Unrestricted Quantum Moduli Algebras, II: Noetherianity and Simple Fraction Rings at Roots of 1

We prove that the quantum graph algebra and the quantum moduli algebra associated to a punctured sphere and complex semisimple Lie algebra $\mathfrak{g}$ are Noetherian rings and finitely generated rings over $\mathbb{C}(q)$. Moreover, we show that these two properties still hold on $\mathbb{C}\big[q,q^{-1}\big]$ for the integral version of the quantum graph algebra. We also study the specializations $\mathcal{L}_{0,n}^\epsilon$ of the quantum graph algebra at a root of unity $\epsilon$ of odd order, and show that $\mathcal{L}_{0,n}^\epsilon$ and its invariant algebra under the quantum group $U_\epsilon(\mathfrak{g})$ have classical fraction algebras which are central simple algebras of PI degrees that we compute.


Introduction
This paper is the second part of our work on the unrestricted quantum moduli algebras, that we initiated in [28]. These algebras, denoted by M A g,n (g) hereafter, are defined over the ground ring A = C[q, q −1 ] and associated to unrestricted quantum groups of complex simple Lie algebras g, and surfaces of genus g with n + 1 punctures (thus, n = −1 corresponds to closed surfaces). We are in particular interested in the specializations M A, g,n (g) of M A g,n (g) at roots of unity q = .
As in [28] we focus in this paper on the algebras M A 0,n (g) associated to punctured spheres. From now on we fix a complex simple Lie algebra g, and when no confusion may arise we omit g from the notation of the various algebras.
The rational form M g,n of M A g,n = M A g,n (g), which is an algebra over C(q), has been introduced in the mid 90s by Alekseev-Grosse-Schomerus [2,3] and Buffenoir-Roche [30,31]. They defined M g,n by a q-deformation of the algebra of functions on the Fock-Rosly lattice models of the moduli spaces M cl g,n of flat g-connections on surfaces of genus g with n + 1 punctures. Because of this geometric input, it is quite natural to expect that the representation theory of the specializations M A, g,n (g) of M A g,n (g) at roots of unity q = provides a (2+1)-dimensional TQFT for 3-manifolds endowed with general flat g-connections, extending the known TQFTs based on quantum groups (where purely topological ones correspond to the trivial connection).
For instance, representations of the semisimplification of M A, g,n have been constructed and classified in [4]; they involve only the irreducible representations of the finite dimensional "small", also called "restricted", quantum group u (g), which is a quotient of U (g) below, and a version of the Frobenius-Lusztig kernel of g at (see [23], III.6.4). Moreover, [4] deduced from their representations of M A, g,n a family of representations of the mapping class groups of surfaces, that is equivalent to the one associated to the Witten-Reshetikin-Turaev TQFT [81,73].
Recently, representations of another quotient of M A, g,n have been constructed in [47]. The corresponding representations of the mapping class groups of surfaces are equivalent to those previously obtained by Lyubashenko-Majid [62], and are associated to the so called nonsemisimple TQFT defined by Geer, Patureau-Mirand and their collaborators (see eg. [44,45]). In the sl(2) case they involve the irreducible and also the principal indecomposable representations of u (sl (2)). The related link and 3-manifold invariants coincide with those of [64] and [19].
In general, the representation theory of M A, g,n is by now far from being completely understood. As mentioned above, it is expected to provide a good framework to construct and study quantum invariants of 3-manifolds equipped with general flat g-connections. A family of such invariants, called quantum hyperbolic invariants, has already been defined for g = sl(2) by means of certain 6j-symbols, Deus ex machina; they are closely connected to classical Chern-Simons theory, provide generalized Volume Conjectures, and contain quantum Teichmüller theory (see [12]- [18]). It is part of our present program, initiated in [9], to shed light on these invariants and to generalize them to arbitrary g by developing the representation theory of M A, g,n . Besides, the quantum moduli algebras are now recognized as central objects from the viewpoints of factorization homology [20] and(stated) skein theory [22,49,34]. As already suggested above, their underlying formalism of combinatorial quantisation is very-well suited to the construction of mapping class group representations [48]. In another direction, one may expect that the equivalence proved in [63] between combinatorial quantisation for the Drinfeld double D(H) of a finite-dimensional semisimple Hopf algebra H, and Kitaev's lattice model in topological quantum computation, can be extended to the setup of quantum moduli algebras.
We introduced M A 0,n and began its study in [28]. Its definition is based on the original combinatorial quantization method of [2,3] and [30,31], and uses also twists of modulealgebras. This allows us to exploit fully the representation theory of quantum groups, by following ideas of classical invariant theory. Namely, as we shall describe more precisely below, M A 0,n can be regarded as the invariant subalgebra of a certain module-algebra L A 0,n , endowed with an action of the unrestricted (De Concini-Kac) integral form U A = U A (g) of the quantum group U q = U q (g). We therefore study L A 0,n and its specializations L 0,n at q = a root of unity. Under such a specialization M A 0,n embeds in (L 0,n ) U , the invariant subalgebra of L 0,n under the action of the specialization U of U A at q = .
In [28], for q a root of unity we focused on the case g = sl(2) and described a Poisson action of the center on M A, 0,n (sl(2)), derived from the quantum coadjoint action of De Concini-Kac-Procesi [38,39,40]. The results we prove in the present paper hold for every complex simple Lie algebra g. The main ones are a proof that L A 0,n and M A 0,n are Noetherian, finitely generated rings (Theorem 1.1), and L 0,n and (L 0,n ) U are maximal orders of their central localizations (Theorem 1.3). We conclude with an application to their representation theories (Corollary 1.4).
Let us now state precisely and comment our results. First we need to fix notations. Let U q be the simply-connected quantum group of g, defined over the field C(q). From U q one can define a U q -module algebra L 0,n , called graph algebra, where U q acts by means of a right coadjoint action. The quantum moduli algebra M 0,n is the subalgebra L Uq 0,n of invariant elements of L 0,n for this action. The unrestricted quantum moduli algebra M A 0,n is an integral form of M 0,n (thus, defined over A = C[q, q −1 ]). As a C(q)-module L 0,n is just O ⊗n q , where O q = O q (G) is the standard quantum function algebra of the connected and simply-connected Lie group G with Lie algebra g. The product of L 0,n is obtained by twisting both the product of each factor O q and the product between them. It is equivariant with respect to a (right) coadjoint action of U q , which defines the structure of U q -module of L 0,n . The module algebra L 0,n has an integral form L A 0,n , defined over A, endowed with a coadjoint action of the unrestricted integral form U A of U q introduced by De Concini-Kac [38]. The algebra L A 0,n is obtained by replacing O q in the construction of L 0,n with the restricted dual O A of the integral form U res A of U q defined by Lusztig [60], or equivalently with the restricted dual of the integral form Γ of U q defined by De Concini-Lyubashenko [42]. The unrestricted integral form M A 0,n of M 0,n is defined as the subalgebra of invariant elements, M A 0,n := (L A 0,n ) U A . A cornerstone of the theory of M A 0,n is a map originally due to Alekseev [1], building on works of Drinfeld [36] and Reshetikhin and Semenov-Tian-Shansky [70]. In [28] we showed that it eventually provides isomorphisms of module algebras and algebras respectively, A is endowed with a right adjoint action of U A , and (U ⊗n A ) lf is the subalgebra of locally finite elements with respect to this action. When n = 1 the algebra U lf A has been studied in great detail by Joseph-Letzter [52,53,51]; their results we use have been greatly simplified in [80].
All the material we need about the results discussed above is described in [28], and recalled in Section 2.1-2.2.
Our first result, proved in Section 3, is: Theorem 1.1. L 0,n , M 0,n and their unrestricted integral forms and specializations at q ∈ C \ {0, 1} are Noetherian rings, and finitely generated rings.
In [28] we proved that these algebras have no non-trivial zero divisors. Also, we deduced Theorem 1.1 in the sl(2) case by using an isomorphism between M 0,n (sl (2)) and the skein algebra of a sphere with n + 1 punctures, which by a result of [66] is Noetherian and finitely generated. Our approach here is completely different. For L 0,n we adapt the proof given by Voigt-Yuncken [80] of a result of Joseph [51], which asserts that U lf q is a Noetherian ring (Theorem 3.1). For M 0,n we deduce the result from the one for L 0,n , by following a line of proof of the Hilbert-Nagata theorem in classical invariant theory (Theorem 3.2).
From Section 4 we consider the specializations L 0,n of L A 0,n at q = , a root of unity of odd order l coprime to 3 if g has G 2 components. In [42], De Concini-Lyubashenko introduced a central subalgebra Z 0 (O ) of O isomorphic to the coordinate ring O(G), and proved that the Z 0 (O )-module O is projective of rank l dimg . As observed by Brown-Gordon-Stafford [25], Bass' Cancellation theorem in K-theory and the fact that K 0 (O(G)) ∼ = Z, proved by Marlin [68], imply that this module is free. Alternatively, this follows also from the fact that O is a cleft extension of O(G) by the dual of the Hopf algebra u (g), as proved by Andruskiewitsch-Garcia (see [6], Remark 2.18(b), and also Section 3.2 of [21]; this argument was explained to us by K. A. Brown).
The section 4 proves the analogous property for L 0,n . Namely: Theorem 1.2. L 0,n has a central subalgebra Z 0 (L 0,n ) isomorphic to O(G) ⊗n , and it is a free Z 0 (L 0,n )-module of rank l n.dimg , isomorphic to the O(G) ⊗n -module O ⊗n .
A similar statement for (L 0,n ) U is in Theorem 1.3 (3) below.
We prove the first and third claims of Theorem 1.2 in Proposition 4.2. Since L 0,n and O ⊗n are the same modules over O(G) ⊗n , at this point we can just deduce the second claim from the results of [42] and [68], or [6], recalled above. Nevertheless we give a self-contained proof that L 0,1 is finite projective of rank l dimg over Z 0 (L 0,1 ) by adapting the original arguments of Theorem 7.2 of De Concini-Lyubashenko [42]. In particular we study the coregular action of the braid group of g on L 0,1 ; by the way, in the Appendix we provide different proofs of some technical facts shown in [42]. Of course, it remains an exciting problem to describe the centralizing extension O(G) ⊗n ⊂ L 0,n (and similarly O(G) ⊗n ⊂ (L 0,n ) U below), aiming at generalizing the results of [6] and finding a direct, more structural proof of freeness in Theorem 1.2.
It is worth noticing that the most natural definition of Z 0 (L 0,1 ) is Φ −1 1 (U lf ∩Z 0 (U )), where Z 0 (U ) is the De Concini-Kac-Procesi central subalgebra of U , and U lf the specialization at q = of the algebra U lf A . Thus it is not directly connected to Z 0 (O ), and the algebra structures of L 0,1 and O are completely different indeed. For arbitrary n we set Z 0 (L 0,n ) = Z 0 (L 0,1 ) ⊗n . The fact that Z 0 (L 0,n ) is central in L 0,n , and Z 0 (L 0,1 ) and Z 0 (O ) coincide and give L 0,1 and O the same module structures over these subalgebras, relies on results of De Concini-Kac [38], De Concini-Procesi [39,40], and De Concini-Lyubashenko [42], that we recall in Section 2.3-2.4.
