Finite presentations for stated skein algebras and lattice gauge field theory
aa r X i v : . [ m a t h . QA ] D ec FINITE PRESENTATIONS FOR STATED SKEIN ALGEBRAS AND LATTICE GAUGEFIELD THEORY
JULIEN KORINMAN
Abstract.
We provide finite presentations for stated skein algebras and deduce that those algebras areKoszul and that they are isomorphic to the quantum moduli algebras appearing in lattice gauge field theory,generalizing previous results of Bullock, Frohman, Kania-Bartoszynska and Faitg. Introduction
Stated skein algebras and lattice gauge field theory . A punctured surface is a pair Σ = (Σ , P ), whereΣ is a compact oriented surface and P is a (possibly empty) finite subset of Σ which intersects non-triviallyeach boundary component. We write Σ P := Σ \ P . The set ∂ Σ \ P consists of a disjoint union of open arcswhich we call boundary arcs . Warning:
In this paper, the punctured surface Σ will be called open if the surface Σ has non emptyboundary and closed otherwise. This convention differs from the traditional one, where some authors referto open surface a punctured surface Σ = (Σ , P ) with Σ closed and P 6 = ∅ (in which case Σ P is not closed).The Kauffman-bracket skein algebras were introduced by Bullock and Turaev as a tool to study theSU(2) Witten-Reshetikhin-Turaev topological quantum field theories ([Wit89, RT91]). They are associativeunitary algebras S ω ( Σ ) indexed by a closed punctured surface Σ and an invertible element ω ∈ k × in somecommutative unital ring k . Bonahon-Wong [BW11] and Lˆe [Le18] generalized the notion of Kauffman-bracketskein algebras to open punctured surfaces, where in addition to closed curves the algebras are generated byarcs whose endpoints are endowed with a sign ± (a state). The motivation for the introduction of theseso-called stated skein algebras is their good behaviour for the operation of gluing two boundary arcs together.This property permitted the authors of [BW11] to define an embedding of the skein algebra into a quantumtorus, named the quantum trace, and offers new tools to study the representation theory of skein algebras.Except for genus 0 and 1 surfaces ([BP00]), no finite presentation for the Kauffman-bracket skein algebrasis known, though a conjecture in that direction was formulated in [San18, Conjecture 1 . Σ . Let us briefly sketch their construction, we refer to Section2.2 for details.The finite presentations we will define depend on the choice of a finite presentation P of some groupoidΠ (Σ P , V ). In brief, for each boundary arc a of Σ , choose a point v a ∈ a and let V be the set of such points.The groupoid Π (Σ P , V ) is the full subcategory of the fundamental groupoid of Σ P whose set of objects is V . A finite presentation P = ( G , RL ) for Π (Σ P , V ) will consist in a finite set G of generating paths relating Department of Mathematics, Faculty of Science and Engineering, Waseda University,3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555, Japan
E-mail address : [email protected] .2020 Mathematics Subject Classification.
Key words and phrases.
Stated skein algebras, Lattice gauge field theory. oints of V and a finite set RL of relations among those paths which satisfy some axioms (see Section 2.2for details). For instance for the triangle T (the disc with three punctures on its boundary), the groupoidΠ ( T , V ) admits the presentation with generators G = { α, β, γ } drawn in Figure 1 and the unique relation αβγ = 1. Figure 1.
The triangle and some paths.A path α ∈ G can be seen as an arc in Σ P and, after choosing some states ε, ε ′ ∈ {− , + } for its endpoints,we get an element α εε ′ ∈ S ω ( Σ ) in the stated skein algebra. We denote by A G ⊂ S ω ( Σ ) the (finite) set ofsuch elements. It was proved in [Kor20] that A G generates S ω ( Σ ) and its elements will be the generators ofour presentations.Concerning the relations, first for each α ∈ G , one has a q-determinant relation between the elements α εε ′ . For each pair ( α, β ) ∈ G we will associate a finite set of arcs exchange relations permitting to expressan element of the form α εε ′ β µµ ′ ∈ S ω ( Σ ) as a linear combination of elements of the form β ab α cd . Eventually,to each relation R ∈ RL in the finite presentation P , we will associate a finite set of so-called trivial loopsrelations . Theorem 1.1.
Let Σ a connected open punctured surface and P a finite presentation of Π (Σ P , V ) . Then thestated skein algebra S ω ( Σ ) is presented by the set of generators A G and by the q-determinant, arcs exchangeand trivial loops relations. For every open punctured surface, we can choose finite presentation P of Π (Σ P , V ) such that the set ofrelations is empty (for instance for the triangle of Figure 1, one might choose the presentation with generators G = { α, β } and no relation). In this case, the presentation of S ω ( Σ ) is quadratic inhomogeneous and, byusing the Diamond Lemma, we prove the Theorem 1.2.
For Σ a connected open punctured surface, the quadratic inhomogeneous algebra S ω ( Σ ) isKoszul and admits a Poincar´e-Birkhoff-Witt basis. Theorem 1.2 implies that S ω ( Σ ) has an explicit minimal projective resolution (the so-called Koszul reso-lution) which permits to compute effectively its cohomology (see [LV12] for details).Let (Γ , c ) be a ciliated graph, that is a finite graph with the data for each vertex of a linear ordering of itsadjacent half-edges. Inspired by Fock and Rosly’s original work in [FR99] on the Poisson structure of char-acter varieties, Alekseev-Grosse-Schomerus ([AGS95, AGS96, AS96]) and Buffenoir-Roche ([BR95, BR96])independently defined the so-called quantum moduli algebras L ω (Γ , c ), which are combinatorial quantizationsof relative character varieties (see Section 4.2 for details). Those algebras arise with some right comodulemap ∆ G : L ω (Γ , c ) → L ω (Γ , c ) ⊗ O q [ G ], where O q [ G ] = O q [SL ] ⊗ ˚ V (Γ) is the so-called quantum gauge groupHopf algebra. The subalgebra L invω (Γ) ⊂ L ω (Γ , c ) of coinvariant vectors plays an important role in combi-natorial quantization. More precisely, as reviewed in Section 4.1, we associate to each ciliated graph (Γ , c )two punctured surfaces: an open one Σ (Γ , c ) and a closed one Σ (Γ) such that the algebras L ω (Γ , c ) and L invω (Γ) are quantization of the SL ( C ) (relative) character varieties of Σ (Γ , c ) and Σ (Γ) respectively withtheir Fock-Rosly Poisson structures. We deduce from Theorem 1.1 the following: Theorem 1.3.
There exist isomorphisms of algebras S ω ( Σ (Γ , c )) ∼ = L ω (Γ , c ) and S ω ( Σ (Γ)) ∼ = L invω (Γ) . Theorem 1.3 is not surprising and was already proved in some cases. First it is well-known that (stated)skein algebras also induces deformation quantizations of (relative) character varieties : it follows from thework in [Bul97, PS00, Tur91] for closed punctured surfaces and is proved in [KQ19, Theorem 1 .
3] and [CL19,Theorem 8 .
12] for open punctured surfaces. So Theorem 1.3 was expected; for instance its statement wasconjectured by Costantino and Lˆe in [CL19]. Next the skein origin of the defining relations of quantum moduli lgebra was discovered by Bullock, Frohman and Kania-Bartoszynska in [BFKB98b] where the authorsalready proved that S ω ( Σ (Γ)) and L invω (Γ) are isomorphic in the particular case where k = C [[ ~ ]] and q := ω − = exp ~ . However, their proof does not extend straightforwardly to arbitrary ring (see item (6) ofSection 5). Eventually, in the special case where (Γ , c ) is the so-called daisy graph (it has only one vertex, so Σ (Γ , c ) has exactly one boundary component with one puncture on it), Theorem 1.3 was proved by Faitg in[Fai20a] in the case where ω is not a root of unity. A detailed comparison between Faitg’s isomorphism andours is made in Section 4.4. Faitg’s result can also be derived indirectly from the works in [BZBJ18, GJS],as detailed in Section 4.4, though the so-obtained isomorphism is less explicit due to a change of duality. Acknowledgments.
The author thanks S.Baseilhac, F.Costantino, M.Faitg, L.Funar , A.Quesney andP.Roche for useful discussions. He acknowledges support from the Japanese Society for Promotion of Science(JSPS) and the Centre National pour la Recherche et l’Enseignement (CNRS).2.
Finite presentations for stated skein algebras
Definitions and first properties of stated skein algebras.Definition 2.1. A punctured surface is a pair Σ = (Σ , P ) where Σ is a compact oriented surface and P isa finite subset of Σ which intersects non-trivially each boundary component. A boundary arc is a connectedcomponent of ∂ Σ \ P . We write Σ P := Σ \ P . Definition of stated skein algebras
Before stating precisely the definition of stated skein algebras, let us sketch it informally. Given a puncturedsurface Σ and an invertible element ω ∈ k × in some commutative unital ring k , the stated skein algebra S ω ( Σ ) is the quotient of the k -module freely spanned by isotopy classes of stated tangles in Σ P × (0 ,
1) bysome local skein relations. The left part of Figure 2 illustrates such a stated tangle: each point of ∂T ⊂ ∂ Σ P is equipped with a sign + or − (the state). Here the stated tangle is the union of three stated arcs and oneclosed curve. In order to work with two-dimensional pictures, we will consider the projection of tangles inΣ P as in the right part of Figure 2; such a projection will be referred to as a diagram. Figure 2.
On the left: a stated tangle. On the right: its associated diagram. The arrowsrepresent the height orders.A tangle in Σ P × (0 ,
1) is a compact framed, properly embedded 1-dimensional manifold T ⊂ Σ P × (0 , ∂T ⊂ ∂ Σ P × (0 ,
1) the framing is parallel to the (0 ,
1) factor and points tothe direction of 1. Here, by framing, we refer to a thickening of T to an oriented surface. The height of ( v, h ) ∈ Σ P × (0 ,
1) is h . If b is a boundary arc and T a tangle, we impose that no two points in ∂ b T := ∂T ∩ b × (0 ,
1) have the same heights, hence the set ∂ b T is totally ordered by the heights. Two tanglesare isotopic if they are isotopic through the class of tangles that preserve the boundary height orders. Byconvention, the empty set is a tangle only isotopic to itself.Let π : Σ P × (0 , → Σ P be the projection with π ( v, h ) = v . A tangle T is in generic position if foreach of its points, the framing is parallel to the (0 ,
1) factor and points in the direction of 1 and is suchthat π (cid:12)(cid:12) T : T → Σ P is an immersion with at most transversal double points in the interior of Σ P . Everytangle is isotopic to a tangle in generic position. We call diagram the image D = π ( T ) of a tangle in genericposition, together with the over/undercrossing information at each double point. An isotopy class of diagram D together with a total order of ∂ b D := ∂D ∩ b for each boundary arc b , define uniquely an isotopy classof tangle. When choosing an orientation o ( b ) of a boundary arc b and a diagram D , the set ∂ b D receives a atural order by setting that the points are increasing when going in the direction of o ( b ). We will representtangles by drawing a diagram and an orientation (an arrow) for each boundary arc, as in Figure 2. When aboundary arc b is oriented we assume that ∂ b D is ordered according to the orientation. A state of a tangleis a map s : ∂T → {− , + } . A pair ( T, s ) is called a stated tangle . We define a stated diagram ( D, s ) in asimilar manner.Let ω ∈ k × an invertible element and write A := ω − . Definition 2.2. [Le18] The stated skein algebra S ω ( Σ ) is the free k -module generated by isotopy classes ofstated tangles in Σ P × (0 ,
1) modulo the following relations (1) and (2),(1) = A + A − and = − ( A + A − ) ;(2) ++ = −− = 0 , + − = ω and ω − − + − ω − − = . The product of two classes of stated tangles [ T , s ] and [ T , s ] is defined by isotoping T and T in Σ P × (1 / ,
1) and Σ P × (0 , /
2) respectively and then setting [ T , s ] · [ T , s ] = [ T ∪ T , s ∪ s ]. Figure 3 illustratesthis product.For a closed punctured surface, S ω ( Σ ) coincides with the classical (Turaev’s) Kauffman-bracket skeinalgebra. Figure 3.
An illustration of the product in stated skein algebras.
Reflexion anti-involution
Suppose that k = Z [ ω ± ] and consider the Z -linear involution x x ∗ on k sending ω to ω − . Let r :Σ P × (0 , ∼ = −→ Σ P be the homeomorphism defined by r ( x, t ) = ( x, − t ). Define an anti-linear map θ : S ω ( Σ ) ∼ = −→ S ω ( Σ ) by setting θ X i x i [ T i , s i ] ! := X i x ∗ i [ r ( T i ) , s i ◦ r ] . Proposition 2.3. ( [Le18, Proposition 2 . ) θ is an anti-morphism of algebras, i.e. θ ( xy ) = θ ( y ) θ ( x ) . Bases for stated skein algebras
A closed component of a diagram D is trivial if it bounds an embedded disc in Σ P . An open component of D is trivial if it can be isotoped, relatively to its boundary, inside some boundary arc. A diagram is simple ifit has neither double point nor trivial component. By convention, the empty set is a simple diagram. Let o denote an arbitrary orientation of the boundary arcs of Σ . For each boundary arc b we write < o the inducedtotal order on ∂ b D . A state s : ∂D → {− , + } is o − increasing if for any boundary arc b and any two points x, y ∈ ∂ b D , then x < o y implies s ( x ) < s ( y ), with the convention − < +. Definition 2.4.
We denote by B o ⊂ S ω ( Σ ) the set of classes of stated diagrams ( D, s ) such that D is simpleand s is o -increasing. Theorem 2.5. ( [Le18, Theorem 2 . ) the set B o is a basis of S ω ( Σ ) .Remark . The basis B o is independent on the choice of the ground ring k and of q ∈ k × . This fact hasthe following useful consequence: let k := Z [ ω ± ] and k ′ be any other commutative unital ring with aninvertible element ω ′ ∈ k ′ × . There is a unique morphism of rings µ : k → k ′ sending ω to ω ′ and the two k ′ algebras S ω ( Σ ) ⊗ k k ′ and S ω ′ ( Σ ) are canonically isomorphic through the isomorphism preserving the basis B o . This fact permits to prove formulas in k using the reflexion anti-involution θ and then apply them toany ring k ′ by changing the coefficients. luing maps Let a , b be two distinct boundary arcs of Σ and let Σ | a b be the punctured surface obtained from Σ bygluing a and b . Denote by π : Σ P → (Σ | a b ) P | a b the projection and c := π ( a ) = π ( b ). Let ( T , s )be a stated framed tangle of Σ | a b P | a b × (0 ,
1) transversed to c × (0 ,
1) and such that the heights of thepoints of T ∩ c × (0 ,
1) are pairwise distinct and the framing of the points of T ∩ c × (0 ,
1) is vertical. Let T ⊂ Σ P × (0 ,
1) be the framed tangle obtained by cutting T along c . Any two states s a : ∂ a T → {− , + } and s b : ∂ b T → {− , + } give rise to a state ( s a , s, s b ) on T . Both the sets ∂ a T and ∂ b T are in canonical bijectionwith the set T ∩ c by the map π . Hence the two sets of states s a and s b are both in canonical bijection withthe set St( c ) := { s : c ∩ T → {− , + }} . Definition 2.7.
