Relating Stated Skein Algebras and Internal Skein Algebras

We give an explicit correspondence between stated skein algebras, which are defined via explicit relations on stated tangles in [Costantino F., Lê T.T.Q., arXiv:1907.11400], and internal skein algebras, which are defined as internal endomorphism algebras in free cocompletions of skein categories in...

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Опубліковано в: :	Symmetry, Integrability and Geometry: Methods and Applications
Дата:	2022
Автор:	Haïoun, Benjamin
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Мова:	Англійська
Опубліковано:	Інститут математики НАН України 2022
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Цитувати:	Relating Stated Skein Algebras and Internal Skein Algebras. Benjamin Haïoun. SIGMA 18 (2022), 042, 39 pages

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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author	Haïoun, Benjamin
author_facet	Haïoun, Benjamin
citation_txt	Relating Stated Skein Algebras and Internal Skein Algebras. Benjamin Haïoun. SIGMA 18 (2022), 042, 39 pages
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container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	We give an explicit correspondence between stated skein algebras, which are defined via explicit relations on stated tangles in [Costantino F., Lê T.T.Q., arXiv:1907.11400], and internal skein algebras, which are defined as internal endomorphism algebras in free cocompletions of skein categories in [Ben-Zvi D., Brochier A., Jordan D., J. Topol. 11 (2018), 874-917, arXiv:1501.04652] or in [Gunningham S., Jordan D., Safronov P., arXiv:1908.05233]. Stated skein algebras are defined on surfaces with multiple boundary edges, and we generalise internal skein algebras in this context. Now, one needs to distinguish between left and right boundary edges, and we explain this phenomenon on stated skein algebras using a half-twist. We prove excision properties of multi-edge internal skein algebras using excision properties of skein categories, and agree with excision properties of stated skein algebras when = q²( ₂)-modᶠⁱⁿ. Our proofs are mostly based on skein theory, and we do not require the reader to be familiar with the formalism of higher categories.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 042, 39 pages Relating Stated Skein Algebras and Internal Skein Algebras Benjamin HAÏOUN Institut de Mathématiques de Toulouse, France E-mail: benjamin.haioun@ens-lyon.fr URL: https://perso.math.univ-toulouse.fr/bhaioun/ Received October 07, 2021, in final form May 25, 2022; Published online June 11, 2022 https://doi.org/10.3842/SIGMA.2022.042 Abstract. We give an explicit correspondence between stated skein algebras, which are de- fined via explicit relations on stated tangles in [Costantino F., Lê T.T.Q., arXiv:1907.11400], and internal skein algebras, which are defined as internal endomorphism algebras in free cocompletions of skein categories in [Ben-Zvi D., Brochier A., Jordan D., J. Topol. 11 (2018), 874–917, arXiv:1501.04652] or in [Gunningham S., Jordan D., Safronov P., arXiv:1908.05233]. Stated skein algebras are defined on surfaces with multiple boundary edges and we generalise internal skein algebras in this context. Now, one needs to distin- guish between left and right boundary edges, and we explain this phenomenon on stated skein algebras using a half-twist. We prove excision properties of multi-edges internal skein algebras using excision properties of skein categories, and agreeing with excision proper- ties of stated skein algebras when V = Uq2(sl2)-modfin. Our proofs are mostly based on skein theory and we do not require the reader to be familiar with the formalism of higher categories. Key words: quantum invariants; skein theory; category theory 2020 Mathematics Subject Classification: 57K16; 18M15 1 Introduction The goal of this paper is to draw a comprehensive link between stated skein algebras of [7] and internal skein algebras of [11] and [2], including their structures and properties. The interest of such a link is to benefit from the nice features of both sides. Stated skein algebras are defined very explicitly, and have numerous applications. Internal skein algebras are much more theoretical, they are defined for all ribbon categories of coefficients, and their basic (in particular, excision) properties derive formally. Skein categories Skein algebras, introduced by Przytycki and Turaev, are a generalisation of the Kauffman bracket polynomial to surfaces. The Kauffman bracket ⟨−⟩ is the quotient map from the vector space freely generated by isotopy classes of framed links in R3 to the quotient of this vector space by the local Kauffman-bracket relations: = q + q−1 and = ( −q2 − q−2 ) . Every link in R3 can be reduced in an essentially unique way to the empty link via some Kauffman-bracket relations, thus ⟨−⟩ takes values in the ground field k. The skein algebra of an oriented surface Σ is the quotient of the vector space freely generated by isotopy classes of mailto:benjamin.haioun@ens-lyon.fr https://perso.math.univ-toulouse.fr/bhaioun/ https://doi.org/10.3842/SIGMA.2022.042 2 B. Häıoun framed links in Σ × (0, 1) by the Kauffman-bracket relations. Its algebra structure is given by stacking in the (0, 1)-coordinate. It is interesting to extend these constructions to form a category, so one can cut and glue pieces of links together. One considers the category of framed tangles instead of the vector space of framed links. Its objects are finite sequences of framed points (in R2, or in Σ) and its morphisms are isotopy classes of framed tangles linking them (in R2 × [0, 1], or in Σ × [0, 1]). Then the skein algebra is the quotient of the endomorphism algebra of the empty set by the Kauffman-bracket relations. Skein categories, introduced in [29] and [12] (see also [6]), are a generalisation of the Resheti- khin–Turaev construction to surfaces, as well as a categorical extension of skein algebras. Given a k-linear ribbon category V, in [23] (see also [27]) one constructs a functor RT from the category TanfrV of framed oriented coloured tangles in R2 × [0, 1] to V. If one adds coupons to framed tangles, this functor is surjective on morphisms. One can quotient by the relations which are true in V after RT, which gives a category equivalent to V. The skein category SkV(Σ) of an oriented surface Σ with coefficients in V is the quotient of the category of framed tangles with coupons in Σ × [0, 1] by the local relations which are true in V after RT on a little disk on Σ. We are particularly interested in the case where V is the category of type 1 finite-dimensional modules over the ribbon Hopf algebra Uq2(sl2) for generic q, which is equivalent to the category of finite-dimensional right comodules over the coribbon Hopf algebra Oq2(SL2). In this case every coupon is a linear combination of tangles so there is no need for coupons and the local relations which are true in V after RT are exactly the Kauffman-bracket relations. Then the endomorphism algebra of the empty set is exactly the skein algebra. This categorical extension comes with much more information and structure than skein alge- bras. A boundary edge c of a surface Σ provides SkV(Σ) with a structure of SkV(c×(0, 1))-module category, and the skein category of a gluing of two surfaces along a boundary edge is equiva- lent to the Tambara relative tensor product of the two module categories. We will recall these definitions and constructions in Section 2.2. The skein category construction moreover satisfies nice functoriality properties with respect to embeddings and their isotopies. One would like to reduce skein categories to an algebra without losing these extra information and structure. This is the point of both stated skein algebras and internal skein algebras, which happen to be isomorphic. Stated skein algebras Without going through this categorical extension/algebraic reduction story, one would like to have nice formulas to compute skein algebras of a gluing of surfaces. Skein algebras are not suited for this kind of operation because one cannot cut a link, and most links on a gluing cannot be described by some links on each part. Therefore, it is natural to consider an extension of skein algebras that allows links to be cut, namely considering tangles, with endpoints above the boundary of the surface (in ∂Σ× (0, 1)), ready to be glued to other tangles. Instead of oriented surfaces one now works with marked surfaces that have boundary intervals above which one allows tangles to end. Now any tangle α on a gluing of two surfaces Σ1 ∪c Σ2 can be cut on the gluing arc c to give a pair of tangles (α ∩ Σ1, α ∩ Σ2), one on each surface. This procedure induces a morphism from the vector space of tangles in the gluing to the tensor product of the vector spaces of tangles in both surfaces. This cutting however seems to highly depend on the choice of the representative α in its isotopy class, thus one has to quotient the vector space of tangles by some boundary relations to make this well-defined on the level of isotopy classes. Stated skein algebras were introduced in [3] and some variants in [21]. They were later refined in [18] and further studied in [7]. The authors define the stated skein algebra S (S) of a marked surface S as the algebra of unoriented framed tangles in S × (0, 1) with states + Relating Stated Skein Algebras and Internal Skein Algebras 3 or − on the endpoints, modulo the Kauffman-bracket relations and two additional boundary relations. It makes the cutting construction well-defined, namely one has an algebra morphism ρc : S (Σ1 ∪c Σ2)→ S (Σ1)⊗S (Σ2) in the situation above, see Theorem 2.30. For a boundary edge c, we want an action on the boundary similar to the SkV(c × (0, 1))- module category structure on SkV(S). This is done by cutting out a bigon B = • • from the surface near this boundary edge. The stated skein algebra of the bigon S (B) has a structure of Hopf algebra given by the cutting construction, and actually of coribbon Hopf algebra by very geometric means. It is isomorphic to the Hopf algebra Oq2(SL2). The stated skein algebra S (S) is then naturally a comodule over S (B). It is either a right or a left S (B)-comodule depending on whether one sees the boundary edge at the right or at the left. For a gluing of two surfaces, stated skein algebras satisfy very simple excision properties expressed by their S (B)-comodule structures. Internal skein algebras Sticking to the skein category, one could ask if it can be completely described by an algebra, namely such that SkV(Σ) is equivalent to a category of modules over this algebra. When one has a boundary edge on a connected surface Σ, one has a SkV ( R2 ) -action � on SkV(Σ) by this boundary edge and it is essentially surjective on objects. The internal Hom object with respect to this action captures the idea of being described by a single object. Namely, one asks whether there exists an object AΣ such that HomSkV (Σ)(V �∅,∅) ≃ HomV(V,AΣ) naturally in V ∈ V. Here V is seen as an object of V in the right hand side, and as an object of SkV(R2), a single point coloured by V , in the left hand side. Actually by rigidity of V this implies that HomSkV (Σ)(V �∅,W�∅) ≃ HomV(V,W⊗AΣ) hence that all morphisms in SkV(Σ) are described by morphisms in V involving this object AΣ. It turns out that this object does exist, though not necessarily as an object of V but rather of its “free cocompletion” E , where one formally adds all colimits. One now has an equivalence of categories between the free cocompletion of SkV(Σ) and the category of right modules over AΣ in E , see Theorem 2.47. In the case where V is the semisimple category Oq2(SL2)-comodfin, in the free cocompletion one merely needs to add infinite direct sums. It means that AΣ is a possibly infinite-dimensional Oq2(SL2)-comodule, and indeed stated skein algebras usually are. The structure of internal Hom object comes with its own composition and endows AΣ with an algebra structure. Note that we use the term internal skein algebra from [11] though we use the definition as internal endomorphism algebra from [2], referred there as moduli algebras. This name refers to Alekseev–Grosse–Schomerus moduli algebras which, though isomorphic, have a very different definition, and the first denomination seemed more appropriate to our context. The notion of a free cocompletion is only defined up to canonical equivalence, but in [11] the authors chose to work over a standard choice, the category of presheaves [Vop,Vect]. Similarly, the internal endomorphism algebra of an object is only defined up to unique isomorphism, but in [11] the authors chose a standard choice, namely the one of Proposition 2.42. In the case of V = Oq2(SL2)-comodfin, there is a canonical equivalence R : Oq2(SL2)-comod →̃ [Vop,Vect], X 7→ HomOq2 (SL2)-comod(−, X), and we will compare the stated skein algebras which are objects of Oq2(SL2)-comod to internal skein algebras which are objects of [Vop,Vect] using this equivalence. Actually, because internal 4 B. Häıoun endomorphism algebras from [2] are only defined up to unique isomorphism, and in any free cocompletion, we can say that the stated skein algebra is the internal endomorphism algebra. Note also that in [11] and [2] one considers a right SkV ( R2 ) -action � on SkV(Σ) which induces a braided opposite algebra structure compared to the left action. We will rather use the left action as the algebra structure then coincides with the one on stated skein algebras (namely α.