Also, we note that basis of L 0,n over Z 0 (L 0,n ) are complicated. The only case we know is g = sl(2), described in [43], and it is far from being obvious (see (43)).
In Section 5 we turn to fraction rings. As mentioned above L 0,n has no non-trivial zero divisors. Therefore its center Z(L 0,n ) is an integral domain. Denote by Q(Z(L 0,n )) its fraction field. Denote by (L 0,n ) U the subring of L 0,n formed by the invariant elements of L 0,n with respect to the right coadjoint action of U . Note that we trivially have an inclusion M A, 0,n ⊂ (L 0,n ) U , and these two algebras are distinct in general; for instance, when n = 1 we have by definition (L 0,1 ) U = Z(L 0,1 ), which is a finite extension of O(G) by Theorem 1.2 and Corollary 5.7 discussed below, whereas M A, 0,n is the specialization at q = of Z(L A 0,1 ), a polynomial algebra which may be identified via Φ 1 with Z(U A ), generated by the quantum Casimir elements. Also the center Z(L 0,n ) of L 0,n is contained in (L 0,n ) U (this follows from [28], Proposition 6.17). Consider the rings In general, given a ring A with center Z an integral domain we reserve the notation Q(A) to the central localization of A, ie. Q(A) := Q(Z) ⊗ Z A. Though the center Z((L 0,n ) U ) of (L 0,n ) U is larger than Z(L 0,n ), the notation Q((L 0,n ) U ) is not ambiguous, for Z((L 0,n ) U ) is an integral domain finite over Z(L 0,n ), and hence the central localization of (L 0,n ) U coincides with Q((L 0,n ) U ) as defined above. Throughout the paper, unless we mention it explicitly we follow the conventions of Mc Connell-Robson [69] as regards the terminology of ring theory; in particular, for the notions of central simple algebras, (maximal) orders and PI degrees, see in [69] the sections 5.3 and 13.3.6-13.6.7.
Denote by m the rank of g, and by N the number of its positive roots. We prove: is a central simple algebra of PI degree l nN , and L 0,n is a maximal order of Q(L 0,n ).
The first claim of the statement (1) means that Q(L 0,n ) is a complex subalgebra of a full matrix algebra M at d (F), where d = l nN and F is a finite extension of Q(Z(L 0,n )) such that We deduce it from Theorem 1.2 and the computation of the degree of Q(Z(L 0,n )) as a field extension of Q(Z 0 (L 0,n )). This computation uses Φ n and the computation of the degree of Q(Z(U )) over Q(Z 0 (U )) by De Concini-Kac [38] (see Proposition 5.3).
The second claim of (1) is proved in Theorem 5.6. More precisely we prove that L 0,n is integrally closed in Q(L 0,n ), in the sense of [38,40]. So, before the theorem we show in Lemma 5.5 that a ring A with no non-trivial zero divisors, Noetherian center, and finite dimensional classical fraction algebra Q(A), which is the case of L 0,n and (L 0,n ) U , is integrally closed in Q(A) if and only if it is maximal as a (classical) order. For the sake of clarity we have included a general discussion of these notions before Theorem 5.6. The proof of that theorem uses the facts that O is a maximal order of its classical fraction algebra, which is Theorem 7.4 of [42], and that the twist which defines the algebra structure of L 0,n from O ⊗n keeps the Z 0 -module structure unchanged. It seems harder to prove directly that L 0,n is a maximal order, without this twist argument, essentially because we know only one localization of L 0,n which is a maximal order (and thus cannot apply the Serre argument as in Theorem 7.4 of [42]), and, in another direction, we lack of a complete set of defining relations, allowing for degeneration arguments as in [40,41]. However, as an example we do it in the sl(2) case when n = 1.
As a consequence of the maximality of L 0,n and the fact that Z(L 0,n ) is Noetherian, it is an integrally closed domain, equal to the trace ring of L 0,n . In fact Z(L 0,n ) = Z(L 0,1 ) ⊗n , and it is a free Z 0 (L 0,n )-module of rank l mn (see Corollary 5.7).
We deduce the first claim of (2) and the second of (3) from the assertion (1), the double centralizer theorem for central simple algebras, a few results of [28] and [42], and Theorem 1.2 again.
The first claim of (3) follows directly from the fact that O(G) and L 0,n are Noetherian rings (the latter by Theorem 1.1; see the proof of Theorem 4.7 for details). Finally, the left regular action of ∆ (n) (L 0,1 ) ⊗ ∆ (n) (Z(L 0,1 )) (L 0,n ) U on L 0,n yields the following decomposition into simple components, From this and our previous results for L 0,n we deduce the last claims of (2) and (3). We note that Z(L 0,1 ) = (L 0,1 ) U , and a certain localization of L 0,1 is a direct summand of U (see Theorem 2.2 (2) and Corollary 2.5 (2)). So one can view the freeness of the Z 0 (L 0,n )-module (L 0,1 ) U as a generalization of the fact that Z(U ) is free of rank l m over Z 0 (U ) (proved in [40], Proposition 20.2).
We conclude with an application of Theorem 1.3, providing a characterization of the irreducible representations of maximal dimension. Recall that given a classical order A of PI degree d and with center Z a Noetherian and integrally closed domain, the discriminant D(A) is the ideal of Z generated by the elements det((t red (x i x j )) 1≤i,j≤d ), where x 1 , . . . x d ∈ A and t red : A → Z is the reduced trace map of Q(A) restricted to A (see [71], Section 10). Given a central character χ ∈ Maxspec(Z) denote by I χ the ideal of A generated by the kernel of χ, and let A χ := A/I χ . Our results show all this applies in particular to A := L 0,n or (L 0,n ) U . Classical arguments then imply that if A has no non-trivial zero divisors, then A χ = 0, and moreover we have (see eg. Lemma 3.7 of [38]): ∈ D(L 0,n ), and if χ ∈ D(L 0,n ) every irreducible representation of (L 0,n ) χ has dimension less than d. Much more can be said on irreducible representations of dimension < d, eg. by using lower discriminant ideals (see the Main Theorem of [26]). Also, it follows from Theorem 7.18 of [28] that L 0,n (sl(2)) is a Poisson order relative to its center, which is a Poisson central finite extension of O(SL(2, C) n ) endowed with the Fock-Rosly Poisson structure. This should extend without difficulty to all g beyond the sl(2) case. By the results of [24], the zero locus of D(L 0,n (sl(2))) is then a union of symplectic leaves in Maxspec(Z(L 0,n (sl(2)))) (a determined, finite covering space of SL(2, C) n ). There is a similar result for M 0,n (sl(2)) (Corollary 7.21 of [28]), in terms of the Atiyah-Bott-Goldman Poisson structure on the invariant coordinate ring O(SL(2, C) n ) SL (2,C) .
In [29] we use all this to describe the subalgebra M A, 0,n ⊂ (L 0,n ) U and its representations, and we give applications to skein algebras (which is the sl(2) case). In [27] we consider the algebras M A, g,n for genus g = 0.
Given a Hopf algebra H with product m and and coproduct ∆, we denote by H cop (resp. H op ) the Hopf algebra with the same algebra (resp. coalgebra) structure as H but the opposite coproduct σ • ∆ (resp. opposite product m • σ), where σ(x ⊗ y) = y ⊗ x, and the antipode S −1 . We use Sweedler's coproduct notation, We let g be a finite dimensional complex simple Lie algebra of rank m, with Cartan matrix (a ij ). We fix a Cartan subalgebra h ⊂ g and a basis of simple roots α i ∈ h * R ; we denote by d 1 , . . . , d m the unique coprime positive integers such that the matrix (d i a ij ) is symmetric, and ( , ) the unique inner product on h * R such that d i a ij = (α i , α j ). For any root α the coroot is αˇ= 2α/(α, α); in particular where i is the fundamental weight dual to the simple coroot αˇi, ie. satisfying ( i , αˇj) = δ i,j . We denote by P + := m i=1 Z ≥0 i the cone of dominant integral weights, by N the number of positive roots of g, by ρ half the sum of the positive roots, and by D the smallest positive integer such that D(λ, µ) ∈ Z for every λ, µ ∈ P . Note that (λ, α) ∈ Z for every λ ∈ P , α ∈ Q, and D is the smallest positive integer such that DP ⊂ Q. We denote by B(g) the braid group of g; we recall its standard defining relations in the Appendix (Section 6.1).
We let G be the connected and simply-connected Lie group with Lie algebra g. We put We let q be an indeterminate, set A = C[q, q −1 ], q i = q d i , and given integers p, k with 0 ≤ k ≤ p we put We denote by a primitive l-th root of unity such that 2d i = 1 is also a primitive l-th root of unity for all i ∈ {1, . . . , m}. This means that l is odd, and coprime to 3 if g has G 2 -components.
In this paper we use the definition of the unrestricted integral form U A (g) given in [40], [42]; in [28] we used the one of [38], [39]. The two are (trivially) isomorphic, and have the same specialization at q = . Also, we denote here by L i the generators of U q (g) we denoted by i in [28].
To facilitate the comparison with [42] we note that their generators, that we will denote byK i ,Ẽ i andF i , can be written respectively as K i , K −1 i E i and F i K i in our notations. They satisfy the same algebra relations.

2.1.
On U q , O q , L 0,n , M 0,n , and Φ n . Except when stated differently, we refer to [28], Sections 2-4 and 6, and the references therein for details about the material of this section.
