FINITE GENERATION OF SU(3)-SKEIN ALGEBRAS CHARLES FROHMAN, ADAM S. SIKORA

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1 FINITE GENERATION OF SU(3)-SKEIN ALGEBRAS CHARLES FROHMAN, ADAM S. SIKORA Abstract. We develop a decomposition theory for SU(3)-webs in surfaces. Using such decompositions we prove that the SU(3)-skein algebra of any surface is finitely generated and Noetherian. 1. Introduction The Kauffman bracket skein module of a 3-manifold M of Przytycki and Turaev, [Prz, Tu], is an algebra for the cylinder M = F I over orientable surfaces F. Recently, there has been a lot of interest in the connections between skein algebras and representation theory. On the representation theory end this has been inspired by the problem of categorification, beginning with the work of Khovanov and Kuperberg[Ku, KK], and being developed by Cautis, Fontaine, Kamnitzer, Morrison, Peters, and Snyder, [FKK, CKM, MPS1, MPS2]. In another direction the explication of Kashaev s invariant, led to the development of Quantum Teichmuller theory, starting with the work of Fock and Chekhov [CF], Fock and Goncharov, [FG], Baseilhac and Benedetti [BB], and Bohahon and Wong,[BW1, BW2, BW3]. In [FKL] a big step towards understanding the representation theory of skein algebras was taken. It depended on the fact that the Kauffman bracket skein algebra of a finite type surface is generated, first proved by Bullock [Bu], and the fact that it had no zero divisors proved in [PS2]. In this paper we begin exploring these properties for other skein algebras. Specifically we prove that the reduced SU(3)-skein of a finite type, marked surface is finitely generated and Noetherian. In particular, these properties hold for the SU(3)-skein algebras of (unmarked) finite type surfaces. 2. SU(3)-skein algebra of a marked surfaces A marked surface is an oriented surface F together with a finite subset B of F of marked points. ( F may be empty). We say that a boundary component of F is marked if it contains a point of B. Connected components of a marked boundary component with points of B removed are the marked boundary intervals of (F, B). The authors acknowledge support from U.S. National Science Foundation grants DMS , , RNMS: GEometric structures And Representation varieties (the GEAR Network). 1

2 2 CHARLES FROHMAN, ADAM S. SIKORA We assume that all marked boundary intervals are oriented. (One can always assume their default orientation induced by the orientation of F.) It is sometimes convenient to consider an alternative approach in which F has a hyperbolic metric, the unmarked components of F are punctures, points of B are ideal (at infinity) and the marked components are composed of infinite geodesics. However, we will not use this approach in this paper. Two generalizations of Kauffman bracket skein algebras of surfaces have been considered: they have been extended to skein algebras of marked surfaces in [PS2], as well as work of Bonahon, Le, Muller and others, built of properly embedded framed 1-submanifolds in F I. The Kauffman bracket skein algebras encapsulate the skein relations of the quantum group U q (sl 2 )). Their generalization to SU(n)-skein algebras, based on the U q (sl n ))-skein relations has been studied in [Si]. It coincides with the one considered by Kamnitzer and Morrison and, for n = 3, with the Kuperberg s webs of type A 2 and his skein relations between them. In this paper we merge these two constructions into the SU(n)-skein algebra of a marked surface S n (F, B). An n-web in (F, B) is an oriented graph W in F all of whose internal vertices are either n-valent sinks or sources or 4-valent crossings. Its external vertices, called endpoints are 1-valent and lie in the marked boundary intervals of (F, B). A web may be empty,, disconnected, and it may contain closed loops and arcs as its components. Figure 1. Types of internal vertices in 3-webs (Alternatively, one can think of webs in (F, B) as framed graphs in F I satisfying certain conditions.) Let W n (F, B) be the set of all webs in (F, B) up to isotopy. Let R be a commutative ring with a specified invertible element q and specified n-th root, denoted by q 1 n R. The SU(n)-skein module S n (F, B) of (F, B) is the quotient of the free R-module with the basis W n (F, B) by the skein relations of [Si, Sec. 3.1]. For n = 3 they are as follows: (1) (q q 2 ), + (q + q 1 ) (2)

