# Unrestricted quantum moduli algebras, II:
Noetherianity and simple fraction rings at roots of $\displaystyle 1$

Stéphane Baseilhac, Philippe Roche

###### Abstract.

We prove that the unrestricted quantum moduli algebra of a punctured sphere
and complex simple Lie algebra $\displaystyle\mathfrak{g}$ is a finitely
generated ring and a Noetherian ring, and that specializations at roots of
unity of odd order $\displaystyle l$ embed in a natural way in a central
simple algebra of PI degree $\displaystyle l^{(n-1)N-m}$, where $\displaystyle
N$ is the number of positive roots of $\displaystyle\mathfrak{g}$,
$\displaystyle m$ its rank, and $\displaystyle n+1\geq 3$ the number of
punctures.

IMAG, Univ Montpellier, CNRS, Montpellier, France

<EMAIL_ADDRESS><EMAIL_ADDRESS>
Keywords: quantum groups, invariant theory, TQFT

AMS subject classification 2020: 16R30, 17B37, 20G42, 57R56

###### Contents

  1. 1 Introduction
    1. 1.1 Basic notations
  2. 2 Background results
    1. 2.1 On $\displaystyle U_{q}$, $\displaystyle{\mathcal{O}}_{q}$, $\displaystyle{\mathcal{L}}_{0,n}$, $\displaystyle{\mathcal{M}}_{0,n}$, and $\displaystyle\Phi_{n}$
    2. 2.2 Integral forms and specializations
    3. 2.3 Perfect pairings
    4. 2.4 Structure theorems for $\displaystyle U_{\epsilon}$ and $\displaystyle{\mathcal{O}}_{\epsilon}$
  3. 3 Noetherianity and finiteness
  4. 4 Proof of Theorem 1.2
  5. 5 Proof of Theorem 1.3
  6. 6 Appendix
    1. 6.1 Quantum Weyl group
    2. 6.2 Regular action on $\displaystyle{\mathcal{O}}_{\epsilon}$

##  1\. Introduction

This paper is the second part of our work on the unrestricted quantum moduli
algebras, that we initiated in [23]. These algebras, denoted by
$\displaystyle{\mathcal{M}}_{g,n}^{A}(\mathfrak{g})$ hereafter, are defined
over the ground ring $\displaystyle A=\mathbb{C}[q,q^{-1}]$ and associated to
unrestricted quantum groups of complex simple Lie algebras
$\displaystyle\mathfrak{g}$, and surfaces of genus $\displaystyle g$ with
$\displaystyle n+1$ punctures (thus, $\displaystyle n=-1$ corresponds to
closed surfaces). We are in particular interested in the specializations
$\displaystyle{\mathcal{M}}_{g,n}^{A,\epsilon}(\mathfrak{g})$ of
$\displaystyle{\mathcal{M}}_{g,n}^{A}(\mathfrak{g})$ at roots of unity
$\displaystyle q=\epsilon$.

As in [23] we focus in this paper on the algebras
$\displaystyle{\mathcal{M}}_{0,n}^{A}(\mathfrak{g})$ associated to punctured
spheres. From now on we fix a complex simple Lie algebra
$\displaystyle\mathfrak{g}$, and when no confusion may arise we omit
$\displaystyle\mathfrak{g}$ from the notation of the various algebras.

The rational form $\displaystyle{\mathcal{M}}_{g,n}$ of
$\displaystyle{\mathcal{M}}_{g,n}^{A}={\mathcal{M}}_{g,n}^{A}(\mathfrak{g})$,
which is an algebra over $\displaystyle\mathbb{C}(q)$, has been introduced in
the mid ${}^{\prime}90$ by Alekseev-Grosse-Schomerus [2, 3] and Buffenoir-
Roche [25, 26]. They defined $\displaystyle{\mathcal{M}}_{g,n}$ by
$\displaystyle q$-deforming the Fock-Rosly lattice models of the moduli spaces
$\displaystyle{\mathcal{M}}_{g,n}^{cl}$ of flat
$\displaystyle\mathfrak{g}$-connections on surfaces of genus $\displaystyle g$
with $\displaystyle n+1$ punctures. It is expected for long that the
representation theory of $\displaystyle{\mathcal{M}}_{g,n}$ at roots of unity
recovers all known $\displaystyle(2+1)$-dimensional TQFTs based on quantum
groups, and also provide new TQFTs for $\displaystyle 3$-manifolds endowed
with flat $\displaystyle\mathfrak{g}$-connections.