The simply-connected quantum group U q = U q (g) is the Hopf algebra over C(q) with j . The coproduct ∆, antipode S, and counit ε of U q are given by We fix a reduced expression s i 1 . . . s i N of the longest element w 0 of the Weyl group of g. It induces a total ordering of the positive roots, The root vectors of U q with respect to such an ordering are defined by where T i is Lusztig's algebra automorphism of U q associated to the simple root α i ( [61,60], see also [35], Ch. 8). In the Appendix we recall the relation between T i and the generator w i of the quantum Weyl group, which we will mostly use. Let us just recall here that the monomials F r 1 form a basis of U q . U q is a pivotal Hopf algebra, with pivotal element The adjoint quantum group U ad q = U ad q (g) is the Hopf subalgebra of U q generated by the elements E i , F i (i = 1, . . . , m) and K α with α ∈ Q; so ∈ U ad q . When g = sl(2), we simply write the above generators We denote by U q (n + ), U q (n − ) and U q (h) the subalgebras of U q generated respectively by the E i , the F i , and the K λ (λ ∈ P ), and by U q (b + ) and U q (b − ) the subalgebras generated by the E i and the K λ , and by the F i and the K λ , respectively (they are the two-sided ideals generated by U q (n ± )). We do similarly with U ad q . U ad q is not a braided Hopf algebra in a strict sense, but it has braided categorical completions. Namely, denote by C the category of type 1 finite dimensional U ad q -modules, by V ect the category of finite dimensional C(q)-vector spaces, and by F C : C → V ect the forgetful functor. The categorical completion U ad q of U ad q is the set of natural transformations F C → F C . Let us recall briefly what this means and implies. For details we refer to the sections 2 and 3 of [28] (see also [80], Section 2.10, where U q below is formulated in terms of multiplier Hopf algebras). An element of U ad q is a collection (a V ) V ∈Ob(C) , where a V ∈ End C(q) (V ) satisfies F C (f ) • a V = a W • F C (f ) for any objects V, W of C and any arrow f ∈ Hom U ad q (V, W ). It is not hard to see that U ad q inherits from C a natural structure of Hopf algebra such that the map is a morphism of Hopf algebras, where π V : U ad q → End(V ) is the representation associated to a module V in C. It is a theorem that this map is injective; U ad q can be understood as a weak- * completion of U ad q by means of the pairing ., . introduced below. From now on, let us extend the coefficient ring of the modules and morphisms in C to C(q 1/D ). Put U q = U ad q ⊗ C(q) C(q 1/D ) One can show that the map ι above extends to an embedding of U q ⊗ C(q) C(q 1/D ) in U q . The category C, with coefficients in C(q 1/D ), is braided and ribbon. We postpone a discussion of that fact to Section 2.3, where it will be developed. As a consequence, U q is a quasitriangular and ribbon Hopf algebra. The R-matrix of U q is the family of morphisms where q = e h , R h is the universal R-matrix of the quantized universal enveloping algebra One defines the categorical tensor product U⊗ 2 q similarly as U q ; it contains all the infinite series of elements of U ⊗2 q having only a finite number of non-zero terms when evaluated on a given module V ⊗ W of C. The expansion of R h as an infinite series in U h (g)⊗ 2 induces an expansion of R as an infinite series in U⊗ 2 q . Adapting Sweedler's coproduct notation ∆(x) = (x) x (1) ⊗ x (2) we find convenient to write this series as We put R + := R, The quantum function Hopf algebra O q = O q (G) is the restricted dual of U ad q , ie. the set of C(q)-linear maps f : U ad q → C(q) such that Ker(f ) contains a cofinite two sided ideal I (ie. such that I ⊕ M = U q for some finite dimensional vector space M ), and r s=−r (K i − q s i ) ∈ I for some r ∈ N and every i. The structure maps of O q are defined dually to those of U ad q . We denote by its product. The algebras O q (T G ), O q (U ± ), O q (B ± ) are defined similarly, by replacing U ad q with U ad q (h), U ad q (n ± ), U ad q (b ± ) respectively. O q is generated as an algebra by the functionals x → w(π V (x)v), x ∈ U ad q , for every object V ∈ Ob(C) and vectors v ∈ V , w ∈ V * . Such functionals are called matrix coefficients. We can uniquely extend the (nondegenerate) evaluation pairing ., . : O q ⊗ U ad q → C(q) to a bilinear pairing ., . : O q ⊗ U q → C(q 1/D ) such that the following diagram is commutative: : : It is a perfect pairing, and reflects the properties of the R-matrix R ∈ U⊗ 2 q in a subtle way. In particular, these properties imply that the maps are well-defined morphisms of Hopf algebras. Here we stress that it is the simply-connected quantum group U cop q that is the range of Φ ± . This will be explained in more details in Section 2.3.
The quantum loop algebra L 0,1 = L 0,1 (g) is defined by twisting the product of O q , keeping the same underlying linear space. The new product is equivariant with respect to the right coadjoint action coad r of U ad q ; noting that coad r extends to an action of the simply-connected quantum group U q , the new product thus gives L 0,1 a structure of U q -module algebra. Recall that for all x ∈ U q and α ∈ O q , where £, ¡ are the left and right coregular actions of U q on O q , defined by Using the fact that U q ⊗ C(q 1/D ) can be regarded as a subspace of U q , these actions extend naturally to actions of U q . The product of L 0,1 is expressed in terms of by the formula ( [28], Proposition 4.1): where (R) R (1) ⊗ R (2) and (R) R (1 ) ⊗ R (2 ) are expansions of two copies of R ∈ U⊗ 2 q . Note that the sum in (3) has only a finite number of non zero terms. This product gives L 0,1 (like O q ) a structure of module algebra for the actions £, ¡, and also for coad r (x). Spelling this out for coad r , this means The relations between O q , L 0,1 and U q (the simply-connected quantum group) are encoded by the map where R = σ • R, and as usual σ : x ⊗ y → y ⊗ x. Note that We call Φ 1 the RSD map, for Drinfeld, Reshetikhin and Semenov-Tian-Shansky introduced it first (see [36,70], [67]). Recall that U q embeds in U q . It is a fundamental result of the theory ( [33,51,11]) that Φ 1 affords an isomorphism of U q -modules For full details on that result we refer to Section 2.12 of [80] (where different conventions are used). Here, U lf q is the set of locally finite elements of U q , endowed with the right adjoint action ad r of U q . It is defined by S(y (1) )xy (2) for every x, y ∈ U q . The action ad r gives in fact U lf q a structure of right U q -module algebra. Moreover, Φ 1 affords an isomorphism of U q -module algebras The centers Z(L 0,1 ) of L 0,1 , and Z(U q ) of U q , coincide respectively with L Uq 0,1 and U Uq q , the subsets of U q -invariants elements of L 0,1 and U q . As a consequence, Φ 1 affords an isomorphism between Z(L 0,1 ) and Z(U q ).
The quantum graph algebra L 0,n = L 0,n (g) is the braided tensor product of n copies of L 0,1 (considered as a U q -module algebra). Thus it coincides with L ⊗n 0,1 as a linear space, and it is a right U q -module algebra, the action of U q (extending coad r on L 0,1 ) being given by for all y ∈ U q and α (1) ⊗ . . . ⊗ α (n) ∈ L 0,n . The algebra structure can be explicited as follows. For every 1 ≤ a ≤ n define i a : L 0,1 → L 0,n by i a (x) = 1 ⊗(a−1) ⊗ x ⊗ 1 ⊗(n−a) ; i a is an embedding of U q -module algebras. We will use the notations 0,n with a < b. Then the product of L 0,n is given by the following formula (see in [28] the proposition 6.2-6.3 and the formulas (13)-(41)-(42)): , are expansions of four copies of R ∈ U⊗ 2 q , and on the right-hand side the product is componentwise that of L 0,1 . Later we will use the fact that the product of L 0,n is obtained from the standard (componentwise) product of L ⊗n 0,1 by a process that may be inverted. Indeed, (6) can be rewritten as where F = (F ) F (1) ⊗ F (2) := (∆ ⊗ ∆)(R ), and the symbol "·" stands for the right action of U ⊗2 q on L 0,1 that may be read from (6). The tensor F is known as a twist. Then, by replacing F with its inverseF = (∆ ⊗ ∆)(R −1 ), one can express the product of L ⊗n 0,1 in terms of the product of L 0,n by We call quantum moduli algebra and denote by M 0,n = M 0,n (g) the subalgebra L Uq 0,n of L 0,n formed by the U q -invariant elements. Consider the following action of U q on the tensor product algebra U ⊗n q , which extends ad r on U q : for all y ∈ U q , x ∈ U ⊗n q . This action gives U ⊗n q a structure of right U q -module algebra. In [1] Alekseev introduced a morphism of U q -module algebras Φ n : L 0,n → U ⊗n q which extends Φ 1 . In Proposition 6.5 and Lemma 6.8 of [28] we showed that Φ n affords isomorphisms where (U ⊗n q ) lf is the set of ad r n -locally finite elements of U ⊗n q . We call Φ n the Alekseev map; we will not use the definition of Φ n in this paper.
It is a key argument of the proof of (10), to be used later, that the set of locally finite elements of U ⊗n q for (ad r ) ⊗n •∆ (n−1) coincides with (U lf q ) ⊗n ; this follows from the main result of [57]. Using that the map (11) ψ n = Φ n • (Φ −1 1 ) ⊗n extends to a linear automorphism of U ⊗n q which intertwines the actions (ad r ) ⊗n • ∆ (n−1) and ad r n of U q , we deduced that ψ n ((U lf q ) ⊗n ) = (U ⊗n q ) lf , whence Im(Φ n ) = (U ⊗n q ) lf .
Let us point out here two important consequences of (10). First, Φ n yields isomorphisms between centers, Z(L 0,n ) ∼ = Z(U q ) ⊗n and Z(L Uq 0,n ) ∼ = Z((U ⊗n q ) Uq ), where one can show that ( [28], Lemma 6.25) Second, we see that L 0,n (and therefore M 0,n ) has no non-trivial zero divisors, by using the isomorphisms Φ n : L 0,n → (U ⊗n q ) lf ⊂ U ⊗n q and U ⊗n q ∼ = U q (g ⊕n ), and the fact that U q (g ⊕n ) has no non-trivial zero divisors (proved eg. in [38]).

2.2.
Integral forms and specializations. An integral form of a (Hopf) C(q)-algebra is a (Hopf) A-subalgebra, where A = C[q, q −1 ], that becomes isomorphic to the algebra after tensoring it with C(q). We consider three integral forms related by the pairing , , one of U q , one of U ad q , and one of O q . The unrestricted integral form of U q is the A-subalgebra U A = U A (g) introduced by De Concini-Kac-Procesi in [40], Section 12 (and in a differently normalized form in [38] and [39]). It is generated by the elements (i = 1, . . . , m) Clearly, the subalgebra of locally finite elements of Note that Γ contains the elements K i , and the unrestricted integral form U ad A . It plays a fundamental rôle in relation with the integral pairings π ± A considered in Section 2.3; it is by this rôle that Γ is more suited to our purposes than the more standard restricted integral form U res A defined by Lusztig, and discussed below. The integral forms U A (h), U A (b ± ) and Γ(h), Γ(b ± ) associated to the subalgebras h, b ± ⊂ g are the subalgebras of U A and Γ defined in the obvious way. For instance the "Cartan" subalgebra Γ(h) is generated by the elements (K i ; t) q i and K −1 i . Denote by C A the category of Γ-modules which are free A-modules of finite rank, and semisimple as Γ(h)-modules; so they have a basis where K i and (K i ; t) q i act diagonally with respective eigenvalues of the form The integral quantum function Hopf algebra It is immediate that the U q -module structure of O q restricts to an U A -module structure on O A .
We note that O A is also the restricted dual of U res A , the Lusztig integral form of U ad q [60,61], defined as Γ except that the (K i ; t) q i (i = 1, . . . , m), are replaced by the elements Indeed, Γ(h) contains U res A (h) strictly, but the restriction functor C A → C res A is an equivalence of categories, where C res A is the category of U res A -modules defined as C A above, but replacing the condition on (K i ; t) q i by its analog for [K i ; t] q i , ie. that it acts diagonally with eigenvalues The integral form L A 0,1 of L 0,1 is defined as the U A -module O A endowed with the product of L 0,1 , and the integral form L A 0,n of L 0,n is the braided tensor product of n copies of L A 0,1 . That these two products are well-defined over A is elementary (see Definition 4.10 and 6.7 of [28] for the details). The integral quantum moduli algebra is Finally, given q = ∈ C × we define U , Γ , O , L 0,n and M A, 0,n as the C-algebras obtained by tensoring U A , Γ, O A , L A 0,n and M A 0,n respectively with C , the A-module C where q acts by multiplication by . They are the specializations of the latter algebras at q = ; they can also be defined as the quotients by the ideal generated by q − . We find convenient to use the notations Let us stress here that when is a root of unity, taking the locally finite part and taking the specialization at are non commuting operations. Indeed, when has odd order, it follows from Theorem 2.14 below that U is finite over Z 0 (U ) and therefore has all its elements locally finite for ad r ; on another hand U lf A ⊗ A C , ie. U lf in the notations above, does not contain the elements L i .