3 FINITE GENERATION OF SU(3)-SKEIN ALGEBRAS 3 (3) q 1 3 q 2 3, q 1 3 q 3 3. (Our q is q 1/2 in [Ku].) It might seem more natural to denote q 1/n by q, but we follow here the standard notation in which the value of the unknot is the quantum n, cf. [Ku, Si]. The reduced SU(3)-skein module RS(F, B) is the quotient of the skein algebra S(F, B) by a submodule generated by webs which contain an arc or a 2-edge path parallel to a marked boundary interval of (F, B), as in the figure below. Figure 2. Additional skein relations of the reduced skein modules W(F, B) admits a product operation W 1 W 2 given by stacking W 1 on top of W 2 and isotoping their endpoints so that in every marked boundary interval the endpoints of W 1 lie after the endpoints of W 2. That operation extends to a product on S(F, B) making it into an associative R-algebra with the identity. It is easy to see that the submodule of S(F, B) generated by the relations of Figure 2 is an ideal and, hence, the above product defines an algebra product on RS(F, B). S(F, B) and RS(F, B) are straightforward SU(3) analogs of the skein algebras considered in [PS2]. 3. The Main result An R-algebra is Noetherian if its every ascending chain of left ideals and its every ascending chain of right ideals stabilizes. For commutative algebras being finitely generated is equivalent being Noetherian. In general, however, these two properties are independent. Theorem 1. For every marked surface (F, B), RS 3 (F, B) is a finitely generated Noetherian R-algebra. One can show that the non-reduced skein algebra S(F, B) is not finitely generated, even for (F, B) = (D 2, ). (This has been observed for the Kauffman bracket skein algebra in [PS1, Example 8].) The SU(n)-skein algebra and of its reduced version generalize to SU(n) in a straightforward manner.

4 4 CHARLES FROHMAN, ADAM S. SIKORA Conjecture 2. For every marked surface (F, B), RS n (F, B) are finitely generated Noetherian R-algebras for all n > 3. For closed F there is a natural epimorphism RS n (F ) RS n (F ) where F is F with an open disk removed. Therefore, it is enough to find finite generating sets and prove the Noetherian property for non-closed surfaces F. Furthermore, we conjecture: Conjecture 3. For every marked surface (F, B) and every n > 2 RS n (F, B) is a domain. 4. Zig-zag subgraph decomposition From now on let n = 3. Consequently, a web is a 3-web and we denote RS 3 (F, B) by RS(F, B). In this section we consider geometric and combinatorial properties of crossingless webs in surfaces F. These webs may have endpoints in F and since we do not discuss the product in the skein algebra, the marked points in F are irrelevant. Suppose that Γ F is a crossingless web, so that its monovalent vertices are exactly Γ F. The closures of the components of F Γ are called faces. We say the web Γ fills F if the faces are all disks. Given a face σ of Γ, its boundary is decomposed into edges and vertices. We say an edge of a face σ is internal if it is shared by two faces, otherwise the edge is external. Similarly we say a vertex of σ is internal if it corresponds to a trivalent vertex of Γ, and it is external if it corresponds to a monovalent vertex of Γ. Given a face σ let ie(σ) denote the number of internal edges of σ, ee(σ) denote the number of external edges. Let iv(σ) denote the number of internal vertices of σ and let ev(σ) denote the number of external vertices of σ. Let χ(σ) denote the Euler characteristic of σ. We define the index of σ to be (4) index(σ) = χ(σ) ie(σ) ee(σ) + iv(σ) + ev(σ) The standard computation of Euler characteristic from the inclusion-exclusion principle results in the formula (5) χ(f ) = σ index(σ). where the sum is over all faces of Γ. A face is internal if all its edges are internal. Since the vertices of a web are all sources or sinks, an internal face of a web that has a vertex has an even number of edges. An n-gon is a face whose underlying surface is a disk, that has n-sides. The only internal faces that have positive index are monogons, bigons, or quadrilaterals. The only internal faces with index 0 are hexagons. External faces are a little more complicated, as they can have an odd number of edges, and can have more than one external edge. The only external faces with positive index are;