Let i | a b : S ω ( Σ | a b ) → S ω ( Σ ) be the linear map given, for any ( T , s ) as above, by: i | a b ([ T , s ]) := X s ∈ St( c ) [ T, ( s, s , s )] . Figure 4.
An illustration of the gluing map i a b . Theorem 2.8. [Le18, Theorem 3 . The linear map i | a b : S ω ( Σ | a b ) → S ω ( Σ ) is an injective morphism ofalgebras. Moreover the gluing operation is coassociative in the sense that if a, b, c, d are four distinct boundaryarcs, then we have i | a b ◦ i | c d = i | c d ◦ i | a b . Relation with U q sl and its restricted dual O q [SL ]Recall that A = ω − and write q := A . The stated skein algebra has deep relations with the quantum group U q sl and its restricted dual O q (SL ), explored in [Le18, KQ19, CL19, Fai20a] that we briefly reproduce herefor later use. Let ρ : U q sl → End( V ) be the standard representation of U q sl , where V is two dimensionalwith basis ( v + , v − ) and ρ ( E ) = (cid:18) (cid:19) , ρ ( F ) = (cid:18) (cid:19) , ρ ( K ) = (cid:18) q q − (cid:19) . Let ρ ∗ : U q sl → End( V ∗ ) be the dual representation of ( ρ, V ), where ρ ∗ ( x ) is the transposed of ρ ( S ( x )).One has a U q sl -equivariant isomorphism V ∗ ∼ = −→ V whose matrix in the bases ( v ∗ + , v ∗− ) and ( v + , v − ) writes C = (cid:18) C ++ C + − C − + C −− (cid:19) := (cid:18) ω − ω (cid:19) . Therefore C − = − A C = (cid:18) − ω − ω − (cid:19) . Define the operators τ, q H ⊗ H ∈ End( V ⊗ ) by τ ( v i ⊗ v j ) := v j ⊗ v i and q H ⊗ H ( v i ⊗ v j ) = A ij v i ⊗ v j for i, j ∈ { + , −} (we identified − with − R ∈ End( V ⊗ ) be the braiding operator R = τ ◦ q H ⊗ H ◦ exp q (cid:0) ( q − q − ) ρ ( E ) ⊗ ρ ( F ) (cid:1) = τ ◦ q H ⊗ H ◦ (cid:0) + ( q − q − ) ρ ( E ) ⊗ ρ ( F ) (cid:1) . In the basis ( v + ⊗ v + , v + ⊗ v − , v − ⊗ v + , v − ⊗ v − ), it writes R = R ++++ R +++ − R ++ − + R ++ −− R + − ++ R + − + − R + −− + R + −−− R − +++ R − ++ − R − + − + R − + −− R −− ++ R −− + − R −−− + R −−−− := A A − A − A − A −
00 0 0 A , so R − = A − A − − A A A A − . We now list three families of skein relations, which are straightforward consequences of the definition, andwill be used in the paper. Let i, j ∈ {− , + } . • The trivial arc relations: ij = C ij , ij = ( C − ) ij . • The cutting arc relations: (4) = X i,j = ± C ij ij , = X i,j = ± ( C − ) ij ij . • The height exchange relations: (5) ij = = X k,l = ± R klij lk , ji = = X k,l = ± ( R − ) klij kl . We refer to [Le18] for proofs.The algebra O q [SL ] is the algebra presented by generators x εε ′ , ε, ε ′ ∈ {− , + } and relations x ++ x + − = q − x + − x ++ x ++ x − + = q − x − + x ++ x −− x + − = qx + − x −− x −− x − + = qx − + x −− x ++ x −− = 1 + q − x + − x − + x −− x ++ = 1 + qx + − x − + x − + x + − = x + − x − + It has a Hopf algebra structured characterized by the formulas (cid:18) ∆( x ++ ) ∆( x + − )∆( x − + ) ∆( x −− ) (cid:19) = (cid:18) x ++ x + − x − + x −− (cid:19) ⊗ (cid:18) x ++ x + − x − + x −− (cid:19)(cid:18) ǫ ( x ++ ) ǫ ( x + − ) ǫ ( x − + ) ǫ ( x −− ) (cid:19) = (cid:18) (cid:19) and (cid:18) S ( x ++ ) S ( x + − ) S ( x − + ) S ( x −− ) (cid:19) = (cid:18) x −− − qx + − − q − x − + x ++ (cid:19) . When q ∈ C ∗ is generic (not a root of unity), O q [SL ] is the restricted dual of U q sl (see [BG02]). The bigon B is the punctured surface made of a disc with two punctures on its boundary. It has two boundary arcs a and b and is generated by the stated arcs α εε , ε, ε ′ = ± made of an arc α linking a to b with state ε on α ∩ a and ε ′ on α ∩ b . Consider a disjoint union B F B of two bigons; by gluing together the boundary arc b of thefirst bigon with the boundary arc a of the second, one obtains a morphism ∆ := i b a : S ω ( B ) → S ω ( B ) ⊗ which endows S ω ( B ) with a structure of Hopf algebra where ∆ is the coproduct. Theorem 2.9 ([Le18, CL19, KQ19]) . There is an isomorphism of Hopf algebras ϕ : O q [SL ] ∼ = S ω ( B ) sending the generator x εε ′ ∈ O q [SL ] to the element α εε ′ ∈ S ω ( B ) . More precisely, the fact that ϕ is an isomorphism of algebras is proved in [Le18] and the fact that itpreserves the coproduct was noticed independently in [CL19, KQ19]. In all the paper, we will (abusively)identify the Hopf algebras O q [SL ] and S ω ( B ) using ϕ . Note that the definition of ϕ depends on an indexingby a and b of the boundary arcs of B .Now consider a punctured surface Σ and a boundary arc c . By gluing a bigon B along Σ while gluing b with c , one obtains a punctured surface isomorphic to Σ , hence a map ∆ Lc := i b c : S ω ( Σ ) → O q [SL ] ⊗ S ω ( Σ )which endows S ω ( Σ ) with a structure of left O q [SL ] comodule. Similarly, gluing c with a induces a rightcomodule morphism ∆ Rc := i c a : S ω ( Σ ) → S ω ( Σ ) ⊗ O q [SL ]. The following theorem characterizes theimage of the gluing map and was proved independently in [CL19] and [KQ19]. Theorem 2.10. ( [CL19, Theorem 4 . , [KQ19, Theorem 1 . ) Let Σ be a punctured surface and a, b twoboundary arcs. The following sequence is exact: → S ω ( Σ | a b ) i a b −−−→ S ω ( Σ ) ∆ La − σ ◦ ∆ Rb −−−−−−−→ O q [SL ] ⊗ S ω ( Σ ) , where σ ( x ⊗ y ) := y ⊗ x . n easy but very important consequence of the fact that ∆ La and ∆ Ra are comodule maps are the following boundary skein relations :(6) ( ǫ ⊗ id ) ◦ ∆ La = id and ( id ⊗ ǫ ) ◦ ∆ Ra = id. The image through the counit ǫ of a stated diagram in B can be computed using the formulas:(7) ǫ (cid:18) ij (cid:19) = C ij , ǫ (cid:18) ij (cid:19) = ( C − ) ij , ǫ (cid:18) (cid:19) = R ijkl , ǫ (cid:18) (cid:19) = ( R − ) ijkl . Figure 5 illustrates an instance of boundary skein relation (6). Here we draw a dotted arrow to illustratewhere we cut the bigon. Note that all the trivial arcs (3), cutting arc (4) and height exchange (5) relationsare particular cases of (6).
Figure 5.
An example of boundary skein relation.2.2.
The small fundamental groupoid and its finite presentations.
During all this section we fix apunctured surface Σ = (Σ , P ) such that Σ is connected and has non empty boundary. For each boundaryarc a of Σ , fix a point v a ∈ a and denote by V the set { v a } a . Definition 2.11.
The small fundamental groupoid Π (Σ P , V ) is the full subcategory of the fundamentalgroupoid Π (Σ P ) generated by V .Said differently, Π (Σ P , V ) is the small groupoid whose set of objects is V and such that a morphism(called path) α : v → v is a homotopy class of continuous map ϕ α : [0 , → Σ P with ϕ α (0) = v and ϕ α (1) = v . The map ϕ α will be referred to as a geometric representative of α . The composition is theconcatenation of paths. For a path α : v → v , we write s ( α ) = v (the source point) and t ( α ) = v (thetarget point) and α − : v → v the path with opposite orientation ( i.e. ϕ α − ( t ) = ϕ α (1 − t )).We will define the notion of finite presentation P of the groupoid Π (Σ P , V ) and attach to each such P a finite presentation of S ω ( Σ ). In order to get some intuition, consider the punctured surface in Figure 6:it is an annulus with two punctures per boundary component, so it has four boundary arcs. The figureshows some paths β , . . . , β and we will say that Π (Σ P , V ) is finitely presented by the set of generators { β , . . . , β } together with the relation β − β β β = 1. We will deduce that S ω ( Σ ) is generated by thestated arcs ( β i ) εε ′ and that the relation β − β β β = 1 induces a relation among them. Alternatively, thesame punctured surface has a presentation with the smaller set of generators { β , . . . , β } and no relation.The induced finite presentation of S ω ( Σ ) will be simpler. Figure 6.
A punctured surface and a set of generators for its small fundamental groupoid. efinition 2.12. (1) A set of generators for Π (Σ P , V ) is a set G of paths in Π (Σ P , V ) such that anypath α ∈ Π (Σ P , V ) decomposes as α = α ε . . . α ε n n with ε i = ± α i ∈ G . We also require thateach path α ∈ G is the homotopy class of some embedding ϕ α : [0 , → Σ P such that the images ofthe ϕ α do not intersect outside V and eventually intersect transversally at V . The generating graph is the oriented ribbon graph Γ ⊂ Σ P whose set of vertices is V and edges are the image of the ϕ . Wewill always assume implicitly that the geometric representatives ϕ α is part of the data defining a setof generators. Moreover, when α ∈ G is a path such that s ( α ) = t ( α ) (i.e. α is a loop) we add theadditional datum of a ”height order” for its endpoints, that is we specify whether h ( s ( α )) < h ( t ( α ))or h ( t ( α )) < h ( s ( α )).(2) For a path α : v → v and ε, ε ′ ∈ {− , + } , we denote by α εε ′ ∈ S ω ( Σ ) the class of the stated arc( α, σ ), where the state σ is given by σ ( v ) = ε and σ ( v ) = ε ′ . When both endpoints lye in the sameboundary arc (i.e. when s ( α ) = t ( α )) we use the chosen height order to specify which endpoint lieson the top. Set A G := { α εε ′ | α ∈ G , ε, ε ′ ∈ {− , + }} ⊂ S ω ( Σ ) . Example . For any connected open punctured surface Σ , the groupoid Π (Σ P , V ) admits a finite set ofgenerators depicted in Figure 7 and defined as follows. Denote by a , . . . , a n the boundary arcs, by ∂ , . . . , ∂ r the boundary components of Σ with a ⊂ ∂ and write v i := a i ∩ V . Let Σ be the surface obtained fromΣ by gluing a disc along each boundary component ∂ i for 1 ≤ i ≤ r , and choose α , β , . . . , α g , β g somepaths in π (Σ P , v )(= End Π (Σ P , V ) ( v )), such that their images in Σ generate the free group π (Σ , v ) (saiddifferently, the α i and β i are longitudes and meridians of Σ). For each inner puncture p choose a peripheralcurve γ p ∈ π (Σ P , v ) encircling p once and for each boundary puncture p ∂ between two boundary arcs a i and a j , consider the path α p ∂ : v i → v j represented by the corner arc in p ∂ . Eventually, for each boundarycomponent ∂ j , with 1 ≤ j ≤ r , containing a boundary arc a k j ⊂ ∂ j , choose a path δ ∂ j : v → v k j . The set G ′ := { α i , β i , α p , δ ∂ j | ≤ i ≤ g, p ∈ P , ≤ j ≤ r } is a generating set for Π (Σ P , V ) and Figure 7 represents a set of geometric representatives for G ′ . Moreovereach of its generators which is not one of the δ ∂ j can be expressed as a composition of the other ones (we willsoon say that there is a relation among those generators), therefore a set G obtained from G ′ by removingone of the element of the form α i , β i or γ p , is still a generating set for Π (Σ P , V ). The height orders can bechosen arbitrarily. Note that G has cardinality 2 g − s + n ∂ , where g is the genus of Σ, s := |P| is thenumber of punctures and n ∂ := | π ( ∂ Σ) | is the number of boundary components.In the particular case where Σ has exactly one boundary component with one puncture on it (and eventualinner punctures), the generating graph of G is called the daisy graph . The daisy graph was first consideredin [AM95] in the context of classical lattice gauge field theory and in [AS96, Fai20b, Fai20a, BR19] in thequantum case. Figure 7.
The geometric representatives of a set of generators for Π (Σ P , V ). Proposition 2.14. [ [Kor20, Proposition 3 . ] If G is a set of generators of Π (Σ P , V ) , then the set A G generates S ω ( Σ ) as an algebra. The proof of Proposition 2.14 is an easy consequence of the cutting arc relations illustrated in Figure 8. igure 8. The figure illustrates how an application of the cutting arc relations permits toexpress any simple stated diagram in terms of the elements of A G . Here G = { β , β , β , β } are the generators of Figure 6. We draw dotted arrows to exhibit where we perform thecutting arc relations.We now define the notion of relations for a generating set G . Let F ( G ) denote the free semi-groupgenerated by the elements of G and let Rel G denote the subset of F ( G ) of elements of the form R = β ⋆. . .⋆β n such that s ( β i ) = t ( β i +1 ) and such that the path β . . . β n is trivial. We write R − := β − n ⋆ . . . ⋆ β − . Arelation R = β ⋆ . . . ⋆ β n ∈ Rel G is called simple if the β i admit representative as embedded curveswhose concatenation forms a contractible simple closed curve γ in Σ P whose orientation coincides withthe orientation of the disc bounded by γ . Note that ”being simple” depends on the choice of geometricrepresentatives of the generators. Definition 2.15.
