Howe Duality and Affine Extensions 11 ∼ , ∼ , (2.8) ∼ ◦ ◦ ◦ 1 2 −1 2 −1 2 (2.9) Definition 2.6. A Kauffman bracket for sln webs is a map Ψ from diagrams of sln web-tangles to nWeb, defined locally by replacing a crossing by a linear combination of smoothings, and subject to relations (2.6)–(2.9). Let us now restrict to the knotted analogue of nWeb+ m, and define a knotted ladder (or web-tangle in ladder position) to be a vertical composition of images of Ei1λ ∈ U̇q(slm), Fi1λ ∈ U̇q(slm) and crossings between two adjacent uprights in the ladder. Interpreting a crossing between the i-th and (i + 1)-th strands as the quantum Weyl group action given by T ′′i , one obtains a smoothing process for crossings: ΨT ′′i  2 2 1 1  = 2 2 1 1 − q 2 2 1 12 Thus, smoothing all crossings in a ladder web-tangle, one obtains a formal sum of non-knotted ladders that one can see as an element of a skein module. To obtain a more powerful skein module allowing less rigidity, we want to forget the 0-labeled strands. Indeed, in ladder position, even if the 0-labeled strands are not depicted, one knows where they are. If we want to start from any diagram and use the same smoothing rules as in the ladder case, we cannot know where 0-strands should be and we want to make sure that crossings involving 0-labeled strands do not play any role. The goal of this section is to obtain a skew-Howe duality process with a conveniently rescaled braiding, so that the smoothing rules derived from this braiding induce a Kauffman bracket for general sln web-tangles. Using [29, Proposition 5.2.2], we can show that the use of the smoothing rules provided by T ′′i gives the expected non-rescaled diagonal strand for the “trivial” positive (k, 0) crossing. Similarly, using T ′i −1 gives the expected result for the negative (0, k) crossing. Proposition 2.7. ΨT ′′i  k 0  = k 0 , ΨT ′i −1  k0  = k0 Note that if we use T ′′i on a (0, k) crossing and T ′i on a (k, 0) crossing, the situation is different. Using Proposition 2.3, we have: ΨT ′i  k 0  = (−1)kqk k 0 , ΨT ′′i −1  k0  = (−1)kq−k k0 12 H. Queffelec It appears that we cannot choose one of the two solutions and apply it in all cases, since there would always be a situation where a trivial crossing would lead to a non-trivially rescaled piece of strand in the associated skein module. A natural idea would be to use a braiding mixing both definitions, which may produce some gaps if we still want to have some instances of Propositions 2.1 and 2.2 (which will prove useful later). In order to avoid these distortions, we introduce additional rescalings that utilize Uq(glm) data that is naturally encoded in the representation we are looking at (namely, the sequence (a1, . . . , am), which is determined by the slm weight and the choice of an integer N). Note that the following definitions are rather symmetric in the T ′i ’s and T ′′i ’s Ti1λ = (−1)−ai+1q−ai+1T ′′i 1λ = (−1)−aiq−aiT ′i1λ, T−1 i 1λ = (−1)aiqaiT ′′i −1 1λ = (−1)ai+1qai+1T ′i −1 1λ. (2.10) It is easy to see from Proposition 2.3 that both definitions agree, and that this definition still provides a braiding. We can check that we still have CT1T2(E1) = E2 as endomorphisms of a given representation appearing in the skew Howe context. 2.2.2 sl2 case Let us now give a complete description of the sl2 case. In [2], Blanchet introduces sl2 webs to be oriented trivalent graphs with two kinds of edges (1 and 2-labeled, we draw the latter doubled), and vertices having two ingoing 1-labeled strands and one outgoing 2-labeled one, or one ingoing 2-labeled strand and two outgoing 1-labeled ones. The sl2 skein module 2Web is the quotient of (linear combinations of) webs with edges labeled 1 or 2 by the next relations: = [2], = 1, = (2.11) = [2] , = (2.12) For non-oriented webs above, the depicted relations hold for any compatible orientation. The definition of the braidings then gives the following smoothing rules: ΨTi ( ) = −q−1 + , ΨTi ( ) = −q + ΨTi ( ) = −q−1 , ΨTi ( ) = −q ΨTi ( ) = −q−1 , ΨTi ( ) = −q ΨTi ( ) = q−2 , ΨTi ( ) = q2 We could check, following [17], that the previous relations (2.11), (2.12), and the above smoothing relations define a framed skein module. Checking directly all formulas is rather long Skein Modules from Skew Howe Duality and Affine Extensions 13 and tedious, and we note that using the description in terms of the Uq(slm)-action gives us an efficient way to considerably simplify the proof, in the general case. Indeed, most formulas we want to check are consequences of Uq(slm)-relations. The previous skein module provides invariants of framed webs. Here are the effects of adding a negative twist on a 1-strand (depicted in a ribbon version in the two left parts of the equation below): ΨTi ( ) = ΨTi ( ) = −q + = −q2 The same computation for a 2-labeled strand gives a q2 coefficient. We may introduce twists with half-integers, assigning to them in the negative case the multiplication by q, and in the positive case the multiplication by q−1, up to fourth roots of the unity. We fix the value to be (−1) k 2 q −2k+k(k−1) 2 , where k stands for the labeling of the strand, and where we have chosen a favorite primitive fourth root of the unity (−1) 1 2 . The same formula will generalize to the sln case, assigning to a half twist on a k-strand (−1) k 2 q −nk+k(k−1) 2 . 2.2.3 sln case The previous process applies as well to any value of n, and produces a skein module in the following sense: Cautis, Kamnitzer and Morrison [6] prove that all web relations come from U̇q(slm) relations, so we just need to extend it to crossings and prove the invariance under Reidemeister moves. This is the purpose of the following theorem, which is the main result of the first part of this article. Theorem 2.8. For all n, the map ΨTi induced by the local smoothing rules defined by Ti (from relation (2.10)) for positive crossings and T−1 i for negative crossings extends to a Kauffman bracket on sln web-tangles. In a diagrammatic version, the smoothing rules for (k, l) crossings are thus given as follows: ΨTi  k l  = ∑ a−b+c=k−l (−1)b−lq−ac+b−l k l c b a ΨTi  k l  = ∑ a−b+c=l−k (−1)b+kqac−b+k k l c b a Proof. Braid-like relations: Braid-like Reidemeister II relation is direct, and braid-like Rei- demeister III relations are consequences of the braiding relation (see [9] for a presentation of all 6 braid-like Reidemeister III relations). Framing: The framed Reidemeister I relation is easy to check. Furthermore, we can deal locally with the framing as we did in the sl2-case. Details will be given in Lemma 2.9. Braid-like web relation 2.8:4 Braid-like relations (2.8) are consequences of the next equa- 4Note that similar relations are studied at the categorified level in [30]. 