In a similar manner, taking invariants and taking the specialization at are non commuting operations when is a root of unity: indeed, it is easily checked that in this case (U ⊗n A ) U A and (U ⊗n ) U , or M A, 0,n = M A 0,n ⊗ A C and (L 0,n ) U , are distinct spaces. As explained in the introduction, when is a root of unity, we will not consider the algebras M A, 0,n in this paper.
The morphism Φ n has also an integral form. In order to define it, we first consider the relations between U A and U lf A . Denote by T ⊂ U A the multiplicative Abelian group formed by the elements K λ , λ ∈ P , and by T 2 ⊂ T the subgroup formed by the K λ , λ ∈ 2P . Consider the subset T 2− ⊂ T 2 formed by the elements K −λ , λ ∈ 2P + . It is easily seen to be an Ore  (1) are finite-dimensional U A -modules (by eg. (14) below), so the action ad r is completely reducible on U lf A . In fact, U lf A is the socle of ad r on U A , and by the theorem of separation of variables ( [53,51,11], see also [80]), U lf A has an U A -invariant subspace H such that the multiplication in U A affords an isomorphism of U A -modules from In particular, U lf A is free over Z(U A ). Moreover, any simple finite dimensional U A -module has in H a multiplicity equal to the dimension of its zero-weight subspace.
From the definition (4) it is quite easy to see that Proof. An elementary computational proof of this result in the sl(2) case is given in Section 5 of [28]. A proof of the general case can be found in Lemma 4.11 of [28]. It uses Theorem 2.2 (1). We point out an alternative proof in Remark 2.13 (1). P 0,1 has no non-trivial zero divisors, d is a regular element. It is enough to show that for all x ∈ L A 0,1 there exists elements y, y ∈ L A 0,1 such that xd = dy and dx = y d.
). Therefore the left Ore condition is satisfied with y = coad r (K 2ρ )(x). Similarly one finds y . ( , which implies the assertion (2). P Remark 2.6. When g = sl(2) the element d is the generator of L 0,1 (sl(2)) appearing in (44) below. In this case we had already shown in [28] that Φ 1 : Denote by C(µ), µ ∈ P + , the linear subspace of L 0,1 generated by the matrix coefficients of V µ , the U q -module of type 1 and highest weight µ. The formula (13) can be used to prove (see Section 7.1.22 in [51], or page 112 of [80]) that Φ 1 yields the following linear isomorphism, which illuminates the claim (1) of Theorem 2.2: Working over the ground ring A one has to consider for V µ the highest weight Γ-module of highest weight µ. In that situation Φ 1 affords an isomorphism from C(µ) By (13) we have Φ 1 (ψ −ρ ) = −1 , where as usual is the pivotal element of U A . Because the latter has the elementary factorization = m j=1 L 2 j , this naturally raises the question of the factorization of ψ −ρ . This question is considered in [54], where L 0,1 (g) for g = gl(r + 1) is analysed and quantum minors are extensively studied. Let us review here some of their results in relation with ψ −ρ . First note that for for g = sl(r + 1) the irreducible representation V −ρ of lowest weight −ρ is isomorphic to the representation of highest weight V ρ because −w 0 (ρ) = ρ. By the Weyl formula the dimension of this representation is α>0 (2ρ,α) (ρ,α) = 2 N . In [58] a presentation of U q (gl(r+1)) is given, which differs from our presentation of U q (sl(r+1)) only by its subalgebra U q (h), generated by r + 1 elements .., r. The quantum minors, properly defined in [54], of the matrix of matrix elements of the natural representation of U q (gl(r+1)) are denoted det q (A ≥k ) for k = 1, ..., r + 1. We have det q (A ≥1 ) = 1 in the case of sl(r + 1). Then [54] proves that det q (A ≥k ) = (K k ...K r+1 ) 2 , and there exists an element K ∈ U q (gl(r + 1)) such that This has to be interpreted in the sl(r + 1) case as . As a result this gives the equality Corollary 2.4 can be extended as follows: The proof relies on (10) and the expression of Φ n in terms of Φ 1 and R-matrices (see [28], Proposition 6.5 and Lemma 6.8).
In the case of g = sl(2) we proved in [30] the existence of elements ξ (i) ∈ L A 0,n (i = 1, ..., n), and we defined an algebra loc L A 0,n generalizing L A 0,1 [d −1 ] above, containing L A 0,n as a subalgebra and the inverses of the elements ξ (i) . We showed that Φ n extends to loc L A 0,n , and that Φ n ( loc L A 0,n ) = U ad A (sl(2)) ⊗n . The key property of ξ (i) is For general g we now describe a partial generalization of this result. Define elements ξ (i) j ∈ L A 0,n , for i = 1, ..., n and j = 1, ..., m, by The elements ξ Let us explain the case n = 2. Since the elements ξ (1) j , j ∈ {1, . . . , m}, are commuting regular Ore elements of L A 0,2 we can define the localisation of L A 0,2 with respect to the multiplicative sets {ξ . They are Ore elements, and we can define similarly the localisation 10 for an explanation of this additional construction). We want to define the inverses of the elements ξ (2) j , j ∈ {1, . . . , m}, and a new algebra and Φ 2 extends naturally to an algebra homomorphism Φ 2 : , . . . , m}. As in the sl(2) case described in [28], this can be done by writing explicitly, for every j 2 ∈ {1, . . . , m}, the exchange relations between the matrices M (1) . . , m} (these matrices are defined in (16)). Similarly, by replacing the elements ξ (1) . . , m}. This morphism of algebras will be shown to be an isomorphism.
For any n ≥ 2 we can proceed in the same way: Definition 2.8. By iterating the above construction we define: In the sequel it will be convenient to define invertible elements δ j (i) ∈ loc L A 0,n , for i = 1, ..., n and j = 1, ..., m, satisfying ν j are invertible, commute and satisfy Proof. We know from Corollary 2.4 that Φ 1 : generates the group T we get the result for n = 1. The result for Φ n is obtained by induction. We have Because the matrix elements of (id ⊗ Φ n )( Since the matrix elements of R 0n and the matrix elements of R 0n−1 R 0n−1 , and therefore the space It is a natural problem to determine the image by Φ n of loc L A 0,n , and it is natural to expect that it would be (T −1 2− U lf A ) ⊗n , because this is true for n = 1, as well as for any n in the sl(2) case, as shown in [28]. Unfortunately this is not so. This comes from the fact, eg. for n = 2, that the matrix elements of R 02 R 01 R 01 R −1 02 do not belong to (T −1 2− U lf A ) ⊗2 as can be shown by an explicit computation in the sl (3) case. This explains the reason why we had to introduce the square roots ν (i) j in the previous theorem. Arguments similar to those mentioned at the end of Section 2.1 imply that the algebras L A 0,n , M A 0,n and L 0,n , M A, 0,n , ∈ C × , have no non-trivial zero divisors (see [28], Proposition 7.1). By Theorem 2.7 the Alekseev map yields isomorphisms of U -module algebras, and of algebras for the latter, (2). We collect their properties in Theorem 2.11 and the discussion thereafter. In order to state it, we recall first a few facts about R-matrices and related pairings.

Perfect pairings. We will need restrictions on the integral forms O
In [60,61] Lusztig proved that the category of U res A -modules C res A ⊗ A C[q ±1/D ] (ie. with coefficients extended to C[q ±1/D ]) is braided and ribbon, with braiding given by the collection of endomorphisms ] on the basis of V ⊗ W formed by the tensor products of the canonical (Kashiwara-Lusztig) basis vectors of V and W . The restriction functor C A → C res A is an equivalence of categories, so C A ⊗ A C[q ±1/D ] has the same braided and ribbon structure. This can be rephrased as follows in Hopf algebra terms. Denote by U Γ the categorical completion of Γ, ie. the Hopf algebra of There are pairings of Hopf algebras naturally related to the R-matrix R ∈ U⊗ 2 q . What follows is standard (see eg. [55,56,59] • There is a unique pairing of Hopf algebras ρ : that, for every α, λ ∈ P and l, k ∈ U q (h), • ρ and τ are perfect pairings; this means that they yield isomorphisms of Hopf algebras ). Note that it is the use of weights α, λ ∈ P that forces the pairings ρ, τ to be defined over C(q 1/D ), instead of C(q). Then, let us consider the restrictions π + q of ρ, and π − q of τ , obtained by taking α ∈ Q and l ∈ U q (h), k ∈ U ad q (h). They take values in C(q), and define pairings . By the same arguments as for ρ and τ (eg. in [80], Proposition 2.92), it follows that π ± q are perfect pairings. Note also that π where κ is the conjugatelinear automorphism of U q , viewed as a Hopf algebra over C(q) with conjugation given by κ(q) = q −1 , defined by (20) κ In [42], De Concini-Lyubashenko described integral forms of π ± q as follows. Denote by is free over A, and that a basis is given by the elements The following theorem summarizes results proved in the sections 3 and 4 of [42]. For the sake of clarity, let us spell out the correspondence between statements. First, π + q , π − q , and J are denoted in [42] respectively by π ,π , , A and µ . Also, the definition of j ± A is implicit in the section 4.2 of [42], and the formulas in Theorem 2.11 (3) are related to those in Lemma 4.5 of [42] by observing that their generatorsẼ i andF i are respectively K −1 i E i and F i K i in our notations; this also explains the appearance of q i , q −1 i in the formulas in (3). Finally, κ in (20) Theorem 2.11. (1) π ± q restricts to a perfect Hopf pairing between the unrestricted and restricted integral forms, cop is an embedding of Hopf algebras, and it extends to an isomorphism J : In particular it satisfies (where λ ∈ P + ): For our purposes it is necessary to reformulate this result. Consider the morphisms of Hopf Thus, the theorem above tells us that Φ ± is an isomorphism of Hopf algebras, such that . Moreover, changing the notation J for Φ, is an embedding of Hopf algebras, and it extends to an isomorphism Φ : O A [ψ −1 −ρ ] → U A (H) which in particular satisfies: Proof of Lemma 2.12. By definitions, for every . By keeping these respective notations for X and Y , we deduce j − q (i + (S −1 (X))) = X and j + q (i − (S −1 (Y ))) = Y , ie.
Also, for every α − ∈ O q (B − ) we have where the first equality is by definition of Φ + (see (2)), the second is (19), the third follows from (24), and the last from the definition of j − q . Similarly, for every α + ∈ O q (B + ) we have These computations imply Φ ± = S •i −1 ∓ = j ± q , and the result follows by taking integral forms. P The dualities of Theorem 2.11 (2) afford a refinement defined over A of the quantum Killing form κ : U q ⊗ C(q) U q → C(q 1/D ) (studied eg. in [80], Section 2.8). This form is the duality realizing the isomorphism ad r (U A )(K −2w 0 (µ) ) ∼ = End A ( A V µ ) * stated after (14).