5 FINITE GENERATION OF SU(3)-SKEIN ALGEBRAS 5 bigons which have index 1 2, trigons which have index 1 3, quadrilaterals having a single external edge which have index 1 6. The only external faces with index 0 are quadrilaterals with two external edges and pentagons with one external edge. We say a web is non-elliptic if it has no internal faces that are bigons or quadrilaterals. Lemma 4. The only non-elliptic webs in a disk with fewer than four boundary vertices are: The empty web. A single arc. A connected web with one, trivalent vertex. Proof. Since there are no internal faces of positive index, if there are any internal faces with negative index there must be enough external faces with positive index so that the sum of the indices of the faces is 1. This rules out any internal faces with negative indices. If there are internal faces that are hexagons, then there are going to be at least 6 vertices on the boundary, so that hexagons are ruled out. Finally, there is only so much you can do with at most three external vertices, and no internal faces. Let W be a crossingless web in (F, B). Lemma 5. Let Γ D 2 be an non-elliptic web. (1) zig-zag paths in Γ do not self-intersect. (2) Different zig-zag paths in Γ share at most one edge. Proof. If the same edge appears twice in the zig-zag path, Z, then by truncating the path we get a simple closed zig-zag path J. Since we are in a disk J is the boundary of a disk B. Since a zig-zag path alternates turning right and left at each edge, the web consisting of the interior edges of Γ J has the property that all of its edges that touch the boundary of B either all point in, or all point out. Ignoring the vertices on the boundary of J that are only adjacent to edges contained in J, the computation of index above is valid, and the sum of the indices of the faces of Γ J is one. This means that there must be a face with positive index. Since the web is non-elliptic, it has no interior faces of positive index. If it has an exterior face that is a trigon, putting the ignored vertex back in, the web Γ has a face that is a quadrilateral. Hence Γ is not non-elliptic. If it has an exterior face that is a quadrilateral, putting the ignored vertex back in Γ has a face which is a pentagon which is ridiculous. Hence Γ does not intersect. If two zig-zag paths share two edges, we can find a disk between them and the same argument above leads to a contradiction.

6 6 CHARLES FROHMAN, ADAM S. SIKORA Let V (W ) denote the set of vertices of a crossingless web W. A zig-zag subgraph W of W is composed of some 3-valent vertices and some 1-valent vertices (endpoints) V of W zig-zag paths of odd length connecting the above vertices. Counting the edges of paths of W connecting vertices V starting from zero leads to the notion of odd and even edges in the paths of W. In particular, every zig-zag path of odd length is a zig-zag subgraph. W is not a web on its own, since it usually contains 2-valent vertices. However, if we ignore its 2-valent vertices and its odd edges (by collapsing them) then W becomes a web W with vertices V and edges given by the paths connecting them. Hence, V ( W ) = V. (We count edges from zero, so that the odd edges are indeed odd and collapsed for the purpose of defining the web W.) Lemma 6. Let W be a zig-zag subgraph of W. Let W be a subgraph of W composed of all edges of W except for the even edges of W. Then W is a zig-zag subgraph of W. We call W the complementary zig-zag subgraph of W. Note that W and W share their odd edges. Note also that and W are zig-zag subgraphs of W. We say that W W is non-trivial if W, W. A crossingless web W is simple if it does not have a non-trivial zig-zag subgraph. Proposition 7. For every B D 2, every simple non-elliptic web in (D 2, B) is either a single edge e with e D 2 or a triad, i.e. a web W with a single 3-valent sink or source and three edges connecting it with D 2. 1 More generally we expect: Conjecture 8. The number of simple non-elliptic webs in every surface F is finite, up to isotopy and the action of the mapping class group of F. The following simple web (in D 2 ) shows that the non-ellipticity assumption in the proposition above is necessary. 1 Note however that each of these graphs requires B. In particular, (D 2, ) has no simple non-elliptic webs.