β is α above β). Relating stated skein algebras and internal skein algebras When V = Oq2(SL2)-comodfin, we show that these two algebras are isomorphic as Oq2(SL2)- comodule-algebras. This result was already known as folklore and stated in a slightly weaker form (only as algebras in Vect) in [20, Theorem 4.4], [19, Theorem 9.1] and [11, Remark 2.21]. Note that in their context the isomorphism a priori cannot be improved to an Oq2(SL2)-comodule- algebra isomorphism because of the product inversion coming from seeing the boundary edge at the right. On the other hand, both algebras are known to be isomorphic to the Alekseev–Grosse– Schomerus moduli algebras, which are deformation quantizations of the representation variety of a punctured surface in the direction of the Fock–Rosly Poisson structure. In [10, Theorem 5.3] and [16] the authors provide an Oq2(SL2)-comodule algebra isomorphism between these AGS algebras and stated skein algebras. In [2, Section 7] the authors provide an isomorphism between AGS algebras and internal skein algebras. Note though that one does not see the braided opposite algebra structure through these isomorphisms, and it might be that one deforms the representation variety in the direction of the opposite Poisson structure. Section 3 gives a more direct proof of the isomorphism using only skein theory. Theorem 1.1 (Theorem 3.4). For V = Oq2(SL2)-comodfin and Σ a compact oriented surface with a boundary edge e seen at the right for stated skein algebras, and at the left for internal skein algebras, one has an isomorphism of Oq2(SL2)-comodule-algebras AΣ ≃ S (Σ). We explicitly give the natural isomorphisms StW : HomSkV (Σ)(W �∅,∅)→̃HomV(W,S (Σ)) exhibiting S (Σ) as the internal endomorphism algebra of the empty set in SkV(Σ), without relying on excision properties. We only give StW for W ∈ SkV(R2) a well-ordered set of points coloured by the standard corepresentation V of basis v+, v− – equivalently, an object of the Temperley–Lieb full subcategory TL – which is sufficient as these objects generate the whole category under colimits. This natural isomorphism is the expected one. Any morphism in the left hand side can be described by a (linear combination of) tangle α with n boundary points coloured by V . Then StV ⊗n(α) : V ⊗n → S (Σ) maps a simple tensor vε1 ⊗ · · · ⊗ vεn to the tangle α with states ε1, . . . , εn from top to bottom. Multi-edges Internal skein algebras are usually defined on connected surfaces with a single boundary edge, and we extend the definition to marked surfaces S with multiple boundary edges, so multiple actions. It is simply the internal endomorphism object of the empty set in SkV(S) with respect to the SkV ( R2 )⊗n -module structure. The multi-edge internal skein algebra AS thus obtained is an algebra object in Free(V⊗n) ≃ E⊠n. Moreover, we now allow boundary edges to be seen either as left or as right edges. The corresponding left or right SkV ( R2 ) -actions differ by rotating the picture by 180 degrees, which is homotopic to the identity by the half twist . We give an explicit way to compare internal skein algebras obtained from right and left edges using the half twist, which induces a braided opposite algebra structure: Proposition 1.2 (Proposition 4.6). Let Σ be a surface with a boundary edge e and AΣ the internal skein algebra of Σ with e seen at the left. Pre-composition with the half twist gives Relating Stated Skein Algebras and Internal Skein Algebras 5 a natural isomorphism σR exhibiting AΣ as the internal skein algebra with e seen at the right. The right algebra structure mR : AΣ ⊗ AΣ → AΣ differs from the left one m : AΣ ⊗ AΣ → AΣ by a braiding: mR = m ◦ cAΣ⊗AΣ . Namely, the “right” internal skein algebras introduced in [11] and [2] are the braided opposites of the “left” ones introduced in Section 2.4. The skein category can still be rebuilt as the category of modules over the internal skein algebra as soon as there is at least one boundary edge per connected component, i.e., when all objects “come from the action”: Theorem 1.3 (Theorem 4.13). The free cocompletion of the full subcategory of SkV(S) gener- ated by points near a boundary edge is equivalent to the category of right AS-modules in E⊠n, where E⊠n has opposite monoidal structure on coordinates associated with right edges. Having multiple edges now, one can state excision for a gluing of two surfaces. This follows from the excision properties of skein categories. Theorem 1.4 (Theorem 4.32). Let S1 be a marked surface with n1 + k boundary edges with a sequence of k right boundary edges c⃗1 numbered k⃗1 and S2 a marked surface with n2 + k boundary edges with a sequence of k left boundary edges c⃗2 numbered k⃗2. One has two thick embeddings S1 ←↩ C ↪→ S2 where C = ⊔k(0, 1). Let S = S1 ∪c⃗1=c⃗2 S2 be their collar gluing, one has an isomorphism AS ≃ (AS1 , AS2) inv k⃗1,k⃗2 in E⊠n1+n2 between the internal skein algebra of the gluing and the invariants of the tensor product of the internal skein algebras of the two surfaces. Hence one has a good candidate for the algebraic reduction of the skein category. When V = Oq2(SL2)-comodfin and all edges are seen at the left, one has an equivalence of categories Free(V⊗n) ≃ Oq2(SL2) ⊗n-comod and multi-edge internal skein algebras are isomorphic to stated skein algebras as Oq2(SL2) ⊗n-comodule algebras by the same arguments as in Section 3. We explain more explicitly what happens when one considers right edges in this context, and use a geometric half-coribbon structure on S (B) which is needed to fully describe the relation with stated skein algebras. Theorem 1.5 (Theorem 4.28). Let S be a marked surface with n boundary edges labelled either as left or as right edges. There is an isomorphism of ( Oq2(SL2) ⊗k,Oq2(SL2) ⊗n−k)-bicomodule algebras A Lk+1,...,Ln S ≃ S (S), between S (S) seen as a left Oq2(SL2)-comodule on right edges, and AS with the comodule structures of the right edges switched at the left using the antipode. We check that the excision properties expressed under this isomorphism are indeed the same. 2 Preliminaries We recall basic facts about the quantum groups Oq2(SL2) and Uq2(sl2), and the needed con- structions and properties of skein categories in [6] and of stated skein algebras in [7]. We finish with a slight modification of the construction of internal skein algebras of [11]. Let k be either Q ( q 1 2 ) or C with q 1 2 ∈ C× generic, i.e., not a root of unity. We work in the sym- metric monoidal (2,1)-category Catk of small k-linear categories, k-linear functors and natural isomorphisms, where the tensor product C ⊗ D has objects Ob(C) × Ob(D) and morphisms HomC⊗D((c, d), (c ′, d′)) := HomC(c, c ′)⊗k HomD(d, d ′). 2.1 The coribbon Hopf algebra Oq2(SL2) We recall the definition of the coribbon Hopf algebra Oq2(SL2), its categories of right comodules and their links with left Uq2(sl2)-modules. 6 B. Häıoun Definition 2.1. The coribbon Hopf algebra Oq2(SL2) is the free k-algebra with (one should read the matrix equations component-wise, and the tensor product of matrices should be computed as a usual matrix product with tensor products of coefficients instead of products) generators : a, b, c, d, relations : ca = q2ac, db = q2bd, ba = q2ab, dc = q2cd, bc = cb, ad− q−2bc = 1 and da− q2cb = 1, coproduct : ∆ ( a b c d ) = ( a b c d ) ⊗ ( a b c d ) , counit : ε ( a b c d ) = ( 1 0 0 1 ) , antipode: S ( a b c d ) = ( d −q2b −q−2c a ) , co-R-matrix : R  a⊗ a a⊗ b b⊗ a b⊗ b a⊗ c a⊗ d b⊗ c b⊗ d c⊗ a c⊗ b d⊗ a d⊗ b c⊗ c b⊗ d d⊗ c d⊗ d  =  q 0 0 0 0 q−1 q − q−3 0 0 0 q−1 0 0 0 0 q , coribbon functional : θ ( a b c d ) = ( −q3 0 0 −q3 ) . Remark 2.2. We took the coribbon functional so that the twist on a comodule V is given by θV : V ∆V→ V ⊗H IdV ⊗θ→ V . In the literature (see [28] for the coribbon case, or [13] for the ribbon case), the coribbon functional is defined to be θ−1 in our definition. Moreover, we use here an unusual coribbon functional, which arises naturally on the stated skein algebra of the bigon in Section 2.3. It is studied in [26] where the author proves that it gives precisely the Kauffman- bracket relation under the Reshetikhin–Turaev functor, whereas the usual coribbon functional gives relations which differ by a sign, and give the Jones polynomial after writhe renormalisation. Definition 2.3. The quantum plane A2 q2 is the free k-algebra generated by x and y modulo the relation yx = q2xy. It is a right Oq2(SL2)-comodule algebra with ∆ ( x y ) = ( x y ) ⊗ ( a b c d ) on generators. Its subspace of homogeneous polynomials of degree n forms a sub-comodule which we denote by Vn. In particular the generators span the comodule V1 which we abbreviate V and call the standard co-representation of Oq2(SL2). It is more conventional to call its basis v+ = x and v− = y. This comodule is self dual. The left dual V ∗ of V has basis v∗+ and v∗−, and one has an isomorphism of Oq2(SL2)-comodules φ :  V → V ∗, v+ 7→ −q 5 2 v∗−, v− 7→ q 1 2 v∗+. Proposition 2.4 ([15, Section 4.2.1]). The Oq2(SL2)-comodules Vn, n ∈ N, are all the sim- ple Oq2(SL2)-comodules. Moreover, the categories Oq2(SL2)-comod and Oq2(SL2)-comodfin are semi-simple. Definition 2.5. The Hopf algebra Uq2(sl2) is the free k-algebra with: generators : E, F, K, relations : KE = q4EK, KF = q−4FK, EF − FE = K −K−1 q2 − q−2 , Relating Stated Skein Algebras and Internal Skein Algebras 7 coproduct : ∆(K) = K ⊗K, ∆(E) = 1⊗ E + E ⊗K, ∆(F ) = K−1 ⊗ F + F ⊗ 1, counit : ε(K) = 1, ε(E) = ε(F ) = 0, antipode: S(K) = K−1, S(E) = −EK−1, S(F ) = −KF. Definition 2.6. The left Uq2(sl2)-module V±,n is the vector space of dimension n+1 with action: K = ±  q2n 0 · · · 0 0 q2(n−2) . . . ... ... . . . . . . 0 0 · · · 0 q−2n , E =  0 [n]q2 0 . . . . . . ... . . . [1]q2 0 · · · 0 , F = ±  0 · · · 0 [1]q2 . . . ... . . . . . . 0 [n]q2 0 , where [n]q2 = q2n−q−2n q2−q−2 . The modules of the form V+,n are called of type 1, and the others are discarded here. The module V+,1 is denoted by V and called the standard representation of Uq2(sl2). A module W is called locally finite (of type 1) if ∀w ∈ W , Uq2(sl2) · w is finite-dimensional (and isomorphic to some V+,n). Proposition 2.7 ([13, Theorem VII.2.2]). The Uq2(sl2)-modules V±,n, n ∈ N are all the locally finite simple Uq2(sl2)-modules. Moreover, the categories Uq2(sl2)-modlf and Uq2(sl2)-modfin, of respectively locally finite and finite-dimensional Uq2(sl2)-modules of type 1, are semi-simple. Definition 2.8. We can define a dual pairing, namely a non-degenerate bilinear form ⟨·, ·⟩ : Uq2(sl2)⊗Oq2(SL2)→ k satisfying ⟨x, y.y′⟩ = ⟨x(1), y⟩⟨x(2), y′⟩, ⟨x.x′, y⟩ = ⟨x, y(1)⟩⟨x′, y(2)⟩, ⟨x, 1⟩ = ε(x), ⟨1, y⟩ = ε(y) and ⟨S(x), y⟩ = ⟨x, S(y)⟩. It is given on generators by〈 K, ( a b c d )〉 = ( q2 0 0 q−2 ) , 〈 E, ( a b c d )〉 = ( 0 1 0 0 ) , 〈 F, ( a b c d )〉 = ( 0 0 1 0 ) . A right Oq2(SL2)-comodule structure on some vector space W induces a left Uq2(sl2)-module structure by x · w = w(1).⟨x,w(2)⟩, x ∈ Uq2(sl2), w ∈W . Proposition 2.9 ([1, equation (3.3), p. 126], [25, Theorem 7.9]). This correspondence in- duces equivalences of categories Oq2(SL2)-comodfin ≃ Uq2(sl2)-modfin and Oq2(SL2)-comod ≃ Uq2(sl2)-modlf . The simple comodules Vn are mapped on the simple modules V+,n. Note that this correspondence also preserves the monoidal (and actually ribbon, though using an unusual twist on Uq2(sl2)-mod) structure. In the following, we will mostly adopt the point of view of comodules over Oq2(SL2), because there are fewer conditions to ask, the braided and ribbon structure of Oq2(SL2) is algebraic and not “topological”, and it has a geometric description using stated skein algebras. 8 B. Häıoun 2.2 Skein categories Tangle invariants obtained from a k-linear ribbon category V are defined in [27, Section I.2]. They can be extended to give tangle invariants on any oriented surface Σ by local relations, and produce the skein category of the surface SkV(Σ), see [29]. We will use and recall their boundary structure and excision properties, see [6]. Definition 2.10. Let [n] denote the set of n points {1, . . . , n} × {0} in R2, equipped with blackboard framing (coming out of the page when we draw ribbon graphs as below) and RibbonV the category whose objects are framed oriented V-coloured points of R2 of the form [n], and morphisms ribbon graphs (i.e., V-coloured oriented framed tangles with coupons coloured by morphisms) between them in R2 × [0, 1]. • (X,+) • (Y,+) • (Z,−) • (Y,−) • (T,+) • (Z,−) < f < < Theorem 2.11 (Reshetikhin–Turaev). Given V a strict ribbon category, there exists a unique strictly monoidal functor RT: RibbonV → V, called the Reshetikhin–Turaev functor, mapping positive single points to their colour, preserving ribbon structures and mapping coupons to their colour. Remark 2.12. For skein categories, we would prefer to consider all points with all possible framings instead of only those of the form [n] as in RibbonV . We denote this bigger category by RibbonV ( R2 ) . The two categories are equivalent as the inclusion of the first in the second is obviously fully faithful and essentially surjective. A quasi-inverse is however not so natural to define. One has to choose an isomorphism from each object of RibbonV ( R2 ) to one of the form [n]. This can be done for example by giving lexicographical order on R2 and pushing all points in good position while preserving this order, then turning the framing clockwise until it is vertical. Working with RibbonV ( R2 ) , and allowing non-strict ribbon categories V, the above theorem still holds, but the functor RT is only essentially unique. We now assume a choice of quasi-inverse as above has been made, so that the functor RT is defined on RibbonV ( R2 ) . The above functor RT describes a way to obtain invariants of coloured tangles, and actually coloured ribbon graphs, from a ribbon category V. We want to study this invariant, and say that two ribbon graphs are identified if they give the same invariant, namely the same morphism in V after evaluation under RT. Skein categories generalize this construction for coloured ribbon graphs on a thickened surface. The idea is to take the relations between ribbon graph which are true locally, on an embedded cube. Definition 2.13. Let Σ be a compact oriented surface possibly with boundary and V a ribbon category. The k-linear category RibbonV(Σ) has objects finite sets of V-coloured oriented framed points of Σ and morphisms k-linear combinations of isotopy classes of ribbon graphs. The skein category SkV(Σ) with coefficients in V is the quotient of RibbonV(Σ) by the local relation ∑ λiFi = 0 if there exists an orientation preserving embedding of a cube ϕ : [0, 1]3 ↪→ Σ× [0, 1] such that all of the Fi’s coincide outside this little cube, intersect ϕ ( ∂[0, 1]3 ) on either the top or the bottom face, transversally, and give the zero morphism in V after evaluation of the functor RT on this little cube, namely ∑ λiRT ( ϕ−1(Fi\|im ϕ) ) = 0. Put differently, let F be a ribbon graph and ϕ a little cube of Σ×(0, 1) such that G := F \|im ϕ is a (possibly complicated) ribbon graph. Then RT(G) is a morphism in V, and we allow ourselves to replace G with a single coupon coloured by this morphism. Relating Stated Skein Algebras and Internal Skein Algebras 9 Remark 2.14. When Σ = R2, since RT is full, via coupons, it induces an equivalence of categories SkV ( R2 ) → V. Its quasi-inverse is given by the inclusion of the full subcategory V ⊆ SkV ( R2 ) mapping an object V ∈ V to the framed point [1] coloured by V , and a morphism to a coupon coloured by this morphism. Note that this inclusion is monoidal but not strictly monoidal, namely one has an isomorphism from the two framed points [2] respectively coloured by V and W to the framed point [1] coloured by V ⊗W , this isomorphism is the coupon IdV⊗W with two incoming strands coloured respectively by V and W and one outgoing strand coloured by V ⊗W . V • • W • V ⊗W > > > IdV⊗W Remark 2.15 ([6, Remark 1.7]). For a general surface Σ, the categories defined above are not monoidal because there is no notion of horizontal juxtaposition, which we used in R2. However, if A = C × [0, 1] for a 1-manifold C, the category SkV(A) is monoidal with tensor product induced by A ⊔A [0, 1 3 ]⊔[ 2 3 ,1] ↪→ A. Remark 2.16 ([6, Remarks 1.6 and 1.18]). An orientation-preserving smooth embedding f : Σ1 → Σ2 induces a functor SkV(f) : SkV(Σ1)→ SkV(Σ2). It maps an object s, which is a bunch of coloured points in Σ1, to f(s), and a ribbon graph T ⊆ Σ1 × [0, 1] to (f × Id)(T ). This defines a symmetric monoidal functor SkV : (Mfldor2 ,⊔)→ (Catk,⊗k). An isotopy of smooth embeddings λ : Σ1 × [0, 1] → Σ2 between f = λ0 and g = λ1 induces a natural isomorphism ribλ : SkV(f)⇒ SkV(g), where ribλ,s : f(s)→ g(s) is the braid in Σ2×[0, 1] drawn by {(λt(s), t), t ∈ [0, 1]}. Homotopic isotopies give isotopic ribbon graphs, and this extends to a symmetric monoidal ∞-functor SkV : (Mfldor2 ,⊔)→ (Catk,⊗k). It is shown in [6] that this functor is the factorisation homology with coefficients in V. Proposition 2.17 ([6, Section 1.3]). For C ⊂ ∂Σ a boundary component (actually, a thick left boundary component, i.e., equipped with an embedding C × [0, 2) ↪→ Σ) the category SkV(Σ) inherits a structure of left SkV(A)-module category, where A = C × [0, 1]. Namely, one has a functor � : SkV(A) ⊗ SkV(Σ) → SkV(Σ) compatible with the monoidal structure of SkV(A). It is given by pushing points or ribbon graphs in A inside Σ. Similarly, a thick right boundary component C × (−1, 1] ↪→ Σ induces a right SkV(A)-module structure on SkV(Σ). Skein categories satisfy a form of excision, namely the skein category of a gluing is obtained as a relative tensor product of the skein categories of the initial surfaces. Theorem 2.18 (see [6, Theorem 1.22] for details). Let Σ1 and Σ2 be two surfaces and C a thick right boundary component of Σ1 and a thick left boundary component of Σ2. Let Σ1∪AΣ2 be the collar gluing of the surfaces along the two thick embeddings. The skein category SkV(Σ1 ∪A Σ2) is the Tambara relative tensor product of the right SkV(A)-module SkV(Σ1) and the left SkV(A)- module SkV(Σ2). Namely, the canonical functor SkV(Σ1)⊗ SkV(Σ2)→ SkV(Σ1 ∪A Σ2) induces an equivalence of categories between k-linear functors out of SkV(Σ1 ∪A Σ2) and SkV(A)-balanced functors out of SkV(Σ1) ⊗ SkV(Σ2) (i.e., equipped with a natural isomorphism ι : (− � −,−)⇒̃(−,− � −) between the actions on Σ1 and on Σ2). Idea of proof. One has an explicit description of the Tambara relative tensor product of two module categories, by formally adding a natural isomorphism ι to the morphisms of the two categories. 