A finite subset RL ⊂ Rel G is called a finite set of relations if its elements are simple andevery word R ∈ Rel G can be decomposed as R = β ⋆ R ε ⋆ . . . ⋆ R ε m m ⋆ β − , where R i ∈ RL , ε i ∈ {± } and β = β ⋆ . . . ⋆ β n ∈ F ( G ) is such that s ( β i ) = t ( β i +1 ). The pair P := ( G , RL ) is called a finite presentation of Π (Σ P , V ).As illustrated in the introduction, the small fundamental groupoid of the triangle T admits the finitepresentation with generating set G = { α, β, γ } and unique relation RL = { α ⋆ β ⋆ γ } .For a general connected open punctured surface Σ , the set G of Example 2.13 is the generating set of apresentation of Π (Σ P , V ) with no relation.2.3. Relations among the generators of the stated skein algebras.
We fix a connected open puncturedsurface Σ , a finite presentation P = ( G , RL ) of Π (Σ P , V ), and look for relations in S ω ( Σ ) among the elementsof A G . Definition 2.16. An oriented arc β is a non-closed connected simple diagram of Σ P together with anorientation plus an eventual height order of its endpoints in the case where they both lye in the sameboundary arc. We will denote by s ( β ) and t ( β ) its endpoints so that β is oriented from s ( β ) towards t ( β ).For ε, ε ′ ∈ {− , + } , we denote by β εε ′ ∈ S ω ( Σ ) the class of the stated diagram ( β, σ ) where σ ( s ( β )) = ε and σ ( t ( β )) = ε ′ .Note that to each oriented arc one can associate a path in Π (Σ P , V ) by first isotoping its endpoints to V and then taking its homotopy class. However a path in Π (Σ P , V ) can be associated to several distinctoriented arcs, so an oriented arc contains more information that a path in the small fundamental groupoid.We want to see the elements of G as pairwise non intersecting oriented arcs as illustrated in Figure 9.Recall that by Definition 2.12, any path α ∈ G is endowed with a geometric representative ϕ α whose image isan oriented arc α ⊂ Σ P so that the α pairwise do not intersect outside of V and their intersect transversallyin V . So each point v a ∈ V is endowed with a total order < v a on the set of its adjacent arcs (so the presentinggraph has a ciliated ribbon graph structure).The orientation of Σ P induces an orientation of its boundary arcs which, in turn, induces a total order < a on each boundary arc a , where v < a v if a is oriented from v towards v . After isotoping the α in a smallneighbourhood of each v a in such a way that the vertex order order < v a matches with the boundary arcorder < a as illustrated in Figure 9, we get a family of pairwise non-intersecting oriented arcs representingthe elements of G . Convention 2.17.
From now on, we consider the elements of G as pairwise non-intersecting oriented arcs. Definition 2.18.
Let α be an oriented arc, set v := s ( α ) and v := t ( α ) and denote by u and v theboundary arcs containing v and v respectively. The arc α is said • of type a if u = v ; igure 9. (1) An illustration of the local isotopy we perform to turn the set of edges ofa (ribbon) presenting graph into a set of pairwise non-intersecting oriented arcs. (2) Anexample in the case of the triangle. • of type b if u = v , h ( v ) < h ( v ) and v < u v ; • of type c if u = v , h ( v ) < h ( v ) and v < u v ; • of type d if u = v , h ( v ) < h ( v ) and v < u v ; • of type e if u = v , h ( v ) < h ( v ) and v < u v .Here h ( v ) represents the height of v ( h is the second projection Σ P × (0 , → (0 , Figure 10.
An illustration of the five types of oriented arcs.
Notations 2.19. (1) For α an oriented arc, write M ( α ) := (cid:18) α ++ α + − α − + α −− (cid:19) the 2 × S ω ( Σ ). The relations among the generators of S ω ( Σ ) that we will soon define are muchmore elegant when written using of the following matrix N ( α ) := M ( α ) , if α is of type a ; M ( α ) C , if α is of type b ; M ( α ) t C , if α is of type c ; C − M ( α ) , if α is of type d ; t C − M ( α ) , if α is of type e ;where t M denotes the transpose of M .(2) Let M a,b ( R ) the ring of a × b matrices with coefficients in some ring R (here R will be S ω ( Σ )).The Kronecker product ⊙ : M a,b ( R ) ⊗ M c,d ( R ) → M ac,bd ( R ) is defined by ( A ⊙ B ) i,kj,l = A ij B kl . Forinstance M ( α ) ⊙ M ( β ) = α ++ β ++ α ++ β + − α + − β ++ α + − β + − α ++ β − + α ++ β −− α + − β − + α + − β −− α − + β ++ α − + β + − α −− β ++ α −− β + − α − + β − + α − + β −− α −− β − + α −− β −− . (3) By abuse of notations, we also denote by τ the matrix of the flip map τ : v i ⊗ v j v j ⊗ v i , V ⊗ → V ⊗ , i.e. τ = . (4) For a 4 × X = ( X ijkl ) i,j,k,l = ± , we define the 2 × L ( X ) and tr R ( X ) by theformulas tr L ( X ) ba := X i = ± X ibia and tr R ( X ) ba := X i = ± X biai .
5) For M = (cid:18) a bc d (cid:19) , we set det q ( M ) := ad − q − bc and det q ( M ) := ad − q − bc . Lemma 2.20 (Orientation reversing formulas) . Let α be an oriented arc and α − be the same arc withopposite orientation. Then one has M ( α − ) = t M ( α ) . Therefore, one has (8) N ( α − ) = t N ( α ) , if α is of type a ; t C − t N ( α ) t C , if α is of type b or d ; C − t N ( α ) C , if α is of type c or e. Proof.
This is a straightforward consequence of the definitions. (cid:3)
Lemma 2.21 (Height reversing formulas) . Let α be an oriented arc with both endpoints in the same boundaryarcs let α be the same arc with reversing height order for its endpoints. Then one has: (9) M ( α ) = tr R (cid:0) R − ( t C − ⊙ M ( α ) t C ) (cid:1) , if α is of type b ;tr L (cid:0) R − ( M ( α ) C ⊙ C − ) (cid:1) , if α is of type c ;tr L (cid:0) ( t C − M ( α ) ⊙ t C ) R (cid:1) , if α is of type d ;tr R (cid:0) ( C ⊙ C − M ( α )) R (cid:1) , if α is of type e. Proof.
Equations (9) are obtained by using the boundary skein relations (6). Figure 11 illustrates the proofin the case where α is of type e. The other cases are similar and left to the reader.In Figure 11, we represented the curve α in blue to emphasize that, despite what the picture suggests,the curve can be arbitrarily complicated. Since the boundary arcs relation only involves the intersection of α with a small neighborhood (a bigon) of the boundary arc (colored in grey), how is the blue part of thefigure does not matter. Figure 11.
An illustration of the proof of Equation (9) in the case where α is of type e. (cid:3) Remark . Reversing the orientation of an arc exchanges (type b) ↔ (type c) and (type d) ↔ (typee) whereas reversing the height order exchanges (type b) ↔ (type e) and (type c) ↔ (type d). ThereforeEquations (8) and (9) permit to switch between the types b, c, d, e ; this will permit us to write the arcsexchange and trivial loops relations in a simpler form by specifying the type of arc. Lemma 2.23 (Trivial loops relations) . Let R = β k ⋆ . . . ⋆ β be a simple relation. Suppose that all arcs β i are either of type a or d . Then (10) = CM ( β k ) C − M ( β k − ) C − . . . C − M ( β ) . Proof.
Equation (10) is a consequence of the trivial arc and cutting arc relations illustrated in Figure 12in the case of the triangle with presentation whose generators are the arcs { α, β, γ } drawn in Figure 1 andthe relation is α ⋆ β ⋆ γ = 1. Figure 12 shows the equality between the matrix coefficients of C − and M ( α ) C − M ( β ) C − M ( γ ).Let us detail the proof in the general case. Since β i is either of type a or d , it can be represented by atangle T ( β i ) such that the height of the source endpoint of β i (say v i ) is smaller than the height of its targetendpoint (say w i ); said differently h ( v i ) < h ( w i ). One can further choose the T ( β i ) so that T ( β i +1 ) lies on igure 12. An illustration of the proof of Equation (10) in the case of the triangle.the top of T ( β i ) (so h ( v ) < h ( w ) < h ( v ) < . . . < h ( w k )). Let T be the tangle made of the disjoint unionof the T ( β i ). By the assumption that R is a simple relation, we can suppose that T is in generic position (inthe sense of Section 2 .
1) and that its projection diagram is simple. Fix i, j ∈ {− , + } and let α be a trivialarc with endpoints s ( α ) = v and t ( α ) = w k so that α can be isotoped (relatively to its boundary) to anarc inside ∂ Σ P . One the one hand, the trivial arc relation (3) gives the equality α ij = ( C − ) ji . On the otherhand, the cutting arc relation (4) gives the equality( C − ) ji = α ij = X s ∈ St( T ) ,s ( v )= i,s ( w k )= j [ T, s ]( C − ) s ( v ) s ( w ) ( C − ) s ( v ) s ( w ) . . . ( C − ) v k s ( w k − ) = X µ ,...µ k − = ± M ( β k ) jµ ( C − ) µ µ M ( β k − ) µ µ . . . M ( β ) µ k − i = (cid:0) M ( β k ) C − M ( β k − ) C − . . . M ( β ) (cid:1) ji . This concludes the proof. (cid:3)
Let α, β be two non-intersecting oriented arcs. Denote by a, b, c, d the boundary arcs containing s ( α ) , t ( α ) , s ( β ) , t ( β )respectively. Reversing the orientation and the height order of α or β if necessary, we have ten differentpossibilities illustrated in Figure 13. Figure 13.
Ten configurations for two non-intersecting oriented arcs.
Lemma 2.24. (i)
If the elements of { a, b, c, d } are pairwise distinct, one has (11) N ( α ) ⊙ N ( β ) = τ ( N ( β ) ⊙ N ( α )) τ. (ii) When a = c , { a, b, d } has cardinal and s ( β ) < a s ( α ) , one has (12) N ( α ) ⊙ N ( β ) = τ ( N ( β ) ⊙ N ( α )) R . (iii) When a = c = b = d and s ( β ) < a s ( α ) , t ( α ) < b t ( β ) , one has (13) N ( α ) ⊙ N ( β ) = R − ( N ( β ) ⊙ N ( α )) R . (iv) When a = c = b = d and s ( β ) < a s ( α ) , t ( β ) < b t ( α ) , (14) N ( α ) ⊙ N ( β ) = R ( N ( β ) ⊙ N ( α )) R . (v) When b = c = d = a and s ( β ) < a t ( β ) < a t ( α ) and h ( s ( β )) < h ( t ( β )) , one has (15) N ( α ) ⊙ N ( β ) = R − ( N ( β ) ⊙ ) R ( N ( α ) ⊙ ) . vi) When b = c = d = a and t ( α ) < a t ( β ) < a s ( β ) and h ( s ( β )) < h ( t ( β )) < h ( t ( α )) , one has (16) N ( α ) ⊙ N ( β ) = R − ( N ( β ) ⊙ ) R ( N ( α ) ⊙ ) . (vii) When b = c = d = a and t ( β ) < a t ( α ) < a s ( β ) and h ( s ( β )) < h ( t ( α )) < h ( t ( β )) , one has (17) N ( α ) ⊙ N ( β ) = R ( N ( β ) ⊙ ) R ( N ( α ) ⊙ ) . (viii) When a = b = c = d and s ( β ) < a s ( α ) < a t ( β ) < a t ( α ) and h ( s ( β )) < h ( s ( α )) < h ( t ( β )) < h ( t ( α )) , one has (18) ( ⊙ N ( α )) R − ( ⊙ N ( β )) R − = R ( ⊙ N ( β )) R − ( ⊙ N ( α )) . (ix) When a = b = c = d and s ( β ) < a t ( β ) < a s ( α ) < a t ( α ) and h ( s ( β )) < h ( t ( β )) < h ( s ( α )) < h ( t ( α )) , one has (19) R − ( ⊙ N ( α )) R ( ⊙ N ( β )) = ( ⊙ N ( β )) R − ( ⊙ N ( α )) R . (x) When a = b = c = d and s ( α ) < a s ( β ) < a t ( β ) < a t ( α ) , and h ( s ( α )) < h ( s ( β )) < h ( t ( β )) < h ( t ( α )) , one has (20) ( ⊙ N ( α )) R − ( ⊙ N ( β )) R = R ( ⊙ N ( β )) R − ( ⊙ N ( α )) . Proof.