7 FINITE GENERATION OF SU(3)-SKEIN ALGEBRAS 7 Proof of Prop. 7: By Lemma 5(1), W has an edge e ending in D 2. If e has its other end in D 2 then W being simple implies that W = e and the statement follows. Hence, assume that the other end of e is a 3-valent vertex v. There are two zig-zag paths P 1, P 2 of maximal length in W containing e. We are going to prove that P 1 and P 2 are of even length: Assume otherwise that one of them, say P 1, is of an odd length. Then it forms a zig-zag subgraph of W. By Lemma 5, P 2 shares edge e with P 1 only. Since P 2 is not a single edge, P 1 W. That implies that W is decomposable, contradicting the assumption of statement. Therefore, P 1 and P 2 are of even length. By Lemma 5, P 1 and P 2 do not intersect outside of e. This implies that v together with e, P 1 e and P 2 e forms a zig-zag subgraph of W. Since W is simple, this subgraph is the entire W. Since W that subgraph cannot have 2-valent vertices, we easily see that it must be a triad. 5. Filtration on skein algebra and the associated graded algebra Let (I, +, 0) be an ordered, commutative monoid with the identity, 0, being its smallest element. A sequence of R-submodules F i of an R-algebra A is an algebra filtration if F i F j for every i < j, F i F j F i+j and F 0 = R, F i = A. Let F <i = i <i F i for i > 0 and F <0 = {0}. An algebra filtration on A defines an associated graded R-algebra GA. As an R-module, GA = i I i I F i /F <i. F i /F <i for i I are the homogenous components of GA. Their non-zero elements are homogenous and are of the form x + F <i for x F i F <i. The product of two non-zero homogeneous elements x + F <i and y + F <j for x F i F <i, y F j F <j is (x + F i ) (y + F j ) = xy + F i+j. That product extends additively onto GA. GA can be thought as a deformed, simplified version of A, which nonetheless shares some important algebraic properties with A. In particular, we are going to prove Theorem 1 by proving its graded version, Theorem 9, first. Filtrations on skein algebras were utilized in [CM, AF, Le1, PS2, Mu]. These constructions can be generalized to SU(3)-skein modules. However, instead of doing

8 8 CHARLES FROHMAN, ADAM S. SIKORA it in full generality here we limit ourselves to the setup needed for the proof of Theorem 1. By the remark at the end of Section 3, it is sufficient to assume that F is bounded and, indeed, that will be our assumption from now on. Let Γ be a maximal collection of disjoint, non parallel, properly embedded arcs γ in (F, B). Hence, the endpoints of γ are in B or in the unmarked components of F. We require that Γ includes all boundary intervals of (F, B). Let F (n,v) RS(F, B) be spanned by crossingless webs W with the geometric intersection number with Γ, W Γ n and the number of vertices #V (W ) v. Let I = Z 0 Z 0 be ordered lexicographically, i.e. (n, v) > (n, v ) iff n > n or n = n and v > v. Note that {F i } ı I is an algebra filtration of RS(F, B). We denote the associated graded algebra by GRS(F, B). Theorem 9. For every marked surface (F, B), GRS(F, B) is Noetherian and finitely generated R-algebra. Theorem 9 implies Theorem 1: (1) By [MR, Thm ], if GA is Noetherian (i.e. left and right Noetherian) then so is A. (To be precise [MR, Thm ] refers to filtrations indexed by integers only, but the proof holds in full generality. Also the cited statement refers to Noetherian rings. However, again the proof extends to R-algebras verbatim.) (2) If I is well ordered and GA is finitely generated then so is A. Can t find a reference, so here is a proof: Suppose that GA is finitely generated. Since every element of GA is a sum of homogeneous elements, one can take a finite homogenous generating set {x g + F <g } g G for GA, with x g F g F <g. (That G stands for generators and it is unrelated to the first letter of GA.) We claim that {x g } g G generate A. Suppose they do not. Since I is well ordered and i I F i = A, there is the smallest i 0 and x F i0 such that x is not a polynomial expression in x g s. By our assumptions, x + F <i0 is a polynomial expression p in x g + F <g for g G. Since all monomials of p are homogeneous elements of GA and, hence, all monomials outside F i0 cancel, we can assume that all monomials of p are in F i0. Consequently, x p F <i0 implying x p F j for some j < i 0. Therefore, by our assumptions x p is a polynomial in x g s and, hence, x is too. Contradiction. 6. Proof of Theorem 9: Let Γ be as in the previous section. In the geometric interpretation of F mentioned in the introduction (with unmarked components of F being punctures of a hyperbolic surface and marked components composed of infinite geodesics), such Γ defines an ideal triangulation of F. However, in our setup, F is compact and the connected components of F Γ are not necessarily triangles, since they may contain segments of F. They are of one of the forms in Figure 3, where b 1, b 2, b 3 B and the dashed lines denote segments of unmarked boundary components of (F, B) and