For instance, representations of the semisimplification of
$\displaystyle{\mathcal{M}}_{g,n}^{A,\epsilon}$ have been constructed and
classified in [4]; they involve only the irreducible representations of the so
called finite quantum groups $\displaystyle U_{\epsilon}^{fin}(\mathfrak{g})$
(in the notations of [30], Section 9.3). Moreover, by using their
representations of $\displaystyle{\mathcal{M}}_{g,n}^{A,\epsilon}$, [4]
deduced representations of the mapping class groups of surfaces, that are
equivalent to those from which one can built the quantum invariants of
3-manifolds of Witten-Reshetikin-Turaev [71, 63].

Recently, representations of another quotient of
$\displaystyle{\mathcal{M}}_{g,n}^{A,\epsilon}$ have been constructed in [39];
in the $\displaystyle sl(2)$ case they involve the irreducible and also the
principal indecomposable representations of $\displaystyle
U_{\epsilon}^{fin}(sl(2))$. The corresponding representations of the mapping
class groups of surfaces are equivalent to those previously obtained by
Lyubashenko-Majid [54]. The related link and $\displaystyle 3$-manifold
invariants coincide with those of [55] and [17].

In general, the representation theory of
$\displaystyle{\mathcal{M}}_{g,n}^{A,\epsilon}$ is far from being completely
understood. As mentioned above, it is expected to provide a good framework to
construct and study quantum invariants of $\displaystyle 3$-manifolds equipped
with flat $\displaystyle\mathfrak{g}$-connections. A family of such
invariants, called quantum hyperbolic invariants, has already been defined for
$\displaystyle\mathfrak{g}=sl(2)$ by means of certain $\displaystyle
6j$-symbols, Deus ex machina (see [10]–[16]). They are closely connected to
classical Chern-Simons theory, provide generalized Volume Conjectures, and
contain quantum Teichmüller theory. It is part of our present program,
initiated in [7], to shed light on these quantum invariants and to generalize
them to arbitrary $\displaystyle\mathfrak{g}$ by developing the representation
theory of $\displaystyle{\mathcal{M}}_{g,n}^{A,\epsilon}$.

Besides, the quantum moduli algebras are very interesting objects in
themselves. They are now recognized as central objects from the viewpoints of
factorization homology [18], (stated) skein theory [19, 41, 29] and, as
already said, the mapping class group representations associated to
topological quantum field theories [40].

We introduced the integral form $\displaystyle{\mathcal{M}}_{0,n}^{A}$ and
began its study in [23]. We presently prove new results about its algebra
structure, especially when $\displaystyle q$ is a root of unity, that hold for
every complex simple Lie algebra $\displaystyle\mathfrak{g}$. We use a
definition of $\displaystyle{\mathcal{M}}_{0,n}^{A}$ that comes from the
original combinatorial quantization method of [2, 3] and [25, 26], using also
twists of module-algebras; 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,
$\displaystyle{\mathcal{M}}_{0,n}^{A}$ can be regarded as the invariant
subalgebra of a certain module-algebra $\displaystyle{\mathcal{L}}_{0,n}^{A}$,
endowed with an action of the unrestricted (De Concini-Kac) integral form
$\displaystyle U_{A}=U_{A}(\mathfrak{g})$ of the quantum group $\displaystyle
U_{q}=U_{q}(\mathfrak{g})$. We therefore study
$\displaystyle{\mathcal{L}}_{0,n}^{A}$ and its specializations
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$ at $\displaystyle q=\epsilon$ a
root of unity. It happens that under such a specialization,
$\displaystyle{\mathcal{M}}_{0,n}^{A}$ embeds in the invariant subalgebra
$\displaystyle({\mathcal{L}}_{0,n}^{\epsilon})^{U_{\epsilon}}$ of
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$ under the action of the
specialization $\displaystyle U_{\epsilon}$ of $\displaystyle U_{A}$ at
$\displaystyle q=\epsilon$. Our results in this paper basically concern
$\displaystyle{\mathcal{L}}_{0,n}^{A}$,
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$ and
$\displaystyle({\mathcal{L}}_{0,n}^{\epsilon})^{U_{\epsilon}}$.