10 B. Häıoun Σ1 C× [0,1] Σ2 ×[0, 1] •s1 • s1 � a •s2 • a� s2 • • It is equivalent to SkV(Σ1∪AΣ2) by sending, for s1, a, s2 some sets of coloured points respectively in Σ1, A, Σ2, the balancing isomorphism ιs1,a,s2 : (s1 � a, s2) → (s1, a � s2) to the morphism depicted hereby. ■ Corollary 2.19. Let s1 ∈ SkV(Σ1) and s2 ∈ SkV(Σ2). Then any morphism α ∈ HomSkV (Σ1∪AΣ2)((s1, s2),∅) can be decomposed in a (linear combination of) pair(s) α1 ∈ HomSkV (Σ1)(s1,∅� a), α2 ∈ HomSkV (Σ2)(a� s2,∅) for some a ∈ SkV(A), with α = (Id∅, α2) ◦ ι∅,a,s2 ◦ (α1, Ids2). This decomposition is unique up to balancing, namely if α2 can be written β2 ◦ (γ � Ids2), with β2 ∈ HomSkV (Σ2)(b � s2,∅) and γ ∈ HomSkV (A)(a, b), for some b ∈ SkV(A), then (α1, β2 ◦ (γ � Ids2)) ∼ ((Id∅ � γ) ◦ α1, β2). Proof. On a drawing one wants to decompose α as α •s1 •s2 = α1 α2 •s1 •s2 a� s2 ∅� a ι which is easily done by pushing the ribbon graph happening in the middle region inside say N leaving only straight lines (namely, ι’s) behind. The relation (α1, β2 ◦ (γ � Ids2)) ∼ ((Id∅ � γ) ◦ α1, β2) is true by sliding γ along the straight lines of ι, and this is the only relation by the above theorem. ■ Note that the asymmetry in this description is purely artificial, and one could have chosen a cup or a cap instead of a slanted line to link the left and right actions. Example 2.20. We are particularly interested in the case V = Oq2(SL2)-comodfin, for which the relations the Reshetikhin–Turaev functor imposes (between framed tangles) are precisely the Kauffman-bracket relations. Remark 2.21. In any skein category, the identity coupon IdX∗ : X∗ → X∗ with entry a down- ward oriented X-coloured ribbon and output an upward oriented X∗-coloured ribbon depicted below gives an identification (X,−) ≃ (X∗,+). (X,−) (X∗,+) IdX∗ < > Relating Stated Skein Algebras and Internal Skein Algebras 11 In the case V = Oq2(SL2)-comodfin and X = V is the standard corepresentation, the object V ∗ is already isomorphic to V in V by φ from Definition 2.3. Thus one gets an identification (V,+) ≃ (V,−) and sliding the coupon coloured by it along a strand changes the orientation of the strand. Consequently, one can switch signs of points and orientations of strands. In the following, we stop mentioning them and talk about unoriented framed tangles. The Reshetikin–Turaev functor is still well-defined on unoriented framed tangles, see [26, Theo- rem 4.2]. An unoriented tangle gives a ribbon graph by choosing an arbitrary orientation, colouring every strand by V and replacing V ∗’s imposed on the boundary points by V ’s using φ or φ−1. Concretely, for the unoriented cap ∩ for example, one can orient it either left or right, so one has to check that RT( ) ◦ (φ⊗ IdV ) = RT( ) ◦ (IdV ⊗ φ). Note that this would not hold with the usual coribbon element in Oq2(SL2). Theorem 2.22 ([5, Theorem 12.3.10] or [4, Theorem 3.3.4] for endomorphisms in R2). Let n.V and m.V be two objects of SkV(Σ) given respectively by n and m points coloured by V . Then any morphism in HomSkV (Σ)(n.V,m.V ) can be represented by a linear combination of unoriented framed tangles (without coupons). Moreover, two linear combinations of unoriented framed tan- gles represent the same morphism in HomSkV (Σ)(n.V,m.V ) if and only if one can get from one to the other by a sequence of isotopies and Kauffman-bracket relations. Corollary 2.23. The algebra HomSkV (R2)(n.V, n.V ) is isomorphic to the Temperley–Lieb alge- bra TLn, defined in [27, Section XII.3]. The full subcategory of SkV ( R2 ) of objects of the form [n] with every point coloured by V is equivalent to the category TL, defined in [27, Section XII.2]. In the following, we will call this full subcategory TL. The category TL is a ribbon full subcategory of V, and the above theorem proves that for any surface Σ the category SkTL(Σ) is a full subcategory of SkV(Σ). 2.3 Stated skein algebras Stated skein algebras generalise skein algebras for tangles on a marked surface with boundary, see [3, 18, 21]. These tangles can be cut and stated skein algebras satisfy nice excision properties. We will recall here the approach of [7]. Stated skein algebras can be defined over integral rings of coefficients, but we will work over a field k being either Q ( q 1 2 ) or C with q 1 2 ∈ C× generic, as our proofs only hold in this context. Definition 2.24. Let S be an oriented surface. The skein algebra S̊ (S) is the k-vector space generated by isotopy classes of unoriented framed links in S× (0, 1) modulo the skein relations: = q + q−1 and = ( −q2 − q−2 ) in a little embedded cube ϕ : D3 ↪→ S× (0, 1). It is an algebra with product given by vertical superposition S× (0, 1)⊔S× (0, 1) ( 1 2 ,1)⊔(0, 12)−→ S× (0, 1) and unit the empty link. Note that by Theorem 2.22, one has an algebra isomorphism S̊ (S) ≃ EndSkO q2 (SL2)-comodfin (Σ)(∅). Definition 2.25. A marked surface is a compact oriented surface with boundary S with a finite set P ⊆ ∂S of boundary points, called marked points. We write S = S ∖ P and call this the 12 B. Häıoun marked surface. We write ∂PS the boundary components of S that contain a point of P and ∂S := ∂PS ∖ P . Namely we only consider boundary components of S that contain a marked point, which we remove in S, so all components of ∂S are intervals. The circular boundary components are discarded and give punctures in S. The boundary structure of stated skein algebras will depend on a way to cut out a bigon out of a boundary edge, and that of internal skein algebras on a way to insert one from a boundary edge. To avoid choices, we suppose that marked surfaces come equipped with a thickening of their boundary edges inside the surface. A stated tangle α on S is an unoriented, framed, compact, properly embedded 1-submanifold ofS×(0, 1) whose boundary ∂α ⊆ ∂S×(0, 1) has positively vertical framing and comes equipped with a state st : ∂α→ {+,−}. We call height the (0, 1)-coordinate of a point, and require that all boundary points of α lying over a same component of ∂S have distinct heights. An isotopy of stated tangles is an isotopy with values in stated tangles, in particular preserving the height order over a same boundary component. A stated tangle in S× (0, 1) can always be represented as a diagram with blackboard framing in S, with its under/over crossing information, such that the height order of boundary points corresponds to a given orientation on the boundary edges, see [3, Section 3.5]. Definition 2.26 ([7, Section 2.5]). The stated skein algebra S (S) of a marked surface S is the k-vector space generated by isotopy classes of stated tangles on S modulo usual skein relations: = q + q−1 , = ( −q2 − q−2 ) and the boundary skein relations: >+ − = q− 1 2 > , >+ + = >− − = 0, >− + = q2 >+ − + q− 1 2 > , where the arrows on the boundary edges represent the relative height order of the two points. It is an algebra with product given by vertical superposition and unit the empty link. We denote by Cµ ν the coefficient such that > µ ν = Cµ ν > , namely C+ + = C−− = 0, C+ − = q− 1 2 and one can compute C−+ = −q− 5 2 , see [7, Lemma 2.3(13)]. We also write C(ν) := C−νν . Proposition 2.27 ([18, Lemmas 2.3 and 2.4]). These relations express equivalently with the boundary at the left, namely: > ν µ = µ ν C > , where + + C= −− C= 0, + − C= −q 5 2 and −+ C= q 1 2 , and > = q− 1 2 > − + − q− 5 2 > + − . Remark 2.28. It is easy to check that S (S⊔S′) ≃ S (S)⊗k S (S′) since all relations happen in a connected disk. Relating Stated Skein Algebras and Internal Skein Algebras 13 Definition 2.29 (see [18, Section 3.1] for details). Let S be a marked surface and c an ideal arc on S, joining two marked points in S. Denote by Cutc(S) the marked surface obtained by cutting S along c. Given a stated tangle α on S, one can cut it along c and get a tangle Cutc(α) on Cutc(S). This tangle has new boundary points, two lifts per points of α ∩ c, and we may give any state to these points. The obtained stated tangle is called a lift of α if the two lifts of a point of α∩ c have same state. This definition is not perfectly innocent and it seems that one could have chosen different state-matching patterns for lifts, see Remark 4.34. Theorem 2.30 ([18, Theorem 3.1]). Let S be a marked surface and c an ideal arc. The map ρc : S (S)→ S (Cutc(S)), α 7→ ∑ lifts α̃ is well-defined (it only depends on the isotopy class of α) and is an injective algebra morphism. Moreover, the splitting morphisms ρc and ρc′ associated to two disjoint ideal arcs c and c′ commute. Example 2.31. The bigon B is the marked surface (D, {±i}), the disk with two marked points. The algebra S (B) is generated as an algebra by the µβν = • • µ ν, µ, ν ∈ {±}, and has basis the µ⃗βν⃗ = > < • • ·· · ·· · µn νn µ1 ν1 , where µ⃗ = (µ1, . . . , µn) and ν⃗ = (ν1, . . . , νn) are decreasing sequences of signs. It is a bialgebra with coproduct given by cutting along the “unique” arc joining the two marked points • • c , ∆ = ρc : S (B)→ S (B ⊔B) ≃ S (B)⊗S (B). Coassociativity comes from the second part of Theorem 2.30. The counit ε : S (B) → k is defined on the basis by ε(µ⃗βν⃗) = δµ⃗,ν⃗ . It is a Hopf algebra with antipode S ( > < • • ·· · ·· · µm νn µ1 ν1 β ) = > < • • ·· · ·· · −ν1 −µ1 −νn −µm β . C(ν⃗) C(µ⃗) , where C(ν⃗) := n∏ i=1 C(νi). It is coquasitriangular with co-R-matrix R(α⊗ β) = ε  • > • < β α , see [7, Theorem 3.5]. It is coribbon with coribbon functional θ(α) = ε  • <• > α . 14 B. Häıoun Proposition 2.32 ([18, Theorem 4.1], [17, Section 2.2], [7, Theorem 3.4]). One has an iso- morphism of coribbon Hopf algebras S (B) ≃ Oq2(SL2) given on the generators by +β+ 7→ a, −β− 7→ d, +β− 7→ b and −β+ 7→ c. Stated skein algebras provide great examples of (possibly infinite-dimensional) Oq2(SL2) or S (B)-comodules. Given a marked surface S and a boundary edge e of S, we can consider an ideal arc c going along e but inside S̊. The piece between e and c is a bigon, and the rest is homeomorphic to the original surface. The splitting morphism along c, ∆ = ρc : S (S)→ S (S⊔B) ≃ S (S)⊗S (B) endows S (S) with right Oq2(SL2)-comodule structure. • • ec It is compatible with its algebra structure, namely S (S) is an Oq2(SL2)-comodule-algebra, see [7, Proposition 4.1]. Definition 2.33. Let rot : B → B be the homeomorphism of marked surfaces given by the planar 180 degree rotation. It induces rot∗ : S (B)→ S (B), with rot∗ ( > < • • ·· · ·· · µm νn µ1 ν1 β ) = < > • • ·· · ·· · ν1 µ1 νn µm β which is an algebra isomorphism. It reverses the coproduct, namely ∆ ◦ rot∗(β) = < > • • β(2) ⊗ < > • • β(1) = (rot∗⊗ rot∗) ◦∆op(β), and preserves the counit, because ε(rot∗(µ⃗βν⃗)) = ε(ν⃗βµ⃗) = δν⃗,µ⃗ = δµ⃗,ν⃗ . On Oq2(SL2), it is given by r ( a b c d ) = ( a c b d ). Remark 2.34. If one sees the edge e at the left instead of the right of the surface, one gets a structure of left Oq2(SL2)-comodule. One can easily get from one to another by rotating the whole picture. Namely, the left coaction ∆l is obtained from the right coaction ∆r by rotating the bigon by 180 degrees. In [7, Proposition 4.1] one gets ∆l = fl ◦(IdS (S) ⊗ rot∗) ◦∆r, where fl denotes the flip of tensors. • • c • • W E S N • • c • • WE S N Actually, one gets such a structure for each boundary edge of S, and if S has n right boundary edges and m left, S (S) is an ( Oq2(SL2) ⊗n,Oq2(SL2) ⊗m)-bicomodule algebra. Remark 2.35. The structure forms on S (B), such as the co-R-matrix or the coribbon func- tional, are often defined using the counit on some transformation of the tangle. This has a direct interpretation on how this form then acts on comodules. Relating Stated Skein Algebras and Internal Skein Algebras 15 Let φ : S (B)→ k be given by some φ  • <• > εn ηm ... ... ε1 η1 α  = ε  • <• > εn ... s m (η 1 ...η m ) ε1 α Tm , where Tm, m ∈ N, is a family of tangles with m right and m left boundary points and sm : {±}m → vectk({±}m). Here, we allowed sm to have values in formal linear combina- tion of m-tuples of states because in the definition of the co-R-matrix for example one needs coefficients depending on the states. What we mean by a tangle α with state a formal linear combination of states ∑ i λiη⃗i is the linear sum of stated tangles α∑ i λiη⃗i := ∑ i λiαη⃗i . Note that the Tm’s and the sm’s should satisfy extra conditions for this to be well-defined on S (B). Then for a marked surface S with a right edge e, the map ΦS (S) : S (S) ∆→ S (S)⊗S (B) Id⊗φ−→ S (S) is given by • <• ηm ... η1 α ∆7−→ ∑ ν⃗  • <• νm ... ν1 α ⊗ • <• > νm ηm ... ... ν1 η1  Id⊗φ7−→ (Id⊗ ε)  • <• νm ... ν1 α ⊗ • <• > νm ... s m (η 1 ...η m ) ν1 Tm  splitting = well def (Id⊗ ε)  • <• νm ... ν1 α Tm ⊗ • <• > νm ... s m (η 1 ...η m ) ν1  counit = • <• s m (η 1 ...η m ) α Tm . The co-R-matrix isn’t exactly of this form, but the same kind of computation applies. Remember that the braiding on S (S)⊗S (S) is defined using the coaction on each S (S), the co-R-matrix on the S (B)⊗2 part thus obtained, and then flipping the two factors, namely cS (S),S (S) = fl ◦R24 ◦ ( ∆S (S) ⊗ ∆S (S) ) . The braided opposite product on S (S) is defined as mbop := m ◦ cS (S),S (S) and it has a nice geometric depiction, namely: mbop(α⊗ β) = m ◦ fl  • <• νn ... ν1 α(1) ⊗ • <• µm ... µ1 β(1) . ε  • > • < ν⃗ µ⃗ β(2) α(2)   = • <• ν⃗ µ⃗β(1) α(1) . ε  • > • < ν⃗ µ⃗ β(2) α(2)  = • <• β α . Stated skein algebras satisfy a form of excision, namely the stated skein algebra of a gluing is obtained as a relative tensor product of the stated skein algebras of the two initial surfaces. Let S′ be a marked surface and c1 and c2 respectively a right and a left boundary edges of S′. Then S (S′) has a structure of (Oq2(SL2),Oq2(SL2))-bicomodule. We consider S = S′/c1=c2 the marked surface obtained by gluing c1 to c2, and c the ideal arc formed by c1 = c2 in S. We 16 B. Häıoun have S′ = Cutc(S), and Theorem 2.30 gives an injective algebra morphism ρc : S (S)→ S (S′). • • c1 S′ • • c2 Definition 2.36. Let H be a Hopf algebra and M an (H,H)-bicomodule, with coproducts denoted by ∆1 : M →M ⊗H and ∆2 : M → H ⊗M . The 0-th Hochschild cohomology of M is the subalgebra of M defined as HH0(M) := {x ∈M / ∆1(x) = fl ◦∆2(x)}. Theorem 2.37 ([17, Section 2.3], see also [7, Theorem 4.8]). Let S = S′/c1=c2 be a gluing. The splitting morphism ρc : S (S)→ S (S′) maps isomorphically S (S) on HH0(S (S′)). Remark 2.38. If S1 and S2 are two marked surfaces, c1 is a right boundary edge of S1, c2 a left boundary edge of S2 and S′ = S1 ⊔ S2. Then S (S′) ≃ S (S1) ⊗k S (S2), note that this is the tensor product as vector spaces and not as comodules, and the 0-th Hochschild cohomology of S (S′) corresponds to the cotensor product of S (S1) and S (S2) over S (B), namely S (S1)□HS (S2) := {x ∈ S (S1)⊗S (S2) / ∆1 ⊗ Id2(x) = Id1 ⊗∆2(x)}. 