9 FINITE GENERATION OF SU(3)-SKEIN ALGEBRAS 9 Figure 3. Types of triangle-like regions the solid lines can be segments of the marked boundary intervals of (F, B). We call these triangle-like regions faces of Γ. Since webs in (F, B) cannot touch the dashed lines, we will collapse them to points, when convenient. In this way each face of Γ has three sides denoted by solid lines above. Remark 10. If two arcs of a web are crossing an arc e of Γ then, by (3), introducing a crossing of these arcs changes the value of the web in GRS(F, B) by the factor of q ±1/3. The dashed line represents an arc in Γ. A crossingless web in (F, B) is reduced if it does not contain an arc or a 2-edge path parallel to a boundary interval of (F, B). By Proposition 7, there are eight reduced simple webs in the triangle, D 2, {b 1, b 2, b 3 }, 6 arcs and the source and sink triads, cf. Figure 4. Figure 4. Seven different reduced simple webs in (D 2, {b 1, b 2, b 3 }): 6 arcs and the source triad. The sink triad not shown. Lemma 11. For every crossingless web W in (D 2, B) with v vertices there are reduced simple webs Y 1,..., Y k in (D 2, B) and d Z such that W and Y 1... Y k have equal number of endpoints on each boundary interval of (D 2, B) and W q d Y 1... Y k is a linear combination of webs in RS(D 2, B) with fewer than v vertices.

10 10 CHARLES FROHMAN, ADAM S. SIKORA Proof. The statement holds for webs W with no 3-valent vertices since then W is a disjoint union of reduced arcs. (The statement also holds for webs W with one 3-valent vertex since then W is a disjoint union of reduced arcs and one reduced triad.) One can continue the proof the statement by induction on the number of 3-valent vertices of W first and then on the number of its connected components. However, we find it simpler to proceed with a proof by contradiction. Let n be the smallest number of 3-valent vertices in a web for which the statement fails. Let W be such a web with a minimal number of connected components. Since the statement holds for simple webs, W is not simple and, hence, it contains a nontrivial zig-zag subgraph W and its complementary subgraph W. Observe that W = q c W W + linear combination of webs with fewer vertices, in RS(F, B), for some c Z where W W denotes W stacked on top of W. That follows from (3). By Remark 10 (with Γ = D 2 ), up to a power of q, we can rearrange the endpoints of W so that they are after those of W in every component of D 2 B. Consequently, W = q d W W + linear combination of webs with fewer vertices. If the statement holds for W and W then it clearly holds for W. Therefore, it fails for one of them. Since W, W are non-trivial zig-zag subgraphs, either W and W are disjoint or they intersect and W, W have fewer vertices. In either case, we get contradiction with the minimality of crossings and of components assumption on W. Fix an orientation of Γ. Than each face of Γ (i.e. connected component of F Γ, as in Figure 3) is a marked surface in which a multiplication of skeins is well defined. (The orientation of the dashed edges in Figure 3 will play no role.) We denote the set of set of faces of Γ by T (Γ). We say that a web is in minimal position with respect to Γ if it has a minimal number of intersections with Γ. (Then it has a minimal number of intersections with each of the arcs of Γ. A labeling of Γ assigns two non-negative integers, called edge labels, to each arc of Γ, and two non-negative integers, called face labels, to each face of Γ. These pairs of integers are denoted with subscripts + and. Hence, (Z 0 Z 0 ) Γ T (Γ) is the set of all labels of Γ. Each web W in a minimal position with respect to Γ defines a labeling l of Γ as follows: for each face T of Γ, l T,+ and l T, are the numbers of 3-valent sources and sinks of W in T respectively. for each arc β in Γ, l β,+ and l β, denote the number of positive and negative intersections of β with W. (These intersections have signs because β, W and F are oriented.)