2.4.
Structure theorems for U and O . As usual we denote by a primitive l-th root of unity, where l is odd, and coprime to 3 if g has G 2 -components.
Let G 0 = B + B − (the big cell of G), and define the group Consider the map The restriction of σ to H is an unramified covering of degree 2 m . It can be seen as the classical analog of the map m Denote by Z 1 (U ) the image of Z(U q ) in Z(U ) under the specialization map U q → U , and by Z 0 (U ) ⊂ U the subalgebra generated by E l β k , F l β k , L ±l i , for k ∈ {1, . . . , N } and i ∈ {1, . . . m}. In [38], Section 1.8-3.3-3.8, and [40], Theorem 14.1 and Section 20-21, the following results are proved: Theorem 2.14. (1) U has no non-trivial zero divisors, Z 0 (U ) is a central Hopf subalgebra of U , and U is a free Z 0 (U )-module of rank l dimg . Moreover U is a maximal order of its classical fraction algebra Q(U ) = Q(Z(U )) ⊗ Z(U ) U , and Q(U ) is a central simple algebra of PI degree l N .
(2) Maxspec(Z 0 (U )) is a group isomorphic to H above, and the multiplication map yields an It follows from (1) and dimg = m+2N that the field Q(Z(U )) is an extension of Q(Z 0 (U )) of degree l m . Conversely, this degree and the rank of U over Z 0 (U ) imply that Q(U ) has PI degree l N .
As for (2), note that Z 0 (U ) being an affine and commutative algebra, Maxspec(Z 0 (U )), viewed as the set of characters of Z 0 (U ), acquires by duality a structure of affine algebraic group. Thus, the first claim means precisely the identification of this group with H.
In addition to (2), Maxspec(Z 0 (U )) and H have natural Poisson structures, that the isomorphism identifies. Moreover we have the following identifications (see [40], Section 21.2). Consider the l m -fold coveringT G → T G . Recall that T is the group formed by the elements K λ ∈ U A , λ ∈ P . We can identify T with the additive group P , The isomorphism of Theorem 2.14 (2) then affords identifications A result similar to Theorem 2.14 holds true for O . Namely, take the specializations at q = in Theorem 2.11. Denote by Z 0 (U (H)) the subalgebra of U (H) generated by the elements (k ∈ {1, . . . , N }, i ∈ {1, . . . m}) It is a central Hopf subalgebra. Recall that O(G) can be realized as a Hopf subalgebra of U (g) • , the restricted dual of the envelopping algebra U (g) over C. In [42] De Concini-Lyubashenko introduced an epimorphism of Hopf algebras η : Γ → U (g) (essentially a version of Lusztig's "Frobenius" epimorphism in [60]). Let us put where η * : U (g) • → Γ • is the monomorphism dual to η. For the proof, see in [42]: the proposition 6.4 for the first claim of (1) (where Z 0 (O ) and Z 0 (U (H)) are denoted F 0 and A 0 respectively), the appendix of Enriquez and [46] for the second claim of (1), the propositions 6.4-6.5 for (2), the theorem 7.2 (where O is shown to be projective over Z 0 (O )) and [25] (which provides the additional K-theoretic arguments to deduce that O is free), or Remark 2.18(b) of [6], for the first claim of (3), and the theorem 7.4 for the second claim.
As above for U , it follows directly from (3) that Q(Z(O )) has degree l m over Q(Z 0 (O )). A complete description of Z(O ) is obtained in [46] and Enriquez' Appendix in [42]. We do not know a basis of O over Z 0 (O ) for general G, but see [43] for the case of SL 2 . We will recall the known results in this case of SL 2 before Lemma 4.3.
There is a natural action of the braid group B(g) on O , that we will use. Namely, let n i ∈ N (T G ) be a representative of the reflection s i ∈ W = N (T G )/T G associated to the simple root α i . In [77,76] Soibelman-Vaksman introduced functionals t i : O A → A which quantize the elements n i . They correspond dually to generators of the quantum Weyl group of g; in the Appendix we recall their main properties (see also [35], Section 8.2, and [55,77,59,56,42]). Denote by ¡ the natural right action of functionals on O A , namely (using Sweedler's notation) h(α (1) )α (2) for every α ∈ O A and h ∈ O A → A. Let us identify Z 0 (O ) with O(G) by means of (25). We have ( [42], Proposition 7.1): We provide an alternative, non computational, proof of this result in the Appendix (Section 6.2).

Noetherianity and finiteness
In this section we prove Theorem 1.1. Recall that by Noetherian we mean right and left Noetherian.
Theorem 3.1. The algebras L 0,n , L A 0,n and L 0,n , ∈ C × , are Noetherian. Let us note that the algebras in this theorem are generated by a finite number of elements over their respective ground rings C(q), A and C. Indeed, by the formula (6) it is enough to verify this for L A 0,1 , but L A 0,1 = O A as a vector space, and O A with its product is well-known to be finitely generated by the matrix coefficients of the fundamental Γ-modules A V k , k ∈ {1, . . . , m}. Then the claim follows from the formula inverse to (3), expressing the product in terms of the product of L 0,1 (see (18) in [28]).
Proof of Theorem 3.1. The result for L 0,1 and L A 0,1 follows immediately from Theorem 2.2 (3) by identifying L A 0,1 with U lf A via Φ 1 . Assume now that n > 1. We are going to develop the proof for L 0,n ; the arguments can be repeated verbatim for L A 0,n , and the result for L 0,n will then follow immediately by lifting ideals by the quotient map L A 0,n → L 0,n = L A 0,n /(q − )L A 0,n . Recall the isomorphism of U q -modules (see (11)): where lf means respectively locally finite for the action ad r n of U q (g) on U q (g) ⊗n , locally finite for the action ad r of U q (g) on U q (g), and locally finite for the action ad r of U lf q (g ⊕n ) on itself. It is a fact that Theorem 2.2 (3) holds true by replacing U lf q (g) with U lf q (g ⊕n ), but one cannot use this to deduce the result because ψ n is not a morphism of algebras. However, one can adapt the arguments of the proof of Theorem 2.2 (3) given in Theorem 2.137 of [80]. Let us begin by recalling these arguments.
(iii) Finally, for every µ ∈ P + fix a basis of weight vectors e µ 1 , . . . , e µ m of V µ . Denote by e 1 µ , . . . , e m µ ∈ V * µ the dual basis, and by w(e µ i ) the weight of e µ i . One can assume that the ordering of e µ 1 , . . . , e µ m is such that w(e µ i ) > w(e µ j ) implies i < j; indeed, e µ 1 generates the subspace of weight µ, then come (in any order) the e µ i such that w(e µ i ) = µ − α s for some s, then those such that w(e µ i ) = µ − α s − α t for some s and t, etc. Consider the matrix coefficients µ φ j i (x) := e i µ (π V (x)(e µ j )), x ∈ U q . By (3), using the explicit form of the R-matrix it can be shown that where q ijkl = q (w(e µ j )+w(e µ i ),w(e ν k )−w(e ν l )) , and γ ijkl rs , δ ijkl rsuv ∈ C(q 1/D ) are such that γ ijkl rs = 0 unless w(e µ r ) < w(e µ i ) and w(e ν s ) > w(e ν k ), and δ ijkl rsuv = 0 unless w(e ν u ) > w(e ν l ), w(e µ v ) < w(e µ j ), w(e µ r ) ≤ w(e µ i ) and w(e ν s ) ≥ w(e ν k ). By (28) (or more simply by using (3), as observed before the proof), Gr F 2 (L 0,1 ) is generated by the matrix coefficients k φ j i of the fundamental representations V k . One can list these matrix coefficients, say M in number, in an ordered sequence u 1 , . . . , u M such that the following condition holds: if w(e s k ) < w(e r i ), or w(e s k ) = w(e r i ) and w(e s l ) < w(e r j ), then u a := r φ j i and u b := s φ l k satisfy b < a. Then denoting µ φ j i , ν φ l k in (29) by u j , u i respectively, and assuming u j < u i , one finds that all terms u s := µ φ v r , µ φ j r in the sums are < u j . Therefore, for all 1 ≤ j < i ≤ M it takes the form: for some q ij ∈ C(q 1/D ) × , α st ij ∈ C(q 1/D ). By Proposition I.8.17 of [23] (see also Proposition 2.133 of [80]) an algebra A over a field K generated by elements u 1 , . . . , u M such that for all 1 ≤ j < i ≤ M and some q ij ∈ K × and α st ij , β st ij ∈ K, is Noetherian. In fact A has an algebra filtration, say F 3 , such that Gr F 3 (A) is a quotient of a skew-polynomial algebra, and thus is Noetherian. Moreover, it is classical that a filtered algebra which graded algebra is Noetherian is Noetherian too (see eg. [69], 1.6.9-1.6.11). Applying this to A = Gr F 2 (L 0,1 ) and going up the filtration F 2 it follows that L 0,1 is Noetherian too.
We are going to extend all these facts to L 0,n . The main point is to generalize the filtration F 2 , which we do first. Consider the semigroup Then L 0,n = . This space is canonically identified with C([µ]) [λ] , so the graded vector space associated to F 2 is We claim that F 2 is an algebra filtration with respect to the product of L 0,n , and therefore Gr F 2 (L 0,n ) is a graded algebra. For notational simplicity let us prove it for n = 2, the general case being strictly similar. Recall that the product of L 0,n is given by the formula (6).
Denoting by q k 2 l 2 k 1 l 1 this scalar, it follows for some scalars α l 1 l 1 l 2 l 2 p k 1 k 2 p ∈ C(q 1/D ), with the dots as above. Moreover α l 1 l 1 l 2 l 2 p k 1 k 2 p = 0 unless w(e µ 2 p ) > w(e µ 2 k 2 ) and w(e ). Now, recall (8). In a similar way we find for all p ∈ {1, . . . , k 2 }, p ∈ {k 1 , . . . , d(µ 1 )} that for some scalars β l 1 l 2 l 1 l 2 p r rp ∈ C(q 1/D ) such that β l 1 l 2 l 1 l 2 p r rp = 0 unless w(e µ 1 r ) > w(e µ 1 k 1 ) and w(e ). Summing up we obtain r =k 2 +1 α l 1 l 1 l 2 l 2 p k 1 k 2 p β l 1 l 2 l 1 l 2 p r rp where at p = k 2 , p = k 1 we set α l 1 l 1 l 2 l 2 k 1 k 1 k 2 k 2 := q k 1 l 1 k 1 l 1 q k 2 l 2 k 2 l 2 q k 2 l 2 k 1 l 1 , and the dots are sums of tensors of the form ( Recall that in (30) we denoted by u 1 , . . . , u M the ordered list of matrix coefficients k φ j i . Let us order in a lexicographic way the elements u i ⊗ u j , ie. as a sequence u (2) 1 , . . . , u (2) M 2 such that the following condition holds: Then, by the conditions ensuring when α l 1 l 1 l 2 l 2 p k 1 k 2 p and β l 1 l 2 l 1 l 2 p r rp are non zero, the last identity takes the form of (30) by replacing u i , u j with u . At the beginning of this computation we , so eventually we find that (30) holds true for all cases 1 ≤ u As in the case of L 0,1 , by using Proposition I.8.17 of [23] one can therefore conclude that there is a filtration F 3 of Gr F 2 (L 0,n ) such that Gr F 3 (Gr F 2 (L 0,n )) is a quotient of a quasipolynomial algebra, and finally that L 0,n is Noetherian. P Theorem 3.2. The algebra M 0,n = L Uq 0,n (respectively M A 0,n , and M A, 0,n , ∈ C × ) is Noetherian, and generated over C(q) (resp. A, C) by a finite number of elements.