2.4 Internal skein algebras When a surface has a boundary edge, one can push a little disk inside the surface from this boundary edge. This process induces an action of the skein category of the disk on the skein category of the surface. Namely, SkV(Σ) is a SkV ( R2 ) -module category, see [6, Sections 3.2]. In this situation there is a notion of internal Hom objects that encode entirely in SkV ( R2 ) the behavior of objects of SkV ( R2 ) seen in SkV(Σ) by the action. Namely for fixed V,W ∈ SkV(Σ) one has HomSkV (Σ)(X � V,W ) ≃ HomSkV (R2)(X,Hom(V,W )) naturally in X, see [9, Section 7.9]. However, such an object does not always exist in SkV ( R2 ) , and actually lives in its free cocompletion. The internal skein algebra AΣ of the surface is the internal endomorphism algebra of the empty set Hom(∅,∅), see [11] or [2] together with [6]. This means one can understand ribbon graphs in SkV(Σ) with boundary points on the bottom and near the boundary edge as morphisms in (the free cocompletion of) SkV ( R2 ) with target AΣ. Note that in [6], [2] and [11] one uses a right SkV ( R2 ) -action and we use a left SkV ( R2 ) -action here. See Section 4 for more details. Definition 2.39. Let A be a k-linear monoidal category and M a left A-module category. Let M1 and M2 be two objects of M. The internal Hom of M1 and M2 with respect to the A-module structure is an object Hom(M1,M2) ∈ A equipped with a natural isomorphism η : HomM(− �M1,M2)→̃HomA(−,Hom(M1,M2)). It is unique up to canonical isomorphism when it exists. When all involved internal Hom objects exist, one can define: the evaluation map evM1,M2 : Hom(M1,M2)�M1 →M2 is the image under η−1 of IdHom(M1,M2) and the composition map c : Hom(M2,M3)⊗ Hom(M1,M2) → Hom(M1,M3) is the image under η of the morphism evM2,M3 ◦ (IdHom(M2,M3) � evM1,M2) : (Hom(M2,M3)⊗Hom(M1,M2))�M1 → Hom(M2,M3)� M2 → M3. In particular, an internal endomorphism End(M) := Hom(M,M) is an algebra object in A, with unit the morphism 1A → End(M) associated with IdM . The functor HomM(−�M1,M2) : Aop → Vectk cannot always be represented in A. However, it is always an object of its free cocompletion. Definition 2.40. The 2-category Cocompk has objects essentially small k-linear cocomplete categories and morphisms k-linear cocontinuous functors between them, i.e., functors preserving Relating Stated Skein Algebras and Internal Skein Algebras 17 colimits and natural transformations between functors. It is symmetric monoidal with the Kelly tensor product ⊠ for cocomplete categories, which is characterized by HomCocompk(A⊠ B, C) ≃ HomCocompk(A,HomCocompk(B, C)) ≃ Cocont(A,B; C), see [14, Section 6.5] and [22, Theorem 2.45]. The free cocompletion of a k-linear category C is a cocomplete category Free(C) ∈ Cocompk together with a functor i : C → Free(C) which is initial among functors to a k-linear cocomplete category. Namely, i∗ : HomCatk(C,D)→ HomCocompk(Free(C),D) is an equivalence of categories for any D ∈ Cocompk. The free cocompletion Free(C) is unique up to essentially unique equiv- alence. Remark 2.41. A standard choice for the free cocompletion is the presheaf category [Cop,Vectk], in which C embeds by the Yoneda embedding. Moreover for any other choice of free cocomple- tion Ĉ, the canonical equivalence is given by R : { Ĉ → [Cop,Vectk], X 7→ HomĈ(i(−), X). See [8, Section 2.2] for more details. From now on, we assume that we made this standard choice and set Free(C) := [Cop,Vectk]. Note that every object C of C seen as an object of Free(C) is compact projective, i.e., the functor HomFree(C)(C,−) is cocontinuous. It is then easy to verify that the free cocompletion Free : Catk → Cocompk is a symmetric monoidal functor. Hence if A ∈ Catk is a monoidal k-linear category and M an A-module category, after free cocompletions Free(A) is again monoidal and Free(M) is a Free(A)-module category. Proposition 2.42. Let A ∈ Catk be a monoidal k-linear category, M an A-module category and M1, M2 two objects ofM. The presheaf F = HomM(−�M1,M2) ∈ Free(A) is the internal Hom object of M1 and M2 (seen as objects of Free(M)) with respect to the Free(A)-module structure. Proof. For the “small” objects A ∈ A ↪→ Free(A), the isomorphism HomFree(A)(A,F ) ≃ F (A) := HomM(A �M1,M2) is given by the Yoneda lemma. Now any object X ∈ Free(A) is obtained as a colimit of such small objects, X = colimiAi, Ai ∈ A, by the co-Yoneda lemma. Then it is straightforward to check that HomFree(A)(X,F ) ≃ lim i HomFree(A)(Ai, F ) ≃ lim i HomFree(M)(Ai �M1,M2) ≃ HomFree(M)(colimi(Ai �M1),M2) ≃ HomFree(M)(X �M1,M2). Here we kept the notation � for its essentially unique cocontinuous extention to free cocomple- tions. ■ This means that when one works with free cocompletions of a module structure on some “small” categories, the internal Hom objects of the free cocompletions are completely described by what happens on the “small” subcategories. Using the same argument one can rephrase the definition of the internal Hom object of M1 and M2 in a free cocompletion Â of A as an object X ∈ Â together with an isomorphism of presheaves HomM(− �M1,M2)→̃R(X), where R is the equivalence from Remark 2.41. Proposition 2.43. The inclusion Oq2(SL2)-comodfin ↪→ Oq2(SL2)-comod is a free cocompletion. 18 B. Häıoun Proof. At generic q, the category Oq2(SL2)-comod is semisimple and its simples are finite- dimensional. Hence any Oq2(SL2)-comodule is a direct sum of these, and any colimit in Oq2(SL2)-comodfin is simply a direct sum. ■ Note that the monoidal structure on Oq2(SL2)-comod is the free cocompletion of the monoidal structure on Oq2(SL2)-comodfin as it extends it and commutes with direct sums in both factors. In the preceding proposition, one could replace Oq2(SL2)-comodfin with its full subcategory TL, as every Oq2(SL2)-comodule is a quotient of direct sums of objects of TL. Actually, the inclu- sion SkTL(Σ) ↪→ Free ( SkOq2 (SL2)-comodfin(Σ) ) is also a free cocompletion by [6, Theorems 2.28 and 3.27], using factorisation homology. We now apply the internal Hom object constructions to the case of skein categories. We choose a k-linear ribbon category V and choose a free cocompletion E . Definition 2.44. We consider an oriented surface with boundary Σ, with a “red” arc chosen on its boundary, seen at the left. This arc can be thickened in the surface, which gives a thick left embedding (0, 1)→ ∂Σ. In particular one has an embedding of surfaces (0, 1)× [0, 1] ⊔ Σ ↪→ Σ depicted hereby. By Proposition 2.17, this gives a structure of left SkV((0, 1)×[0, 1]) ≃ SkV ( R2 ) - module category on SkV(Σ). We denote the action functor by � : SkV ( R2 ) ⊗ SkV(Σ)→ SkV(Σ). Σ ⊔ To simplify notation, we write SKV(−) := Free(SkV(−)) and still denote the action functor on free cocompletions by � : E ⊠ SKV(Σ) → SKV(Σ). For M1 and M2 two objects of SkV(Σ), one has an internal Hom object Hom(M1,M2) ∈ E . Definition 2.45. Let Σ be a surface with boundary with a chosen thickened arc on its boundary. The internal skein algebra AΣ := Hom(∅,∅) is the endomorphism algebra of ∅ ∈ SkV(Σ) ⊆ SKV(Σ) with respect to the E-module structure. Its algebra structure is recalled in Defini- tion 2.39. Explicitly, AΣ comes equipped with a natural isomorphism σ : HomSKV (Σ)(− � ∅,∅)⇒̃HomE(−, AΣ) between (contravariant) functors E → Vectk. If V = Oq2(SL2)-comodfin with E = Oq2(SL2)-comod, then AΣ is an Oq2(SL2)-comodule algebra. Remark 2.46. The defining properties of AΣ enables one to describe morphisms V �∅→ ∅ in SkV(Σ), so where the boundary points of tangles are at the bottom, as morphisms V → S (Σ) in E . We would also like to describe morphisms W � ∅ → V � ∅ in SkV(Σ). By duality in V, the functor V � − is right adjoint to V ∗ � −, see [9, Proposition 7.1.6]. So one has natural isomorphisms: HomSkV (Σ)(W �∅, V �∅) →̃ HomSkV (Σ)(V ∗ � (W �∅),∅) σV ∗⊗W→ HomE(V ∗ ⊗W,AΣ) →̃ HomE(W,V ⊗AΣ). When Σ is connected, every object of SkV(Σ) is isomorphic to one of the form V �∅, and the above natural isomorphism suggests that the algebra AΣ ∈ E is enough to fully describe SKV(Σ). Relating Stated Skein Algebras and Internal Skein Algebras 19 Theorem 2.47 ([2, Theorem 5.14]). Suppose that Σ is connected, then there is an equivalence of categories SKV(Σ) →̃ modE −AΣ, M 7→ Hom(∅,M) between the free cocompletion of SkV(Σ) and the category of right modules over AΣ in E. For M of the form V � ∅, V ∈ E, which is always the case for M ∈ SkV(Σ), this functor is given by V �∅ 7→ V ⊗AΣ. Proof. For the last statement, one has Hom(∅, V �∅) ≃ V ⊗AΣ by Remark 2.46. This is the general idea of the proof, as morphisms of AΣ-modules from W ⊗AΣ to V ⊗AΣ are equivalent to morphisms in E from W to V ⊗AΣ, which are equivalent by the above Remark to morphisms from W �∅ to V �∅ in SKV(Σ). For the details, one uses Barr–Beck reconstruction, or more precisely its reformulation in [2, Theorem 4.6]. One has to check that ∅ ∈ SKV(Σ) is a progenerator. It is projective: actR∅ = Hom(∅,−) : { SKV(Σ)→ [Vop,Vectk] ≃ E , M 7→ HomSKV (Σ)(−�∅,M) is cocontinuous because − � ∅ : V → SkV(Σ) ⊆ SKV(Σ) takes values in compact projective objects. It is a generator: actR∅ being faithful is equivalent to − � ∅ : E → SKV(Σ) being dominant by [2, Remark 4.9], which is the case because −�∅ is essentially surjective on SkV(Σ) which generate SKV(Σ) under colimits. One gets right modules over AΣ because we considered left module categories, see [2, Remark 4.7]. ■ Remark 2.48. We are in the same context as [2] and [11]. In [11, Definition 2.18], the internal skein algebra is defined similarly as HomSkV (Σ)(∅ � −,∅) ∈ Free(V) for V a k-linear ribbon category whose unit 1V is simple, which is the case for V = Oq2(SL2)-comodfin. In [2, Definition 5.3], the moduli algebra AΣ is defined to be the endomorphism algebra of the distinguished object OE,Σ of the factorization homology over Σ of a presentable abelian balanced k-linear category E generated under filtered colimits by rigid objects, with respect to the E- module structure. The factorization homology of V is computed by SkV , see [6, Theorem 2.28], and the factorization homology of E = Free(V) by SKV = Free(SkV), see [6, Theorem 3.27]. The distinguished object is ∅ ∈ SkV(Σ) ⊆ SKV(Σ), and a k-linear free cocompletion is always abelian and generated under filtered colimit by rigid objects. There is one difference with our paper though: we consider a left E-action, and [2, 11] consider a right E-action, with adapted definitions of internal Hom objects. In our description, it would mean seeing the red arc on the right instead of on the left, thus SkV(R2) acting from the right. This gives a braided opposite product, and we need left actions to have the same product than the one on S (Σ). Right actions and more generally multiple right/left actions and how they interact will be studied in Section 4. 3 The relation We show in this section that the stated skein algebra of a surface with a single boundary edge is isomorphic to AΣ. We consider the algebra AΣ from last section with V = Oq2(SL2)-comodfin and E = Oq2(SL2)-comod at generic q, and prove that S (Σ) ≃ AΣ as Oq2(SL2)-comodule algebras. Actually since AΣ is only defined up to canonical isomorphism one may take an equality here, so we prove that S (Σ) satisfies the defining properties of AΣ, namely that it is the internal endomorphism algebra of the empty set in SKV(Σ) with respect to the E-module structure. 20 B. Häıoun This result is not new and was stated in a weaker form in [20, Theorem 4.4], [19, Theorem 9.1] and [11, Remark 2.21], namely as algebras in Vect. However, in these references one considers right E-actions and therefore the internal skein algebra is isomorphic to the braided opposite of the stated skein algebra. The full result can still be recovered using [10, Theorem 5.3] or [16], which give an isomorphism between the opposite of the stated skein algebras and Alekseev– Grosse–Schomerus-algebras, which are themselves isomorphic to internal skein algebras by [2]. Our approach here is more direct and uses only skein theory. We need a natural isomorphism StW : HomSKV (Σ)(W � ∅,∅)⇒̃HomE(W,S (Σ)), for W ∈ E = Oq2(SL2)-comod. In the case where W = V ⊗n ∈ TL is a tensor product of standard corepresentations, and the element on the left hand side is a tangle α with n boundary points, we want a morphism V ⊗n → S (Σ) associated to it. This is done by setting the entries, elements of V ⊗n, as states of the tangle α. Recall that V has basis v+, v− and we identify the states + and − with these elements. As in Remark 2.35, we allow formal linear combination of states for stated tangles. In this context, the defining relations of stated skein algebras correspond exactly to naturality conditions, see Proposition 3.3. This gives the idea of how to deal with the objects of the full subcategory TL ⊆ V of objects of the form V ⊗n, and this extends to E ≃ Free(TL) by Proposition 2.42. Note that one still has an action � : TL ⊗ SkTL(Σ) → SkTL(Σ) which is the restriction of � : V ⊗ SkV(Σ)→ SkV(Σ). Definition 3.1. Let W = V ⊗n ∈ TL and α ∈ HomSkTL(Σ)(W � ∅,∅). By Theorem 2.22, α can be represented by a (linear combination of) tangle(s), still denoted by α, with n ordered boundary points near the boundary edge, which is well defined up to isotopy and Kauffman- bracket relations. Graphically, we set StW ( α ∂Σ Σ [0, 1] 1 · · · n ) (vε1 ⊗ · · · ⊗ vεn) := α ε1 ·· · εn , εi ∈ {±}. For brevity we denote this last stated tangle by ε⃗ α and vε⃗ = vε1 ⊗ · · · ⊗ vεn . Note that in this figure there are two implicit sums, α is a linear combination of tangles and an element of V ⊗n is a linear combination of vε⃗’s. They will remain implicit in the following. In the context of stated skein algebras one needs the boundary points of the tangle α to be above the boundary arc though they are on the bottom at its right in the context of morphisms in SkTL(Σ). One wants to simply push the boundary points through the boundary edge and up above it, but without braiding the strands with one another. Therefore we use a global automorphism of Σ× [0, 1]. Consider an isotopy of the identity on Σ× [0, 1] which is trivial far from the corner and in it pushes the bottom boundary through and up above the edge. It results in the global automorphism ψhv of Σ× [0, 1] which maps a tangle “from skein categories” to one “from stated skein algebras” and preserves the order as desired, namely the bottom points on the leftmost will end up above. We will still denote the modified tangle ψhv(α) by α. Now, it only needs states to give an element of S (Σ). We set StW (α) := { V ⊗n → S (Σ), vε⃗ 7→ ε⃗ α, so where ε⃗ α is the tangle ψhv(α) with states ε1, . . . , εn from top to bottom. It is well defined because the tangle representing α is well defined up to isotopy and Kauffman-bracket relations, which are quotiented out in S (Σ). Relating Stated Skein Algebras and Internal Skein Algebras 21 Proposition 3.2. Given a tangle α, the map StW (α) : V ⊗n → S (Σ) is an Oq2(SL2)-comodule morphism. Proof. Note that we still see S (Σ) as a right Oq2(SL2)-comodule even though we draw the edge at the left. Its comodule structure is given by ∆(ε⃗ α) = ∑ η⃗∈{±}n η⃗α ⊗ η⃗βε⃗, and here η⃗βε⃗ is a product ∏ i ηiβεi in S (B). The comodule structure on V ⊗n is given by ∆(vε⃗) = ∑ η⃗ vη⃗ ⊗∏ i xηi,εi , where xη,ε is the vη part of ∆(vε). Namely, x+,+ = a, x+,− = b, x−,+ = c and x−,− = d. Under the isomorphism Oq2(SL2) ≃ S (B), xη,ε 7→ ηβε. Thus ∆(StW (α)(vε⃗)) = ∆(ε⃗ α) = ∑ η⃗ StW (α)(vη⃗)⊗ η⃗βε⃗ = (StW (α)⊗ IdS (B))(∆(vε⃗)). ■ Proposition 3.3. The maps StW : HomSkTL(Σ)(W � ∅,∅) → HomOq2 (SL2)-comod(W,S (Σ)), W ∈ TL, define a natural transformation St: HomSkTL(Σ)(−�∅,∅)⇒ HomE(−,S (Σ)) between functors TLop → Vectk. Proof. For g ∈ HomTL(V ⊗n, V ⊗m), α ∈ HomSkTL(Σ)(V ⊗m�∅,∅) and vε⃗ ∈ V ⊗n , one needs to check that StV ⊗m(α)(g(vε⃗)) = StV ⊗n(α ◦ (g � Id∅))(vε⃗). Morphisms of TL are generated under composition and juxtaposition by identities, caps and cups, so one only needs to prove the result for these last two. We will need some explicit computations of these caps and cups. Recall from Remark 2.21, or [26, Theorem 4.2], that to do so one uses the isomorphism φ :  V → V ∗, v+ 7→ −q 5 2 v∗−, v− 7→ q 1 2 v∗+ of Definition 2.3 and sets ∩ = ◦ (φ ⊗ IdV ) and ∪ = (IdV ⊗ φ−1) ◦ , where and are the usual ev and coev in Vectfink . Let g = = Id⊗kV ⊗ ∩ ⊗ Id⊗n−kV : V ⊗n+2 → V ⊗n, α ∈ HomSkTL(Σ)(V ⊗n � ∅,∅) and v = (vε1 ⊗ · · · ⊗ vεk)⊗ vµ ⊗ vν ⊗ (vεk+1 ⊗ · · · ⊗ vεn) ∈ V ⊗n+2. We want to compare StV ⊗n(α)(g(v)) = StV ⊗n(α)(vε⃗ . ∩ (vµ ⊗ vν)) = α ε1 ·· · εn . ∩ (vµ ⊗ vν) and StV ⊗n+2(α ◦ (g � Id∅))(v) = StV ⊗n+2 ( α ) (v) = α ε1 ·· · µ ν εn = α ε1 ·· · εn .