11 FINITE GENERATION OF SU(3)-SKEIN ALGEBRAS 11 Figure 5. (a) Positive crossing between a web and γ Γ (b) Negative crossing (c) Positive web endpoint in γ We call such labelings for crossingless webs admissible and denote their set by L(Γ). Consider a labeling assigning l T,± to a face T of Γ and l 1,±, l 2,±, l 3,± to its sides. We say that this labeling is admissible in face T if there is a crossingless web in T with l T,+ 3-valent sources, l T,+ 3-valent sinks, and l i,+ positive and l i, negative endpoints on the i-th side of T. (A web endpoint is positive in T with the edge containing it extended further would have positive crossing with the side of T.) Clearly a labeling of Γ is admissible iff it is admissible in every face of Γ. Lemma 12. (1) L(Γ) is the solution set of a finite set of inequalities on the coordinates of (Z 0 Z 0 ) Γ T (Γ). (2) For each l L(Γ) and each face T of Γ bounded by sides l 1,±, l 2,±, l 3,± there is a unique collection of reduced, disjoint arcs in T which together with l T,+ reduced source triads and l T, reduced sink triads (yielding altogether a web possibly with crossings) has l i,+ positive and l i, negative intersections with the i-th side, for i = 1, 2, 3. Proof. (1) Since a labeling is admissible iff it is admissible in each face and there are finitely many of them, it is enough to prove that the set of labelings admissible in a face T is solution set of a finite set of inequalities on the coordinates l T,±, l 1,±, l 2,±, l 3,± associated to the face T and its sides. Assume first that the sides of T (being arcs of Γ) are oriented according to the orientation of T (induced by that of F ). Then each source triad (respectively: sink triad) has a positive (respectively: negative) endpoint on each side of T. Let l i,+ = l i,+ l T,+, l i, = l i, l T, for i = 1, 2, 3. Then a labeling is realizable in T if l i,+, l i, are non-negative and l i,± + l i+1,± l i+2,± is even and non-negative for i = 1, 2, 3 (mod 3). This completes the proof in the case that the sides of T are oriented according to the orientation of T. If the i-th side has the opposite orientation then l i,± should be negated in the above inequalities. (2) follows from the proof of (1).