14 H. Queffelec lity, or similar ones: a1 a2 a3 = a1 a2 a3 The previous relation is a diagrammatically depicted consequence of CT1T2(E1) = E2, a rela- tion from Propositions 2.1 and 2.2, which still holds after rescalings. For obtaining the general case, one needs a straightforward generalization of the previous relation: CT1T2(E (k) 1 ) = E (k) 2 . Star relations using duality: Following [9], it suffices to have the Reidemeister II relation with opposite orientations to deduce the following Reidemeister III star relations: ∼ , ∼ We obtain the Reidemeister II case from the braid-like one as follows: ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ We used in the previous computation the star Reidemeister II relation in the n-n case, which is easy to prove: ∼ ∼ ∼ Then, we want to obtain the missing forms of relation (2.8). We proceed as follows in one case, the other ones being similar: ∼ ∼ ∼ ∼ Skein Modules from Skew Howe Duality and Affine Extensions 15 Relation (2.9) and framing: The last relation, for which we need a better understanding of the framing, will be proved separately in Lemma 2.10 below. � It is easy to see that a positive framing on a k-strand is equivalent to multiplication by a polynomial in q and q−1. Denote tk this polynomial: ΨTi  ◦1 k  = tk k Lemma 2.9. tk = (−1)kq−knqk(k−1). Note that this formula explains the choice for the half twists. Proof. We claim that tk+1t −1 k = −q2k−n. Indeed, the following relation holds from already proven Kauffman relations: k + 1 1 k = = ◦ ◦ The equality of the l.h.s. and r.h.s parts implies the recurrence relation. The general solution follows then from the computation of the value for a 1-strand. An explicit computation for this gives t1 = −q−n. � We are now ready to prove the last relation by induction: Lemma 2.10. ΨTi  k r l  = ΨTi  k r l ◦ ◦ ◦ 1 2 −1 2 −1 2  Proof. The computation is easy for r = 1 or l = 1. Then we use: ΨTi  k r l  = 1 [l] ΨTi  l − 1 1 k  = 1 [l] ΨTi  k r l  = 1 [l] ΨTi  k r l  = a 1 2 r+1a −1 2 r a −1 2 1 [l] ΨTi  k r l  = a 1 2 r+1a −1 2 r a −1 2 1 [l] ΨTi  k r l  = a 1 2 r+1a −1 2 r a −1 2 1 a 1 2 k−1a −1 2 l−1a −1 2 r [l] k r l = a 1 2 r+1a −1 2 r a −1 2 1 a 1 2 k−1a −1 2 l−1a −1 2 r k r l 16 H. Queffelec An explicit computation of the coefficient shows that the previous term equals: ΨTi  k r l ◦ 1 2 ◦−1 2 ◦−1 2  , which completes the proof. � We therefore obtain a well-defined skein module providing an invariant of knotted webs. Note that in the sl3 case, 2-strands are usually translated into 1-strands by reversing the orientation. In this case, smoothings of crossings would be defined only up to a power of q, and understanding a skew-Howe based way to fix this power seems difficult. We choose here not to apply this duality process and keep distinct 1- and 2-strands with their own orientations, and more generally to keep all strands numbered 1, . . . , n in the sln case, with their orientation. So, we have seen that to a web-tangle in ladder position, we can assign a Uq(sln) morphism of tensor product of minuscule representations. The diagrammatic form of this morphism corre- sponds to the image of the same web-tangle in the sln skein module. If we start with a non-ladder web, we can assign to it its skein element, but the skew-Howe process does not directly apply. Cautis, Kamnitzer and Morrison [6] explain a process for turning upward webs to ladder form, which we summarize in the following section. 2.2.4 Turning a knot to a ladder Let us now consider a tangle T (possibly a web-tangle) with only upward boundaries. Following Cautis–Kamnitzer–Morrison, we can present it as: T = T ′ The left part of the r.h.s. is easily presentable as a ladder. The tangle T ′ is assumed to be presented as a horizontal grid diagram generated by caps, cups and crossings (plus 3-valent vertices for webs). We request to have all crossings vertical, which is possible up to some isotopy. So, two caps or cups cannot lie one over the other one, and we determine the number of n-strands we will have to add as the number of elementary pieces that contain a downward strand: we will then put a strand on the right of this place. Let this number be denoted α. This being done, start over, but adjoining on the right of T α upward n-strands placed at the right place T = T ′ By performing some moves and simplifications near the downward strands, we get a ladder L. These local changes Cautis–Kamnitzer–Morrison perform are smoothings and simplifications of Skein Modules from Skew Howe Duality and Affine Extensions 17 some Reidemeister moves, and so the image of the tangle is equal in the previous skein module to T with α disjoint n-strands added to it. For example, if we start from the elementary web we considered in the introduction: k + l kl we can add on the right one n-labeled strand and perform Reidemeister–Kauffman moves: k + l kl n ∼ k + l kln ∼ k + l kln ∼ n l k n − k − l The last isomorphism above is a digon removal, which can be found in [6] for example. So, from any upward web-tangle union n-strands, we can obtain by a succession of Reide- meister moves and equivalences a ladder diagram. The morphism of representation we compute by the skew Howe process has then a diagrammatic depiction equivalent to the skein element associated to the web-tangle we started from union the n-strands. Note now that instead of pulling n-strands from far away, we could have performed a Jones– Kauffman product. Recall that the skein module may be endowed with an algebra structure by defining α ∗ β to be the smoothing of the superposition of diagrams of α over β. This superpo- sition is usually assumed to be a knotted web diagram, meaning that the only singularities are crossings. However, we can allow a singular case: k ∗ n → n− k ∼ ∼ where the dashed line on the left above indicates the place we want to put the n strand: this allows not to perform any simplification on the diagram. This re-interpretation of the process will show useful when we turn to the annular case, where we have no free space where to put the n-strands before pulling them on the place they are needed. We have seen here only the case where all the boundary of the tangle is upward. First, notice that this is enough for dealing with knots. However, as explained in [6], any tangle is actually isomorphic to such an upward tangle. 