Our method of proof follows closely that of the Hilbert-Nagata theorem (see [37]). Let us recall one version of this theorem, which is enough for our purposes. Let A = K[a 1 , ..., a n ] be a finitely generated commutative algebra over an arbitrary field K, and G a group of algebra automorphisms of A. Theorem 3.3. If the action of G on A is completely reducible on finite dimensional representations, then the ring A G of invariants of A with respect to G is Noetherian and a finitely generated algebra over K.
We recall here the main steps of the proof that we will adapt in order to prove Theorem 3.2: (a) From the complete reducibility of the action of G on A one can define a linear map namely the projection onto the space of invariants along the space of non-trivial isotypical components of A. This linear map is called the Reynolds operator; it satisfies (b) Let I be an ideal of A G . Then I = R(AI) = AI ∩ A G . Because AI is an ideal of A, and A is Noetherian, there exist elements b 1 , ..., b s , that can be chosen in I ⊂ A G , such that AI = Ab 1 + . . . + Ab s . Since I = R(AI) = R(Ab 1 + . . . + Ab s ) = A G b 1 + . . . + A G b s , I is finitely generated over A G . Therefore A G is Noetherian.
(c) Let B be an algebra graded over N (for simplicity of notations): B = +∞ i=0 B n , with B m .B n ⊂ B m+n . The augmentation ideal of B is B + = +∞ i=1 B n . If B + is a Noetherian ideal of B, then B is a finitely generated algebra over B 0 . This is Lemma 2.4.5 of [75] (in that statement B is commutative, but this hypothesis is not necessary for the proof).
(d) Assume that A G is graded over N (for simplicity of notations): A G n is an ideal of A G , which is Noetherian by (b) above. Applying (c) we deduce that A G is a finitely generated algebra over K.
Proof of Theorem 3.2. As for Theorem 3.1 the result for M A, 0,n follows from that for M A 0,n and M 0,n , which are proved in the same way. Let us consider M 0,n . Consider the filtration F of L 0,n by the subspaces where P n + is given the lexicographic partial order induced from [Λ]. It is easily seen that F is an algebra filtration: the coregular actions £, ¡ fix globally each component C(µ) of L 0,1 , so the claim follows from (3), (6) and the fact that C(µ) C(ν) ⊂ C(µ + ν) for all µ, ν ∈ P + . Denote by Gr F (L 0,n ) the corresponding graded algebra. Again (38) Gr Because each space C([µ]) is stabilized by the coadjoint action of U q , the decomposition (38) has a key advantage on (33). Indeed, since L 0,n is a U q -module algebra, the action of U q is well-defined on Gr F (L 0,n ) and it gives it a structure of U q -module algebra. As vector spaces we have Now we can adapt the differents steps (a)-(d) recalled above: (a') The action of U q on L 0,n is completely reducible. This follows from Theorem 2.2 (1) (noting that the summands, being isomorphic by (14) to spaces C(µ), are finite-dimensional and thus completely reducible U q -modules), and the isomorphism of U q -modules (see (11)): (40) L 0,n Φn −→ (U q (g) ⊗n ) lf ψ −1 n −→ U lf q (g) ⊗n where lf means respectively locally finite for the action ad r n of U q (g) on U q (g) ⊗n , and locally finite for the action ad r of U q (g) on U q (g). By (38) it follows that Gr F (L 0,n ) is also completely reducible. We can therefore define the Reynolds operator R : Gr F (L 0,n ) → Gr F (L 0,n ) Uq as in (a).
(b') In the proof of Theorem 3.1 we showed that Gr F 2 (L 0,n ) is Noetherian, and then deduced that L 0,n is Noetherian by a classical argument (see eg. [69], 1.6.9). This same argument implies that Gr F (L 0,n ) is Noetherian, because (38) shows it is filtered by F 2 , and Gr F 2 (Gr F (L 0,n )) = Gr F 2 (L 0,n ) is Noetherian. As in (b) we deduce that Gr F (L 0,n ) Uq is Noetherian. But Gr F (L 0,n ) Uq = Gr F (L Uq 0,n ), which implies that L Uq 0,n is Noetherian. (c'-d') Then we can apply the steps (c)-(d). As a result Gr F (L 0,n ) Uq is finitely generated, say by k homogeneous elementsx (e') From (39) we deduce that L Uq 0,n is generated by the x i ∈ C([µ i ]) with leading terms x 1 , . . . ,x k . This follows from the following elementary fact: if A is a filtered K-algebra (K a field) which graded algebra Gr(A) is finitely generated, then A is finitely generated by elements which leading terms generate Gr(A). Indeed, let A have the algebra filtration (A i ) i∈N (we take a filtration over N to simplify notations). Put Gr(A) = ⊕ i∈N A (i) , A (i+1) := A i+1 /A i . We haveā +b = a + b andāb = ab + A n+m−1 ∈ A (m+n) , soāb = 0 if ab ∈ A n+m−1 , andāb = ab otherwise. Now assume that Gr(A) is finitely generated over K.
The conclusion follows by an easy induction. P Remark 3.4. (1) Because L Uq 0,1 is the center of L 0,1 , (e') proves it is finitely generated. Of course this follows also from the isomorphism L 0,1 ∼ = U lf q and the fact that the center of U lf q is the center of U q (by Theorem 2.2), plus the well-known description of the latter. But the argument here is elementary and it applies to L Uq 0,n for any n ≥ 1. (2) In spite of the isomorphism Φ n : M 0,n → (U ⊗n q ) Uq , in order to prove Theorem 3.2 one cannot bypass the hard study of (U ⊗n q ) lf , whence of L 0,1 ∼ = U lf q , by working directly with U q . Indeed the adjoint action is not completely reducible thereon. In fact, U lf q is exactly the socle of this action (see [51], Lemma 7.1.24).
(3) In the sl(2) case the filtration F on L Uq 0,n should correspond via the Wilson loop isomorphism (defined in [28], Section 8.2) to the filtration of skein algebras of spheres with n + 1 punctures used in [66].

Proof of Theorem 1.2
As usual we let be a primitive l-th root of unity with l odd and l > d i for all i ∈ {1, . . . , m}.
Recall that Z 0 (U ) ⊂ U is the central polynomial subalgebra generated by E l β k , F l β k , L ±l i , for k ∈ {1, . . . , N } and i ∈ {1, . . . m}. Define Examples show that generating sets of Z 0 (U lf ) have complicated expressions in general. Nevertheless, specializing q at in Theorem 2.2 (2) we get where T (l) , T 2 are the subsets of T , T 2− and T 2 formed by the elements K λl with λ ∈ P , λ ∈ −2P + and λ ∈ 2P respectively. Define (1)).
Here is an alternative proof of the isomorphism Z 0 (L 0,1 ) ∼ = O(G), not using η * . Recall the notations introduced before Theorem 2.14. As varieties H = U + T G U − = G 0 , so the map σ Consider the space V = ∧ N g, endowed with the action of G given on each factor by the adjoint representation. Put on g a basis consisting of one element e α per root space g α , along with a basis of h. Let v ∈ V be the exterior power of the e α 's for α negative, and v * a dual vector such that v * (v) = 1 and v * vanishes on a T G -invariant complement of v. It is classical that G \ G 0 has defining equation δ(g) = 0, where δ is the matrix coefficient δ(g) = v * (π V (g)v) (see eg. [50], page 174). Hence where χ −ρ is the character of T G associated to the root −ρ. Now we can make the connection with U . The isomorphism Consider the linear subspace of L 0,n defined by By Proposition 4.1 we have an isomorphism of algebras (η * −1 ) ⊗n : Z 0 (L 0,n ) → O(G) ⊗n .
Note that we had already obtained independently the Noetherianity of the ring L 0,n as a consequence of the Noetherianity of L A 0,n (Proposition 3.1). We need below explicit descriptions of the Z-and Z 0 -centers for g = sl(2). Let us recall a few facts in this case. Denote by a, b, c, d the standard generators of O q (SL 2 ), ie. the matrix coefficients in the basis of weight vectors v 0 , v 1 = F.v 0 of the 2-dimensional irreducible representation V 2 of U q (sl(2)). Denote by x k , k ∈ N, the k-th power of an element x ∈ O A (SL 2 ). The algebra O A (SL 2 ) is generated by a, b, c, d; the monomials a i b j d r and a i c k d r , i, j, k, r ∈ N, k > 0, form an A-basis of O A (SL 2 ). The algebra Z 0 (O (SL 2 )) is generated by a l , b l , c l , d l ; the monomials a il b jl d rl and a il c kl d rl form a basis of Z 0 (O (SL 2 )), and Z(O (SL 2 )) is generated by Z 0 (O (SL 2 )) and the elements b (l−k) c k , k = 0, . . . , l (see [42], Proposition 1.4 and the Appendix). We have the relation (42) a l d l − b l c l = 1 and the Frobenius isomorphism of Parshall-Wang (see [65], Chapter 7) coincides with the map F r P W : O(SL 2 ) → Z 0 (O (SL 2 )) induced by η * ; it sends the standard generators a, b, c, d of O(SL 2 ) = O 1 (SL 2 ) respectively to a l , b l , c l , d l . Finally, let us quote from [43] that a basis of the rank l 3 free Z 0 (O(SL 2 ))module O (SL 2 ) (see Theorem 2.15 (3)) is formed by the monomials a m b n c s and b n c s d r , with the integers m, n, r, s , s in the range Now consider L A 0,1 (sl(2)). Recall that L A 0,1 = O A as U A -modules. The algebra L A 0,1 (sl(2)) is also generated by a, b, c, d; a set of defining relations is (see [28], Section 5): The element ω := qa + q −1 d is central. Let T k , k ∈ N, be such that T k (x)/2 is the k-th Chebyshev polynomial of the first type in the variable x/2. We have (see [28], Proposition 7.3, for the generalization to L 0,n (sl(2))): Here b l , c l , d l are the l-th powers of b, c, d computed using the product of L A 0,1 (sl(2)), not the product of Z 0 (O (SL 2 )). The above generator of I can be interpreted as a determinant, and ω as a quantum trace on V 2 . Proof. Let α and be the simple root and fundamental weight of sl (2). In the notations of (22) (32) in [28]). By passing to the localization O A (SL 2 )[d −1 ], and using Parshall-Wang's relation (42), one deduces easily Φ 1 (a l ) where Ω is (q − q −1 ) 2l times the Casimir element of U q (sl(2)), and T l (x)/2 is the l-th Chebyshev polynomial of the first type in the variable x/2. We have Φ 1 (ω) = Ω, so Φ 1 (a l ) = T l (ω) − d l . The conclusion follows from the injectivity of Φ 1 . P

This lemma proves that we have a commutative diagram
Z 0 (L 0,1 (sl(2))) / / L 0,1 (sl (2)) where F r P W is Parshall-Wang's Frobenius isomorphism recalled above, F r is the Frobenius isomorphism introduced in [28], Definition 7.2, and the vertical arrows are the identifications as vector spaces (the middle one proved by Proposition 4.2).