12 12 CHARLES FROHMAN, ADAM S. SIKORA For each l L(Γ) we will define a web (possibly with crossings) W l,t in the face T as the product of reduced triads and reduced arcs (like those in Figure 4) specified in Lemma 12(2), in some order. By this construction, for any two faces T, T sharing a side γ Γ, the number of endpoints of W l,t in γ of certain sign coincides with the number endpoints of W l,t in γ of the same sign. That means that the number of outward endpoints of T in γ equals the number of inward endpoints of T starting in γ. Consequently, rearranging the endpoints of T in γ so their consecutive signs coincide with those of T, webs W l,t, W l,t concatenate along γ into a web in T T. By applying that step for all arcs in Γ we obtain a web in (F, B) which we denote by W l. Note that the above construction defines W l uniquely up to rearranging endpoints of webs W l,t along edges. By Remark 10, the choices made in the construction W l is define it uniquely in GRS(F, B), up to a power of q. Note that W 0 =. Corollary 13. Let W be a web (possibly with crossings) such that in every face T of Γ it is a union of reduced simple webs in T stacked one on top of the other. Then W = q e W l, for the labeling of Γ induced by W. Theorem 14. {W l } l L(Γ) spans GRS(F, B). Proof. We will show that every crossingless web W is a linear combination of webs {W l } l L(Γ) in GRS(F, B) by induction on the number of 3-valent vertices of W. Assume that W has the minimal position with respect to Γ. The statement holds for W with no 3-valent vertices since then W T is a disjoint union of reduced arcs for every face T of Γ. Then W = W l for a labeling l with all face labels zero. Assume that the statement holds for all crossingless webs with n vertices and that W has n + 1 vertices. Since all bigons and 4-gons can be resolved without increasing the number of vertices, we can assume that W is non-elliptic. Lemma 11 implies that in every face T of Γ there are simple webs Y T,1,..., Y T,kT such that W T q d T Y T,1... Y T,kT has fewer 3-valent vertices than W T for some d Z. Note that by concatenating the webs Y T,1... Y T,Tk for all faces T of Γ we obtain a web W l in (F, B) form some l L(Γ). Then W q T d T W l is a combination of webs with fewer 3-valent vertices. Now the statement follows from by inductive assumption. The map L(Γ) GRS(F, B) sending l to W l is a homomorphism up to a power of q. Specifically Proposition 15. For any l, l L(Γ), W l W l = q d W l+l for some d Z. Proof. By construction, W l+l and W l W l are concatenations of webs in faces of Γ. In each such component, T, W l+l T and (W l W l ) T are products of the same simple webs in T. However, these simple webs may appear in different orders. A change of order of the product of two simple webs can be achieved by the operation of Remark 10.

13 FINITE GENERATION OF SU(3)-SKEIN ALGEBRAS 13 Let q ±1 be central in A. We say that a, a A q-commute iff a a = q n a a for some n Q. A skew-polynomial extension A[x, α] denotes the set of polynomials p(x) with coefficients in A with the product ax i bx j = aα(b)x i+j for some automorphism α of A. We say that an iterated skew-polynomial extension of R, R[x 0 ][x 1, α 1 ]...[x n, α n ] is a generalized quantum affine space if each α i multiplies each monomial in x 0,.., x i 1 by a power of q. Lemma 16. If a 0,..., a n pairwise q-commute in an R-algebra A then there is a homomorphism from generalized quantum affine space R[x 0 ][x 1, α 1 ]...[x n, α n ] to A from a sending x i to a i for i = 0, 1,..., n. Proof. The statement is obvious for n = 0, 1 and it follows by induction for higher n. Proof of Theorem 9: (1) Let (F, B) and Γ be as above. Note that L(Γ) is an additive Γ T (Γ) submonoid of Z. Since it is defined by linear inequalities, it is a polyhedral cone Γ T (Γ) 0 in Z 0 and, consequently, it has a finite generating set {l 1,..., l r }, cf. [Co]. Then by Theorem 14 and Proposition 15 W l1,..., W lr generate GRS(F, B). (2) Since W l1,..., W lr q-commute, GRS(F, B) is the image of a generalized quantum affine space, by Lemma 16. Such algebras are Noetherian, by [GW, Thm. I.2.6]. Finally, the image of a Noetherian algebra is Noetherian. [AF] [BB] [BL] References N. Abdiel, C. Frohman, The localized skein algebra is Frobenius, arxiv: Baseilhac, Stephane; Benedetti, Riccardo, Quantum hyperbolic geometry, Algebr. Geom. Topol. 