3 Affine extensions We have seen how the skew-Howe duality process, that involves two commuting actions of Uq(sln) and Uq(slm), helps redefine Reshetikhin–Turaev sln invariants for knots and links, that extend to invariants of knotted webs. The first quantum group controls the invariant we are looking at, and we therefore want to keep it unchanged. But the second one plays the role of a parameter related to the topology of the space we are working in. We can thus try to modify the topology of this space. 18 H. Queffelec One of the easiest extensions we can perform starting from Uq(slm) is to pass to its affine version Uq(ŝlm), and we will show that the topological analogue of this is to close the square the knots were drawn in into an annulus. We begin by defining different versions of the quantum affine algebra Uq(ŝlm) that we will use here, before turning toward easy representations of it. We then relate this extension to knots, and study the invariants we can deduce from it. 3.1 Affine slm Uq(ŝlm) is the quantum affine algebra corresponding to Uq(slm), that is the Kac–Moody algebra described by the following Dynkin diagram: ◦ 1 ◦ 2 · · · ◦ m− 2 ◦ m− 1 ◦ 0 (3.1) Following [15], we consider the algebra Uq(ŝlm) as generated by Chevalley generators Ei, Fi and K±1 i for 0 ≤ i ≤ m − 1, and extra generators K±1 d corresponding to the null root. The elements Ei, Fi and K±1 i are subject to slm relations, where we identify m and 0, so that the quantum Serre relations hold for pairs (E0, E1), (F0, F1), (E0, Em−1) and (F0, Fm−1): KiK −1 i = K−1 i Ki = 1, KiKj = KjKi, KiEjK −1 i = qaijEj , KiFjK −1 i = q−aijFj , EiFj − FjEi = δij Ki −K−1 i q − q−1 , E2 i Ej − ( q + q−1 ) EiEjEi + EjE 2 i = 0 if j = i± 1, F 2 i Fj − ( q + q−1 ) FiFjFi + FjF 2 i = 0 if j = i± 1, EiEj = EjEi, FiFj = FjFi if \|i− j\| > 1. Furthermore, Kd and K−1 d are subject to the following relations: KdKi = KiKd ∀ i ∈ {0, . . . ,m− 1}, KdEiK −1 d = qδ0,iEi ∀ i ∈ {0, . . . ,m− 1}, KdFiK −1 d = q−δ0,iFi ∀ i ∈ {0, . . . ,m− 1}. If we restrict to the sub-algebra generated by Ei, Fi and K±1 i , we produce a quantum group usually denoted U ′q(ŝlm). A key difference between the two versions is that the second one has finite dimensional irreducible modules, while the first one admits no non-trivial finite dimen- sional representations. We will also use an idempotented version of U ′q(ŝlm), that we denote U̇′q(ŝlm), generated by 1λ, Ei1λ and Fi1λ with the obvious generalization of the relations of the slm case. Weights here are m-tuples λ = (λ0, . . . , λm−1), which in our case, with N fixed, will be related to the sequences (a1, . . . , am) by λi = ai+1−ai for i 6= 0 and λ0 = a1−am. Note that we have ∑ λi = 0. 3.2 Evaluation representations Uq(slm)-representations may be extended to representations of U ′q(ŝlm). The complete formulas (that require a step through Uq(glm)) can be found in [8, p. 400]. These formulas seem at first sight a bit mysterious, and rather than directly using them, we choose to define differently Skein Modules from Skew Howe Duality and Affine Extensions 19 extensions based on the use of the braidings T ′′i and Ti. We will denote by ρa and ρ̃a the two morphisms used in these definitions. We will then investigate a posteriori the relation with usual formulas as written in [8] in Proposition 3.3. First, recall that CT ′′i is the automorphism of Uq(slm) (or its idempotented version) given by conjugation by T ′′i (that is, X ∈ Uq(slm) 7→ T ′′i XT ′′ i −1). We assume here that whenever we refer to a weight λ, a number N is fixed, and we freely refer to the associated m-tuple (a1, . . . , am). For a complex number, define ρa : U̇q(ŝlm) 7→ U̇q(slm) by: ρa(E01λ) = aq−(a1+am)CT ′′m−1 · · ·CT ′′2 (F1)1λ, ρa(F01λ) = a−1qa1+amCT ′′m−1 · · ·CT ′′2 (E1)1λ. ρa is extended to other generators Ei (i 6= 0) by sending Ei ∈ Uq(ŝlm) to Ei ∈ Uq(slm), and similarly for Fi. In terms of the graphical calculus previously defined, this can be drawn as: E01λ 7→ aq−(a1+am)ΨT ′′i  · · · · · · · · · a1 a2 am−1 am  F01λ 7→ a−1qa1+amΨT ′′i  · · · · · · · · · a1 a2 am−1 am  (3.2) The above depiction corresponds to letting U̇q(ŝlm) act on a tensor product of minuscule representations of Uq(slm) via the map ρa. This diagrammatic definition gives us an easy way to reprove (in Proposition 3.1) in a diagrammatic fashion the results of [8, Proposition 12.2.10] applied to these particular representations ∧ q(Cn⊗Cm). However, since all fundamental Uq(slm) representations ∧k q (Cm) appear as summands of a ∧N (Cn ⊗ Cm), using the coproduct, we can recover from this construction all evaluation representations associated to finite-dimensional representations. The relations in Propositions 2.1 and 2.2 together with the fact that the T ′′i ’s provide a brai- ding give the next useful diagrammatic relations. In order to simplify the notation, we omit here to write ΨT ′′i in all relations below: = (3.3) 20 H. Queffelec ai ai+1 = −qai−ai+1 ai ai+1 , ai ai+1 = −qai−ai+1 ai ai+1 (3.4) ai ai+1 = −qai−ai+1 ai ai+1 , ai ai+1 = −qai−ai+1 ai ai+1 (3.5) Proposition 3.1. The action of U̇q(slm) on ∧ q(Cn ⊗ Cm) extends via ρa to an action of U̇′q(ŝlm). Proof. The proof consists in checking the defining relations of U̇′q(ŝlm) that involve E0 or F0, that is [E0, F0]1λ = [λ0]1λ, [E0, Fj ]1λ = 0, [F0, Ej ]1λ = 0, E2 0Ej1λ − ( q + q−1 ) E0EjE01λ + EjE 2 01λ = 0 if j = 1 or j = m− 1, E2 jE01λ − ( q + q−1 ) EjE0Ej1λ + E0E 2 j 1λ = 0 if j = 1 or j = m− 1, F 2 0Fj1λ − ( q + q−1 ) F0FjF01λ + FjF 2 0 1λ = 0 if j = 1 or j = m− 1, F 2 j F01λ − ( q + q−1 ) FjF0Fj1λ + F0F 2 j 1λ = 0 if j = 1 or j = m− 1, E0Ej = EjE0, F0Fj = FjF0 if j 6= 