Remark 4.5. By Lemma 4.3 the monomials T l (ω) i b jl d rl and T l (ω) i c kl d rl , for i, j, k, r ∈ N and k > 0, form an A-basis of Z 0 (L 0,1 (sl(2))). It is straightforward (though cumbersome) to express these basis elements in terms of the basis elements a il b jl d rl and a il c kl d rl of Z 0 (O (SL 2 )), and conversely; this can be done by using Lemma 4.4, the formula (3) or the inverse one (expressing in terms of the product of L 0,1 , see (18) in [28]), and the formula of the coproduct ∆ : L 0,1 (sl(2))) → L 0,2 (sl(2))) restricted to Z 0 (L 0,1 (sl(2))) (given in Proposition 6.15 and Lemma 7.7 of [28]).
Since L A 0,1 = O A as an A-module, the functionals t i in Proposition 2.16 can be seen as maps t i : L A 0,1 → A. Though the algebra structures of O and L 0,1 are very different, we have the analogous result: Proposition 4.6. The maps ¡t i preserve Z 0 (L 0,1 ), and they satisfy (f ¡ t i )(a) = f (n i a) and Proof. The first two claims follow from Proposition 2.16 and the equality Z 0 (L 0,1 ) = Z 0 (O ) in Proposition 4.1.
It is enough to prove the identity t(f g) = t(f )t(g) when f ranges in a set of generators of the algebra Z 0 (L 0,1 (sl(2))). So one can take f among, say, T l (ω) − d l = a l and x l = x l , x ∈ {b, c, d} (using Lemma 4.3). By (3) and Proposition 6.1 in the Appendix we have Expanding coproducts and using that R −1 = (S ⊗ id)(R) we deduce where w ∈ U Γ is the quantum Weyl group element dual to t (see Section 6.1), and in the last equality we used that Φ − maps Z 0 (O ) into Z 0 (U ) (see Theorem 2.15 (2)), which acts on Γ-modules by the trivial character (the counit) ε : U → C. By (58)- (59) in the Appendix we have t(a l ) = t(d l ) = 0 and t(b l ) = 1, t(c l ) = −1. Now the computation of t(f g) follows easily. For instance, taking The other cases f = T l (ω) − d l , c l , d l are similar. P Theorem 4.7. L 0,n is a free Z 0 (L 0,n )-module of rank l n.dimg , and (L 0,n ) U is a Noetherian ring and a finite, whence Noetherian, Z 0 (L 0,n )-module.
We will see in Section 5 (proof of Theorem 1.3 (2) and (3)) that in fact (L 0,n ) U is finite free of rank l (n−1).dimg+m over Z 0 (L 0,n ).
Proof of Theorem 4.7. By Proposition 4.2 (1), L 0,n and O ⊗n coincide as modules over Z 0 (L 0,n ) = Z 0 (O ⊗n ), so the first claim follows immediately from Theorem 7.2 of [42], which shows that O is a finitely generated projective module of rank l dimg over Z 0 (O ), and the arguments of [68] and [25], which imply that this module is free. Alternatively, it follows from the fact that O is a cleft extension of O(G) (see [6] and [21]). For the second claim, since L 0,n is a Noetherian Z 0 (L 0,n )-module, the Z 0 (L 0,n )-submodule (L 0,n ) U is necessarily finitely generated. But Z 0 (L 0,n ) being Noetherian, (L 0,n ) U is then a Noetherian Z 0 (L 0,n )-module and a Noetherian ring.
For the sake of clarity let us provide a self-contained proof of the first claim, not invoking directly [42,25] or [6,21]. Since L 0,n and L ⊗n 0,1 coincide as modules over Z 0 (L 0,n ) = Z 0 (L 0,1 ) ⊗n , the result follows from the case n = 1. Then we argue in four steps. First, using Theorem 2.2 we show that a certain localization of L 0,1 is a free module of rank l dimg . Then, assuming that L 0,1 is finitely generated and projective, we explain why it has constant rank l dimg (this is very classical). Thirdly, we prove that L 0,1 is finitely generated and projective as in Theorem 7.2 of [42]. Finally we obtain that it is a free module as in [25].
We have T , where is the pivotal element. Then Theorem 2.14 (1) and (41) imply that U is a free Z 0 (U lf )[ l ]-module of rank 2 m l dimg , and Theorem 2.2 (2) says that it is also the direct sum of 2 m copies of the (free) . Here we note that, by the first formula of Theorem 2.11 (3), taking powers with respect to the product of L 0,1 we have ψ l −ρ = ψ −lρ . Assume that L 0,1 is finitely generated and projective. Let us show that its rank is l dimg . The localization (L 0,1 ) P of L 0,1 at any prime ideal P of Z 0 (L 0,1 ) is a free module over Z 0 (L 0,1 ) P ( [72], Proposition 2.12.15); the ranks of such modules are finite in number ( [72], Proposition 2.12.20). If these ranks are all equal, then, by definition, it is the rank of L 0,1 over Z 0 (L 0,1 ). This happens if Z 0 (L 0,1 ) has no non-trivial (ie. = 1) idempotent ( [72], Corollary 2.12.23), which is the case since it has non non-trivial zero divisors. To compute the rank, suppose P does not contain d l . Such ideals P are in 1-1 correspondence with the prime ideals of Z 0 (L 0,1 )[d −l ] by the natural ring monomorphism Z 0 (L 0,1 ) → Z 0 (L 0,1 )[d −l ]. The set S = Z 0 (L 0,1 ) \ P is multiplicatively closed, and we have also a ring morphism Z 0 (L 0,1 )[d −l ] → S −1 Z 0 (L 0,1 ), which is also an injection (there are no zero divisors in Z 0 (L 0,1 ), whence in S). Then (45) ( shows that (L 0,1 ) P has over Z 0 (L 0,1 ) P = S −1 Z 0 (L 0,1 ) the same rank l dimg as L 0,1 [d −l ] over Z 0 (L 0,1 )[d −l ]. This proves our claim.
In order to show that L 0,1 is finitely generated and projective over Z 0 (L 0,1 ) it is enough to show it is finite locally free, ie. there are elements d i ∈ Z 0 (L 0,1 ) such that the localizations We have seen above that L 0,1 [d −l ] is free of rank l dimg over Z 0 (L 0,1 )[d −l ]. In the proof of Proposition 4.1 we saw that there are isomorphisms Z 0 Consider the Bruhat decomposition of G: any g ∈ G can be written in the form g = b 1 nb 2 , where b 1 , b 2 ∈ B − , n ∈ W . Hence we have g = nn −1 b 1 nb 2 ∈ nB + B − = nG 0 , and therefore Under the isomorphism of Z 0 (L 0,1 ) with G, we thus get that Maxspec(Z 0 (L 0,1 )) is covered by the open sets U (d l w ) ∼ = n w G 0 , and L 0, . Therefore L 0,1 is finitely generated and projective over Z 0 (L 0,1 ).
Finally, let us explain why L 0,1 is free over Z 0 (L 0,1 ), following the arguments of [25]. Let R be a commutative Noetherian ring, put X = Maxspec(R), and let P be an R-module. Denote by R I , P I the localizations of R, P at a maximal ideal I ∈ X. Define the f-rank of P as frank(P ) = inf I∈X { f-rank R I (P I )}, where f-rank R I (P I ) = sup{r ∈ N, R ⊗r I ⊂ P I } ∈ N ∪ {+∞} (ie. the maximal dimension of a free summand of P I ). Bass' Cancellation theorem asserts that if P is projective and f-rank(P ) > dim(X), and P ⊕Q ∼ = M ⊕Q for some R-modules Q and M such that Q is finitely generated and projective, then P ∼ = M (see [10], IV.3.5 and pages 167 and 170, taking A = R, or [69], section 11.7.13). Let us apply this to R = O(G) and P = L 0,1 . We proved above that f-rank R I (P I ) = l dimg , a constant, and we have l dimg > dimg = dim(G). By a result of Marlin [68], the Grothendieck ring K 0 (O(G)) is isomorphic to Z. Therefore L 0,1 ⊕ Q ∼ = O(G) r for some free O(G)-module Q and r ∈ N. Then Bass' Cancellation implies L 0,1 is free over Z 0 (L 0,1 ) ∼ = O(G). P

Proof of Theorem 1.3
We begin with two lemmas, interesting in themselves.
Lemma 5.1. Z(L 0,n ) is a finite Z 0 (L 0,n )-module and a Noetherian ring. Therefore the ring Z(L 0,n ) is integral over Z 0 (L 0,n ).
Proof. We know that L 0,n is finite over Z 0 (L 0,n ) (Theorem 4.7), and Z 0 (L 0,n ) is a Noetherian ring (Proposition 4.2). Therefore L 0,n is a Noetherian Z 0 (L 0,n )-module. This implies that the submodule Z(L 0,n ) is finitely generated. But being finite over the Noetherian ring Z 0 (L 0,n ), it is a Noetherian ring (by eg. Proposition 7.2 of [7]). Let x ∈ Z(L 0,n ). The Z 0 (L 0,n )-submodule Z 0 (L 0,n )[x] of L 0,n is finitely generated by the same argument. Using the fact that an element x is integral over Z 0 (L 0,n ) if and only if Z 0 (L 0,n )[x] is a finitely generated Z 0 (L 0,n )-module (by eg. Proposition 5.1 of [7]), this proves the last claim. P As usual, denote by Q(Z) the quotient field of a commutative integral domain Z. Then, consider the fields Q(Z(L 0,n )) and Q(Z 0 (L 0,n )). Since Z(L 0,n ) is finite over Z 0 (L 0,n ) and has no non-trivial zero divisors, the ring Z(L 0,n ) ⊗ Z 0 (L 0,n ) Q(Z 0 (L 0,n )) is a field. Therefore it is equal to Q(Z(L 0,n )).
Recall that we denote by N the number of positive roots of g.
Proposition 5.3. Q(L 0,n ) is a central simple algebra of dimension l 2N n over Q(Z(L 0,n )).
Proof. From its definition Q(L 0,n ) is a vector space over Q(Z(L 0,n )). Because L 0,n has no non-trivial divisors and Q(L 0,n ) is finite-dimensional over Q(Z(L 0,n )), it is a division algebra over Q(Z(L 0,n )), whence a simple algebra. Its center being Q(Z(L 0,n )), this proves the first part of the statement. By classical theory (see eg. Section 13.3.5 of [69], or [72], Corollary 2.3.25), it then follows that there is a finite extension (a splitting field) F of Q(Z(L 0,n )) such that where d ∈ N, the PI degree of Q(L 0,n ), is given by d 2 = [Q(L 0,n ) : Q(Z(L 0,n ))]. Therefore where we use Theorem 4.7 and Lemma 5.2, and we recall that dimg = m + 2N . P Let us recall for the sake of clarity different notions of ring theory, bearing in mind that we will apply them to the case where A = L 0,n . Let A be a ring with no non-trivial zero divisors. The center Z = Z(A) is a commutative integral domain. Denote by Q(Z) its field of fractions, and let Q(A) = Q(Z) ⊗ Z A. It is an algebra over its center Q(Z).