µν C. One simply needs to check that the coefficient µ ν Cfrom the left boundary skein relations of Proposition 2.27 coincides with ∩(vµ⊗vν) and indeed ∩(v+⊗v+) = ev ( −q 5 2 v∗−⊗v+ ) = 0 = + + C, ∩(v− ⊗ v−) = ev ( q 1 2 v∗+ ⊗ v− ) = 0 = − − C, ∩(v+ ⊗ v−) = ev ( −q 5 2 v∗− ⊗ v− ) = −q 5 2 = + − Cand ∩(v− ⊗ v+) = ev ( q 1 2 v∗+ ⊗ v+ ) = q 1 2 = −+ C. Now let g = = Id⊗kV ⊗∪⊗ Id⊗n−kV : V ⊗n → V ⊗n+2, α ∈ HomSkTL(Σ) ( V ⊗n+2 �∅,∅ ) and v = vε1 ⊗ · · · ⊗ vεn ∈ V ⊗n. One can directly compute ∪(1) = ( IdV ⊗ φ−1 ) ◦ coev(1) =( IdV ⊗ φ−1 ) (v− ⊗ v∗− + v+ ⊗ v∗+) = −q− 5 2 v− ⊗ v+ + q− 1 2 v+ ⊗ v−. We want to compare StV ⊗n+2(α)(g(v)) = StV ⊗n+2(α)((vε1 ⊗ · · · vεk)⊗ ( −q− 5 2 v− ⊗ v+ + q− 1 2 v+ ⊗ v− ) 22 B. Häıoun ⊗ (vεk+1 ⊗ · · · vεn)) = −q− 5 2 α ε1 ·· · − + εn + q− 1 2 α ε1 ·· · + − εn and StV ⊗n(α ◦ (g � Id∅))(v) = StV ⊗n ( α ) (v) = α ε1 ·· · εn . They are equal by the last left boundary skein relation of Proposition 2.27. ■ Recall from Proposition 2.42 and the discussion below that the internal endomorphism object of ∅ with respect to the E-action is an object X equipped with an isomorphism HomSkTL(Σ)(−� ∅,∅)⇒̃HomE(−, X) in [TLop,Vectk] Theorem 3.4. The natural transformation St is a natural isomorphism and exhibits S (Σ) as the internal endomorphism object of the empty set. Namely one can take AΣ = S (Σ) as Oq2(SL2)-comodule. Proof. We exhibit an inverse to St. Let W = V ⊗n ∈ TL, which decomposes as a direct sum of simples Wi. Let f : W → S (Σ) be a morphism in E , and denote by fi its restriction on Wi. We want a morphism St−1W (f) ∈ HomSkTL(Σ)(W �∅,∅), which is equivalent to a collection of morphisms St−1W (fi) ∈ HomSkV (Σ)(Wi � ∅,∅). Each fi is determined by its value on a single element wi ∈ Wi (pick a highest weight element for example), it extends to all Wi by applying the Uq2(sl2)-action (recall from Proposition 2.9 that Oq2(SL2)-comodules correspond to Uq2(sl2)-modules). Choose any stated tangle ai representing the element fi(wi) ∈ S (Σ). Denote by αi its underlying tangle, ni = \|∂αi\| its number of boundary points and ε⃗i its states. This representative is well defined up to the boundary skein relations, the usual skein relations and isotopy. The assignment wi 7→ vε⃗i extends to a unique Oq2(SL2)-morphism gi : Wi → V ⊗ni by applying the Uq2(sl2)-action. We then set St−1W (fi) = αi◦(gi�Id∅). Note that here αi denotes the tangle αi seen as a morphism in SkV(Σ), so we actually mean ψ−1hv (αi), the same tangle but with boundary points at the bottom instead of the left. We have to check that this definition does not depend on the representative ai. Usual skein relations and isotopy do not change αi seen as a morphism in the skein category. The boundary skein relations are equivalent to naturality using Proposition 3.3. Namely, another representative a′i has to be of the form α′i = αi◦(g�Id∅), for some g ∈ TL, with states ε⃗ ′ i such that g(ε⃗ ′ i ) = ε⃗i. Therefore the g′i for a ′ i will be such that gi = g ◦g′i. Then we simply check that αi ◦(gi� Id∅) = αi ◦(g� Id∅)◦(g′i� Id∅) = α′i ◦(g′i� Id∅), and St−1W (fi) is well defined. For simplicity we assume that St−1W (fi) and gi are actually defined on all W but are 0 ex- cept on Wi, namely we precompose by the projection W ↠ Wi, and thus we set St−1W (f) =∑ i St −1 W (fi). It is now easy to check that St−1W is the inverse of StW . For vε⃗ ∈ W , suppose vε⃗ ∈ Wi0 ⊆ W lies in a simple, or decompose it in the Wi’s, and write vε⃗ = X · wi0 , X ∈ Uq2(sl2). One has gi0(vε⃗) = X · vε⃗i0 and for j ̸= i0, gj(vε⃗) = 0. Thus: StW ( St−1W (f) ) (vε⃗) = StW (∑ i αi ◦ (gi � Id∅) ) (vε⃗) = ∑ i StW (αi ◦ (gi � Id∅))(vε⃗) Proposition 3.3 = ∑ i StW (αi)(gi(vε⃗)) = StW (αi0)(X · vε⃗i0 ) Relating Stated Skein Algebras and Internal Skein Algebras 23 Proposition 3.2 = X · StW (αi0)(vε⃗i0 ) = X · ai0 = X · fi0(wi0) = fi0(vε⃗) = f(vε⃗). Symmetrically, let α ∈ HomSkTL(Σ)(W �∅,∅) and vε⃗ ∈ W . Set f = StW (α) : V ⊗\|∂α\| → S (Σ), in the definition of St−1W (f) one has αi = α and vε⃗i = wi, so gi is the inclusion Wi ↪→ W . If vε⃗ ∈Wi ⊆W , one has St−1W (StW (α)) = ∑ i α ◦ (gi � Id∅) = α ◦ (IdW � Id∅) = α. ■ Proposition 3.5. The algebra structure inherited from the internal endomorphism object struc- ture on S (Σ) coincides with its usual algebra structure. Namely AΣ = S (Σ) as Oq2(SL2)- comodule algebras. Proof. Recall that the product on S (Σ) is given by stacking the left tangle a above the right one b, and the product on AΣ is defined by evaluation and composition maps on internal Hom objects, in Definition 2.39. Graphically, the stated tangles a and b, which we see as morphisms α and β in SkV(Σ) with prescribed inputs vε⃗ and vη⃗, map to the morphism α ◦ (IdV ⊗\|∂α\| � β) in SkV(Σ) with prescribed input vε⃗ ⊗ vη⃗, which we see as the stated tangle a above b: α ε1 ·· · εn , β η1 ·· · ηm α ε1 ·· · εn β η1 ·· · ηm ↕ ↕ α 1 · · · n , vε⃗, β 1 · · · m , vη⃗ 7−→ α 1 · · · n β 1 · · · m , vε⃗ ⊗ vη⃗. The evaluation map ev∅,∅ : S (Σ) � ∅ → ∅ is the image under St−1 of IdS (Σ). We have not constructed St−1 on all E above, but only on TL, and it extends by cocontinuity in Proposi- tion 2.42. The comodule S (Σ) decomposes as simples as S (Σ) = ⊕ α∈B+ Uq2(sl2) ·α where B+ is the set of o-ordered simple stated tangles with only + states, see [7, Theorem 4.6(b)]. These stated tangles with only + states are simply a way to represent canonically a tangle without state information, and again in the following we will see α as a morphism in HomSkV (Σ) ( V ⊗\|∂α\|,∅ ) , which is actually ψ−1hv (α). We denote by Wα = Uq2(sl2) · α and gα : Wα → V ⊗\|∂α\| the inclusion mapping α to its states v−−−→ +···+. Then: ev∅,∅ = St−1(IdS (Σ)) = ⊕α∈B+ St−1(Wα ↪→ S (Σ)) = ⊕α∈B+α ◦ (gα � Id∅). The composition map c : S (Σ)⊗S (Σ)→ S (Σ) is the image under St of the morphism ev∅,∅ ◦ (IdS (Σ) � ev∅,∅) : (S (Σ)⊗S (Σ))�∅→ S (Σ)�∅→ ∅. This morphism is the double sum: ⊕α∈B+ ⊕β∈B+ (α ◦ (gα � Id∅)) ◦ (IdS (Σ) � (β ◦ (gβ � Id∅))) = ⊕α∈B+ ⊕β∈B+ α ◦ (IdV ⊗\|∂α\| � β) ◦ (gα � gβ � Id∅). The product is obtained by applying St to this morphism. For a, b ∈ S (Σ), write a = X ·α and b = Y · β with X,Y ∈ Uq2(sl2) and α, β ∈ B+. Thus a has states vε⃗ = X · v−−−→ +···+ and b has states vη⃗ = Y · v−−−→ +···+. By naturality: c(a⊗ b) := StS (Σ)⊗S (Σ)(ev∅,∅ ◦ (IdS (Σ) � ev∅,∅))(a⊗ b) 24 B. Häıoun = ⊕α′∈B+ ⊕β′∈B+ St(α′ ◦ (Id V ⊗\|∂α′\| � β′))((gα′ ⊗ gβ′)(a⊗ b)) = St(α ◦ (IdV ⊗\|∂α\| � β))((gα ⊗ gβ)(a⊗ b)) = St(α ◦ (IdV ⊗\|∂α\| � β))(vε⃗ ⊗ vη⃗) which is precisely the usual product of a and b in S (Σ). ■ 4 Multi-edges We define internal skein algebras for surfaces with more than one boundary, and possibly left or right boundary edges. We show they are isomorphic to stated skein algebras when V = Oq2(SL2)-comodfin, and re-prove their excision properties using excision properties of skein categories. 4.1 Right internal skein algebras In order to extend the definition of internal skein algebras to surfaces with multiple boundary edges, we would need a notion of left and right action to be able to glue surfaces together, such that internal skein algebras satisfy excision properties, just like stated skein algebras. One subtlety though is that one is only allowed to talk about right Oq2(SL2)-comodules in the context of internal skein algebras, to stay in the category E (as opposed to the stated skein algebra context). Definition 4.1 ([6, Section 3.2]). Let Σ be a surface with a chosen boundary interval, which we see at the right of the surface. One can make a construction similar to Definition 2.44 to have a right action functor � : SkV(Σ) ⊗ SkV ( R2 ) → SkV(Σ). It differs the one � of Definition 2.44 only by rotating the disk by 180 degrees. The right moduli algebra AR Σ of [2, Section 5.2], or right internal skein algebra of [11], is the internal endomorphism algebra of the empty set in SkV(Σ) with respect to this SkV ( R2 )⊗-op - module structure. ⊔ Definition 4.2. Denote by rot the diffeomorphism of the disk given by 180◦ rotation. By Remark 2.16 it induces an automorphism (−)ht := SkV(rot) : SkV ( R2 ) → SkV ( R2 ) which squares to the identity. For X ∈ SkV ( R2 ) , we call Xht := SkV(rot)(X) the half-twisted X. One easily checks that (−)ht is anti-monoidal, namely (X ⊗ Y )ht = Y ht ⊗ Xht, because rot reverses left- right order. The diffeomorphism rot is isotopic to the identity by rotating from 0 to 180◦. This isotopy induces a natural isomorphism ht: IdSkV (R2)⇒̃(−)ht called the half twist, which squares to the twist (the 360◦ rotation). Remember from Remark 2.16 that ht is given on n blackboard framed points on the real axis by n parallelly half-twisted vertical strands, namely drawn on the half twisted ribbon . Naturality, namely htW ◦ f ◦ht−1V = fht, expresses the fact that one can untwist a top half twist Relating Stated Skein Algebras and Internal Skein Algebras 25 and a bottom anti-half-twist by half twisting the middle. For f = htV one gets htV ht = hthtV . Note too that htV⊗W = (htW ⊗ htV ) ◦ cV,W by = . Now, one simply has � = � ◦ fl ◦((−)ht ⊗ IdSkV (Σ)) and � ◦ fl = � ◦ ((−)ht ⊗ IdSkV (Σ)), where fl is the flip of tensors. And indeed a left action turns into a right action under an anti-monoidal functor. Moreover, the natural isomorphism ht: IdSkV (R2)⇒̃(−)ht gives a natural isomorphism ht�− := � ◦ (ht⊗ IdSkV (Σ)) : � ⇒̃� ◦ fl. Remark 4.3. One can give an explicit relation between left and right internal skein algebras. Internal skein algebras are only defined up to isomorphism, so this description is just one choice. Actually, we will give another one below. Consider the internal skein algebra AΣ defined as in Section 2.4 by seeing the red arc at the left, with a natural isomorphism σ : HomSkV (Σ)(− � ∅,∅)⇒̃HomE(−, AΣ). Then one has natural isomorphisms HomSkV (Σ)(∅� V,∅) = HomSkV (Σ) ( V ht �∅,∅ ) σ V ht→ HomE ( V ht, AΣ ) (−)ht→ HomE ( V,Aht Σ ) . Namely, AR Σ ≃ Aht Σ as object of SKV ( R2 ) , but with natural isomorphism (σ(−)ht) ht. Remark 4.4. Note that the half twist is defined on SkV ( R2 ) and not on V, which is its full subcategory of objects with only one coloured point, with blackboard framing. The half twist will map such an object to a point with anti-blackboard framing, so it does not stabilise V. The equivalence of categories SkV ( R2 ) ≃ V preserves properties of the half twist only up to natural isomorphism, and depends on the choice of a quasi-inverse of the inclusion. The one described in Remark 2.12 will map the half twist on SkV ( R2 ) to the identity on V, but if we had chosen to restore the framing counter-clockwise it would map it to the full twist. In V = Oq2(SL2)-comodfin an actual half twist exists, see [24], and we will study it below. However, in general we will prefer a construction that uses the half twist only on the disk inserted in SkV(Σ), where it is well defined, and not on V. In particular, the above description AR Σ ≃ Aht Σ holds in SKV ( R2 ) but has an unclear meaning in E (it depends on the choice of some quasi-inverses). Definition 4.5. There is another explicit relation between left and right internal skein algebras using ht�− to relate the right action to the left one. Set σR = σ◦(ht�∅) : HomSkV (Σ)(∅�−,∅) ⇒̃HomE(−, AΣ). What we mean is that for V ∈ V and α ∈ HomSkV (Σ)(∅� V,∅), we have α ◦ (htV � Id∅) ∈ HomSkV (Σ)(V �∅,∅), and σRV (α) := σV (α ◦ (htV � Id∅)). Proposition 4.6. The natural isomorphism σR exhibits AΣ as the right internal skein algebra. The algebra structure mR : AΣ ⊗ AΣ → AΣ inherited from this right internal endomorphism object structure differs from the left one m : AΣ ⊗AΣ → AΣ by a braiding: mR = m ◦ cAΣ⊗AΣ . Namely, the right internal skein algebras introduced in [11] and [2] are the braided opposites of the left ones introduced in Section 2.4. Proof. The σRV ’s form a natural isomorphism: let f ∈ HomV(V,W ) and α ∈ HomSkV (Σ)(∅ � W,∅), one checks: σRV (α ◦ (Id∅ � f)) = σRV ( α ◦ ( fht � Id∅ )) := σV ( α ◦ ( fht � Id∅ ) ◦ (htV � Id∅) ) = σV (α ◦ (htW � Id∅) ◦ (f � Id∅)) = σRW (α) ◦ f by using naturality of σ and of ht � −. For f ∈ HomE(V,AΣ) one has ( σRV )−1 (f) = σ−1V (f) ◦ (ht−1V � Id∅). Therefore ( AΣ, σ R ) is the internal endomorphism object of the empty set in 26 B. Häıoun SKV(Σ) with respect to the right SKV ( R2 ) -action by Proposition 2.42. We now study its algebra structure. The evaluation map is evR : = ( σRAΣ )−1 (IdAΣ ) = σ−1AΣ (IdAΣ ) ◦ ( ht−1AΣ � Id∅ ) = ev ◦ ( ht−1AΣ � Id∅ ) ∈ HomSKV (Σ)(∅�AΣ,∅). The product, or composition map, is: mR : = σRAΣ⊗AΣ ( evR ◦ ( evR � IdAΣ )) = σAΣ⊗AΣ ( ev ◦ ( ht−1AΣ � Id∅ ) ◦ IdAht Σ � ( ev ◦ ( ht−1AΣ � Id∅ )) ◦ (htAΣ⊗AΣ � Id∅) ) = σAΣ⊗AΣ ( ev ◦ (IdAΣ � ev) ◦ ( ht−1AΣ ⊗ ht−1AΣ � Id∅ ) ◦ (htAΣ⊗AΣ � Id∅) ) = σAΣ⊗AΣ (ev ◦ (IdAΣ � ev) ◦ (cAΣ⊗AΣ � Id∅)) = m ◦ cAΣ⊗AΣ . The units σR1V (Id∅) := σ1V (Id∅ ◦ (ht1V � Id∅)) = σ1V (Id∅) coincide. ■ Remark 4.7. In SKV ( R2 ) , one has σRV (α) := σV (α ◦ (htV � Id∅)) = σV ht(α) ◦ htV . If one post-composes with ht−1 Aht Σ this is exactly (σV ht)ht. Hence the two descriptions of right internal skein algebras we gave, ( Aht Σ , σ(−)ht ht ) in Remark 4.3 and (AΣ, σ ◦ (ht�−)) in Proposition 4.6, are isomorphic (as they should) by ht−1 Aht Σ : AΣ → Aht Σ . The product on Aht Σ is