1 and j 6= m− 1. By symmetry between the E′s and the F ’s, it is enough to check only relations involving E0 and Ej or Fj , plus the first one. The proof mostly relies on a straightforward use of relations (3.3), (3.4), (3.5), and the braiding relation. We present some of them below, where we identify an element of U̇′q(slm) with its image under the representation [E0, F0]1λ = E0F01λ − F0E01λ = q0[CT ′′w(F1), CT ′′w(E1)]1λ = CT ′′w([F1, E1]1w·λ) = CT ′′w ( − [( w−1 · λ ) 1 ]) 1λ = −(−[λ0])1λ = [λ0]1λ. Under the braiding relation, we can reduce [E0, F1] to (we again omit to write ΨT ′′i ): [E0, F1]1λ = aq−a−1−c a b c − aq−a−c a b c Skein Modules from Skew Howe Duality and Affine Extensions 21 which equals = aq−a−1−c a b c − aq−a−c a b c = aq−a−1−c(− q−a−1+b−1 ) a b c − aq−a−c ( −q−a−2+b−1 ) a b c We then obtain: −aq−2a+b−c−3  a b c − a b c  = 0 using the Uq(slm) relation [E1, F2] = 0. We now turn to E2 0E1 − ( q + q−1 ) E0E1E0 + E1E 2 0 = a2q−2a−2c+2 a b c 22 H. Queffelec − a2 ( q + q−1 ) q−2a−2c+1 a b c + a2q−2a−2c a b c = a2q−2a−2c+2 a b c − ( q + q−1 ) a2q−2a−2c+1 a b c + a2q−2a−2c a b c = −a2q−a−b−2c+2 a b c + a2 ( q + q−1 ) q−a−b−2c+2 a b c − a2q−a−b−2c+2 a b c The latter equals 0 since F 2 2F1 − (q + q−1)F2F1F2 + F1F 2 2 = 0. � Note that the usual definition of the evaluation representations requires to have a Uq(glm) action. In our case, the Uq(slm) action, with a choice of N made, is actually a disguised Uq(glm)- representation, and the factor q±(a1+am) utilizes this glm-information. It will be useful when we turn to skein modules to rescale the braidings and replace the T ′′i ’s by the Ti’s. Doing so, we have analogues of relations (3.3), (3.4) and (3.5). This time, we omit ΨTi everywhere = Skein Modules from Skew Howe Duality and Affine Extensions 23 ai ai+1 = −qai−ai+1+1 ai ai+1 , ai ai+1 = −qai−ai+1−1 ai ai+1 ai ai+1 = −qai−ai+1+1 ai ai+1 , ai ai+1 = −qai−ai+1−1 ai ai+1 We denote ρ̃a, for a complex number, the analogue of ρa: ρ̃a(E01λ) = aq−(a1+am)CTm−1 · · ·CT2(F1)1λ, ρ̃a(F01λ) = a−1qa1+amCTm−1 · · ·CT2(E1)1λ. ρa and ρ̃a are defined similarly for other generators. The proof of Proposition 3.1 remains valid if we replace the braidings T ′′i ’s by the (glm) rescaled Ti’s. Proposition 3.2. The action of U̇q(slm) on ∧ q(Cn ⊗ Cm) extends via ρ̃a to an action of U̇′q(ŝlm). Now, using the fact that the rescaled braidings Ti allow some liberty, we can give a posteriori a formula very close to Chari-Pressley original evaluation representations [8]. Assume that we have chosen a square root q 1 2 of q, and define the bracket [ · , · ] q 1 2 by [u, v] q 1 2 = q 1 2uv − q− 1 2 vu. Proposition 3.3. We have ρ̃a(E01λ) = (−1)m−2aq−a1−am+m−2 2 [ Fm−1, [ Fm−2, . . . , [F2, F1] q 1 2 · · · ] q 1 2 ] q 1 2 = (−1)m−2aq−a1−am+m−2 2 [[ · · · [Fm−1, Fm−2] q 1 2 · · · , F2 ] q 1 2 , F1 ] q 1 2 . Proof. The core of the identification is to understand the bracket process in the diagrammatic definition. By Reidemeister or Kauffman-type moves, we can rewrite ρ̃a(E01λ) as (which doesn’t make sense in Uq(slm) anymore): E01λ 7→ aq−(a1+am) · · · · · · a1 a2 am−1 am Then, we can smooth the rightmost crossing using: ΨTi  k 1 1 k  = −q k 1 k − 1 1 k + k 1 1 k . 24 H. Queffelec This gives: E01λ 7→ −qeaq−(a1+am) · · · · · · a1 a2 am−1 am + aq−(a1+am) · · · · · · a1 a2 am−1 am = −qaq−(a1+am)Fm−1 · · · · · · a1 a2 am−1 am + aq−(a1+am) · · · · · · a1 a2 am−1 am Fm−1 = −aq−a1−am+ 1 2  Fm−1, · · · · · · a1 a2 am−1 am  q 1 2 . We can then iterate by smoothing successive crossings. This proves the first part of the equality. For the second part, we perform the same process, but we start from the left-most crossing. � Because of the similarity of this definition to the usual one, we will from now on refer to the representations induced by ρ̃a as evaluation representations. This process helps us extend ∧ q(Cn ⊗ Cm) to a U ′q(ŝlm) representation. The U ′q(ŝlm) and Uq(sln) actions still commute, but this certainly cannot provide new information, since the new representation is entirely built on the old one. An extension of this representation will be studied later. Since the usual skew Howe duality context is closely related to skein modules (or spider categories), it is natural to wonder if we can understand a skein analogue for the evaluation representations. It appears that closing the Dynkin diagram of sln corresponds to gluing two opposite sides of the box on which one usually looks at tangles: this produces an annulus. 3.3 Annular knots Represent ∧N q (Cn⊗Cm) ' ⊕ a1+···+am=N ∧a1 q (Cn)⊗ · · ·⊗ ∧am q (Cn) by a sequence (a1, . . . , am), as before, but now drawn on a circle instead of drawing it on a segment, so that a1 and am lie next to each other. E0 is a map: ∧a1 q (Cn) ⊗ · · · ⊗ ∧am q (Cn) → ∧a1+1 q (Cn) ⊗ · · · ⊗ ∧am−1 q (Cn). This idea gives a diagrammatic presentation of the affine extension of the skew Howe duality phenomenon. See below the diagram corresponding to E0E1 acting on a sequence (2, 0), with Skein Modules from Skew Howe Duality and Affine Extensions 25 m = 2, n = 2, N = 2: Annular webs can be defined in a very similar fashion as in the plane case. One just embeds trivalent graphs in the annulus instead of the plane, and all relations being local, they look the same for webs considered in any surface. Definition 3.4. Let nAWeb, the sln annular web skein module, be the Z[q, q−1]-module gene- rated by annular webs (oriented trivalent graphs with preserved flow embedded in an annulus) possibly with boundary (embedded in the two boundary circles of the annulus) up to isotopy and the local sln web relations (2.3), (2.4) and (2.5). All web relations and all ŝlm relations being “local relations”, it is easy to observe that web relations are implied by ŝlm. However, if we consider the evaluation representation to extend the slm action, there are other relations than web relations: for example, the above diagram corresponds to a scalar action, while in the skein module, this would rather correspond to a generator. We shall first identify here a skein module that corresponds to these representations, before seeking for a situation closer to the skein module of the annulus. For this purpose, let us introduce the annular extension of the category nWeb+ m. Definition 3.5. Define nAWeb+ m to be the category with objects, sequences (a1, . . . , am) (0 ≤ ai ≤ n) labeling points regularly drawn on a circle, together with a zero object. Points labeled by zero can be erased, but, as in [2] and [28], we will keep the n-strands as well. Morphisms are formal sums over Z[q±1] of upward sln-webs (in the sense of Definition 3.4) drawn on a cylinder, generated by5: Ei1λ = ai+1ai ai+1 + 1ai − 1 and Fi1λ = ai+1ai ai+1 − 1ai + 1 In the above pictures, we have drawn only strands i and i+ 1. nAWeb+ m(N) will be the full subcategory with objects such that ∑ ai = N . Given m, n, N , we can define on weights a map Φ that sends λ = (λ0, . . . , λm−1) to a sequence (a1, . . . , am) such that ai+1 − ai = λi and a1 − am−1 = λ0, with ∑ ai = N . If such a solution doesn’t exist, the weight is sent to the zero object. 