7 (2007), Bonahon, Francis; Liu, Xiaobo, Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms, Geom. Topol. 11 (2007), [BW1] Bonahon, Francis; Wong Helen, Representations of the Kauffman skein algebra I: invariants and miraculous cancellations, arxiv: [math.gt]. [BW2] Bonahon, Francis; Wong, Helen, Representations of the Kauffman Bracket Skein Algebra II: Punctured Surfaces, math.gt/ [BW3] Bonahon, Francis; Wong, Helen, Representations of the Kauffman bracket skein algebra III: closed surfaces and naturality, math.gt/ arxiv: [Bu] Bullock, Doug, A finite set of generators for the Kauffman bracket skein algebra Math. Z. 231 (1999), [CKM] Cautis, Sabin; Kamnitzer, Joel; Morrison, Scott, Webs and quantum skew Howe duality, Math. Ann. 360 (2014), no. 1-2, [CM] L. Charles, J. Marché, Multicurves and regular functions on the representation variety of a surface in SU(2), Comm. Math. Helv. 87 (2012), no. 2, , arxiv: [Co] J. G. van der Corput, Über Systeme von linear-homogenen Gleichungen und Ungleichungen, Proceedings Koninklijke Akademie van Wetenschappen te Amsterdam 34 (1931), [CF] Fock, V. V.; Chekhov, L. O., Quantum Teichmüller spaces, (Russian) Teoret. Mat. Fiz. 120 (1999), no. 3, ; translation in Theoret. and Math. Phys. 120 (1999), no. 3, [FG] Fock, V. V.; Goncharov, A. B., Moduli spaces of convex projective structures on surfaces, Adv. Math. 208 (2007), no. 1,

14 14 CHARLES FROHMAN, ADAM S. SIKORA [FKL] Frohman, Charles; Kania-Bartoszynska, Joanna; Lê, T.T.Q, Unicity for Representations of the Kauffman Bracket Skein Algebra, arxiv: [FKK] Fontaine, Bruce; Kamnitzer, Joel; Kuperberg, Greg, Buildings, spiders, and geometric Satake, Compos. Math. 149 (2013), no. 11, [GW] K.R. Goodearl, R.B. Warfield, Introduction to noncommutative noetherian rings, London Math. Soc, 61, 2nd ed., Cambridge University Press, [KK] Khovanov, Mikhail; Kuperberg, Greg, Web bases for sl(3) are not dual canonical, Pacific J. Math. 188 (1999), no. 1, [Ku] G. Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180 (1996), no. 1, , arxiv: q-alg/ [Le1] T.T.Q. Lê, On Kauffman bracket skein modules at roots of unity, Algebr. Geom. Topol. 15 (2015) , arxiv: [MPS1] Morrison, Scott; Peters, Emily; Snyder, Noah, Categories generated by a trivalent vertex, Selecta Math. (N.S.) 23 (2017), no. 2, [MPS2] Morrison, Scott; Peters, Emily; Snyder, Noah, Knot polynomial identities and quantum group coincidences, Quantum Topol. 2 (2011), no. 2, [MR] McConnell, J. C.; Robson, J. C., Noncommutative Noetherian rings. With the cooperation of L. W. Small. Revised edition., Graduate Studies in Mathematics, 30.American Mathematical Society, Providence, RI, xx+636 pp. ISBN: [Mu] G. Muller, Skein algebras and cluster algebras of marked surfaces, Quan. Topol. to appear (2014), arxiv: [Prz] J.H. Przytycki, Fundamentals of Kauffman bracket skein modules, Kobe Math. J., 16(1), 1999, 45 66, arxiv:math/ [PS1] J. H. Przytycki, A. S. Sikora, On Skein Algebras And Sl 2 (C)-Character Varieties, Topology, 39(1) 2000, , arxiv:q-alg/ [PS2] J. H. Przytycki, A. S. Sikora, Skein algebras of surfaces, Trans. of AMS, to appear. [Si] A. S. Sikora, Skein theory for SU(n)-quantum invariants, Alg. Geom. Topol. 5 (2005) arxiv: math/ [Tu] V. G. Turaev, The Confway and Kauffman modules of the solid torus, Zap. Nauchn. Sem. Lomi 167 (1988), English translation: J. Soviet Math. 52, (1990), University of Iowa, and 244 Math Bldg, University at Buffalo, SUNY, Buffalo, NY address: charles-frohman@uiowa.edu, asikora@buffalo.edu