given by ht−1 Aht Σ ◦mR ◦ htAht Σ ⊗ htAht Σ = ht−1 Aht Σ ◦m ◦ cAΣ⊗AΣ ◦ htAht Σ ⊗ htAht Σ = ht−1 Aht Σ ◦m ◦ htAΣ⊗AΣ = mht. 4.2 Multi-edges internal skein algebras We extend the definition of internal skein algebras to the multi-edge context, and define them as internal endomorphism algebras of the empty set in the skein category with multiple boundary actions, as expected. We check that they still describe skein categories well-enough. Definition 4.8. LetS be a marked surface with n boundary edges labelled either as left (numbe- red 1 to k) or as right (numbered k+1 to n) edges. Each left (resp. right) boundary edge induces a left SkV ( R2 ) -action (resp. left SkV ( R2 )⊗-op -action) on SkV(S), which all commute, so one has a left SkV ( R2 )⊗k ⊗ (SkV(R2 )⊗-op)⊗n−k -action ▷ on SkV(S). We denote its components by �i or �i ◦ fl: SkV ( R2 ) ⊗ SkV(S)→ SkV(S), 1 ≤ i ≤ n, though we forget the indices when they are understood. When there are missing components they will be implicitly filled by 1V . We may also write (V1, . . . , Vk)�∅� (Vk+1, . . . , Vn) instead of (V1, . . . , Vn) ▷∅. The internal skein algebra AS is the internal endomorphism object of the empty set in SkV(S) with respect to the SkV ( R2 )⊗k ⊗ (SkV ( R2 )⊗-op )⊗n−k-action. It is an algebra object in E⊠n ≃ Free ( V⊗n ) ≃ Free ( SkV ( R2 )⊗n) , where E⊠n has opposite tensor products on the last n − k components. We denote this monoidal structure by ⊗. We denote by ⊗i the tensor product on coordinate i, and adopt the same convention as with ▷ filling with missing 1V ’s and eventually writing (V1, . . . , Vk) ⊗W ⊗ (Vk+1, . . . , Vn) instead of (V1, . . . , Vn) ⊗W . Explicitly, AS comes equipped with natural isomorphisms σV⃗ : HomSkV (S)(V⃗ ▷∅,∅)⇒̃HomE⊠n(V⃗ , AS) for V⃗ = (V1, . . . , Vn) ∈ V⊗n, between functors (Vop)⊗n → Vectk. Remark 4.9. Objects and morphisms of tensor product of categories (e.g., V⊗n) are sometimes denoted by tensor product of objects and morphisms (e.g., V1 ⊗ · · · ⊗ Vn and f1 ⊗ · · · ⊗ fn). To avoid confusion with the monoidal structures on the categories (e.g., ⊗ on V), we prefer to use commas (e.g., (V1, . . . , Vn) and (f1, . . . , fn)). Relating Stated Skein Algebras and Internal Skein Algebras 27 Remark 4.10. A legitimate worry about this extended definition of internal skein algebras is that when one has multiple boundary actions on a same connected component one cannot keep track of where did an object come from. Namely for V ∈ V and c1, c2 two boundary edges on a same connected component of S, one has an isomorphism V �1 ∅→ V �2 ∅ which sounds surprising because (V, 1V) and (1V , V ) are hardly isomorphic in V⊗2. The internal skein algebra actually keeps track of such identifications, and one has an isomorphism (V, 1V)⊗AS → (1V , V )⊗AS. Note that the above definition makes sense for n = 0, where we want endomorphisms of the empty set in SkV(S) to be described by morphisms k → AS in E⊠0 = Vectk. For V = Oq2(SL2)-comodfin one gets AS = S̊ (S) is the usual skein algebra. It is no longer true, however, that all objects of SkV(S) are described as modules over AS, because the (trivial) action of E⊠0 on ∅ is no longer dominant. Definition 4.11. Let S be a marked surface and V a ribbon category. The reduced skein category SkredV (S) is the full subcategory of SkV(S) spanned by objects of the form V⃗ ▷ ∅, namely in the image of the action of SkV ( R2 )⊗n on the empty set. It is equivalent to SkV(S) if S has at least one boundary edge per connected component. Remark 4.12. One can still apply Remark 2.46, slightly modified because for right edges the left adjoint of − � V is given by acting by the left dual − � ∗V . For V⃗ = (V1, . . . , Vn) ∈ V⊗n write V⃗ ∗ = (Vi ∗ or ∗Vi)1≤i≤n with right duals for left edges and left duals for right edges. Then, as the notation suggests, V⃗ ∗ is the left dual of V⃗ in V⊗n for the monoidal structure ⊗, and V⃗ ▷− has left adjoint V⃗ ∗ ▷−. For W⃗ ∈ V⊗n, one has natural isomorphisms: HomSkV (S) ( W⃗ ▷∅, V⃗ ▷∅ ) ≃ HomSkV (S) ( V⃗ ∗ ▷ (W⃗ ▷∅),∅ ) σ V⃗ ∗⊗W⃗→ HomE⊠n ( V⃗ ∗⊗ W⃗ ,AS ) ≃ HomE⊠n ( W⃗ , V⃗ ⊗AS ) . Theorem 4.13. The free cocompletion of the reduced skein category SKred V (S) is equivalent to the category of right AS-modules in E⊠n, with monoidal structure ⊗, by SKred V (S) →̃ modE −AS, M 7→ Hom(∅,M). For M of the form V⃗ ▷∅, one has Hom(∅, V⃗ ▷∅) ≃ V⃗ ⊗AS. Proof. We follow the proof of Theorem 2.47, namely we use [2, Theorem 4.6] on ∅ ∈ SKred V (S). It is projective by the same arguments and −�∅ : E⊠n → SKred V (S) is dominant by construction so actR∅ is faithful and ∅ is a generator. For the last statement, one has Hom(∅, V⃗ ▷∅) ≃ V⃗ ⊗AS by Remark 4.12. ■ 4.3 Relation for multiple left edges Let V = Oq2(SL2)-comodfin, E = Oq2(SL2)-comod ≃ Free(V) and S be a marked surface with all boundary edges labelled left. We show that AS ≃ S (S) as Oq2(SL2) ⊗n-comodule-algebras. Proposition 4.14. There is an equivalence of categories E⊠n ≃ Oq2(SL2) ⊗n-comod. Proof. The category Oq2(SL2) ⊗n-comod is semi-simple with simples tensor products of simples Oq2(SL2)-comodules. It implies that the cocontinuous extension of ⊗n : V⊗n → Oq2(SL2) ⊗n- comod to E⊠n → Oq2(SL2) ⊗n-comod is an equivalence. ■ In particular Free ( TL⊗n ) ≃ Free(TL)⊠n ≃ E⊠n ≃ Oq2(SL2) ⊗n-comod. 28 B. Häıoun Theorem 4.15. Let S be a marked surface with all boundary edges labelled left, then AS ≃ S (S) as Oq2(SL2) ⊗n-comodule-algebras. Proof. We give a natural isomorphism St exhibiting S (S) as the internal endomorphism object of ∅ ∈ SKV(S) with respect to the Oq2(SL2) ⊗n-comod-module structure. For X ∈ Oq2(SL2) ⊗n-comod, we want StX : HomSKV (S)(X � ∅,∅) → HomOq2 (SL2)⊗n-comod(X,S (S)). Let X ∈ TL⊗n and α ∈ HomSkV (S)(X �∅,∅) represented by a tangle, we set StX(α) : { X → S (S), v1ε⃗1 ⊗ · · · ⊗ v n ε⃗n 7→ ε⃗1···ε⃗nα, where ε⃗1···ε⃗nα is the tangle α with endpoints pushed over the boundary edges and states the εij over the i-th edge and in j-th position from top to bottom, as in Section 3 but with more than one edge. It is an Oq2(SL2) ⊗n-comodule morphism because it is an Oq2(SL2)-comodule morphism on each coordinate by the same calculations as in Proposition 3.2. It is natural in TL⊗n because it is a natural in each coordinate by the same calculations as in Proposition 3.3. It is an isomorphism by the same arguments as in Theorem 3.4. Namely, let W = V1 ⊗ · · · ⊗ Vn ∈ TL⊗n and f : W → S (S), split W = ⊕iWi and f = ⊕fi with Wi = Vi,1 ⊗ · · · ⊗ Vi,n simple and choose wi ∈Wi∖{0}. Choose ai representing fi(wi) and denote by αi its underlying tangle and ε⃗i,1, . . . , ε⃗i,n its states. Include Wi gi ↪→ V ⊗ni,1 ⊗ · · · ⊗ V ⊗ni,n by mapping wi to vε⃗i,1 ⊗ · · ·⊗ vε⃗i,n , and set St−1Wi (fi) = αi ◦ (gi ▷ Id∅) ∈ HomSkV (S)(Wi ▷∅,∅). Then the inverse of St is given by St−1W (f) = ⊕i St −1 Wi (fi) ∈ HomSkV (S)(W ▷∅,∅) and does not depend on the choice of representative. As in Proposition 3.5, because every boundary edge of S is labelled left, the product inherited from the internal endomorphism object structure is still given by α with prescribed inputs vε⃗i,1 ⊗ · · · ⊗ vε⃗i,n times β with prescribed inputs vη⃗i,1 ⊗ · · · ⊗ vη⃗i,n equals α ◦ (Id ▷ β) with prescribed inputs (vε⃗i,1 ⊗ vη⃗i,1)⊗ · · · ⊗ (vε⃗i,n ⊗ vη⃗i,n) which is the usual product on S (S). ■ 4.4 The half twist on Oq2(SL2)-comod In the last subsection, we only allowed left SkV ( R2 ) -actions. We study here how to change from left to right actions using the half twist in the case V = Oq2(SL2)-comodfin. Remark 4.16. As we saw, a half twist on V is usually not necessary to the general study of internal skein algebras, but it is needed to relate them to stated skein algebras when there are right edges, and to mirror their excision properties. When one sees a boundary edge at the right instead of the left, it has very different effects on both sides. For stated skein algebras, it does not change the vector space, nor the algebra structure, but switches the right comodule structure to a left one using rot∗. For internal skein algebras, it does not change the vector space, one keeps right comodules (AR Σ is still an object of E) though slightly changed: it is half-twisted, and the algebra structure is opposed. To make both sides agree, one needs to switch the comodule structure of the internal skein algebra while taking the opposite of its algebra structure. This is done very naturally by using S. Using the other known comparisons, one gets that the half twist on Oq2(SL2)-comod should be the difference between switching the comodule structure using rot∗ and switching it using S. This is Proposition 4.26, but we give a more complete and algebraic definition below. Definition 4.17 ([24, Section 4.1], for categories of comodules). A half-coribbon Hopf algebra is a coribbon Hopf algebra H equipped with a half-coribbon functional, i.e., a map t : H → k such that: Relating Stated Skein Algebras and Internal Skein Algebras 29 (1) t is invertible by convolution: ∃ t−1 : H → k such that t(a(1))t −1(a(2)) = t−1(a(1))t(a(2)) = ε(a), (2) t squares to the twist: t(a(1))t(a(2)) = θ(a), (3) compatibility with product: t(a.b) = t(b(1))t(a(1))R(a(2) ⊗ b(2)). Definition 4.18. The half-coribbon functional induces a half twist ht on the category H-comod by htV : V ∆→ V ⊗H Id⊗t−→ V . It is an isomorphism of vector space with ht−1V : V ∆→ V ⊗H Id⊗t−1 −→ V . The half-coribbon functional is not supposed to be central though, and this means that htV is not an H-comodule morphism. There is a unique comodule structure on (the target) V that makes it a comodule morphism, namely ∆ht := (htV ⊗ IdH) ◦ ∆ ◦ ht−1V . We denote by V ht the vector space V equipped with the coaction ∆ht. Now, htV : V → V ht is an isomorphism of H-comodules. One has a functor (−)ht : H-comod → H-comod which sends an object V to the half- twisted V ht and a morphism f : V → W to fht = htW ◦ f ◦ ht−1V : V ht → W ht. It is defined so that ht : Id⇒ (−)ht is a natural isomorphism. As maps of vector spaces, one simply has fht = (IdW ⊗ t) ◦∆W ◦ f ◦ ( IdV ⊗ t−1 ) ◦∆V = (IdW ⊗ t) ◦∆W ◦ ( IdW ⊗ t−1 ) ◦ (f ⊗ IdH) ◦∆V = (IdW ⊗ t) ◦∆W ◦ ( IdW ⊗ t−1 ) ◦∆W ◦ f = htW ◦ ht−1W ◦ f = f. In particular htV ht = hthtV = htV . The square of t is θ so htV ht ◦ htV = θV , and θ is central so (V ht)ht = V , and (−)ht ◦ (−)ht = Id. Regarding the monoidal structure, let V and W be two H-comodules. Remember that the braiding is defined as cV,W : V ⊗W R24◦(∆V ⊗∆W )−→ V ⊗W fl→W ⊗ V. The third condition gives that htV⊗W = htV ⊗htW ◦ (fl ◦cV,W ) so fl ◦htV⊗W = htW ⊗htV ◦ cV,W . In particular, fl : (V ⊗W )ht →W ht ⊗ V ht is an H-comodule isomorphism. Definition 4.19. For V a ribbon category, let ht-RibbonV be the full subcategory of RibbonV ( R2 ) spanned by objects of the form [n] ⊆ R2 but now allowing either blackboard or anti-blackboard framing for every point. This subcategory is still ribbon and is stable by the functor (−)ht, and equipped with ht : Id⇒ (−)ht. It also contains the category RibbonV of blackboard framed points. Theorem 4.20 ([24, Theorem 4.11]). Let H be a half-coribbon Hopf algebra and V a finite- dimensional comodule. There is a unique monoidal functor ht-RTV : ht-RibbonH-comodfin → H-comodfin extending RTV and commuting with both (−)ht and ht, so preserving the “half- ribbon structure”. Remark 4.21. This half-twisted Reshetikhin–Turaev functor also gives an equivalence of cat- egories SkV ( R2 ) ≃ V but this time with much nicer properties regarding the half twist. In the usual Reshetikhin–Turaev functor one only prescribes where to send points with blackboard framing and well-placed on the real line. For framed points not of this form, one has to choose an isomorphism with one of these, like in Remark 2.12, but these choices are quite arbitrary. Then the half twist sends blackboard framed points to anti-blackboard framed points which are re-identified with blackboard framed points via these arbitrary isomorphisms. With the half- twisted Reshetikhin–Turaev functor one also prescribes where to send anti-blackboard framed points, so one controls closely what happens with the half twist, namely the half twist on SkV(R2) is mapped to the half twist on V. 30 B. Häıoun Note however that unlike on SkV ( R2 ) , the half twist on V is not strictly anti-monoidal (indeed it is the identity on underlying vector spaces) but (X ⊗Y )ht ≃ Y ht⊗Xht is simply given by the flip of tensors. A bit like the R-matrix, the half twist gives the difference between the monoidal structure on H-comod and the symmetric one on Vectk. Formally, this error lies in the fact that the inclusion of V in SkV ( R2 ) is only monoidal up to natural isomorphism, and this isomorphism, given in Remark 2.14, maps by the half twist to the flip of tensors. In the case of Oq2(SL2), we can define a half-coribbon functional on the generators by t ( a b c d ) = ( 0 −q 5 2 q 1 2 0 ) . This tells in particular how the half twist acts on the standard corep- resentation V , namely htV (v+) = q 1 2 v− = C(−)−1v− and htV (v−) = −q 5 2 v+ = C(+)−1v+. For states η ∈ {±}, we will write htV (η) = −η.C(−η)−1. Note that [24] introduce another half-coribbon element corresponding to the matrix ( 0 q 3 2 −q 3 2 0 ) , but our choice is imposed by conventions from stated skein algebras. To define it on all Oq2(SL2) we prefer a geometric description on S (B). We would like to give the same definition as for the coribbon functional θ with a half twist instead of a full twist, but in the definition of stated skein algebras one only allows upward-framed boundary points, which would map to downward-framed points after the half twist. Still, we know how to do the “global” half twist on many strands (without twisting the framing), and we only need to add a “local” half twist on each strand, which are implicitly coloured by V , on which we know how the half twist acts. Proposition 4.22. The coribbon Hopf algebra S (B) is half-coribbon with half-coribbon func- tional t  • <• > εn ηm ... ... ε1 η1 α  := ε  • <• > εn htV (ηm) ... ... ε1 htV (η1) α  = ε  • > • < η1 −ε1.C(ε1)−1 ... ... ηm −εn.C(εn)−1 α . By Remark 2.35, htS (B)(α) = (Id⊗ t) ◦∆(α) is the stated tangle represented in the middle, and by a left version of Remark 2.35, (t⊗ Id) ◦∆(α) is the stated tangle represented in the right. Proof. These two formulations prove that t is well defined on S (B) as it respects the boundary relations on the left edge by the first and on the right one by the second. So we begin by proving that these two formulations actually coincide. Let β ∈ S (B), then ε  • <• > εn −ηm.C(−ηm)−1 ... ... ε1 −η1.C(−η1)−1 β  ε=ε◦S = ε  • > • < −εn.C(εn)−1 ηm.C(−ηm)−1.C(−ηm) ... ... −ε1.C(ε1)−1 η1.C(−η1)−1.C(−η1) β  = ε  • < • >η1 −ε1.C(ε1)−1 ... ... ηm −εn.C(εn)−1 β  ε=ε◦rot∗= ε  • > • < η1 −ε1.C(ε1)−1 ... ... ηm −εn.C(εn)−1 β , where the second equality is only a change of picture representation, not of stated tangles, coming from switching the orientation of the edges, see [3, Section 3.5]. The convolution inverse t−1 of t is obtained the same way as the middle term but with the inverse half twist and ht−1V on states. Indeed by Remark 2.35, ( Id ⊗ t−1 ) ◦ ∆(α) is α with an inverse half twist at the right and ht−1V on right states. Thus ( t ⊗ t−1 ) ◦ ∆ is the counit of α with an inverse half twist and a half twist at the right, and htV ◦ ht−1V on right states, namely the counit of α. Similarly, ( t−1 ⊗ t ) ◦∆ = ε. Relating Stated Skein Algebras and Internal Skein Algebras 31 One directly checks that htV ◦ htV = θV = −q3IdV on the standard corepresentation. Then (Id⊗ t)◦∆(α) is α with a half twist at the right and htV on right states, and (t⊗ t)◦∆(α) is the counit of α with a full twist (without framing twist) at the right and θV on right states. This is exactly the full twist by separating the unframed full twist and the full twists on framings: θ(α) = ε  • <• > α  ε=(ε⊗ε)◦∆ = ε  • <• > ν1 ... νm α  .