5Note that this a priori doesn’t recover the complete braid group of the annulus. We miss in the braid group associated to Uq(ŝlm) the element given by sending each point on one boundary of the annulus to the one corresponding to its right (for example) neighbor in the other boundary of the annulus. In case of ladders, this can be artificially solved by adding a 0-strand. 26 H. Queffelec Proposition 3.6. For all m, n, N , there is a functor Φ : U̇′q(ŝlm) 7→ nAWeb+ m(N) defined on weights as above and on morphisms by Ei1λ 7→ Ei1λ and Fi1λ 7→ Fi1λ. ŝlm relations are locally slm relations: the proof is straightforward. [4, Proposition 7.4] will receive a direct translation. Let us consider a knotted annular tangle as a composition of Ei1λ, Fi1λ and crossings between the i-th and (i + 1)-th strands. From such a presentation, we can read off an element of ˜ U̇′q(ŝlm) as the corresponding product of Ei1λ, Fi1λ and T±1 i . Let us call Xw this element corresponding to a presentation of a web-tangle w. Proposition 3.7. The evaluation representations produce annular web invariants for ladder- type webs. In other words, if Xw ∈ ˜ U̇′q(ŝlm) corresponds to a presentation w of a web-tangle, then the morphism of Uq(sln) representations given by Xw is an invariant of the web-tangle. ˜ U̇′q(ŝlm) above denotes the completion of U̇′q(ŝlm) given by the quotient of the ring of series of elements of U̇′q(ŝlm) acting by evaluation representation on each irreducible U̇′q(slm) repre- sentation Vλ by zero but for finitely many terms, mod out by the two-sided ideal of elements acting by zero on all Vλ. As explained previously, the above process produces an invariant of annular web-tangles. However, it appears to come with only little information about the topology of the annulus, and we can indeed explicitly identify this invariant. It may be useful for this purpose to see the annulus as a cylinder rather than flattened on a plane: this way, we can fill it. Proposition 3.8. The evaluation representation with a = −qn+1 recovers the sln skein module of the filled cylinder. In other words, for each fixed value of N we have the following commutative diagram: U̇′q(ŝlm) Φ // ev �� nAWeb+ m Filling �� U̇q(slm) Φ // nWeb+ m Proof. We just have to check that the action of E01λ and F01λ corresponds to E01λ and F01λ seen in the skein module of the filled cylinder. For E0 for example, we have the following situation: Filling(Φ(E01λ)) = ΨTi  am am − 1 a1 a1 + 1 ◦−1  = ΨTi  am am − 1 a1 a1+1 ◦ 1 2 ◦ − 1 2 ◦ − 1 2 ◦ 1 2  The negative twist on the l.h.s. of the above equation comes from the fact that the strand goes along the cylinder with framing parallel to the cylinder. When filling the cylinder and projecting it on the back side, this produces a twist. Skein Modules from Skew Howe Duality and Affine Extensions 27 We have depicted here only the leftmost and the rightmost strands. This whole process is to be understood in front of the other strands. Then, a succession of Reidemeister II moves presents this piece of tangle as the elements defining the evaluation representation, presented in (3.2). In the previous computation, the twists produce the following coefficient: t 1 2 a1t − 1 2 a1+1t − 1 2 am t 1 2 am−1 = −q−(a1+am)+n+1, while the evaluation representation provides aq−(a1+am). Choosing a = −qn+1 adjusts the coefficients. Checking the results for F0 is similar. � So, it appears that extending the skew Howe duality phenomenon to the affine case by evaluation representation gives a coherent process, but is too weak to recover the skein module of the annulus. We miss the fact that acting by E1E2 · · ·Em−1E0, although it does not change anything on the weight, has no reason to be something trivial in the skein module. This is a well-known phenomenon in the study of ŝlm: if we want to understand the non-triviality of this action, we have to keep the whole data coming from the Dynkin diagram and not only generators Ei, Fi, K ± i . We should therefore work in the whole Uq(ŝlm) and not only with U ′q(ŝlm). Nonetheless, we want to keep working with an analogue of Howe duality, and it would be convenient to have a process built on these particular representations. It turns out that there is an easy way to do it, called af f inization, as explained for example in [15, p.233]. 3.4 Affinization Following [15], we now consider C(q)[z, z−1] ⊗C(q) ∧N q (Cn ⊗ Cm). The Uq(sln) action can be extended by acting by an identity on the z-part, and the previous U ′q(ŝlm) action may be extended to an Uq(ŝlm) action by the following rules: E0 ( zm ⊗ v ) = zm+1 ⊗ (E0v), Ei ( zm ⊗ v ) = zm ⊗ (Eiv) for i 6= 0, F0 ( zm ⊗ v ) = zm−1 ⊗ (F0v), Fi ( zm ⊗ v ) = zm ⊗ (Fiv) for i 6= 0, Ki ( zm ⊗ v ) = zm ⊗ (Kiv), Kd ( zm ⊗ v ) = qmzm ⊗ v. Here, Kd is the derivation element, that was neglected in the previous subsection. We have the same decomposition as before:⊕ a1+···+am=N C(q) [ z, z−1 ] ⊗C(q) ∧a1 q (Cn)⊗ · · · ⊗ ∧am q (Cn), and more precisely:⊕ a1+···+am=N,k C(q) · zk ⊗ ∧a1 q (Cn)⊗ · · · ⊗ ∧am q (Cn). The Uq(ŝlm)-weight of one summand is (a1 − am, a2 − a1, . . . , am − am−1) + kδ, where δ is the null root (that was neglected in the previous subsections). Note that these representations are of level 0. We obtain here a new process: given a knotted ladder, we can turn it into an element of U̇q(ŝlm) acting on ∧N q (Cn⊗Cm), and extend this into an action on