Equation (11) says that in case ( i ) any α ij commutes with any β kl , which is obvious. Equations (12),(13), (14) in cases ( ii ) , ( iii ) and ( iv ) are straightforward consequences of the height exchange relation (5).All other cases will be derived using the boundary skein relations (6). As in the proof of Lemma 2.21, wewill color the arcs α and β in red and blue to remind the reader that they might be much more complicatedthan what they look in the picture: in the computations we perform while using the boundary skein relation,we only care about the restriction of the diagrams (depicted in grey) in a small bigon in the neighborhoodof the boundary arc a and not of the actual shape of the blue and red parts.Equations (15) and (16) in cases ( v ) and ( vi ) are proved in a very similar way; we detail the proof of (16)and leave (17) to the reader. In case ( vi ), one has:( M ( α ) ⊙ M ( β )) ijkl = α ki β lj = == X a,b,c,d,e,f = ± ( R − ) ijfd M ( β ) fe C ec R cdab M ( α ) ak ( C − ) bl = (cid:0) R − ( M ( β ) C ⊙ ) R ( M ( α ) ⊙ C − ) (cid:1) ijkl . To handle cases ( vii ) to ( x ), we introduce the 4 × V = ( V ijkl ) i,j,k,l ∈{− , + } , where V ijkl = [ α ∪ β, σ ijkl ] ∈ S ω ( Σ ) is the class of the simple diagram α ∪ β with state σ ijkl sending t ( α ) , t ( β ) , s ( α ) and s ( β )to i, j, k and l respectively. Here the height order of the points of ∂ ( α ∪ β ) is given by the boundary arcorientation drawn in Figure 13. The trick is to compute V in two different ways and then equating the twoobtained formulas.In case ( vii ), on the one hand, we first prove the equality V = τ ( M ( β ) C ⊙ ) R ( M ( α ) ⊙ C − ) as follows: V ijkl = = = (cid:0) ( M ( β ) C ⊙ ) R ( M ( α ) ⊙ C − ) (cid:1) jikl . On the other hand, we prove the equality V = τ R − ( M ( α ) ⊙ M ( β )) as follows: V ijkl = = = (cid:0) R − ( M ( α ) ⊙ M ( β )) (cid:1) jikl . So we get the equality R − ( M ( α ) ⊙ M ( β )) = ( M ( β ) C ⊙ ) R ( M ( α ) ⊙ C − )(= τ V ) and Equation (17)follows. n case ( vii ), on the one hand, we first prove the equality V = τ ( C ⊙ M ( α )) R − ( ⊙ C − M ( β )) asfollows: V ijkl = = = (cid:0) ( C ⊙ M ( α )) R − ( ⊙ C − M ( β )) (cid:1) jikl . On the other hand, we prove the equality V = τ ( C ⊙ C ) R ( ⊙ C − M ( β )) R − ( ⊙ C − M ( α )) R as follows: V ijkl = = = (cid:0) ( C ⊙ C ) R ( ⊙ C − M ( β )) R − ( ⊙ C − M ( α )) R (cid:1) jikl . Equation (18) follows by equating the two obtained expressions for V .In case ( x ), on the one hand, we first prove the equality V = ( C ⊙ M ( α )) R − ( ⊙ C − M ( β )) R asfollows: V ijkl = = = (cid:0) ( C ⊙ M ( α )) R − ( ⊙ C − M ( β )) R (cid:1) ijkl . On the other hand, we prove the equality V = ( C ⊙ C ) R ( ⊙ C − M ( β )) R − ( ⊙ C − M ( α )) as follows: V ijkl = = = (cid:0) ( C ⊙ C ) R ( ⊙ C − M ( β )) R − ( ⊙ C − M ( α )) (cid:1) ijkl . Therefore, we obtain the following equality that will be used in the proof of Lemma 2.25:(21) V = ( C ⊙ M ( α )) R − ( ⊙ C − M ( β )) R = ( C ⊙ C ) R ( ⊙ C − M ( β )) R − ( ⊙ C − M ( α )) . Equation (20) follows.In case ( ix ), we slightly change the strategy. We define the 4 × W = ( W ijkl ) i,j,k,l ∈{− , + } by W ijkl := . We first prove the equality W = ( C ⊙ M ( β )) R − ( ⊙ C − M ( α )) as follows: W ijkl = = = (cid:0) ( C ⊙ M ( β )) R − ( ⊙ C − M ( α )) (cid:1) ijkl . Next, we prove the equality W = ( C ⊙ C ) R − ( ⊙ C − M ( α )) R ( ⊙ C − M ( β )) R − as follows: W ijkl = = = (cid:0) ( C ⊙ C ) R − ( ⊙ C − M ( α )) R ( ⊙ C − M ( β )) R − (cid:1) ijkl . Equation (19) follows by equating the two obtained expressions for W . This concludes the proof. (cid:3) emma 2.25 (q-determinant relations) . Let α be an oriented arc. Then (22) det q ( N ( α )) = 1 , if α is of type a, and det q ( N ( α )) = 1 , else . Proof.
First suppose that α is of type a. Applying the trivial arc and cutting arc relation, we obtain:( C − ) − + = − + = ( C − ) + − + ( C − ) − + , which is equivalent to the equation α ++ α −− − q − α + − α − + = 1 as claimed. Next we suppose that α isof type d. Let β be an arc isotope to and disjoint from α , placed as in the configuration (x) of Figure 13.Consider the matrix V = ( V ijkl ) i,j,k,l ∈{− , + } , where V ijkl = [ α ∪ β, σ ijkl ] ∈ S ω ( Σ ) is the class of the simplediagram α ∪ β with state σ ijkl sending t ( α ) , t ( β ) , s ( α ) and s ( β ) to i, j, k and l respectively, like in the proofof Lemma 2.24 (i.e. V ijkl = ). Again, using the trivial arc and cutting arc relation, we obtain: C + − = = C − + + C + − ⇔ V + −− + − qV + − + − = 1 . Next, by developing the matrix coefficients in the equalities (21), we find the equalities V + −− + = qα + − α − + + A − and V + − + − = q − α −− α ++ − A − . Putting these equalities together, we find(23) α −− α ++ − q α + − α − + = A. Now developing Equation (20), we obtain α + − α − + = α − + α + − , therefore:det q ( N ( α )) = det q (cid:18) − ω − α − + − ω − α −− ω − α ++ ω − α + − (cid:19) = − A α − + α + − + A − α −− α ++ = 1 . Now, if α is of type e, then α − is of type d. A simple computation shows that if M = (cid:18) a bc d (cid:19) is suchthat ad = da then det q ( M ) = det q (cid:0) C − t M C (cid:1) , so we deduce the q-determinant formula for α of type efrom the facts that it holds for α − , from the orientation reversing formula in Lemma 2.20 and from theequality α + − α − + = α − + α + − .Suppose that α is of type c and choose k = Z [ ω ± ]. Recall from Section 2 . θ . The image θ ( α ) is of type d, so applying θ to Equation (23), we obtain that:(24) α ++ α −− − q − α − + α + − = A − . By Remark 2.6, since Equation 24 holds for k = Z [ ω ± ], it also holds for any other ring. Also using θ , wefind that α + − α − + = α − + α + − and the equation det q ( N ( α )) = 1 follows. Eventually, when α is of type b,we deduce the q-determinant relation from the facts that it holds for α − (of type c), from the orientationreversing formulas of Lemma 2.20 and from the identity α + − α − + = α − + α + − . (cid:3) Definition 2.26.
Let P = ( G , RL ) be a finite presentation of Π (Σ P , V ). By Proposition 2.14, the set A G generates S ω ( Σ ) and we have found three families of relations:(1) For each α ∈ G we have the either the relation det q ( N ( α )) = 1 or det q ( N ( α )) = 1 by Equation (22)in Lemma 2.25; we call them the q-determinant relations .(2) For each R ∈ RL , we have four relations obtained by considering the matrix coefficients in Equation(10) in Lemma 2.23; we call them trivial loops relations .(3) For each pair ( α, β ) of elements in G , we have 16 relations obtained by considering the matrixcoefficients in one of the Equations (11), . . . , (20) of Lemma 2.24 after having possibly replaced α or β by α − or β − , if necessary, and using the inversion formula (8); we call them arcs exchangerelations . . Proof of Theorems 1.1 and 1.2
In this section, we prove Theorems 1.1 and 1.2. Let L ω ( P ) be the algebra generated by the elements of G modulo the q-determinant, trivial loops and arcs exchange relations and write Ψ : L ω ( P ) → S ω ( Σ ) theobvious algebra morphism. By Proposition 2.14, Ψ is surjective and we need to show that Ψ is injective toprove Theorem 1.1. We cut the proof of Theorem 1.1 in three steps: (1) first in Step 1, we show that it issufficient to make the proof in the case where P has no relation (as in Example 2.13); (2) in this particularcase, the finite presentation defining L ω ( P ) is inhomogeneous quadratic and we will use the Diamond Lemmato extract PBW bases of L ω ( P ) and to prove it is Koszul; in Step 2 we extract the re-written rules and theirleading terms from the q-determinant and arc exchange relations and exhibit the associated spanning family B G ⊂ L ω ( P ); (3) eventually in Step 3, we show that the image by Ψ of B G is a basis, this will prove both theinjectivity of Ψ and the fact that B G is a Poincar´e-Birkhoff-Witt basis and conclude the proofs of Theorems1.1 and 1.2.3.1. Step : Reduction to the case where P has no relation. Let Γ be the presenting graph of P and consider its fundamental groupoid Π (Γ): the objects of Π (Γ) are the vertices of Γ ( i.e. the set V )and the morphisms are compositions α ε k k . . . α ε where α i ∈ G . The inclusion Γ ⊂ Σ P induces a functor F : Π (Γ) → Π (Σ P , V ) which is the identity on the objects. The fact that G is a set of generators impliesthat F is full and P has no relations if and only if F is faithful. Fix v ∈ V . For a relation R ∈ RL ofthe form R = β k ⋆ . . . ⋆ β , the base point of R is s ( β ) = t ( β k ). By inspecting the trivial loop relation(10), we see that changing a relation R by a relation β ⋆ R ⋆ β − does not change the algebra L ω ( P ). SinceΣ P is assumed to be connected, we can suppose that all relations in RL have the same base point v , soeach relation R = β k ⋆ . . . ⋆ β induces an element [ R ] = β k . . . β ∈ π (Γ , v ). The functor F induces asurjective group morphism F v : π (Γ , v ) → π (Σ P , v ) and the fact that RL is a set of relations impliesthat { [ R ] , R ∈ RL } generates ker( F v ). Since π (Γ , v ) is a free group, so is ker( F v ). Let R , . . . , R m ∈ RL be such that { [ R ] , . . . , [ R m ] } is a minimal set of generators for the free group ker( F v ). For each R i , choosean element β i ∈ G such that either β i or β − i appears in the expression of R i such that the set G ′ obtainedfrom G by removing the β i ’s is a generating set. So if Γ ′ is the presenting graph of G ′ , the morphism F ′ v : π (Γ ′ , v ) → π (Σ P , v ) is injective, so the functor F ′ : Π (Γ ′ ) → Π (Σ P , V ) is faithful and P ′ := ( G ′ , ∅ )is a finite presentation of Π (Σ P , V ) with no relations.The inclusion G ′ ⊂ G induces an algebra morphism e ϕ : T [ G ′ ] ֒ → T [ G ] on the free tensor algebras generatedby G ′ and G respectively and e ϕ sends q-determinant and arc exchange relations to q-determinant and arcexchange relations, so it induces an algebra morphism ϕ : L ω ( P ′ ) → L ω ( P ). Lemma 3.1.
The morphism ϕ is an isomorphism.Proof. To prove the surjectivity, we need to show that for each removed path β i ∈ G \ G ′ , the stated arcs( β i ) εε ′ can be expressed as a polynomial in the stated arcs ( α ± ) µµ ′ for α ∈ G ′ . This follows from the trivialloop relation (10) associated to the relation R i ∈ RL containing β ± i . The injectivity of ϕ is a straightforwardconsequence of the definition. (cid:3) Step : Poincar´e-Birkhoff-Witt bases and Koszulness.Convention 3.2. In the rest of the section, we now suppose that P = ( G , ∅ ) is a presentation with norelations and that every arc in G is either of type a, c or d .Note that the convention on the type of the generators is not restrictive but purely conventional since wecan always replace a generator α by α − without changing the set A G of generators of S ω ( Σ ).Since P has no relation, the defining presentation of L ω ( P ) contains only q-determinant and arc exchangerelations. All these relations are quadratic (inhomogeneous) in the generators A G and we want to apply theDiamond Lemma to prove that L ω ( P ) is Koszul. Reminder on the Diamond Lemma for PBW bases ollowing the exposition in Section 4 of [LV12], we briefly recall the statement of the Diamond Lemmafor PBW bases.Let V be a free finite rank k -module, denote by T ( V ) := ⊕ n ≥ V ⊗ n the tensor algebra and fix R ⊂ V ⊗ a finite subset. The quotient algebra A := T ( V ) . ( R ) is called a quadratic algebra . Let { v i } i ∈ I be a totallyordered basis of V and write I = { , . . . , k } so that v i < v i +1 . Then the set J := F n ≥ I n (where I = { } )is totally ordered by the lexicographic order and the set of elements v i = v i . . . v i n , for i = ( i , . . . , i k ), formsa basis of T ( V ). We suppose that the elements r ∈ R (named relators) have the form r = v i v j − X ( k,l ) < ( i,j ) λ ijkl v k v l . The term v i v j is called the leading term of r . We assume that two distinct relators have distinct leadingterms. Define the family(25) B := { v i . . . v i n | so that v i k v i k +1 is not a leading term , ∀ ≤ k ≤ n − } , and denote by B (3) ⊂ B the subset of elements of length 3 (of the form v i v i v i ). Obviously the set B spans A . Theorem 3.3 (Diamond Lemma for PBW bases: Bergman [Ber78], see also [LV12] Theorem 4 . . . If B (3) is free, then B is a (Poincar´e-Birkhoff-Witt) basis and A is Koszul. The arc exchange relations defining L ω ( P ) are quadratic, however the q -determinant relations are not(because of the 1 in det q ( N ( α )) = 1), so L ω ( P ) is not quadratic but rather inhomogeneous quadratic. An inhomogeneous quadratic algebra is an algebra of the form A := T ( V ) . ( R ) , where R ⊂ V ⊗ ⊕ V ⊕ k ⊂ T ( V ).We further make the assumptions that ( ql ) : R ∩ V = { } and ( ql ) : ( R ⊗ V + V ⊗ R ) ∩ V ⊗ ⊂ R ∩ V ⊗ . Thehypothesis ( ql ) says that one cannot create new relations by adding an element to R , so it is not restrictive.Like before, we fix an ordered basis { v i } i ∈ I of V and suppose that the relators of R have the form(26) r = v i v j − X ( k,l ) < ( i,j ) λ ijkl v k v l − c i,j , where c i,j are some scalars and we suppose that two distinct relators have distinct leading terms. Theassociated quadratic algebra q A is the algebra with same generators v i but where the relators have beenchanged by replacing the scalars c i,j by 0. Let q B ⊂ q A and B ⊂ A be the two generating families definedby Equation (25).