An element a ∈ A is integral over Z if Z[a] is a finitely generated Z-module. A is integral over Z if every element of A is integral over Z. An element a ∈ A is c-integral over Z if Z[a] is contained in a finitely generated Z-module. A is c-integral over Z if every element of A is c-integral over Z. When Z is a Noetherian ring these two notions are equivalent ([69], Lemma 5.3.2).
If Q(A) has finite dimension over Q(Z), it is a division algebra and therefore a central simple algebra. Moreover Q(A) is integral over Q(Z).
There are different notions of orders, that are equivalent in our context. Let B ⊂ Q(A) be a subring. B is said to be an order of In particular, because A is always an order of Q(A) these equivalences imply that A is a classical order, whence a finitely generated Z-module. Moreover A is Noetherian by Proposition 5.3.14 of [69].
There are two standard notions of maximality for Z-orders B of Q(A). One applies strictly to Z-orders ([69], 5.3.13): namely, a Z-order B of Q(A) is a maximal Z-order if B ⊂ C with C a Z-order implies B = C. The other notion of maximality applies to arbitrary orders of Q(A) (see [69], 5.1.1); when B is a Z-order with center Z, B is a maximal Z-order if and only if it is maximal in this latter sense.
If A is a maximal order, then Z contains all the c-integral elements over it, ie. Z is cintegrally closed ([69], 5.3.13). Since Z is Noetherian, it is then integrally closed (in the usual sense, see [69], Lemma 5.3.2), and by [71], Theorem 10.1, it therefore coincides with the trace ring of A (ie. the subring of Q(Z) generated over Z by the coefficients of the characteristic polynomials of elements of A, represented by left multiplication as elements of the matrix algebra Q(A) ⊗ Z F, where F is a splitting field of Q(A)).
Finally, we say that A is DCK-integrally closed if the following condition holds: for every subring R of Q(A) such that A ⊂ R ⊂ z −1 A for some non zero z ∈ Z(A), we have R = A. We borrow this notion from [38]; it is closely related to that of fractional ideal of Q(A) (see [69], 3.1.11-3.1.12 and 5.1.4), but simpler. Its relevance comes from the following lemma, which shows that in the commutative and Noetherian case it is equivalent to the usual definition of integrally closure. Conversely, let R be a subring of Q(B) such that B ⊂ R ⊂ z −1 B, and let x ∈ R. Then M = B[x] is a B-submodule of Q(B) such that xM ⊂ M . It is also a B-submodule of z −1 B, which is free with basis z −1 over B. Since B is Noetherian, z −1 B is a Noetherian B-module, and therefore M is a finite B-module. It follows that x is integral over B (Proposition 5.1 of [7]), whence x ∈ B. P Lemma 5.5. Assume that A is a ring with no non-trivial zero divisors, with center Z Noetherian and such that Q(A) has finite dimension over Q(Z). Then A is DCK-integrally closed if and only if A is a maximal order.
Proof. Assume that A is DCK-integrally closed. Let B a Z-order of Q(A) such that A ⊂ B, and let b ∈ B. Since b is c-integral over Z and Z is Noetherian, b is integral and thus A[b] is a finitely generated A-module. Let e i = a i /z i ∈ Q(A), z i ∈ Z and i = 1, . . . , n, be the This proves that A is a maximal order.
Conversely, assume that A is maximal order, and let B be a subring of Q(A) such that A ⊂ B ⊂ z −1 A. Since A is a finitely generated Z-module, B is contained in a finitely generated Z-module, which is necessarily a Noetherian Z-module because Z is Noetherian. Therefore B is a finitely generated Z-module. As clearly Z ⊂ B and Q(Z)B = Q(A), it is in fact a classical order of Q(A). Because A is maximal, we have A = B, which proves A is DCK-integrally closed. P Theorem 5.6. L 0,n is a maximal order of Q(L 0,n ).
Proof. We derive the result by "twisting" the analogous statement for O , obtained in Theorem 7.4 of [42]. We have already proved that L 0,n satisfies the hypothesis on A in Lemma 5.5. So let R ⊂ Q(L 0,n ) be a subring such that L 0,n ⊂ R ⊂ x −1 L 0,n for some non zero x ∈ Z 0 (L 0,n ). We have to show that L 0,n = R. We know L 0,n = O ⊗n as Z 0 (L 0,n )modules (Proposition 4.2 (1)), so x −1 L 0,n = x −1 O ⊗n . Also, the product of R is inherited from that of L 0,n (for, given r 1 , r 2 ∈ R we have r 1 r 2 = x −2 (xr 1 )(xr 2 )), and the latter is defined from the one of O ⊗n by two consecutive twists (see the formulas (3) and (6)). Therefore, by applying the inverse twists on the inclusions L 0,n ⊂ R ⊂ x −1 L 0,n we get O ⊗n ⊂ R ⊂ x −1 O ⊗n where R is the vector space R endowed with the ring structure inherited from O ⊗n . Now O ⊗n = O (G n ) is a maximal order of Q(O (G n )) by Theorem 7.4 in [42]. Therefore O ⊗n = R , and finally L 0,n = R. P Corollary 5.7. The ring Z(L 0,n ) is integrally closed and coincides with the trace ring of L 0,n . Moreover Z(L 0,n ) = Z(L 0,1 ) ⊗n , and it is a free Z 0 (L 0,n )-module of rank l mn .
Proof. The first two claims follow from the last theorem and the discussion before Lemma 5.4. The third follows from the proof of Proposition 5.2 (ie. the inclusion Z(L 0,n ) ⊃ Z(L 0,1 ) ⊗n , and the fact that both rings have quotient fields of the same degree l mn over Q(Z 0 (L 0,n ))). Finally, it is enough to show the last claim for n = 1. Denote by t red : Q(L 0,1 ) → Q(Z(L 0,1 )) the reduced trace map of the central simple algebra Q(L 0,1 ) (see eg. [71], Section 9). Because Z(L 0,1 ) is the trace ring of L 0,1 , we have Z(L 0,1 ) = t red (L 0,1 ). Trivially the inclusion map i : Z(L 0,1 ) → Q(L 0,1 ) satisfies t red • i = id, so Z(L 0,1 ) is a direct summand of L 0,1 as a Z 0 (L 0,1 )-module. But L 0,1 is free over Z 0 (L 0,1 ), so Z(L 0,1 ) is a projective Z 0 (L 0,1 )module. Arguing as in Theorem 4.7, one deduces that the module is free. The rank is, again, given by Proposition 5.2. P A proof of Theorem 5.6 independent of Theorem 7.4 of [42] seems to be difficult for arbitrary g, even for n = 1. In the case of L 0,1 (sl(2)) we can however apply a similar reasoning. Let us explain the details.
In [42] another quantum Weyl group element w is defined. It is dual to the Vaksman-Soibelman functional t : O q (SL 2 ) → C(q) of [77,76], so t(α) = α, w , α ∈ O q (SL 2 ). By comparing (57) with the formulas defining the action of t in Section 1.7 of [42], we find w = ξŵK and the basis vectors w p r of [42] are related to the vectors v j above as follows: v j = λ j w p r where k = 2p, j = p − r, λ 0 = 1, λ 1 = [k]q −k , and Explicit formulas of the evaluation of t on basis vectors of O q (SL 2 ) can be computed. We get: t(ã m c n d p ) = (−1) n δ m,p q −n(p+1) where (60)ã = a ,b = qb ,c = q −1 c ,d = d and as usual a, b, c, d are the standard generators of O q (SL 2 ), ie. the matrix coefficients in the basis of weight vectors v 0 , v 1 = F.v 0 of the 2-dimensional irreducible representation V 2 of U q (sl (2)). Here we have introduced the generatorsã, . . . ,d to facilitate the comparison with the formulas in [42]; these generators come naturally in their setup because they use different generators E i and F i of U q (g), which in our notations can be written respectively as K −1 i E i and F i K i .
The formulas (58)-(59) can be shown by two independent methods. The first uses a definition of t as a GN S state associated to an infinite dimensional representation of O q (SL 2 ), as recalled in Section 1.6 of [42]. The second is to write eg. By (57) or (58)-(59) we see thatŵ (or w) and t are well-defined on the integral forms, We now consider the case where g is of rank m ≥ 2. To each simple root α i , 1 ≤ i ≤ m, it is associated the subalgebra of U q generated by E i , F i , L i , L −1 i . It is a copy of U q i (sl(2)), where q i = q d i . Letŵ i be the corresponding quantum Weyl group element in U q = U q (g), defined by Saito's formula (53), replacing H, E, F by H i , E i and F i . Also, denote by ν i : O q → O q i (SL 2 ) the projection map dual to the inclusion U q i (sl(2)) ⊗ C(q i ) C(q) → U q associated to α i , and put t i = t • ν i . Let w i be the corresponding quantum Weyl group element in U q , ie. t i (α) = α, w i for all α ∈ O q . On integral forms they yield well-defined elementsŵ i , w i ∈ U Γ and t i : O A → A (see [42], Proposition 5.1). They satisfy the defining relations of the braid group B(g) of g [55]: w iŵjŵi =ŵ jŵiŵj if a ij a ji = 1 (ŵ iŵj ) k = (ŵ jŵi ) k for k = 1, 2, 3 if a ij a ji = 0, 2, 3 and similarly by replacingŵ i with w i , or with t i (see [76] for the latter). The Weyl group W = W (g) = N (T G )/T G is generated by the reflexions s i associated to the simple roots α i . Denote by n i ∈ N (T G ) a representative of s i . Let w ∈ W and denote by w = s i 1 . . . s i k a reduced expression. Because of the braid group relations the elementsŵ =ŵ i 1 . . .ŵ i k , w = w i 1 . . . w i k and the functional t w = t i 1 . . . t i k do not depend on the choice of reduced expression. The Lusztig [60] braid group automorphism T w : Γ → Γ associated to w satisfies (see [42]): T w (x) =ŵxŵ −1 , x ∈ Γ.
Let w 0 be the longest element in W . We have Proof. It is sufficient to prove the results for SL 2 because ν i : O → O (SL 2 ) is a morphism of Hopf algebras and ν i (Z 0 (O )) ⊂ Z 0 (O (SL 2 )). In this case (64) can be proved by using (58)-(59), evaluating t on basis elements of Z 0 (O (SL 2 )) as is done in Lemma 1.5 (a) of [42]. Such a basis is formed by monomials like in (58) (2)). Here is an alternative proof of (64): (65) shows that t is a homomorphism on Z 0 (O (SL 2 )), so by proving (65) at first one is reduced to check (64) on the generators a l , . . . , d l , which is easy by means of (61) and (63).