C(q)[z±1]⊗C(q) ∧N q (Cn⊗Cm). If we restrict to knots drawn on a cylinder, that is ladders with a boundary sequence with only 0’s and n’s, we obtain an element of End(C(q)[z±1]⊗C(q) C[q±1]) = C(q)[z±1]. Since all relations are local, [4, Proposition 7.4] receives a direct translation: 28 H. Queffelec Proposition 3.9. The previous process defines a web tangle invariant. In other words, if 1λ′Xw1λ ∈ ˜ U̇′q(ŝlm) corresponds to a presentation w of an annular web-tangle mapping se- quences (aλ1 , . . . , a λ m) to (aλ ′ 1 , . . . , a λ′ m), then the morphism of Uq(sln) representations mapping C(q)[z, z−1]⊗C(q) ∧aλ1 q (Cn)⊗· · ·⊗ ∧aλm q (Cn) to C(q)[z, z−1]⊗C(q) ∧aλ ′ 1 q (Cn)⊗· · ·⊗ ∧aλ ′ m q (Cn) given by Xw is an invariant of the web-tangle. So, to an annular web-tangle presented in a ladder form, we can assign a morphism of Uq(sln) representations which is an invariant of the web-tangle. This morphism may be expressed in a diagrammatic way, producing a skein element. A natural question would be to know whether the web relations alone are sufficient to identify if two diagrams yield the same skein element (see for example [13] for the planar case). Unfortunately, this process is still not faithful enough: E1F0F1E01λ acting for example on a 2-strands sequence (0, 2) acts as [2]2 · 1λ, while the skein element corresponding to both these elements are not equal: It would be interesting to compute the kernel of the map, but we do not know how to address this question. This also suggests to look for richer Uq(ŝlm) representations extending the skew-Howe duality phenomenon. The previous invariant contains two pieces of information: the same as the evaluation repre- sentation, that is, the skein element associated to the web tangle in the skein module of the filled annulus, and an information given by the action on the z-part. This traces a kind of algebraic linking number with the core of the annulus. The problem is that this algebraic number doesn’t detect the possible non-triviality of an algebraically non-linked web, as explained above: the rep- resentations we have been working with are still too weak to give a full representation-theoretic counterpart of the annular webs. In the sl2-case, the unoriented skein module of the annulus is well understood, isomorphic to Z[q±1][z], with z the generator given by a circle around the hole. Note that, if we try to compare the obtained invariant with the usual unoriented sl2 skein module of the annulus, the first difference comes from the fact that the 2-labeled strands don’t play any role in the unoriented version, while when wrapped around the hole, they produce a coefficient z±2 in the oriented version. A second difference comes from the issue explained above. An easy way to compare the (usual, unoriented) Kauffman bracket skein module of the annulus and the invariant obtained using enhanced sl2 webs and the affine representation would thus be to mod both out by the ideal generated by (z2−1), after some renormalizations (since z in the Kauffman bracket version corresponds in spirit to [2]z in the oriented case). More refined comparisons seem to non-trivially involve the orientation and the behavior of the 2-labeled lines, which makes it very hard to control the power of z. 3.5 Forgetting about sln . . . Note that the issue that prevents us to obtain an algebraic object that would mimic the behavior of the skein module comes from the fact that we consider particular Uq(sln) representations that Skein Modules from Skew Howe Duality and Affine Extensions 29 are not powerful enough to detect all the topological data. A kind of virtual analogue would be to only keep the Uq(slm)-part in the duality, mod out by information extracted from the usual case, and extend only this to the annular case. Recall from [6] that we can understand the quotient of Uq(slm) which corresponds to classes of sln-webs. Fix N (from the ladders we are looking at) and a dominant weight λ (corresponding to a sequence (aλ1 , . . . , a λ m)), and denote Iλ the ideal of U̇q(slm) generated by all weights which do not lie in the Weyl orbit of any weight µ so that λ dominates µ. Furthermore, denote U̇q(slm)n the quotient of U̇q(slm) by the set of weights whose associated sequence (a1, . . . , am) either does not exist or has at least one coefficient ai < 0 or ai > n. Then, Cautis, Kamnitzer and Morrison [6, Theorem 4.4.1, Lemma 4.4.2] tell us that the morphism U̇q(slm)n/Iλ 7→ nWeb+ m(aλ1 , . . . , a λ m) is an isomorphism6, where nWeb+ m(aλ1 , . . . , a λ m) is the algebra of ladders that can be reached starting from the sequence (aλ1 , . . . , a λ m). In other words, these are the ladders W : (a1 1, . . . , a 1 m) 7→ (a2 1, . . . , a 2 m), so that the set of ladders between (aλ1 , . . . , a λ m) and (a1 1, . . . , a 1 m) is non-empty. Let us now denote nAWeb+ m(a1, . . . , am) the algebra of annular ladder webs built from a sequence (a1, . . . , am). Each Iλ extends in the affine case to a module I λ̂ simply given by assigning to any µ ∈ Iλ the affine weight µ̂ so that if µ = (aµ2 − aµ1 , . . . , a µ m − aµm−1), µ̂ = (aµ1 − a µ m, a µ 2 − a µ 1 , . . . , a µ m− aµm−1). We can then consider the quotient U̇′q(ŝlm)n of U̇′q(ŝlm) by weights whose associated sequence has indices lower than 0 or bigger than n, and we have the quotient U̇′q(ŝlm)n/I λ̂ . The next result then gives us somehow the result we hoped to find with an explicit Uq(sln)-representation but just by looking on the dual side of the picture! Proposition 3.10. For λ a dominant slm weight, the map: U̇′q ( ŝlm )n /I λ̂ 7→ nAWeb+ m(aλ1 , . . . , a λ m) is an isomorphism. The pre-image of a knotted ladder in U̇′q(ŝlm)n/I λ̂ is therefore an invariant of the knot. Proof. First, note that since E0 acts on weights as F1 · · ·Fn, the weights µ̂ are those that cannot be reached from λ̂. On objects, the statement is therefore obvious. The surjectivity on morphisms comes from the definition. For morphisms, the injectivity argument is the same as before: since all relations are local (and elementary relations involve at most three strands), either the generators E0 and F0 are not involved and we can assume we work with slm, or they are but (for m ≥ 3) there exists i so that Ei and