Theorem 3.4 ([LV12] Theorem 4 . . . Suppose that q B (3) ⊂ q A is free, then both q B and B are (PBW)bases of q A and A respectively and both q A and A are Koszul. There exists a linear surjective morphism ϕ : q A → A sending the generating family q B to B (see [LV12,Section 4 . . B is a basis of A , then q B is free, therefore Theorem 3.4 implies that A is Koszul.Therefore, we have the Theorem 3.5. If B is a basis of A , then A is Koszul. The relators of the stated skein presentations and PBW bases
For α ∈ G , we write B ( α ) = { ( α ++ ) a ( α + − ) b ( α −− ) c , a, b, c ≥ } ∪ { ( α ++ ) a ( α − + ) b ( α −− ) c , a, b, c ≥ } ⊂ L ω ( P ) . Fix a total order < on the set G of generators and index its elements as G = { α , . . . , α n } , where α i < α i +1 .Let B G := { m m . . . m n | m i ∈ B ( α i ) } ⊂ L ω ( P ) . We want to apply Theorem 3.5 to prove that L ω ( P ) is Koszul. By definition, L ω ( P ) is an inhomogeneousquadratic algebra whose set of generators is A G = { α ij | α ∈ G , i, j = ±} and whose relations are the arcsexchange and q -determinant relations.We first define a total order ≺ on A G by imposing that α ab ≺ β cd if α < β and that α ++ ≺ α + − ≺ α − + ≺ α −− . he goal of this subsection is to rewrite the q-determinant and arc exchange relations such that theydefine a set of relators of the form (26) whose leading terms are pairwise distinct, satisfying ( ql ) and ( ql )and such that the set of leading terms is(27) LeadingTerms := { α ab β cd | such that either ( α > β ) or ( α = β and either a < c or b < d ) } . The set B G is the generating set defined by (25) with this set of leading terms ( i.e. B G is the set ofelements v . . . v n where v i ∈ A G and v i v i +1 is not in LeadingTerms). At this stage, it will become clear that B G spans L ω ( P ). Once we’ll perform this task, we will prove in Step 3 that B G if free by showing that itsimage through Ψ : L ω ( P ) → S ω ( Σ ) is a basis of S ω ( Σ ). This will imply that Ψ is an isomorphism (so willprove Theorem 1.1) and Theorem 3.5 will imply that L ω ( P ) is Koszul (so it will prove Theorem 1.2).Consider two distinct generators α, β ∈ G such that α > β . For each a, b, c, d ∈ {±} , we have an arcexchange relation of the form α ab β cd = X ijkl = ± c i,j,k,la,b,c,d β ij α kl , where c i,j,k,la,b,c,d are some scalars. We associate the relator r = α ab β cd − P ijkl = ± c i,j,k,la,b,c,d β ij α kl , whose leadingterm is α ab β cd (because α > β implies that α ab β cd ≻ β ij α kl ) and denote by R α,β the set (of cardinal 16) ofsuch relators.Now suppose that α ∈ G is of type a . The set of relations between the generators α ij are given by M ( α ) ⊙ M ( α ) = R − ( M ( α ) ⊙ M ( α )) R , and det q ( M ( α )) = 1 . Note that in this case, the subalgebra of L ω ( P ) generated by the α ij is isomorphic to O q [SL ] ∼ = S ω ( B ). Werewrite those relations as follows:(Ra) α + − α ++ = qα ++ α + − , α − + α ++ = qα ++ α − + ,α −− α + − = qα + − α −− , α −− α − + = qα − + α −− ,α + − α − + = qα ++ α −− − q, α − + α + − = qα ++ α −− − q,α −− α ++ = q α ++ α −− + 1 − q . The associated set of relators R α is defined by assigning to each of the seven equalities of the form x = y in the system Ra , the relator r := x − y with leading term x . Note that the set of leading terms of theelements of R α is the set of elements α ab α cd such that either a < c or b < d .Now suppose that α ∈ G is of type d . The set of relations between the generators α ij are given by( ⊙ N ( α )) R − ( ⊙ N ( α )) R = R ( ⊙ N ( α )) R − ( ⊙ N ( α )) , and det q ( N ( α )) = 1 , where N ( α ) = C − M ( α ). These relations generate the same ideal as the following set of relations:(Rd) α − + α ++ = α ++ α − + + ( q − q − ) q α + − α −− , α + − α ++ = q α ++ α + − ,α −− α − + = α − + α −− + ( q − q − ) q α + − α −− , α −− α + − = q α + − α −− ,α + − α − + = α ++ α −− − ( q − q − ) α − − A, α − + α + − = α ++ α −− − ( q − q − ) α − − A,α −− α ++ = q α ++ α −− − q ( q − q − ) α − + A (1 − q ) . As before, we denote by R α the set of relators obtained from system (Rd) by assigning to each of theseven equalities of the form x = y in the system Ra , the relator r := x − y with leading term x . Again, theset of leading terms of the elements of R α is the set of elements α ab α cd such that either a < c or b < d .For α ∈ G of type c , the set of relations between the elements α ij can be obtained from the system (Rd)using the reflection anti-involution. Once re-arranging the terms, we get the system of relations: Rc) α − + α ++ = α ++ α − + + ( q − q − ) α + − α −− , α + − α ++ = q α ++ α + − ,α −− α − + = α − + α −− + ( q − q − ) α + − α −− , α −− α + − = q α + − α −− ,α + − α − + = q α ++ α −− − A , α − + α + − = q α ++ α −− − A ,α −− α ++ = q α ++ α −− + ( q − q − ) α − + A − (1 − q ) . Like previously, we denote by R α the associated set of relators and note that the set of leading terms isthe set of elements α ab α cd such that either a < c or b < d .Let V be the free k -module with basis A G and R ⊂ k ⊕ V ⊗ ⊂ T ( V ) be the union of the sets ofrelators R α,β and R α , where α, β ∈ G and α > β . Then L ω ( P ) = T ( V ) . ( R ) , the leading terms of R arepairwise distinct and they form the set LeadingTerms of Equation (27) and the hypotheses ( ql ) and ( ql )are obviously satisfied. Therefore, if we prove that B G is a basis of L ω ( P ) then Theorem 3.5 would implythat L ω ( P ) is Koszul.3.3. Step : Injectivity of Ψ . Denote by B G ⊂ S ω ( P ) the image of B G by Ψ : L ω ( P ) → S ω ( Σ ). Theorem 3.6.
The set B G is a basis of S ω ( Σ ) . Corollary 3.7. (1)
The morphism
Ψ : L ω ( P ) → S ω ( Σ ) is an isomorphism. (2) The family B G is a PBW basis and S ω ( Σ ) is Koszul. The fact that B G spans linearly S ω ( Σ ) follows from the surjectivity of Ψ (so follows from Proposition2.14), however we will reprove this fact. The proof of Theorem 3.6 is divided in two steps: first we introduceanother family B G + ⊂ S ω ( Σ ) and prove that B G + is free by relating it to the basis B . Next we use a filtrationof S ω ( Σ ) to deduce that B G is free from the fact that B G + is free.For α ∈ G and n ≥
0, we denote by α h n i the simple diagram made of n pairwise non-intersecting copies of α . For n ∈ N G , we denote by D ( n ) the simple diagram F α ∈ G α h n ( α ) i . Denote by v and w the two endpointsof α and by a and b the (non necessary distinct) boundary arcs containing v and w respectively. Write v , . . . , v n and w , . . . , w n the endpoints of α h n i such that v i < a v i +1 and w i < b w i +1 (so v i and w i are notnecessary the boundary points of the same component of α h n i ). A state s ∈ St( D ( n )) is positive if for all α ∈ G and for all i ≤ j one has s ( v i ) ≤ s ( v j ) and s ( w i ) ≤ s ( w j ); we let St + ( D ( n )) denote the set of positivestates. Definition 3.8.
We denote by B G + ⊂ S ω ( Σ ) the set of classes [ D ( n ) , s ] for n ∈ N G and s ∈ St + ( D ( n )). Notations 3.9.
Let (
D, s ) be a stated diagram and a a boundary arc. We denote by d a ([ D, s ]) ∈ N thenumber of pairs ( v, w ) in ∂ a D such that v < a w and ( s ( v ) , s ( w )) = (+ , − ) (recall that the orientation of Σ P induces an orientation of a which, in turns, induces the order < a ). We also write d ([ D, s ]) = P a d a ([ D, s ]).Note that s is o + -increasing if and only if d ([ D, s ]) = 0.Let us develop an element b ∈ B G + in the basis B as b = P i a i [ D i , s i ]. We denote by S ( b ) ⊂ B the set ofbasis elements [ D i , s i ] such that a i = 0. Lemma 3.10. (1)
There exists a unique element m ( b ) = [ D , s ] ∈ S ( b ) such that for all [ D i , s i ] ∈ S ( b ) such that [ D i , s i ] = [ D , s ] , one has | ∂D i | > | ∂D | . (2) The map m : B G + → B is injective. (3) The family B G + is a basis.Proof. (1) Let b = [ D ( n ) , s ] ∈ B G + and let us define m ( b ). If s is o + -increasing, then b ∈ B so m ( b ) = b satisfiesthe property. Else d := d ( b ) > a and a pair ( v, w ) of consecutive points of ∂ a D ( n ) such that ( s ( v ) , s ( w )) = (+ , − ) and v < a w . By gluing the points v and w together and then pushingit in the interior of Σ P (that is by performing the local move − + ), we get a new stated diagram( D , s ) such that d ([ D , s ]) = d ( b ) −
1. By performing the above move consecutively, we get a sequenceof stated diagrams ( D ( n ) , s ) = ( D , s ) ( D , s ) . . . ( D d , s d ) with d [ D i +1 , s i +1 ] = d ( D i , s i ) −
1, so m ( b ) := [ D d , s d ] ∈ B (see Figure 14 for an example). The skein relation + = q + − + ω , shows that [ D i , s i ] = q [ D ′ i , s ′ i ] + ω [ D i +1 , s i +1 ], where | ∂D ′ i | = | ∂D i | > | ∂D i +1 | so the first assertion followsby induction.(2) To prove that m : B G + → B is injective, we construct a right inverse g : B → B G + such that g ◦ m = id .Let [ D, s ] ∈ B and let us define g ([ D, s ]). Obviously, g sends the class of the empty diagram to itself, sowe suppose that D is not empty. First suppose that D is connected and consider a path α D ∈ Π (Σ P , V )representing D . Since P = ( G , ∅ ) is a finite presentation of Π (Σ P , V ) without relation, the path α D decomposes in a unique way as α D = α ε i . . . α ε n i n , where α i k ∈ G . Let n D ∈ N G be such that n D ( α ) isthe number of times α appears in the decomposition of α D . Now, if D is not connected and has connectedcomponents D , . . . , D n , we set n D := P ni =1 n D i . By applying the cutting arc relation= ω − − + − q + − consecutively, we write [ D, s ] as a linear combination:[
D, s ] = X s ∈ St( D ( n D )) α s [ D ( n D ) , s ] . Note that because [
D, s ] ∈ B , the stated diagram ( D, s ) contains no trivial arc by definition, so only thepositive states s ∈ St + ( D ( n D )) have a non-vanishing coefficient α s = 0. In particular, we have proved that B G + generates S ω ( Σ ). We define g ([ D, s ]) as the element [ D ( n D ) , s ] such that α s = 0 and d ([ D ( n D ) , s ]) isminimal. Note that it obtained from [ D, s ] by a series of local moves − + which are the inverse ofthe local moves used to define m ([ D ( n D ) , s ]), so g ◦ m = id . Compare the Examples of Figure 8 and 14 foran illustration.(3) It remains to prove that B G + is free. Consider an arbitrary total order ≺ on B such that if | ∂D | < | ∂D ′ | then [ D, s ] ≺ [ D ′ , s ′ ] for any [ D, s ] , [ D ′ , s ′ ] in B . By contradiction, suppose there exists a non empty finitefamily { b + i } i ∈ I of elements of B G + and a family of non-vanishing scalars { x i } i ∈ I such that(28) X i ∈ I x i b + i = 0 . Let i ∈ I be such that m ( b i ) is the minimum for ≺ of the set { m ( b i ) , i ∈ I } . By developing each b + i inEquation (28) in the basis B , we get a vanishing linear combination P j y j b j = 0 of elements b j ∈ B . Since B is free, we have y j = 0 for all j . Let j be such that b j = m ( b i ). It follows from assertions (1) and (2)that y j = x i , so x i = 0 and we have a contradiction. Figure 14.
An element b ∈ B G + and its associated element m ( b ) ∈ B . Here G = { β , β , β , β } are the generators of Figure 6 (cid:3) We now want to deduce that B G is a basis from the fact that B G + is a basis. The argument is based onthe use of an algebra filtration of S ω ( Σ ) that we now introduce. Definition 3.11.
For n ∈ N G , we let | n | := P α ∈ G n ( α ). For a class [ D ( n ) , s ], we set k [ D ( n ) , s ] k :=( | n | , − d ([ D ( n ) , s ])) ∈ N × Z . Denote by < the lexicographic order on N × Z , i.e. ( k , k ) < ( k ′ , k ′ ) if either k < k ′ or k = k ′ and k < k ′ . Eventually, to k = ( k , k ) ∈ N × Z we associate the submodule F k := Span ([ D ( n ) , s ] , such that k [ D ( n ) , s ] k ≤ k ) . n order to prove that the {F k } form an algebra filtration, the following elementary observation will bequite useful: Lemma 3.12.