ε  • <• > νm ηm ... ... ν1 η1  = ε  • <• > −q3η1 ... −q3ηm α . Finally, t(α.β) = ε  • > • < −←−ε .C(ε⃗)−1 −←−η .C(η⃗)−1 β α  ε=(ε⊗ε)◦∆ = ε  • <• > −←−ε .C(ε⃗)−1 ν⃗ µ⃗−←−η .C(η⃗)−1 .ε • > • < ν⃗ µ⃗ β α  ε◦m=ε⊗ε = t(β(1)).t(α(1)).R(α(2) ⊗ β(2)). ■ Definition 4.23. LetS be a marked surface and e a boundary edge, with orientation induced by the one of S. The inversion along the edge e is the morphism of k-vector-spaces inve : S (S)→ S (S) given on a stated tangle α by ordering the heights according to the orientation of e, then switching height order vertically, then taking opposite states and some coefficients, namely: inve  • <• ηm ... η1 α  := • >• −ηm.C(ηm) ... −η1.C(η1) α = • <• ht−1 V (η1) ... ht−1 V (ηm) α . It is well-defined by [7, Proposition 2.7]. Note that inve is neither an algebra morphism nor a comodule morphism. Proposition 4.24. Let er be the right edge of the bigon and α ∈ S (B), then t(α) = ε◦inv−1er (α). In particular, by Remark 2.35, for S a marked surface with a right edge e, the half twist acts on S (S) as inv−1e . Proof. Indeed, t ◦ inver(α) = t  • <• > ht−1 V (η1) ... ht−1 V (ηm) α  = ε  • <• > htV ◦ ht−1 V (η1) ... htV ◦ ht−1 V (ηm) α  = ε(α). ■ Remark 4.25. In [7, Section 3.4] the counit is defined as ε = i∗ ◦ inver , where i is the inclusion of the bigon in the monogon. Surprisingly enough, the half coribbon functional is then t = i∗ and is simpler to write. This suggests that there is a half twist built in the construction of stated skein algebras. We claim this comes from the passage from right to left comodule structure. If A is a right comodule over a Hopf algebra H, it is naturally a left comodule with ∆L = fl ◦(IdA ⊗ S) ◦∆, and we will consider these two as the “same” comodule. Indeed, one has an isomorphism of categories (−)L : H-comod → comod-H which is the identity on vector spaces, 32 B. Häıoun switches the action as above, and is the identity on morphisms. However, when passing from right to left edges – and comodule structures – on stated skein algebras, one uses another way to see a right comodule A as a left, namely with ∆l = fl ◦(IdA ⊗ rot∗) ◦ ∆. Again one has (−)l : H-comod→ comod-H with the identity on morphisms. So we have two functors (−)L and (−)l and we claim that the difference between them is precisely a half twist: Proposition 4.26. One has (−)l = (−)L ◦ (−)ht and (−)L = (−)l ◦ (−)ht. Equivalently the map htlA : Al → AL is an isomorphism of left Oq2(SL2)-comodules. Proof. All these functors are the identity on morphisms and only change the comodule struc- ture. The map htlA : Al → ( Aht )l is just htA as a map of vector spaces and is an isomor- phism of vector spaces. The comodule structure on ( Aht )l is the unique so that htlA is a co- module morphism. We show that it is a comodule morphism Al → AL, and hence that AL = ( Aht )l . Let a ∈ A, one compares ∆L ◦ htA(a) = ∆L(a(1) ⊗ t(a(2))) = Sa(2).t(a(3)) ⊗ a(1) and ( Id⊗ htA ) ◦∆l = (Id⊗ htA)(rot∗(a(2))⊗ a(1)) = rot∗(a(3)).t(a(2))⊗ a(1). We show directly that for β ∈ S (B), Sβ(1).t(β(2)) = rot∗(β(2)).t(β(1)): S ◦ (Id⊗ t) ◦∆(β) = S  • <• > εn −ηm.C(−ηm)−1 ... ... ε1 −η1.C(−η1)−1 β  =  • > • < −εn.C(εn)−1 ηm.C(−ηm)−1.C(−ηm) ... ... −ε1.C(ε1)−1 η1.C(−η1)−1.C(−η1) β  =  • < • >η1 −ε1.C(ε1)−1 ... ... ηm −εn.C(εn)−1 β  rot2∗=Id = rot∗  • > • < η1 −ε1.C(ε1)−1 ... ... ηm −εn.C(εn)−1 β  = rot∗ ◦(t⊗ Id) ◦∆(β). ■ Proposition 4.27. Given two right Oq2(SL2)-comodules A and B one has (A ⊗ B)inv = HH0 ( AL ⊗B ) = HH0 (( Aht )l ⊗B). Proof. The first equality is true in any Hopf algebra by a direct computation, and the second is just the above proposition. ■ 4.5 The general relation We can now express the full correspondence between stated skein algebras and internal skein algebras, with both right and left boundary edges. On a single edge, by Remark 4.3 one gets AR Σ = Aht Σ = S (Σ)ht equipped with the natural isomorphism (σ(−)ht) ht, using the half twist in Oq2(SL2)-comod. More precisely, for α ∈ HomSkV (S)(∅� V,∅) one has σV ht(α)ht = ht−1 Aht Σ ◦ σV ht(α) ◦ htV = ht−1 Aht Σ ◦ σV (α ◦ (htV �∅)) = ht−1 Aht Σ ◦ σRV (α). As in Remark 4.7 its algebra structure is ht−1 Aht Σ ◦mR ◦ htAht Σ ⊗ htAht Σ = ht−1 Aht Σ ◦m ◦ cAΣ⊗AΣ ◦ htAht Σ ⊗ htAht Σ = ht−1 Aht Σ ◦m ◦ fl ◦htAΣ⊗AΣ = mht ◦ fl, so mop as maps of vector spaces. Note that because S is an anti-algebra morphism, the functor (−)L is (almost strictly) anti- monoidal (like the half twist, it is the identity on vector spaces but is anti-monoidal on the Relating Stated Skein Algebras and Internal Skein Algebras 33 comodule structure) namely fl: (V ⊗W )L →WL ⊗ V L is an isomorphism of left H-comodules. Thus a right H-comodule algebra A induces a left H-comodule algebra AL with AL ⊗ AL fl≃ (A⊗ A)L m→ AL, namely with product mop. When one has multi-edges one can switch the i-th right Oq2(SL2)-comodule structure to a left using either S or rot∗ and we denote the associated functors by (−)Li and (−)li , one can take opposite product on the i-th coordinate which we denote by mopi , and there are half twists on each coordinates, which we denote by hti. Theorem 4.28. Let S be a marked surface with n boundary edges labelled either as left (num- bered 1 to k) or as right (numbered k+1 to n) edges. There is an isomorphism of ( Oq2(SL2) ⊗k, Oq2(SL2) ⊗n−k)-bicomodules algebras A Lk+1,...,Ln S ≃ S (S). Proof. To avoid confusion we denote the stated skein algebra of the marked surface S by S R(S) when it is seen as a right Oq2(SL2) ⊗n-comodule and by S (S) = S R(S)lk+1,...,ln when it is seen as an ( Oq2(SL2) ⊗k,Oq2(SL2) ⊗n−k)-bicomodule. We denote by m its product, which is the same in both cases. By Theorem 4.15, S R(S) is the internal skein algebra of S with every edge labelled as left, and then by Remark 4.3 on coordinates k + 1, . . . , n one may take AS := S R(S)htk+1,...,htn as algebra object in E⊠n with skew monoidal structure ⊗. The algebra structure on S R(S)htk+1,...,htn is mopk+1,...,opn . Thus by Proposition 4.26, A Lk+1,...,Ln S := ( S R(S)htk+1,...,htn )Lk+1,...,Ln = S R(S)lk+1,...,ln = S (S) as ( Oq2(SL2) ⊗k,Oq2(SL2) ⊗n−k)-bicomodules, and the algebra structure on( S R(S)htk+1,...,htn )Lk+1,...,Ln is (mopk+1,...,opn)opk+1,...,opn = m. ■ Remark 4.29. A nice miracle with stated skein algebras is that the quantum group Oq2(SL2), which is used to define the tangle invariants used to define stated skein algebras, is re-obtained as the stated skein algebra of the bigon. One can see why this should be true in internal skein algebras. By Definition 4.8, the internal skein algebra of the bigon is an object AB ∈ Oq2(SL2) ⊗2-comod together with a natural isomorphism HomSK(R2)((X,Y )�∅,∅) = HomOq2 (SL2)-comod(X ⊗ Y, k) →̃HomOq2 (SL2)⊗2-comod(X ⊗ Y,AB) for X,Y ∈ Oq2(SL2)-comod. We set AB = Oq2(SL2) with usual first right comodule structure ∆1 = ∆ and with second comodule strucure its left one switched using L−12 namely ∆2 = fl ◦ ( S−1 ⊗ Id ) ◦∆. The demanded isomorphism is given by f 7→ f̃ where f̃(x⊗ y) = x(2).f(x(1) ⊗ y) = S(y(2)).f(x⊗ y(1)). Its inverse is given by f̃ 7→ ε ◦ f̃ . Remark 4.30. Despite this theorem, it is still annoying that in the simplest case one wants to see the boundary at the right for stated skein algebras and at the left for internal skein algebras. This should be solvable by considering the category of left (instead of right) Oq2(SL2)-comodules as coefficients, so it is a minor issue. 4.6 Excision properties of multi-edges internal skein algebras Let S1 ←↩ C ↪→ S2 be a right and a left thick embeddings in two marked surfaces and S their collar gluing. Namely, C embeds as a sequence of k right boundary edges c⃗1 of ∂S1 and as k left boundary edges c⃗2 of ∂S2, and S is the gluing S1 ∪c⃗1=c⃗2 S2. We show how to compute AS 34 B. Häıoun from AS1 and AS2 . The general idea goes as follows. In the case where S1 and S2 both have a single boundary edge, so k = 1, one wants to describe endomorphisms α of the empty set in SkV(S). By Corollary 2.19 they are described by a morphism α1 : ∅ → ∅ � V in SkV(S1) and a morphism α2 : V � ∅ → ∅ in SkV(S2), linked by an isomorphism ιV : ∅ � V → V � ∅, which is just a slanted skein crossing over C in SkV(S), see the idea of proof of Theorem 2.18. One can reconstruct α as α = (Id∅, α2) ◦ ιV ◦ (α1, Id∅). The morphisms α1,2 are well defined up to balancing, namely naturality of ι. Now, by definition of AS1 and AS2 , they are described by some f1 ∈ HomE(1V , AS1 ⊗ V ) and f2 ∈ HomE(V,AS2). Composing them mimicking the reconstruction of α gives a morphism f = (IdAS1 ⊗f2)◦f1 : 1V → AS1⊗AS2 , namely an invariant inside AS1⊗AS2 . This suggests AS ≃ (AS1⊗AS2) inv, which we will prove below. Now we need to define what we mean by invariants of a tensor product in any ribbon category V. Definition 4.31. Let V be a ribbon category, E = Free(V) and n ≥ 2. For 1 ≤ i < j ≤ n − 1 we denote the tensor product of coordinates i and j by ⊗i,j : { V⊗n → V⊗n−1, (V1, . . . , Vn) 7→ (V1, . . . , Vi−1, Vi ⊗ Vj , Vi+1, . . . , Vj−1, Vj+1, . . . , Vn). For k⃗1 < k⃗2 two sequences of k distinct indices we denote the tensor product of coordinates k⃗1 with coordinates k⃗2 by ⊗ k⃗1 ,⃗k2 : V⊗n → V⊗n−k. It extends to ⊗ k⃗1 ,⃗k2 : E⊠n → E⊠n−k by coconti- nuity. We denote the unit on i-th coordinate by ηi : { V⊗n−2 → V⊗n−1,( V1, . . . , ∨ Vi, . . . , ∨ Vj , . . . , Vn ) 7→ ( V1, . . . , 1V , . . . , ∨ Vj , . . . , Vn ) and the unit on k⃗1-th coordinates as η k⃗1 : V⊗n−2k → V⊗n−k. Let X ∈ E⊠n, we want to define its ( k⃗1, k⃗2 ) -invariants X inv k⃗1,k⃗2 ∈ E⊠n−2k. One only needs to describe morphisms from any V⃗ ∈ V⊗n−2k to it. We set X inv k⃗1,k⃗2 ( V⃗ ) = HomE⊠n−2k ( V⃗ , X inv k⃗1,k⃗2 ) := HomE⊠n−k ( η k⃗1 (V⃗ ),⊗ k⃗1 ,⃗k2 (X) ) . For X ∈ E⊠n1 ⊠ E⊠k⃗1 and Y ∈ E⊠k⃗2 ⊠ E⊠n2 we write X ⊗ k⃗1 ,⃗k2 Y := ⊗ k⃗1 ,⃗k2 (X,Y ). Then (X,Y ) inv k⃗1,k⃗2 ((Vn⃗1 , Vn⃗2 )) : = HomE⊠n1+n2+k ( (Vn⃗1 , 1V⊗k , Vn⃗2 ), X ⊗ k⃗1 ,⃗k2 Y ) = HomE⊠n1+n2+k ( (Vn⃗1 , 1V⊗k)⊗k⃗1 ,⃗k2 (1V⊗k , Vn⃗2 ), X ⊗ k⃗1 ,⃗k2 Y ) . For V = Oq2(SL2)-comodfin we get a notion of invariants for bicomodules (Xi, Xj) ∈ V⊗2 where we first “merge” the two comodule structures (by the product, in the definition of the tensor product) and then take invariants in the usual sense, namely maps k → Xi ⊗Xj . Theorem 4.32. Let S1 be a marked surface with n1 + k boundary edges with a sequence of k right boundary edges c⃗1 (numbered k⃗1 = {n1 + 1, . . . , n1 + k}) and S2 a marked surface with n2 + k boundary edges with a sequence of k left boundary edges c⃗2 (numbered k⃗2 = {n1 + k + 1, . . . , n1 + 2k}). We write n⃗1 = {1, . . . , n1} and n⃗2 = {n1 + 2k + 1, . . . , n1 + 2k + n2} the indices of the other edges of S1 and S2. Let S = S1 ∪c⃗1=c⃗2 S2, then one has an isomorphism AS ≃ (AS1 , AS2) inv k⃗1,k⃗2 in E⊠n1+n2. Note that one has two thick embeddings S1 ←↩ C ↪→ S2 where C = ⊔k(0, 1) and S is their collar gluing. Proof. We describe a natural isomorphism between functors V⊗n1+n2 → Vectk σ : HomSkV (S)(− ▷∅,∅) ⇒̃ HomE⊠n1+n2 ( −, (AS1 , AS2) inv k⃗1,k⃗2 ) : = HomE⊠n1+n2+k ( η k⃗1 (−), AS1 ⊗k⃗1 ,⃗k2 AS2 ) . Relating Stated Skein Algebras and Internal Skein Algebras 35 We write (σ1)(Vn⃗1 ,V k⃗1 ) : HomSkV (S1)(Vn⃗1 ▷ ∅,∅ � V k⃗1 ) ⇒̃ HomE⊠n1+k((Vn⃗1 , 1V⊗k), AS1 ⊗k⃗1 V k⃗1 ) for Vn⃗1 ∈ V⊗n1 and V k⃗1 ∈ V⊗k, obtained from the defining natural isomorphism of AS1 by Remark 4.12. We write σ2 : HomSkV (S2)(− ▷ ∅,∅) ⇒̃ HomE⊠n2+k(−, AS2) the defining natural isomorphism of AS2 . Step 1 (decomposition in S). Let V⃗ = (Vn⃗1 , Vn⃗2 ) ∈ V⊗n1+n2 and α ∈ HomSkV (S) ( V⃗ ▷ ∅,∅ ) a morphism from Vn⃗1 ▷∅ in S1 and Vn⃗2 ▷∅ in S2 to the empty set in S. By Corollary 2.19, α decomposes into a pair (α1, α2) as α = (Id∅, α2) ◦ ι∅,V k⃗ ,Vn⃗2 ▷∅ ◦ (α1, IdVn⃗2 ▷∅) with α1 ∈ HomSkV (S1)(Vn⃗1 ▷ ∅,∅ � k⃗1 V k⃗ ) and α2 ∈ HomSkV (S2)(Vk⃗ � k⃗2 (Vn⃗2 ▷ ∅),∅) for some V k⃗ ∈ SkV(C×(0, 1)) ≃ V⊗k, with an implicit sum. This decomposition is unique up to balancing, namely if α2 can be written β2 ◦ (γ � IdVn⃗2 �∅), with β2 ∈ HomSkV (S2)(Wk⃗ � (Vn⃗2 ▷ ∅),∅) and γ ∈ HomSkV (C×(0,1))(Vk⃗,Wk⃗ ) for some W k⃗ ∈ SkV(C × (0, 1)), then: (α1, β2 ◦ (γ � IdVn⃗2 �∅)) ∼ ((Id∅ � γ) ◦ α1, β2). α1 α2 • Vn⃗1 ▷∅ • Vn⃗2 ▷∅ • �Vk⃗ ι • Vk⃗� Step 2 ( re-composition in R2 ) . The morphism α1 is described by a morphism f1 = (σ1)(Vn⃗1 ,V k⃗ )(α1) ∈ HomE⊠n1+k((Vn⃗1 , 1V⊗k), AS1 ⊗k⃗1 V k⃗ ) and α2 is described by f2 = (σ2)(V k⃗ ,Vn⃗2 )(α2) ∈ HomE⊠n2+k((Vk⃗, Vn⃗2 ), AS2). These morphisms are well-defined (depend only on α) up to balancing, namely if α2 = β2 ◦ (γ� IdVn⃗2 �∅) with β2 described by g2 = σ2(β2) ∈ HomE⊠n2+k((Wk⃗ , Vn⃗2 ), AS2), then by naturality of σ1 and σ2, f2 = g2◦(γ, IdVn⃗2 ) and the above relation becomes (f1, g2◦(γ, IdVn⃗2 )) ∼ ((IdAS1 ⊗ k⃗1 γ) ◦ f1, g2). f1 f2 • Vn⃗1 • Vn⃗2 • ⊗k⃗1 Vk⃗ • Vk⃗⊗k⃗2 • AS1 • AS1 • AS2 ⊗k⃗1 ,⃗k2 1V⊗k Thus the map σV⃗ (α) := (IdAS1 ⊗ k⃗1 ,⃗k2 f2) ◦ (f1, IdVn⃗2 ) ∈ HomE⊠n1+n2+k((Vn⃗1 , 1V⊗k , Vn⃗2 ), AS1 ⊗k⃗1 ,⃗k2 AS2) is well defined, because this relation is killed. 