Fi are not involved. There is then an inclusion of Uq(slm) in Uq(ŝlm) that does not involve Ei and Fi and we can assume we work in this one. � 3.6 . . . to better recover it? The objects that appear in the previous paragraph are closely related to q-Schur algebras and affine versions of them. We refer to [11] for a clear presentation of the context in which they appear. Following [11], we will consider in what follows an extension7 of U ′q(ŝlm) with two extra generators R and R−1 subject to relations (indices are to be understood modulo m): RR−1 = R−1R = 0, R−1Ki+1R = Ki, R−1K−1 i+1R = K−1 i , R−1Ei+1R = Ei, R−1Fi+1R = Fi. 6Again, note that the result they state uses glm and not slm. The dependency in the glm weights in our case is hidden in the n-bounded quotient: we need N to know the value of the ai’s. 7This extension has the nice property that it gives us the missing generator of the braid group of the annulus that was previously discussed. 30 H. Queffelec This algebra will be denoted Ûq(ŝlm), and its idempotented version ̂̇Uq(ŝlm). Note that the previous relations in particular give us that E0 = R−1E1R and F0 = R−1F1R. Doty and Green suggest us to replace the fundamental representation Cm of Uq(slm) by an infinite-dimensional version V∞ = C∞ = 〈Xi, i ∈ Z〉 with action: EiXj = Xj+1 if i = j mod (m), EiXj = 0 otherwise, FiKj+1 = Xj if i = j mod (m), FiXj = 0 otherwise, KiXj = q−1Xj and KiXj+1 = qXj+1 if i = j mod (m), KiXj = Xj otherwise, RXj = Xj+1. As in the linear case, we can endow Cn ⊗ V∞ with two commutative actions of Uq(sln) and Ûq(ŝlm). We now wish to consider the quantum exterior power of this tensor product of representations and perform the same type of process as before. However, this quantum exterior power is more complicated to define in the affine case than in the linear one. We refer to [36, 38, 39] and references therein for details about these representations and tools one could use to study them. We intend here to sketch a process allowing us to relate m-uprights annular ladders to sln-representation theory, but the question of completely understanding this relation, and also relating higher-level analogous phenomenon to knot theory remains open. Denote Vm = 〈X1, . . . , Xm〉 that we see as a subspace of V∞. Vm is the vector representation of Uq(slm) (but is not a Uq(ŝlm) module), and its affinization Vm ⊗C[z±1] is isomorphic to V∞. The precise definition of the quantum exterior power of V∞ can be found in [39, Section 2.1]. We have in particular: Xi+km ∧Xi+lm +Xi+lm ∧Xi+km = 0. Uglov’s process consists in making ∧ q ( Cn ⊗ C[z±1]⊗ Vm ) into a U ′q(ŝln)⊗ U ′q(ŝlm) module. In particular, this implies that ∧ q (Cn ⊗ V∞) has two commuting actions of Uq(sln) and U ′q(ŝlm), and it is not hard to see that the latter can be extended into a Ûq(ŝlm) action. We now want to relate annular webs, the algebra ̂̇Uq(ŝlm) and the morphisms of the Uq(sln) representation ∧ q (Cn ⊗ V∞). Let us slightly extend the definition of nAWeb+ m(N) into nÂWeb + m(N) by adding to it the image of R as the next elementary annular ladder web: a1 a2 am−1 am , and similarly for R−1. It is a direct extension of the usual case that we have an isomorphism ⊕N ̂̇Uq(ŝlm)n 7→ nÂWeb + m. Usually, when we relate Uq(slm) and Uq(sln) endomorphisms, the fact that we can kill in Uq(slm) all weights corresponding to sequences (a1, . . . , am) with an ai > n comes from the fact that exterior powers of Cn of degree more than n are zero. Here, we can decompose:∧N q ( Cn ⊗ V∞ ) ' ∧N q ( Cn ⊗ C [ z±1 ] ⊗ Vm ) ' ∧N q ( Cn ⊗ C [ z±1 ] ⊗ (C⊕ C⊕ · · · ⊕ C) ) ' ⊕ a1+···+am=N ∧a1 q ( Cn ⊗ C [ z±1 ]) ⊗ · · · ⊗ ∧am q ( Cn ⊗ C [ z±1 ]) . Skein Modules from Skew Howe Duality and Affine Extensions 31 However, the relationXi+km∧Xi+lm+Xi+lm∧Xi+km = 0 does not ensure that ∧n q (Cn⊗C[z±1]) ' C nor that it is zero after. There is nonetheless a case where this issue does not arise: it is when N ≤ n (this corresponds to looking at a “generic” version, where the choice of n actually doesn’t matter). Let us assume that we are in that situation. We then have: nÂWeb + m(N) ' ̂̇Uq(ŝlm)n(N) 7→ EndUq(sln) (∧N q ( Cn ⊗ V∞ )) , where ̂̇Uq(ŝlm)n(N) denotes the full subcategory of ̂̇Uq(ŝlm)n whose lifts (a1, . . . , am) of the weights (λ1, . . . , λm−1) are such that ∑ ai = N (note again that here, taking the n-bounded quotient only kills weight whose lifts have negative entries, as all entries will be less than or equal to n). We want to show that this map is an inclusion. A very useful tool for this is provided by [31, Proposition 3.14]. Assume that N < m (we can restrict to that case by adding 0-labeled strands), and deno- te 1r the idempotent corresponding to the sequence (a1, . . . , am) = (1, . . . , 1, 0, . . . , 0) contai- ning r times the number 1. Then, 1r ̂̇Uq(ŝlm)1r is isomorphic to the affine Hecke algebra Ĥ Âr−1 (see [11, 31]). We can present it as generated by b1, . . . , br and Tρ, T −1 ρ subject to the following relations: b2i = ( q + q−1 ) bi for i = 1, . . . , r, bibj = bjbi for distant i, j = 1, . . . , r, bibi+1bi + bi+1 = bi+1bibi+1 + bi for i = 1, . . . , r, TρbiT −1 ρ = bi+1 for i = 1, . . . , r. The last equation is to be understood with indices modulo r. Note that we can obtain the generator br as br = Tρbr−1T −1 ρ . Mackaay and Thiel then state, where Ŝ(m,N) is the affine Schur algebra: Proposition 3.11 ([31, Proposition 3.14]). Let N < m. Suppose that A is a Q(q) algebra and f : Ŝ(m,N) 7→ A is a surjective Q(q)-algebra homomorphism which is an embedding when restricted to 1rŜ(m,N)1r ' ĤÂr−1 . Then f is a Q(q)-algebra isomorphism A ' Ŝ(m,N). The idea is that it is enough to check the injectivity on the Hecke algebra for deducing it for the whole Schur algebra. Now, it is proven in [14, Theorem 3.4.8] that the action of Ŝ(m,N) on V ⊗N∞ is faithful. This is the central ingredient in the following proposition. Proposition 3.12. If N < m and N ≤ n, the map nÂWeb + m(N) 7→ EndUq(sln) (∧N q (Cn⊗V∞) ) is injective. Proof. From [14], the action of Ŝ(m,N) ' nÂWeb + m on V ⊗N∞ is faithful. In particular, for all x ∈ 1N Ŝ(m,N)1N , there exists a vector θ which is acted on non-trivially. We have a map V ⊗N∞ 7→ ∧N (Cn ⊗ V∞) sending Xj1 ⊗ · · · ⊗ XjN onto (v1 ⊗ Xj1) ∧ · · · ∧ (vN ⊗ XjN ) (where v1, . . . , vn are the generators of Cn). Then, x acts (on ∧ q(Cn ⊗ V∞)) on the image of x under this map and the result is the image of x(θ) under the same map. We would like to ensure that we can find a θ so that this image is non-zero. 32 H. Queffelec To prove this, assign to vi ⊗ (Xj ⊗ zr), with Xj ∈ Vm, the integer k = i + nj − nmr. This defines a bijective correspondence between Z and vectors of Cn⊗V∞. Denote uk = vi⊗(Xj⊗zr). Then Uglov [39] proves that ordered wedges, that is wedges of the kind uk1 ∧ · · · ∧ ukN with k1 > k2 > · · · > kN , form a basis of the wedge product ∧N (Cn ⊗ V∞). Note now that the action of Ŝ(m,N) on V ⊗N∞ commutes with the multiplication by z on any of the factors. Hence, up to multiplication of the θ by (zp1 , . . . , zpN ), we can assume that all wedges that appear in x(θ) are ordered: they all survive (and are linearly independent) in∧N (Cn ⊗ V∞). We thus obtain that 1N Ŝ(m,N)1N injects into EndUq(sln) (∧ q(Cn⊗ V∞) ) , and using Propo- sition 3.11, we obtain the desired claim. � As explained earlier, when trying to extend this result for any n and N , one goes into trouble because the n-bounded quotient does not naturally translate in ∧ q(Cn ⊗ V∞). A natural way to solve this issue is to consider the quotient of ∧N q (Cn ⊗ V∞) by the following subspace: ⊕ λ\|1λ=0 ∈ ̂̇Uq(ŝlm)n ̂̇Uq ( ŝlm ) 1λ · ∧N q ( Cn ⊗ V∞ ) . This space is clearly a sub ̂̇Uq(ŝlm) and Uq(sln) representation, so the quotient is acted on by both quantum groups. Furthermore, the action of ̂̇Uq(ŝlm) descends to an action of ̂̇Uq(ŝlm)n, precisely because the ideal by which we quotient ̂̇Uq(ŝlm) acts by zero. We denote a ∧N q (Cn⊗V∞) this quotient, and let a ∧ q(Cn ⊗ V∞) = ⊕ N a ∧N q (Cn ⊗ V∞). It is interesting to note that since the proof of Proposition 3.11 consists in factorizing elements of the Hecke algebra through other weights, it can be extended to the n-bounded case as well. In our case, we have to consider that if the weight is killed, then the associated element will be killed. We can thus state: Proposition 3.13. Let N < m. Suppose that A is a Q(q) algebra and f : Ŝ(m,N)n 7→ A is a surjective Q(q)-algebra homomorphism which is an embedding when restricted to 1rŜ(m,N)n1r ' Ĥ2 Âr−1 . Then f is a Q(q)-algebra isomorphism A ' Ŝ(m,N)n. However, it is now not clear anymore that the Hecke algebra will inject into: EndUq(sln) ( a ∧ q ( Cn ⊗ V∞ )) . Such a result would be very interesting, as it would give to annular webs a representation- theory-based interpretation in the same flavor as the original definition of webs as an algebra of intertwiners for minuscule Uq(sln) representations. More generally, it would also be very interesting to better understand the relations between this representation and knot theory, and to see whether there is any translation in the knot theory side of the more general phenomenons studied in [39]. Skein Modules from Skew Howe Duality and Affine Extensions 33 3.7 Turning an annular knot to a ladder So far, we saw that to a ladder annular web-tangle, we can assign a Uq(sln) morphism of tensor product of minuscule representations (possibly tensorized with C(q)[z, z−1]), whose diagram- matic depiction equals the skein element associated to the web-tangle. This holds for any annular web-tangle isotopic to a ladder, but we can extend the process used in the case of usual webs for turning webs to ladder webs. We can present any upward annular web-tangle in a similar form as in [6]: T = T ′ The above pictures are to be understood on an annulus. Then, taking the Jones–Kauffman product with a set of well-placed n-strands, we can apply to T ′ the same process as usual and turn T to a ladder form. Note that taking the Jones– Kauffman product with n-strands is an invertible process, by taking the product with the same number of n-strands oriented downward and pairing the obtained couples of oppositely oriented n-strands. 4 Categorification From the Dynkin diagram (3.1), one can build a categorified quantum group UQ(ŝlm) follo- wing [7], which generalizes works by Khovanov and Lauda [21, 22, 23, 27]. Similarly, it is a straightforward generalization of nBFoamm(N) to consider foams on an annulus nABFoamm(N): local generators are the same ones as in the disk case, but instead of embedding the foams in the thickening of a disk, we embed them in the thickening of an annulus. Then, the main result from [28] generalizes at no cost: Proposition 4.1. For n = 2, 3, for each N > 0 there is a 2-representation Φn : UQ(ŝlm) → nABFoamm(N) defined on single strand 2-morphisms by: Φn ( OO λ i ) = , Φn ( OO • λ i ) = • on crossings by: Φn ( OOOO i i λ ) = 34 H. Queffelec Φn ( OOOO i i+1 λ ) = , Φn ( OOOO i+1 i λ ) = Φn ( OOOO j i λ ) = , Φn ( OOOO i j λ ) = where j − i > 1, and on caps and cups by: Φn ( WW i λ ) = , Φn ( GG �� i λ ) = (−1)ai Φn ( �� JJ i λ ) = (−1)ai+1 , Φn ( ��TT i λ ) = where in the above diagrams the ith sheet is always in the front. Then, following [28], we can build from any annular knot, turned into an annular entangled ladder, a complex over the category UQ(slm). 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Math., Vol. 191, Birkhäuser Boston, Boston, MA, 2000, 249– 299, math.QA/9905196. http://dx.doi.org/10.4171/QT/46 http://arxiv.org/abs/1109.1745 http://dx.doi.org/10.1155/S107379289500016X http://arxiv.org/abs/q-alg/9504013 http://arxiv.org/abs/1105.3038 http://dx.doi.org/10.1007/s002200050252 http://arxiv.org/abs/q-alg/9607031 http://arxiv.org/abs/math.QA/9905196 1 Introduction 1.1 Webs and skew-Howe duality 1.2 Obtaining a skein module 1.3 Affine extensions 2 Skew Howe duality and skein modules 2.1 Skew Howe duality 2.1.1 Context 2.1.2 Quantum Weyl group action 2.1.3 Skew Howe duality and quantum invariants for knots 2.2 Skein modules 2.2.1 Braidings for skein modules 2.2.2 sl2 case 2.2.3 sln case 2.2.4 Turning a knot to a ladder 3 Affine extensions 3.1 Affine slm 3.2 Evaluation representations 3.3 Annular knots 3.4 Affinization 3.5 Forgetting about sln ... 3.6 … to better recover it? 3.7 Turning an annular knot to a ladder 4 Categorification References

Skein Modules from Skew Howe Duality and Affine Extensions

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