Let
T, T ′ be two tangles in Σ P × (0 , which are isotopic through an isotopy that does notpreserves the height orders. Let s ∈ St( T ) and s ′ ∈ St( T ′ ) be two states such that for all boundary arc a , if ∂ a T = { v , . . . , v n } and ∂ a T ′ = { w , . . . , w n } are ordered such that h ( v i ) < h ( v i +1 ) and h ( w i ) < h ( w i +1 ) ,then one has s ( v i ) = s ′ ( w i ) for all i ∈ { , . . . , n } . Then one has (29) [ T, s ] = ω n [ T ′ , s ′ ] + X σ ∈ St( T ′ ) ,d ([ T ′ ,σ ]) Let b ∈ B G , so by definition b = b α . . . b α n , where b α i ∈ B ( α i ), that is one has either b α i = α a i ++ α b i + − α c i −− or b α i = α a i ++ α b i − + α c i −− , for some a i , b i , c i ≥ 0. Let n ∈ N G be defined by n ( α i ) := a i + b i + c i .Let T ( n ) be the tangle underlying D ( n ). Let ( T, s ) be a stated tangle (unique up to isotopy) such that b = [ T, s ], so that T ( n ) is obtained from T by an isotopy that does not necessary preserve the height order.Eventually we define the element b + := [ T ( n ) , s + ] ∈ B G + , where s + ∈ St + ( T ( n )) is the unique state such that( T, s ) and ( T ( n ) , s + ) satisfies the assumption of Lemma 29. Note that the induced map ( · ) + : B G → B G + ,sending b to b + , is a bijection. Lemma 3.14. (1) For k , k ′ ∈ N × Z , one has F k · F k ′ ⊂ F k + k ′ . (2) For b ∈ B G , one has (30) b = ω n b + + lower terms , where n ∈ Z and ’lower terms’ is a linear combination of basis elements b + i ∈ B G + such that k b + i k < k b + k . Note that the second assertion of Lemma 3.14 implies that B G spans S ω ( Σ ) so reproves Proposition 2.14. Proof. (1) Let x := [ T ( n ) , s ] and y := [ T ( n ′ ) , s ′ ] and denote by ( T ( n ) ∪ T ( n ′ ) , s ∪ s ′ ) the stated tangle obtainedby stacking ( T ( n ) , s ) on top of ( T ( n ′ ) , s ′ ), so that x · y = [ T ( n ) ∪ T ( n ′ ) , s ∪ s ′ ]. The tangles T ( n ) ∪ T ( n ′ ) and T ( n + n ′ ) differ by an isotopy that does not necessary preserve the height orders, so Lemma 3.12 implies that x · y is a linear combination of elements of the form [ D ( n + n ′ ) , σ ] such that k [ D ( n + n ′ ) , σ ] k ≤ k x k + k y k .This proves the first assertion.(2) Using Notations 3.13, we apply Lemma 3.12 to b = [ T, s ] and b + = [ T ( n ) , s + ], and Equation 30 is justa rewriting of Equation (29). (cid:3) roof of Theorem 3.6. The fact that B G generates S ω ( Σ ) follows both from Proposition 2.14 or from thesecond assertion of Lemma 3.14. Let us prove that B G is free. Consider a vanishing linear combination P i ∈ I x i b i = 0, where b i ∈ B G and the x i are non-vanishing scalars and suppose by contradiction that I isnot empty. Let i ∈ I be such that k b + i k is the maximum, for < , of the set {k b + i k} ⊂ N × Z . The factthat ( · ) + : B G → B G + is a bijection implies that this maximum is unique. Developing each b i in the basis B G + (using Equation (30)), the equation P i ∈ I x i b i = 0 induces an equation of the form P b + ∈B G + y b + b + andthe fact that B G + is free (Lemma 3.10) implies that each y b + vanishes. Lemma 3.14 implies that x i = y b + i .Therefore x i = 0 and we have a contradiction. (cid:3) Lattice gauge field theory Ciliated graphs and quantum gauge group coaction. Since the pioneer work of Fock and Rosly[FR99], constructions in lattice gauge field theory are based on ciliated graphs. As we now explain, to aciliated graph (Γ , c ) one can associate a punctured surface Σ together with a finite presentation P of itsassociated groupoid. Definition 4.1. (1) A ribbon graph Γ is a finite graph together with the data, for each vertex, of a cyclicordering of its adjacent half-edges. An orientation for a ribbon graph is the choice of an orientationfor each of its edges.(2) A ciliated ribbon graph (Γ , c ) is a ribbon graph Γ together with a lift, for each vertex, of the cyclicordering of the adjacent half-edges, to a linear ordering. In pictures, if the half-edges adjacent toa vertex have the cyclic ordering e < e < . . . < e n < e that we lift to the linear ordering e < e < . . . < e n , we draw a cilium between e n and e .(3) We associate surfaces to ribbon graphs as follows.(i) Place a a disc D v on top of each vertex v and a band B e on top of each edge e , then glue thediscs to the band using the cyclic ordering: we thus get a surface S (Γ) named the fattening of Γ.(ii) The closed punctured surface Σ (Γ) = (Σ(Γ) , P ) associated to Γ is the closed punctured surfaceobtained from S (Γ) by gluing a disc to each boundary component and placing a puncture insideeach added disc. So S (Σ) retracts by deformations to Σ P (Γ).(iii) The open punctured surface Σ (Γ , c ) = (Σ (Γ , c ) , P ) associated to (Γ , c ) is obtained from S (Γ)by first pushing each vertex v to the boundary of S (Γ) in the direction of the associated cilium.Said differently, if the ordered half-edges adjacent to v are e < e < . . . < e n , we push v inthe boundary of D v such that it lies between the band B e n and the band B e . Next place apuncture p v next to v (in the counterclockwise direction) on the same boundary component than v . Eventually, to each boundary component of S (Γ) which does not contain any puncture p v ,glue a disc and place a puncture inside the disc. In the so-obtained punctured surface Σ (Γ , c ),each boundary arc contains exactly one vertex v of Γ, so we denote by a v the boundary arccontaining v . Suppose that Γ is oriented. Then the oriented edges of Γ form a set G of generatorsof Π (Σ P , V ) such that P (Γ , c ) := ( G , ∅ ) is a finite presentation without relations.(4) For v , v two distinct vertices of (Γ , c ), the ciliated graph (Γ v v , c v v ) is obtained by gluing thevertices v and v together to a vertex v in such a way that if e < . . . < e n and f < . . . < f m are the ordered half-edges adjacent to v and v respectively, then the linear order of the half-edgesadjacent to v is e < . . . < e n < f < . . . < f m . Note that c v v = c v v .Figure 15 illustrates two examples having the same ribbon graph but different ciliated structures: thepunctured surface Σ (Γ , c ) is a disc with two inner punctures and two boundary punctures whereas Σ (Γ , c ′ )is an annulus with one puncture per boundary component and one inner puncture. Remark . Costantino and Lˆe made in [CL19] the following important remark: the punctured surface Σ (Γ v v , c v v ) is obtained from Σ (Γ , c ) F T by gluing the boundary arcs a v and a v to two faces ofthe triangle T . In particular, when Γ = Γ F Γ with v ∈ Γ and v ∈ Γ , this property, together withTheorem 2.10 permitted the authors of [CL19] to prove that S ω ( Σ (Γ v v , c v v )) is the cobraided tensor igure 15. (1) From left to right: a ciliated graph (Γ , c ), its fattening S (Γ), its openpunctured surface Σ (Γ , c ) and its closed punctured surface Σ (Γ). (2) The same ribbongraph with a different ciliated structure c ′ and the associated open punctured surface Σ (Γ , c ′ ).product of S ω ( Σ (Γ , c )) with S ω ( Σ (Γ , c )). The same gluing property were discovered by Alekseev-Grosse-Schomerus in [AGS95, AGS96] for the quantum moduli spaces.For an oriented ciliated graph (Γ , c ), we denote by V (Γ) its set of vertices and E (Γ) its set of (oriented)edges. Like in the previous section, we see the elements of E (Γ) as oriented arcs. Denote by D thepunctured surface made of a disc with a single puncture on its boundary. The closed punctured surface Σ (Γ) is obtained from the open one Σ (Γ , c ) by gluing a copy D along each boundary arc a v . Therefore,writing b D := F v ∈ V (Γ) D , by Theorem 2.10 one has an exact sequence:(31) 0 → S ω ( Σ (Γ)) i −→ S ω ( Σ (Γ , c ) G b D ) ∆ R − σ ◦ ∆ L −−−−−−−→ S ω ( Σ (Γ , c ) G b D ) ⊗ O q [SL ] ⊗ V (Γ) , where i represents the gluing map.Using the isomorphism S ω ( D ) ∼ = k sending the class of the empty stated tangle to the neutral element1 ∈ k , we define an isomorphism κ : S ω (cid:16) Σ (Γ , c ) G b D (cid:17) ∼ = S ω ( Σ (Γ , c )) ⊗ ⊗ v ∈ V (Γ) S ω ( D ) ∼ = S ω ( Σ (Γ , c )) . Denote by ι : S ω ( Σ (Γ)) ֒ → S ω ( Σ (Γ , c )) the injective morphism ι := κ ◦ i . Also denote by ∆ G : S ω ( Σ (Γ , c )) → S ω ( Σ (Γ , c )) ⊗ O q [SL ] ⊗ V (Γ) the (unique) morphism making the following diagram com-muting: S ω (cid:16) Σ (Γ , c ) F b D (cid:17) S ω (cid:16) Σ (Γ , c ) F b D (cid:17) ⊗ O q [SL ] ⊗ V (Γ) S ω ( Σ (Γ , c )) S ω ( Σ (Γ , c )) ⊗ O q [SL ] ⊗ V (Γ)∆ R κ ∼ = κ ⊗ id ∼ =∆ G Definition 4.3. The quantum gauge group is the Hopf algebra O q [ G ] := O q [SL ] ⊗ V (Γ) . The (right) Hopf-comodule map ∆ G : S ω ( Σ (Γ , c )) → S ω ( Σ (Γ , c )) ⊗ O q [ G ] is called the quantum gauge group coaction .Note that, by definition, the following diagram commutes: S ω (cid:16) Σ (Γ , c ) F b D (cid:17) S ω (cid:16) Σ (Γ , c ) F b D (cid:17) ⊗ O q [ G ] S ω ( Σ (Γ , c )) S ω ( Σ (Γ , c )) ⊗ O q [ G ] σ ◦ ∆ L κ ∼ = κ ⊗ id ∼ = id ⊗ ǫ Therefore the exactness of the sequence (32) implies that we have the following exact sequence:(32) 0 → S ω ( Σ (Γ)) ι −→ S ω ( Σ (Γ , c )) ∆ G − id ⊗ ǫ −−−−−−→ S ω ( Σ (Γ , c )) ⊗ O q [ G ] . Said differently, ι ( S ω ( Σ (Γ))) is the subalgebra of S ω ( Σ (Γ , c )) of coinvariant vectors for the quantumgauge group coaction. otations 4.4. For x ∈ O q [SL ] and v ∈ ˚ V , we denote by x ( v ) ∈ O q [ G ] = O q [SL ] ⊗ V (Γ) the element ofthe form ⊗ v y v , where y v = 1 for v = v and y v = x .Let α be an arc of type either a or d and write v and v the elements of V corresponding to the boundaryarcs containing s ( α ) and t ( α ) respectively. The quantum gauge group coaction is characterised by thefollowing formula illustrated in Figure 16:(33) ∆ G ( α ij ) = X a,b = ± α ab ⊗ x ( v ) jb x ( v ) ia . Figure 16. An illustration of Equation (33).In order to prepare the comparison between stated skein algebras at ω = +1 and relative charactervarieties in the next subsection, let us derive from 1.1 an alternative presentation of S ω ( Σ ). During all therest of the section, we fix a finite presentation P = ( G , RL ) of Π (Σ P , V ) such that every arc of G is eitherof type a or d .When comparing skein algebras with character varieties, there is a well known sign issue which requiressome attention. When Σ is closed, the skein algebra S +1 ( Σ ) is generated by the classes of closed curves γ whereas the algebra C [ X SL ( Σ )] of regular functions of the character variety is generated by curve functions τ γ , sending a class [ ρ ] of representation ρ : π (Σ P ) → SL ( C ) to τ γ ([ ρ ]) := tr( ρ ( γ )). However there is noisomorphism S +1 ( Σ ) ∼ = C [ X SL ( Σ )] sending γ to τ γ . Instead, we fix a spin structure on Σ P with associatedJohnson quadratic form ω : H (Σ P ; Z / Z ) → Z / Z and define w ( γ ) := 1 + ω ([ γ ]). Then it follows from[Bul97, PS00, Bar99] that we have an isomorphism S +1 ( Σ ) ∼ = C [ X SL ( Σ )] sending γ to ( − w ( γ ) τ γ . A similarsign issue appears when dealing with stated skein algebras and relative character varieties; this was studiedin [KQ19] to which we refer for further details (see also [Thu14, CL19] for an elegant interpretation of thissign issue in term of twisted character variety ).In short, the authors defined in [KQ19] the notion of relative spin structure to which one can associate amap w : G → Z / Z having the property that for any simple relation R = β k ⋆. . .⋆β , one has P ki =1 w ( β i ) = 1.We will call spin function a map w : G → Z / Z satisfying this property. Notations 4.5. Let w be a spin function. For α ∈ G , we denote by U ( α ) the 2 × S ω ( Σ ) defined by(34) U ( α ) := (cid:26) ( − w ( α ) ωC − M ( α ) , if α is of type a;( − w ( α ) C − M ( α ) = ( − w ( α ) N ( α ) , if α is of type d. Proposition 4.6. (1) The stated skein algebra S ω ( Σ ) admits the alternative presentation with genera-tors the elements U ( α ) ji , with α ∈ G and i, j = ± , together with the following relations: • the q -determinant relations det q ( U ( α )) = 1 , when α is of type a , and det q ( U ( α )) = 1 , when α is of type d ; • for R = β k ⋆ . . .