36 B. Häıoun Step 3 (naturalilty). Naturality is quite obvious from the picture: one can insert morphisms from below. For g1 ∈ HomE⊠n1 (Wn⃗1 , Vn⃗1 ) and g2 ∈ HomE⊠n2 (Wn⃗2 , Vn⃗2 ), by naturality of σ1 and σ2, one has σ1(α1◦(g1▷Id∅)) = f1◦g1 and σ2(α2◦(g2▷Id∅)) = f2◦g2. Now α◦((g1, g2)▷Id∅) splits in Step 1 as (Id∅, α2) ◦ ι∅,V k⃗ ,Vn⃗2 ▷∅ ◦ (α1, IdVn⃗2 ▷∅) ◦ (g1 ▷ Id∅, g2 ▷ Id∅) = (Id∅, α2 ◦ (g2 ▷ Id∅)) ◦ ι∅,V k⃗ ,Wn⃗2 ▷∅ ◦ (α1 ◦ (g1 ▷ Id∅), IdWn⃗2 ▷∅). Thus σ(α ◦ ((g1, g2) ▷ Id∅)) = (IdAS1 ⊗ k⃗1 ,⃗k2 (f2 ◦ g2)) ◦ ((f1 ◦ g1), IdWn⃗2 ) = σ(α) ◦ (g1, g2). We now construct an inverse to σ by the same steps in reverse order. Step 2−1 ( decomposition in R2 ) . We want to decompose a morphism f ∈ HomE⊠n1+n2+k ( η k⃗1 ( V⃗ ) , AS1 ⊗k⃗1 ,⃗k2 AS2 ) as f = ( IdAS1 ⊗ k⃗1 ,⃗k2 f2 ) ◦ ( f1, IdVn⃗2 ) with f1 ∈ HomE⊠n1+k((Vn⃗1 , 1V⊗k), AS1 ⊗k⃗1 V k⃗ ) and f2 ∈ HomE⊠n2+k((Vk⃗, Vn⃗2 ), AS2). This is easy in V⊗n1+n2+k, as all maps split on each coordinates. For A⃗1 = (An⃗1 , A k⃗1 ) ∈ V⊗n1+k and A⃗2 = (A k⃗2 , An⃗2 ) ∈ V⊗n2+k, a morphism f ∈ HomV⊗n1+n2+k((Vn⃗1 , 1V⊗k , Vn⃗2 ), A⃗1 ⊗k⃗1 ,⃗k2 A⃗2) is, up to a linear combination, of the form (gn⃗1 , g k⃗1 , gn⃗2 ) with gn⃗1 : Vn⃗1 → An⃗1 , g k⃗1 : 1V⊗k → A k⃗1 ⊗ k⃗1 ,⃗k2 A k⃗2 and gn⃗2 : Vn⃗2 → An⃗2 . Then, set V k⃗ = A k⃗2 , f1 = gn⃗1 ⊗ g k⃗1 ∈ HomV⊗n1+k((Vn⃗1 , 1V⊗k), A⃗1 ⊗k⃗1 V k⃗ ) and f2 = IdV k⃗ ⊗ gn⃗2 ∈ HomV⊗n2+k((Vk⃗, Vn⃗2 ), A⃗2), one has f = (IdA⃗1 ⊗ k⃗1 ,⃗k2 f2) ◦ (f1, IdVn⃗2 ). This decomposition is unique up to balancing, if f = (IdA⃗1 ⊗ k⃗1 ,⃗k2 f ′2) ◦ (f ′1, IdVn⃗2 ) one can split f ′2, which has to coincide with f2 on n⃗2 coordi- nates, and is some γ : W k⃗ → A k⃗2 on k⃗2 coordinates (which are now k⃗1 coordinates after the ⊗ k⃗1 ,⃗k2 ). Similarly, f ′1 coincides with f1 on n⃗1 coordinates, and is some δ : 1V → A k⃗1 ⊗ W k⃗ on k⃗1 coordinates. On k⃗1 coordinates one has (IdA k⃗1 ⊗ k⃗1 γ) ◦ δ = g k⃗1 , so the only relation is ((−, (IdA k⃗1 ⊗ k⃗1 γ) ◦ δ), (IdV k⃗ ,−)) ∼ ((−, δ), (γ,−)). Now, AS1 and AS2 are not objects of V⊗n1+k and V⊗n2+k, but are obtained as canoni- cal colimits of such objects, AS1 = colimi A⃗1,i and AS2 = colimj A⃗2,j , so AS1 ⊗k⃗1 ,⃗k2 AS2 = colimi,j A⃗1,i ⊗k⃗1 ,⃗k2 A⃗2,j by cocontinuity. The object η k⃗1 ( V⃗ ) = ( Vn⃗1 , 1V⊗k , Vn⃗2 ) is compact pro- jective in E⊠n1+n2+k therefore HomE⊠n1+n2+k ( η k⃗1 ( V⃗ ) , AS1 ⊗k⃗1 ,⃗k2 AS2 ) = colimi,j HomV⊗n1+n2+k ( η k⃗1 ( V⃗ ) , A⃗1,i ⊗k⃗1 ,⃗k2 A⃗2,j ) . So a morphism f ∈ HomE⊠n1+n2+k ( η k⃗1 ( V⃗ ) , AS1⊗k⃗1 ,⃗k2 AS2 ) factorises through a single (actually, a linear combination of) A⃗1,i ⊗k⃗1 ,⃗k2 A⃗2,j as f : η k⃗1 ( V⃗ ) fi,j→ A⃗1,i ⊗k⃗1 ,⃗k2 A⃗2,j can1,i⊗k⃗1,k⃗2 can2,j −→ AS1 ⊗k⃗1 ,⃗k2 AS2 . There it splits as fi,j = ( IdA⃗1,i ⊗ k⃗1 ,⃗k2 f2 ) ◦ (f1, IdVn⃗2 ), and f = (IdAS1 ⊗ k⃗1 ,⃗k2 (can2,j ◦ f2)) ◦ ((can1,i ⊗k⃗1 IdV k⃗ ) ◦ f1, IdVn⃗2 ). This fi,j is unique up to the relations in the above colimit, namely for h1 ⊗k⃗1 ,⃗k2 h2 : A⃗1,i ⊗k⃗1 ,⃗k2 A⃗2,j → A⃗1,i′ ⊗k⃗1 ,⃗k2 A⃗2,j′ over AS1 ⊗k⃗1 ,⃗k2 AS2 one has fi′,j′ = (h1 ⊗k⃗1 ,⃗k2 h2) ◦ fi,j and f = can1,i′ ⊗k⃗1 ,⃗k2 can2,j′ ◦ fi′,j′ . Split h2 as (h k⃗2 , hn⃗2 ), then f decomposes through fi′,j′ as f = (IdAS1 ⊗ k⃗1 ,⃗k2 (can2,j′ ◦ (IdV k⃗ ′ , hn⃗2 ) ◦ f2)) ◦ ((can1,i′ ⊗k⃗1 IdV k⃗ ′) ◦ (h1 ⊗k⃗1 h k⃗2 ) ◦ f1, IdVn⃗2 ). Relating Stated Skein Algebras and Internal Skein Algebras 37 Set F1 = (can1,i⊗k⃗1 IdV k⃗ ) ◦ f1 = ((can1,i′ ◦h1)⊗k⃗1 IdV k⃗ ) ◦ f1 and F2 = can2,j′ ◦ (IdV k⃗ ′ , hn⃗2 ) ◦ f2 = can2,j′ ◦ (IdV k⃗ ′ , hn⃗2 ◦ gn⃗2 ). The first decomposition was (F1, can2,j ◦ f2) = (F1, can2,j′ ◦ (hk⃗2 , hn⃗2 ) ◦ (IdV k⃗ , gn⃗2 )) = (F1, F2 ◦ (hk⃗2 , IdVn⃗2 )) and the second is ((can1,i′ ⊗k⃗1 IdV k⃗ ′) ◦ (h1 ⊗k⃗1 h k⃗2 ) ◦ f1, F2) = ((IdAS1 ⊗ k⃗1 h k⃗2 ) ◦ F1, F2) so the only relation is (F1, F2 ◦ (hk⃗2 , IdVn⃗2 )) ∼ ((IdAS1 ⊗ k⃗1 h k⃗2 ) ◦ F1, F2). Step 1−1 (re-composition in S). We decompose f using last step as f = (IdAS1 ⊗ k⃗1 ,⃗k2 f2) ◦ (f1, IdVn⃗2 ) with f1 ∈ HomE⊠n1+k((Vn⃗1 , 1V⊗k), AS1 ⊗k⃗1 V k⃗ ) and f2 ∈ HomE⊠n2+k((Vk⃗, Vn⃗2 ), AS2). They are described by morphisms α1 := (σ1) −1 (Vn⃗1 ,V k⃗ )(f1) ∈ HomSkV (S1)(Vn⃗1 ▷ ∅,∅ � k⃗1 V k⃗ ) and α2 := (σ2) −1 (V k⃗ ,Vn⃗2 )(f2) ∈ HomSkV (S2)(Vk⃗ �k⃗2 (Vn⃗2 ▷∅),∅). The above relation becomes (α1, β2 ◦ (γ � IdVn⃗2 �∅)) ∼ ((Id∅ � γ) ◦ α1, β2). So the morphism σ−1 V⃗ (f) := (Id∅, α2) ◦ ι∅,V k⃗ ,Vn⃗2 ▷∅ ◦ (α1, IdVn⃗2 ▷∅) is well defined, because this relation is killed. Step 4 (isomorphism). One easily checks that σ−1 defined this way is an inverse to σ. Let α ∈ HomSkV (S) ( V⃗ ▷ ∅,∅ ) that decomposes as α = (Id∅, α2) ◦ ι∅,V k⃗ ,Vn⃗2 ▷∅ ◦ (α1, IdVn⃗2 ▷∅), then σV⃗ (α) := (IdAS1 ⊗ k⃗1 ,⃗k2 f2) ◦ (f1, IdVn⃗2 ) is already decomposed with α1 = (σ1) −1 (Vn⃗1 ,V k⃗ )(f1) and α2 = (σ2) −1 (V k⃗ ,Vn⃗2 )(f2), so σ−1 V⃗ (σV⃗ (α)) := (Id∅, α2) ◦ ι∅,V k⃗ ,Vn⃗2 ▷∅ ◦ (α1, IdVn⃗2 ▷∅) = α. Similarly, let f ∈ HomE⊠n1+n2+k ( η k⃗1 ( V⃗ ) , AS1 ⊗k⃗1 ,⃗k2 AS2 ) that decomposes as f = (IdAS1 ⊗ k⃗1 ,⃗k2 f2) ◦ (f1, IdVn⃗2 ) then σ−1 V⃗ (f) := (Id∅, α2) ◦ ι∅,V k⃗ ,Vn⃗2 ▷∅ ◦ (α1, IdVn⃗2 ▷∅) is already decomposed with f1 = (σ1)(Vn⃗1 ,V k⃗ )(α1) and f2 = (σ2)(V k⃗ ,Vn⃗2 )(α2), so σV⃗ ( σ−1 V⃗ (f) ) := (IdAS1 ⊗ k⃗1 ,⃗k2 f2) ◦ (f1, IdVn⃗2 ) = f and σ−1 is indeed an inverse to σ. ■ Remark 4.33. When V = Oq2(SL2)-comodfin, with AS ≃ S (S), one obtains the same excision properties as in Theorem 2.37. One uses repeatedly Theorem 2.37 on S1 ⊔S2 on each couple of boundary edges to glue. This gives S (S) ≃ HH0 k⃗1,k⃗2 (S (S1) ⊗S (S2)) := {x ∈ S (S1) ⊗ S (S2) / ∀1 ≤ i ≤ k, ∆n1+i(x) = fl ◦∆l n1+n2+k+i(x)}. By Proposition 4.27 on all couples of edges to glue one gets HH0 k⃗1,k⃗2 ((S (S1),S (S2))) = ( S R(S1) ht k⃗1 ,S (S2) )inv k⃗1,k⃗2 . By Theorem 4.28 S R(S1) ht k⃗1 is the internal skein algebra of S1, and one obtains exactly the formulation of Theorem 4.32. Note that we described how to glue two surfaces along many edges at once and Theorem 2.37 describes how to glue only two edges but possibly of the same surface. The two forms of excision are equivalent, in one way by applying it repeatedly as above and in the other way by gluing a bigon to the two edges of the surface that one wants to glue together. 38 B. Häıoun Remark 4.34. This remark answers a natural question arising at the sight of the cutting prop- erty of stated skein algebras: why is it not a coevaluation one sees on newly created states when one cuts along an ideal arc? Indeed in the definition one uses ∑ µ⃗ vµ⃗ ⊗ vµ⃗ though the coeval- uation would give coev(1) = ∑ µ⃗ vµ⃗ ⊗ v∗µ⃗ Id⊗φ−1 7−→ ∑ µ⃗ vµ⃗ ⊗ v−←−µC(−µ⃗) in particular matching + states to − states. The answer is that it is indeed given by a coevaluation, but the stated skein algebra of the surface at the right is not the good object: one must take its half-twisted version. Then the half twist re-exchanges + signs to − signs and kills the coefficients appearing. In particular we see that there has been a choice in the way the splitting morphism of stated skein algebras is defined, and that this choice seems to determine both the half twist and the identification V ≃ V ∗. This is to be put in light with the unicity of stated skein coefficients proved in [18, Section 3.4]. Remark 4.35. Internal skein algebras are defined for any ribbon category V, and coincide with stated skein algebras when V = Oq2(SL2)-comodfin. Stated skein algebras for SLn were very recently introduced in [19], and one can expect to prove they coincide with internal skein algebras for V = Oqn(SLn)-comodfin for generic q with the very same proof. The authors actually showed it for surfaces with a single boundary interval using excision properties with respect to gluing patterns from both theories. The constructions and arguments of this paper work more generally with any semisimple coribbon Hopf algebras H, using the equivalence H-comod ≃ Free(H-comodfin), and are actually [11]’s candidate for the generalisation of stated skein algebras. The results of this section show that this generalisation extends to multiple markings, and that one obtains excision properties immediately. Internal skein algebras are defined more generally in [2] for any E2-algebra A ∈ Pr under the name moduli algebras, and the skein-theoretic description holds for A = Free(V). As both moduli algebras and stated skein algebras can be defined integrally, or at roots of unity, it would be very interesting to understand how they compare in greater generality. So far, there is no skein-theoretic description of the factorization homology used in the construction of moduli algebras, but it seems credible that with extra work one could rewrite this whole theory in these integral or non-semisimple contexts. Acknowledgements I would like to thank warmly my three advisors: Francesco Costantino for his guidance through- out the discovery of this subject and the editing of this article; Joan Bellier-Millès for all the time he spent in explanations and David Jordan for very helpful conversations and comments. I am grateful to Patrick Kinnear for his remarks and advice. I would also like to thank the anony- mous referees for their exceptionally detailed feedback. This research took place in the Institut Mathématique de Toulouse and was supported by the École Normale Supérieure de Lyon. References [1] Abe E., Hopf algebras, Cambridge Tracts in Mathematics, Vol. 74, Cambridge University Press, Cambridge – New York, 1980. [2] Ben-Zvi D., Brochier A., Jordan D., Integrating quantum groups over surfaces, J. Topol. 11 (2018), 874–917, arXiv:1501.04652. [3] Bonahon F., Wong H., Quantum traces for representations of surface groups in SL2(C), Geom. Topol. 15 (2011), 1569–1615, arXiv:1003.5250. [4] Carter J.S., Flath D.E., Saito M., The classical and quantum 6j-symbols, Mathematical Notes, Vol. 43, Princeton University Press, Princeton, NJ, 1995. [5] Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994. https://doi.org/10.1112/topo.12072 http://arxiv.org/abs/1501.04652 https://doi.org/10.2140/gt.2011.15.1569 http://arxiv.org/abs/1003.5250 Relating Stated Skein Algebras and Internal Skein Algebras 39 [6] Cooke J., Excision of skein categories and factorisation homology, arXiv:1910.02630. [7] Costantino F., Lê T.T.Q., Stated skein algebras of surfaces, arXiv:1907.11400. [8] Dugger D., Sheaves and homotopy theory, 1998, Incomplete draft, https://ncatlab.org/nlab/files/cech. pdf. [9] Etingof P., Gelaki S., Nikshych D., Ostrik V., Tensor categories, Mathematical Surveys and Monographs, Vol. 205, Amer. Math. Soc., Providence, RI, 2015. [10] Faitg M., Holonomy and (stated) skein algebras in combinatorial quantization, arXiv:2003.08992. [11] Gunningham S., Jordan D., Safronov P., The finiteness conjecture for skein modules, arXiv:1908.05233. [12] Johnson-Freyd T., Heisenberg-picture quantum field theory, in Representation Theory, Mathematical Physics, and Integrable Systems, Progr. Math., Vol. 340, Birkhäuser/Springer, Cham, 2021, 371–409, arXiv:1508.05908. [13] Kassel C., Quantum groups, Graduate Texts in Mathematics, Vol. 155, Springer-Verlag, New York, 1995. [14] Kelly G.M., Basic concepts of enriched category theory, Repr. Theory Appl. Categ. (2005), vi+137 pages. [15] Klimyk A., Schmüdgen K., Quantum groups and their representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997. [16] Korinman J., Finite presentations for stated skein algebras and lattice gauge field theory, arXiv:2012.03237. [17] Korinman J., Quesney A., Classical shadows of stated skein representations at odd roots of unity, arXiv:1905.03441. [18] Lê T.T.Q., Triangular decomposition of skein algebras, Quantum Topol. 9 (2018), 591–632, arXiv:1609.04987. [19] Lê T.T.Q., Sikora A.S., Stated SL(n)-skein modules and algebras, arXiv:2201.00045. [20] Lê T.T.Q., Yu T., Stated skein modules of marked 3-manifolds/surfaces, a survey, Acta Math. Vietnam. 46 (2021), 265–287, arXiv:2005.14577. [21] Muller G., Skein algebras and cluster algebras of marked surfaces, arXiv:1204.0020. [22] Ramos González J., On the tensor product of large categories, Ph.D. Thesis, University of Antwerp, 2017. [23] Reshetikhin N.Yu., Turaev V.G., Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), 1–26. [24] Snyder N., Tingley P., The half-twist for Uq(g) representations, Algebra Number Theory 3 (2009), 809–834, arXiv:0810.0084. [25] Takeuchi M., A short course on quantum matrices, in New Directions in Hopf Algebras, Math. Sci. Res. Inst. Publ., Vol. 43, Cambridge University Press, Cambridge, 2002, 383–435. [26] Tingley P., A minus sign that used to annoy me but now I know why it is there, arXiv:1002.0555. [27] Turaev V.G., Quantum invariants of knots and 3-manifolds, De Gruyter Studies in Mathematics, Vol. 18, Walter de Gruyter & Co., Berlin, 2010. [28] Wakui M., The coribbon structures of some finite dimensional braided Hopf algebras generated by 2 × 2- matrix coalgebras, in Noncommutative Geometry and Quantum Groups (Warsaw, 2001), Banach Center Publ., Vol. 61, Polish Acad. Sci. Inst. Math., Warsaw, 2003, 333–344. [29] Walker K., TQFTs, Early incomplete draft, 2006, http://canyon23.net/math/tc.pdf. http://arxiv.org/abs/1910.02630 http://arxiv.org/abs/1907.11400 https://ncatlab.org/nlab/files/cech.pdf https://ncatlab.org/nlab/files/cech.pdf https://doi.org/10.1090/surv/205 http://arxiv.org/abs/2003.08992 http://arxiv.org/abs/1908.05233 https://doi.org/10.1007/978-3-030-78148-4_13 http://arxiv.org/abs/1508.05908 https://doi.org/10.1007/978-1-4612-0783-2 https://doi.org/10.1007/978-3-642-60896-4 http://arxiv.org/abs/2012.03237 http://arxiv.org/abs/1905.03441 https://doi.org/10.4171/QT/115 http://arxiv.org/abs/1609.04987 http://arxiv.org/abs/2201.00045 https://doi.org/10.1007/s40306-021-00417-2 http://arxiv.org/abs/2005.14577 http://arxiv.org/abs/1204.0020 https://doi.org/10.1007/BF02096491 https://doi.org/10.1007/BF02096491 https://doi.org/10.2140/ant.2009.3.809 http://arxiv.org/abs/0810.0084 http://arxiv.org/abs/1002.0555 https://doi.org/10.1515/9783110221848 https://doi.org/10.4064/bc61-0-20 http://canyon23.net/math/tc.pdf 1 Introduction 2 Preliminaries 2.1 The coribbon Hopf algebra O_{q^2}(SL_2) 2.2 Skein categories 2.3 Stated skein algebras 2.4 Internal skein algebras 3 The relation 4 Multi-edges 4.1 Right internal skein algebras 4.2 Multi-edges internal skein algebras 4.3 Relation for multiple left edges 4.4 The half twist on O_{q^2}(SL_2)-comod 4.5 The general relation 4.6 Excision properties of multi-edges internal skein algebras References
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publisher	Інститут математики НАН України
record_format	dspace
spelling	Haïoun, Benjamin 2026-01-07T13:40:29Z 2022 Relating Stated Skein Algebras and Internal Skein Algebras. Benjamin Haïoun. SIGMA 18 (2022), 042, 39 pages 1815-0659 2020 Mathematics Subject Classification: 57K16; 18M15 arXiv:2104.13848 https://nasplib.isofts.kiev.ua/handle/123456789/211626 https://doi.org/10.3842/SIGMA.2022.042 We give an explicit correspondence between stated skein algebras, which are defined via explicit relations on stated tangles in [Costantino F., Lê T.T.Q., arXiv:1907.11400], and internal skein algebras, which are defined as internal endomorphism algebras in free cocompletions of skein categories in [Ben-Zvi D., Brochier A., Jordan D., J. Topol. 11 (2018), 874-917, arXiv:1501.04652] or in [Gunningham S., Jordan D., Safronov P., arXiv:1908.05233]. Stated skein algebras are defined on surfaces with multiple boundary edges, and we generalise internal skein algebras in this context. Now, one needs to distinguish between left and right boundary edges, and we explain this phenomenon on stated skein algebras using a half-twist. We prove excision properties of multi-edge internal skein algebras using excision properties of skein categories, and agree with excision properties of stated skein algebras when = q²( ₂)-modᶠⁱⁿ. Our proofs are mostly based on skein theory, and we do not require the reader to be familiar with the formalism of higher categories. I would like to thank my three advisors: Francesco Costantino for his guidance throughout the discovery of this subject and the editing of this article; Joan Bellier-Milles for all the time he spent on explanations, and David Jordan for his very helpful conversations and comments. I am grateful to Patrick Kinnear for his remarks and advice. I would also like to thank the anonymous referees for their exceptionally detailed feedback. This research took place in the Institut Mathématique de Toulouse and was supported by the École Normale Supérieure de Lyon. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Relating Stated Skein Algebras and Internal Skein Algebras Article published earlier
spellingShingle	Relating Stated Skein Algebras and Internal Skein Algebras Haïoun, Benjamin
title	Relating Stated Skein Algebras and Internal Skein Algebras
title_full	Relating Stated Skein Algebras and Internal Skein Algebras
title_fullStr	Relating Stated Skein Algebras and Internal Skein Algebras
title_full_unstemmed	Relating Stated Skein Algebras and Internal Skein Algebras
title_short	Relating Stated Skein Algebras and Internal Skein Algebras
title_sort	relating stated skein algebras and internal skein algebras
url	https://nasplib.isofts.kiev.ua/handle/123456789/211626
work_keys_str_mv	AT haiounbenjamin relatingstatedskeinalgebrasandinternalskeinalgebras

Relating Stated Skein Algebras and Internal Skein Algebras

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