⋆ β ∈ RL a relation, where l generators β i are of type a , the trivial loop relation: (35) U ( β k ) . . . U ( β ) = A ω l . • for each pair ( α, β ) of generators in G , the arc exchange relations obtained from the relationsin Lemma 2.24 by replacing N ( α ) and N ( β ) by U ( α ) and U ( β ) respectively. The quantum gauge group coaction is characterized by the formula: (36) ∆ G (cid:16) U ( α ) ji (cid:17) = X a,b = ± U ( α ) ba ⊗ S ( x bj ) ( v ) x ( v ) ia , where we used the same notations than in Equation (33) .Proof. It is clear from Equation (34) that the matrix elements U ( α ) ji generate the same algebra than theelements M ( α ) ji = α ij , so they generate S ω ( Σ ). We need to check that the q -determinant, trivial loopand arcs exchange relations for the elements α ij are equivalent to the relations of the proposition for theelements U ( α ) ji . When α ∈ G is of type d , clearly the relation det q ( N ( α )) = 1 is equivalent to the relationdet q ( U ( α )) = 1. When α ∈ G is of type a , the equivalence det q ( M ( α )) = 1 ⇔ det q ( U ( α )) = 1 follows froma straightforward computation (and is the reason for the ω in the expression U ( α ) = ( − w ( α ) ωC − M ( α )).The equivalence between Equations (10) and (35) is straightforward (and is responsible for the introductionof the spin function and for the ( − w ( α ) factor in the definition of U ( α )). The fact that the arcs exchangerelations are equivalent to the same relations with N ( α ) , N ( β ) replaced by U ( α ) , U ( β ) follows from the factthat C − ⊙ C − commutes with τ , R and R − . Indeed, that C − ⊙ C − commutes with τ is obvious. Recallthat R is the matrix of τ ◦ q H ⊗ H/ ◦ ( + ( q − q − ) ρ ( E ) ⊗ ρ ( F )) and C − ⊙ C − is the matrix of C − ⊗ C − .That C − ⊗ C − commutes with q H ⊗ H/ and ( + ( q − q − ) ρ ( E ) ⊗ ρ ( F )) follows from the fact that C − is U q sl equivariant, so C − ⊙ C − commutes with R . The arguments for R − is similar.It remains to derive the formula (36) from (33). This is done by direct computation, left to the reader,using the following fact: for the two 2 × X = (cid:18) x ++ x + − x − + x −− (cid:19) and S ( X ) = (cid:18) S ( x ++ ) S ( x + − ) S ( x − + ) S ( x −− ) (cid:19) with coefficients in O q [SL ], one has S ( X ) = C − t XC . Figure 17 illustrates Equation (36). In Figure17, we used a special convention: we drawn stated diagrams that go ”outside” of Σ P in some small bigonneighbourhoods of the boundary arcs; it must be understood that we need to apply a boundary skein relationin those neighborhood. This convention permits to draw pictorially the matrix coefficients ( C − M ( α )) ji . Notealso that in Figure 17, we dropped the scalar factor ( − w ( α ) . Figure 17. An illustration of Equation (36). (cid:3) Relative character varieties. Since the quantum moduli algebras are deformation quantizations ofthe (relative) character varieties studied by Fock and Rosly in [FR99], we briefly recall their constructionand refer to [Aud97] for a detailed survey.Let us first consider a closed connected punctured surface Σ and denote by M SL ( Σ ) the set of isomor-phism classes of SL ( C ) flat structures on Σ P , that is the set of isomorphism classes of pairs ( P, ∇ ) where P is a principal SL ( C )-bundle and ∇ a flat connection. Since any principal SL ( C )-bundle on a surface istrivializable, one can restrict to the classes of pairs ( P, ∇ ) where P = Σ P × SL ( C ) is trivial and ∇ = d + A ,where A ∈ Ω (Σ P , sl ). The flatness of ∇ then translates to the equation dA + [ A ∧ A ] = 0 and we denoteby A F ⊂ Ω (Σ P , sl ) the subspace of such 1-forms. The gauge group G := Aut( P ) ∼ = C ∞ (Σ P , SL ( C )) isthe group of automorphisms and the moduli space can be identified with the quotient M SL ( Σ ) ∼ = A F / G .By imposing some Sobolev regularity, as done by Atiyah-Bott in [AB83], one can give to A F a structure ofBanach space however the gauge group action is far from been free; as a result the quotient A F / G does not nherit a geometric structure (it is not even Haussdorf as a topological space). One possibility to remedythis problem is to restrict to the (open dense) subspace A F ⊂ A F of principal orbits and consider onlythe subset M ( Σ ) := A F / G ⊂ M SL ( Σ ) which can be endowed with the structure of smooth manifold.When P = ∅ , Atiyah-Bott defined in [AB83] a symplectic structure on M ( Σ ). A second possibility toget a geometric object out of M SL ( Σ ) is to consider character varieties. Fix a base point v ∈ Σ P . TheRiemann-Hilbert correspondence asserts that the holonomy map induces a bijectionHol : M SL ( Σ ) ∼ = −→ Hom( π (Σ P , v ) , SL ( C )) . SL ( C ) , where SL ( C ) acts on the set R SL ( C ) ( Σ ) := Hom( π (Σ P , v ) , SL ( C )) of representations by conjugacy. Now R SL ( C ) ( Σ ) is an affine variety and the action of (the reducible algebraic group) SL ( C ) is algebraic, thereforewe might consider the algebraic quotient (familiar in Geometric Invariant Theory): X SL ( Σ ) := R SL ( Σ ) // SL ( C )named the character variety , which is an affine (irreducible) variety. One has a surjective (so-called Reynolds)set-theoretical map p : M SL ( Σ ) → X SL ( Σ ) which restricts to a bijection p : M ( Σ ) ∼ = −→ X ( Σ ) on thesmooth locus X ( Σ ). Therefore X SL ( Σ ) might be regarded as a good approximation of the moduli space M SL ( Σ ) which has a geometric structure of affine variety. Goldman showed in [Gol86] that the Atiyah-Bottsymplectic structure on M ( Σ ) induces a Poisson structure on the algebra of regular functions of X SL ( Σ )by giving an explicit formula for the Poisson bracket of two curve functions.Now, let us consider an open punctured surface Σ . We would like to change the definition of M SL ( Σ ) insuch a way that the moduli space would have a nice behaviour for the operation of gluing two boundary arcstogether, more precisely, we would like to have a surjective gluing map π a b : M SL ( Σ ) → M SL ( Σ a b ).The trick is to consider the subset A ∂F ⊂ A F of 1-forms whose holonomy along any subarc of a boundaryarc is null and to restricts to the smaller group G ∂ ⊂ G corresponding to those maps in C ∞ (Σ P , SL ( C ))whose restriction to ∂ Σ P is the constant map with value the neutral element of SL ( C ). We then define M SL ( Σ ) := A ∂F / G ∂ (which coincides with the previous definition when the punctured surface Σ is closed).If c denotes the common image of a and b in Σ | a b , any flat connection is gauge equivalent to a connectionwhose restriction to subarcs of c has trivial holonomy, therefore we have a surjective gluing map π a b : M SL ( Σ ) → M SL ( Σ a b ) as desired. To turn M SL ( Σ ) into a geometric (algebraic) object, one consider aspreviously the holonomy map, though we need more than one base point now. Let V ⊂ Σ P be a finite subsetwhich intersects each boundary arc exactly once, denote by ˚ V := V ∩ ˚Σ P its (possibly empty) subset of innerpoints and consider the discrete gauge group G V := SL ( C ) ˚ V . The holonomy map induces a bijectionHol : M SL ( Σ ) ∼ = −→ R SL ( Σ , V ) (cid:14) G V , where R SL ( Σ , V ) is the set of functors ρ : Π (Σ P , V ) → SL ( C ) and the discrete gauge group acts on theright by: ( ρ • g )( α ) := g ( t ( α )) − ρ ( α ) g ( s ( α )) , for all ρ ∈ R SL ( Σ , V ) , g ∈ G V , α ∈ Π (Σ P , V ) . We claim that R SL ( Σ , V ) can be given a structure of affine variety in such a way that the action of thereducible algebraic group G V is algebraic, so we can define the GIT quotient X SL ( Σ ) := R SL ( Σ , V ) // G V , which we call the relative character variety . To prove the claim, consider a finite presentation P = ( G , RL ) ofΠ (Σ P , V ) and write G = ( α , . . . , α n ) and RL = ( R , . . . , R m ). Consider the regular map R : SL ( C ) G → SL ( C ) RL written R = ( R , . . . , R m ), where the coordinate R i associated to a relation R i = α ε i ⋆ . . . ⋆ α ε k i k is the polynomial function R i ( g , . . . , g n ) = g ε i . . . g ε k i k . Clearly, one has R SL ( Σ , V ) = R − ( , . . . , ) (where is the identity matrix), so R SL ( Σ , V ) is a subva-riety of SL ( C ) G .Note that the algebra C [ R SL ( Σ , V )] of regular functions lies in the following exact sequence C [SL ( C )] ⊗ RL R ∗ − η ⊗ G ◦ ǫ ⊗ RL −−−−−−−−−→ C [SL ( C )] ⊗ G → C [ R SL ( Σ , V )] → . So we have turned R SL ( Σ , V ) into an affine variety. Now the discrete gauge group action is induced bythe Hopf comodule map ∆ G : C [ R SL ( Σ , V )] → C [ R SL ( Σ , V )] ⊗ C [ G V ], which is the restriction of the rightco-module map e ∆ G : C [SL ( C )] ⊗ G → C [SL ( C )] ⊗ G ⊗ C [SL ( C )] ⊗ ˚ V defined by e ∆ G (cid:16) x ( α ) (cid:17) = X x ′′ ( α ) ⊗ S ( x ′′′ ) ( v ) x ′ ( v ) , for x ∈ O q [SL ], α : v → v in G and we used Sweedler’s notation ∆ (2) ( x ) = P x ′ ⊗ x ′′ ⊗ x ′′′ . Inparticular, when x = x ij with i, j ∈ {− , + } , the formula gives:(38) ∆ G (cid:16) x ( α ) ij (cid:17) = X a,b = ± x ( α ) ab ⊗ S ( x bj ) ( v ) x ( v ) ia . Note the analogy between Equations (38) and (36).Eventually, the algebra of regular functions of the relative character variety is defined as the set ofcoinvariant vectors for this coaction, that is by the exact sequence(39) 0 → C [ X SL ( Σ )] → C [ R SL ( Σ )] ∆ G − id ⊗ ǫ −−−−−−→ C [ R SL ( Σ )] ⊗ C [ G V ] . The relative character variety X SL ( Σ ) does not depend (up to unique isomorphism) on the choice of thetriple ( V , G , RL ) used to define it but only on Σ ; we refer to [Kor19b] for a proof. Note that in the particularcase where V ⊂ ∂ Σ P , the gauge group is trivial so X SL ( Σ ) = R SL ( Σ ). Moreover, if the presentation P does not have any relation, then R SL ( Σ ) = SL ( C ) G . As we saw in Example 2.13, such a presentation P always exists when Σ is a connected punctured surface with non trivial boundary, therefore in that case onehas X SL ( Σ ) = SL ( C ) G . Now consider an oriented ciliated graph (Γ , c ) and consider the associated finite presentation ( V , G , RL ) ofthe groupoid Π (Σ P (Γ , c ) , V ) associated to the open punctured surface defined in the previous subsection.The same triple ( V , G , RL ) gives also a finite presentation of Π (Σ P (Γ) , V ) associated to the closed puncturedsurface, where this time, all elements of V are inner vertices of Σ P (Γ). Therefore one has X SL ( Σ (Γ , c )) = R SL ( Σ (Γ)) = SL ( C ) E (Γ) . So the exact sequence (39) can be rewritten as(40) 0 → C [ X SL ( Σ (Γ))] → C [ X SL ( Σ (Γ , c ))] ∆ G − id ⊗ ǫ −−−−−−→ C [ X SL ( Σ (Γ , c ))] ⊗ C [ G V ] . Note the analogy with the exact sequence (32). The main achievement in the work of Fock Rosly in [FR99]is the construction of Poisson structures on C [ X SL ( Σ (Γ , c ))] = C [SL ] ⊗E (Γ) and C [ G V ] = C [SL ] ⊗ ˚ V (Γ) suchthat the coaction ∆ G is a Poisson morphism. Therefore, using the exact sequence (40), the affine variety X SL ( Σ (Γ)) received a (quotient) Poisson structure. A great deal then is to show that this Poisson structureonly depends on the surface Σ P (Γ) and not on (Γ , c ). This strategy permitted the authors of [FR99] to extendthe Atiyah-Bott-Goldman Poisson structure from unpunctured closed surfaces to closed general puncturedsurfaces (see also [Kor19b] for a general treatment in the language of punctured surfaces rather than ciliatedgraphs and using groupoid cohomology).Let us conclude this subsection by the following observation. It is well known that the (stated) skeinalgebra S +1 ( Σ ) is isomorphic (though non canonically) to the algebra C [ X SL ( Σ )] of regular functions ofthe (relative) character variety. For closed punctured surfaces, this was shown by Bullock [Bul97] underthe assumption that S +1 ( Σ ) is reduced; this assumption was proved in [PS00] (see also [CM09] for analternative proof). For open punctured surface, this was proved independently in [KQ19, Theorem 1.3] and[CL19, Theorem 8.12] using triangulations of surfaces. Let us note that Theorem 1.1 gives a straightforwardalternative proof of this result. heorem 4.7 ([Bul97, PS00, KQ19, CL19]) . The algebras S +1 ( Σ ) (where k = C ) and C [ X SL ( Σ )] areisomorphic.Alternative proof using Theorem 1.1 with the additional assumption that P 6 = ∅ . First suppose that Σ is anopen connected punctured surface, let V be such that each of its vertices are on the boundary (so therepresentation and relative character varieties are the same) and let P = ( G , RL ) be a finite presentation ofΠ (Σ P , V ) whose generators are either of type a or d and fix a spin function w . By Equation (37), the algebra C [ X SL ( Σ )] is presented by the generators x ( α ) ij for α ∈ G and i, j ∈ {− , + } with the following relations,where we set X ( α ) := x ( α )++ x ( α )+ − x ( α ) − + x ( α ) −− ! :(i) the exchange relations x ( α ) ij x ( β ) kl = x ( β ) kl x ( α ) ij for all α, β ∈ G , i, j ∈ {− , + } ;(ii) the determinant relations det( X ( α )) = 1, for all α ∈ G ;(iii) the trivial loops relations X ( β k ) . . . X ( β ) = , for R = β k ⋆ . . . ⋆ β ∈ RL .By comparing this presentation of C [ X SL ( Σ )] with the presentation of S ω ( Σ ) obtained in Proposition 4.6by setting ω = +1, we see that one has an isomorphism of algebras Θ : S +1 ( Σ ) ∼ = −→ C [ X SL ( Σ )] sending U ( α )to X ( α ) (note that when ω = +1, R = τ so all arc exchange relations become U ( α ) ⊙ U ( β ) = τ U ( α ) ⊙ U ( β ) τ giving relations α ij β kl = β kl α ij ). Moreover, by comparing Equations (38) and (36), we see that Θ isequivariant for the gauge group coactions.Now suppose that Σ is closed and connected with P 6 = ∅ and let (Γ , c ) be a ciliated fat graph suchthat Σ (Γ) = Σ . By the preceding case, one has an equivariant isomorphism Θ : S +1 ( Σ (Γ , c )) ∼ = −→ C [ X SL ( Σ (Γ , c ))], so one has a commutative diagram0 S +1 ( Σ (Γ))] S +1 ( Σ (Γ , c ))] S +1 ( Σ (Γ , c ))] ⊗ C [ G V ]0 C [ X SL ( Σ (Γ))] C [ X SL ( Σ (Γ , c ))] C [ X SL ( Σ (Γ , c ))] ⊗ C [ G V ] . ∃ ! ∼ = ∆ G − id ⊗ ǫ Θ ∼ = Θ ⊗ id ∼ =∆ G − id ⊗ ǫ The fact that both lines are exact implies the existence of an isomorphism S +1 ( Σ (Γ)) ∼ = −→ C [ X SL ( Σ (Γ))]obtained by restriction of Θ. (cid:3) Combinatorial quantizations of (relative) character varieties. The work of Fock and Roslysuggests a natural way of quantizing character varieties. The following problem was raised and solved inde-pendently by Alekseev-Grosse-Schomerus [AGS95, AGS96] and Buffenoir-Roche [BR95] (see also [BFKB98a]for a survey): Problem 4.8. Associate to each oriented ciliated graph (Γ , c ) an (associative unital) algebra L ω (Γ , c ) overthe ring k := C [ ω ± ] satisfying the following properties:(A1) As a k -module, L ω (Γ , c ) is just the (free) module C [ R SL ( Σ (Γ , c ))] ⊗ C k ∼ = C [SL ] ⊗E (Γ) ⊗ C k ,(A2) As before, write O q [ G ] := O q [SL ] ⊗ V (Γ) . The linear map ∆ G : L ω (Γ , c ) → L ω (Γ , c ) ⊗ O q [ G ] definedby the formulas ∆ G (cid:16) x ( α ) ij (cid:17) = X a,b = ± x ( α ) ab ⊗ S ( x bj ) ( v ) x ( v ) ia . is a Hopf-comodule map. In particular, it is a morphism of algebras.(Inv) The subalgebra L invω (Γ) ⊂ L ω (Γ , c ) defined by the exact sequence0 → L invω (Γ) → L ω (Γ , c ) ∆ G − id ⊗ ǫ −−−−−−→ L ω (Γ , c ) ⊗ O q [ G ] , only depends (up to canonical isomorphism) on the (homeomorphism class of) surface S (Γ).(Q) Let k ~ := C [[ ~ ]] and write ω ~ := exp( − iπ ~ ) ∈ k ~ so that µ : k → k ~ defined by µ ( ω ) := ω ~ is a ringmorphism. Then the k ~ algebra L invω (Γ) ⊗ µ k ~ is a deformation quantization of the Poisson algebra C [ X SL ( Σ (Γ))] equipped with its Fock-Rosly Poisson structure. heorem 4.9 (Alekseev-Grosse-Schomerus ([AGS95, AGS96, AS96]), Buffenoir-Roche ([BR95, BR96])) . Problem 4.8 admits the solution L ω (Γ , c ) := L ω ( Σ (Γ , c )) , where the k -module isomorphism L ω (Γ , c ) ∼ = C [ R SL ( Σ (Γ , c ))] ⊗ C k is given by sending U ( α ) to X ( α ) . The algebras L ω (Γ , c ) are the so-called quantum moduli algebras and Theorem 1.3 is an obvious conse-quence of Theorem 1.1.More precisely, the ciliated graphs considered in [BR95, BR96] are those whose underlying graph is the1-skeleton of some combinatorial triangulation of a Riemann surface. By combinatorial we mean that eachedges has two distinct endpoints, so every arcs is of type a and the only arcs exchange relations among distinctarcs are in configurations ( i ) or ( ii ) (in the notations of Lemma 2.24). In [AGS95, AGS96, AS96] generalciliated graphs are considered, though in [AGS96, AS96] a special attention is given to the quantum modulialgebras of the daisy graphs defined in Example 2.13 (they are called standard graphs in [AGS96, AS96]) andhave been further studied and related to stated skein algebras in [Fai20a]. In those daisy graphs, the arcsare of type d and the more complicated arcs exchange relations in configurations ( viii ) , ( ix ) , ( x ) appearedunder the name of braid relations (see [AGS96, Definition 12]).Note that, except for the study of the Poisson structure (which could have been easily done), we didreproved Theorem 4.9 in this paper. In [MW15], Meusburger and Wise proved that the solution of Problem4.8 is unique, provided that we add some natural axioms for the operation of gluing graphs together. Actuallythe authors of [MW15] consider quantum moduli algebras associated to finite dimensional ribbon algebras,whereas here we consider the infinite dimensional one U q sl , but their proof extends word-by-word to ourcontext.4.4. Comparison with previous works. Let Σ be a connected punctured surface with one boundarycomponent, one puncture on its boundary and eventually some inner punctures. Let (Γ , c ) be its daisygraph and P = ( G , ∅ ) be the associated finite presentation as defined in Example 2.13 (so Σ = Σ (Γ , c )).In this case, since the presentation has no relation, one can consider the spin function w sending everygenerator to 0 ∈ Z / Z . Since every generator α ∈ G is of type d , the isomorphism Ψ : S ω ( Σ ) ∼ = −→ L ω (Γ , c )sends U ( α ) = C − M ( α ) to X ( α ). By precomposing with the reflection anti-involution θ , one obtains anisomorphism Ψ ′ : S ω − ( Σ ) op ∼ = −→ L ω (Γ , c ) , which corresponds to Faitg’s isomorphism in [Fai20a]. Let us stress that our notations are quite differentfrom the ones in [Fai20a]; in particular: • the letter q in [Fai20a] is what we denoted by A (so our q corresponds to q in [Fai20a]), • the letter R in [Fai20a] is related to our R by R = τ ◦ R , • Faitg actually considered S ω − ( Σ ) op , the opposite of the stated skein algebra.As Faitg kindly explained to the author, the existence of an isomorphism Ψ : S ω ( Σ ) ∼ = −→ L ω (Γ , c ) couldhave been derived from the works in [BZBJ18, GJS] as we now briefly explain using the notations in [GJS] towhich we refer for further details. Set k = C [ ω ± ] and fix a structure of Riemann surface Σ. To any k -ribboncategory A , one can associate a skein category SkCat A (Σ) whose objects are oriented embeddings of finitelymany disjoint discs D → Σ colored by objects in A and morphisms are framed A -colored ribbon graphs inΣ × [0 , 1] considered up to skein relations (see [Coo19, Section 4 . 2] for a precise definition). We denote by ∈ SkCat A (Σ ) the empty set. Let Σ be obtained from a connected closed oriented surface Σ by removingan open disc. Fixing an arbitrary disc embedding D → Σ , one get a functor P : A → SkCat A (Σ ) in anobvious way. Let b A := Fun( A op , Vect) be the free cocompletion of A (which inherits a monoidal structurefrom A ). The internal skein algebra is defined as the coend:SkCat int A (Σ ) := Z x ∈A Hom SkCat A (Σ ) ( P ( x ) , ) ⊗ x ∈ b A . The functor Hom SkCat A (Σ ) ( P ( • ) , ) : A op → Vect has a natural Lax monoidal structure, given by stackingribbon graphs on top of each others, which endows SkCat int A (Σ ) with the structure of an algebra object in A . If A is Tannakian, that is if is equipped with a fully faithful monoidal functor for : A → Vect, then S A (Σ ) := for(SkCat int A (Σ )) = Z x ∈A Hom SkCat A (Σ ) ( P ( x ) , ) ⊗ for( x ) ∈ Vectis a unital associative algebra, that we might call the stated skein algebra associated to A and Σ . Letus consider two Tannakian ribbon categories: the (Cauchy closure of the) Temperley Lieb category TLand the category of finite dimensional U q sl left modules Rep fdq (SL ) (recall that q is generic here). TheTannakian structure forget : Rep fdq (SL ) → Vect is just the forgetful functor. It is well known that one hasa monoidal braided equivalence of categories G : TL → Rep fdq (SL ) sending the one strand ribbon [1] ∈ TLto the fundamental representations V of Section 2.1 with basis { v + , v − } , thus we get a Tannakian structureforget ◦ G : TL → Vect. Let us underline that G does not preserve the duality ! This fact is what willmake the present argument ambiguous.On the one hand, there is a natural algebra morphismΨ : S ω ( Σ ) → S T L (Σ )sending the class [ T, s ] of a stated tangle, where ∂T has n elements, to the class of T ⊗ v s ∈ Hom SkCat TL (Σ ) ( P ([1] ⊗ n ) , ) ⊗ V ⊗ n , where v s ∈ V ⊗ n is obtained from the state s by identifying thesigns + and − with the basis vectors v + and v − of V . As noted in Remark [GJS, Remark 2 . is an isomorphism.On the other hand, thanks to Cooke’s excision theorem in [Coo19], and as proved in [GJS, Proposition2 . int A (Σ ) is isomorphic to the so-called moduli algebra A Σ = End( ) ∈ b A introduced in [BZBJ18, Definition 5 . . h Rep fdq (SL ) i Σ ∼ = L ω (Γ), so by composing the two isomorphisms, one get an isomorphismΨ : S Rep fdq (SL ) (Σ ) ∼ = −→ L ω (Γ) . Now, even though the equivalence of categories G : TL → Rep fdq (SL ) does not preserve the duality,one should be able to get a (non-canonical) algebra isomorphism Ψ : S Rep fdq (SL ) (Σ ) ∼ = −→ S TL (Σ ). So bycomposing the isomorphisms Ψ , Ψ and Ψ , one would get an isomorphism S ω ( Σ ) ∼ = −→ L ω (Γ , c ), whichwould recover Faitg result and is a particular case of our Theorem 1.3. Note that the definition of Ψ is notvery clear and certainly not canonical: it is in this definition that the non-canonical choice of a spin functionshould appear. Remark . The above construction of the isomorphism S ω ( Σ ) ∼ = −→ L ω (Γ , c ) is very indirect and not soenlightening. However, it generalises the notion of stated skein algebra S C (Σ ) to an arbitrary Tannakianribbon category C (how to replace Σ to an arbitrary punctured surface is obvious), and [BZBJ18, Theorem5 . 14] seems to permit to give explicit finite presentations for S C (Σ ). The detailed study of these generalizedstated skein algebras will appear in a separate publication [Kor].5. Concluding remarks We conclude the paper by making some remarks concerning the usefulness of relating stated skein algebrasand quantum moduli spaces (Theorem 1.3). We can see the stated skein algebras as defined by a huge set ofgenerators (all stated tangles) and a huge set of relations (isotopy and skein relations) whereas the quantummoduli algebra is defined by a finite subset of generators and by a finite subset of relations. Both presentationshave their own advantages.(1) The fact that the quantum moduli algebra L invω (Γ) only depends, up to canonical isomorphism, onthe thickened surface S (Γ) (or equivalently Σ (Γ)) is usually proved by defining elementary moves on graphsthat preserve the thickened surface and showing that those elementary moves induce isomorphism on thealgebras. This strategy was pioneered by Fock and Rosly in the classical case of relative character varieties[FR99] and latter carried on in [AGS96, BR96] for quantum moduli algebras (see also [MW15] for verydetailed study). Thanks to the isomorphism L invω (Γ) ∼ = S ω ( Σ (Γ)) (and the fact that stated skein algebradepends on surfaces rather than graphs), this fact is also an immediate consequence of Theorem 1.3. Also he image of a closed curve γ through the reverse isomorphism Ψ − : Σ (Γ) → L invω (Γ) is usually called its holonomy Hol( γ ) or Wilson loop operators and the expression of this holonomy in terms of generators andproof of some composition properties is the subject of long and technical computations in [AGS95, AGS96,BR95, BR96, MW15, Fai20b], whereas they become easy in the skein algebra setting.(2) Since the quantum moduli algebra L ω (Γ , c ) is quadratic homogeneous, we might have tried to provethat it is Koszul (so prove that B G is free) without the help of the stated skein algebra. The standardtechnique to prove that the family B of Equation (25) is a PBW basis consists in examining the set of criticalmonomials of the form v i v j v k (we use the notations of Section 3.2) where both v i v j and v j v k are leadingterms. To such a critical monomial, we associate a finite graph (which might have the shape of a Diamond)and the Diamond Lemma implies that if each of these graph is confluent (has a terminal object) then B isa basis so the quadratic algebra is Koszul (see [LV12, Section 4] for details). In our case, due to the hugeamount of different kind of relations in our presentation, this strategy would require to verify the confluenceof 6578 different graphs ! This is way too much to be handled by hand. It is thanks to the fact that statedskein algebras have a lot of relations and generators that Lˆe was able to successfully use the Diamond Lemmain [Le18] to prove that B is basis, and our proof of the fact that B G is a basis is directly derived from thisfact. So proving the Koszulness of L ω (Γ , c ) without the help of stated skein algebra could have been a verydifficult problem.(3) Even if we could find PBW bases for the algebras L ω (Γ , c ) without the help of skein algebras, findingbases for L invω (Γ) would be extremely difficult, since it is only defined as a kernel and no presentation wasknown. However, skein algebras S ω ( Σ (Γ)) ∼ = L invω (Γ) have well known bases (of multi-curves).(4) As we saw in Section 4.2, the fact that L +1 ( Σ , P ) is isomorphic to the algebra of regular functionsof the (relative) character variety X SL ( Σ ) is very easy to prove, whereas relating the (stated) skein algebra S +1 ( Σ ) to C [ X SL ( Σ )] is not so obvious (see [Bul97, PS00] for closed surfaces and [KQ19, CL19] for openones).(5) In [BW16], Bonahon and Wong proved that the Kauffman-bracket skein algebra S +1 ( Σ ), with de-forming parameter +1, embeds into the center of the skein algebra S ζ ( Σ ) with deforming parameter ζ aroot of unity of odd order (see also [Le17] for an alternative proof). This result was generalized in [KQ19] tostated skein algebras as well (see also the forthcoming paper [BL] for generalizations). In [BR19], Baseilhacand Roche showed that the construction of this so-called Chebyshev-Frobenius morphism is much easierin the context of quantum moduli algebras (that is using the finite presentations of Theorem 1.1). Eventhough their study only concerns genus 0 surfaces, their proofs seems to generalize easily to general surfaces,providing simpler proofs for the results in [BW16, KQ19].(6) Bullock, Frohman and Kania-Bartoszynska already proved in [BFKB98b, Theorem 10] that L invω (Γ)and S ω ( Σ (Γ)) are isomorphic in the particular case where k = C [[ ~ ]] and ω = − exp( − ~ / L invω (Γ) → S ω ( Σ (Γ)) (by techniques similar to what wedid in Section 2.2) and (2) noting that under the (mod ~ ) identifications L invω (Γ) . ( ~ ) ∼ = C [ X SL ( Σ )] and S ω ( Σ (Γ)) . ( ~ ) ∼ = S − ( Σ (Γ)), the morphism Ψ reduces modulo ~ to Bullock’s isomorphism C [ X SL ( Σ )] ∼ = S − ( Σ (Γ)). 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