By using some standard tools of representation theory our results allow one to
build a vector bundle of rank $\displaystyle l^{2(N(n-1)-m)}$ ($\displaystyle
N$ being the number of positive roots of $\displaystyle\mathfrak{g}$, and
$\displaystyle m$ its rank) over a Zariski open subset of the maximal spectrum
of the center
$\displaystyle\mathcal{Z}(({\mathcal{L}}_{0,n}^{\epsilon})^{U_{\epsilon}})$ of
$\displaystyle({\mathcal{L}}_{0,n}^{\epsilon})^{U_{\epsilon}}$, for
$\displaystyle n\geq 2$. In [24] we describe the inclusion
$\displaystyle{\mathcal{M}}_{0,n}^{A,\epsilon}\subset({\mathcal{L}}_{0,n}^{\epsilon})^{U_{\epsilon}}$
and the representations of $\displaystyle{\mathcal{M}}_{0,n}^{A,\epsilon}$,
and we give applications to skein algebras (which is the $\displaystyle sl(2)$
case). In [22] we consider the algebras
$\displaystyle{\mathcal{M}}_{g,n}^{A,\epsilon}$ for genus $\displaystyle g\neq
0$.

Let us now state our results. First we need to fix the terminology more
precisely. Let $\displaystyle U_{q}$ be the simply-connected quantum group of
$\displaystyle\mathfrak{g}$, defined over the field
$\displaystyle\mathbb{C}(q)$. From $\displaystyle U_{q}$ one can define a
$\displaystyle U_{q}$-module algebra $\displaystyle{\mathcal{L}}_{0,n}$,
called graph algebra, where $\displaystyle U_{q}$ acts by means of a right
coadjoint action. The quantum moduli algebra
$\displaystyle{\mathcal{M}}_{0,n}$ is the subalgebra
$\displaystyle{\mathcal{L}}_{0,n}^{U_{q}}$ of invariant elements of
$\displaystyle{\mathcal{L}}_{0,n}$ for this action. The unrestricted quantum
moduli algebra $\displaystyle{\mathcal{M}}_{0,n}^{A}$ is an integral form of
$\displaystyle{\mathcal{M}}_{0,n}$ (thus, defined over $\displaystyle
A=\mathbb{C}[q,q^{-1}]$). As a $\displaystyle\mathbb{C}(q)$-module
$\displaystyle{\mathcal{L}}_{0,n}$ is just
$\displaystyle\mathcal{O}_{q}^{\otimes n}$, where
$\displaystyle\mathcal{O}_{q}=\mathcal{O}_{q}(G)$ is the standard quantum
function algebra of the connected and simply-connected Lie group
$\displaystyle G$ with Lie algebra $\displaystyle\mathfrak{g}$. The product of
$\displaystyle{\mathcal{L}}_{0,n}$ is obtained by twisting both the product of
each factor $\displaystyle{\mathcal{O}}_{q}$ and the product between them. It
is equivariant with respect to a (right) coadjoint action of $\displaystyle
U_{q}$, which defines the structure of $\displaystyle U_{q}$-module of
$\displaystyle{\mathcal{L}}_{0,n}$. The module algebra
$\displaystyle{\mathcal{L}}_{0,n}$ has an integral form
$\displaystyle{\mathcal{L}}_{0,n}^{A}$, defined over $\displaystyle A$,
endowed with a coadjoint action of the unrestricted integral form
$\displaystyle U_{A}$ of $\displaystyle U_{q}$ introduced by De Concini-Kac
[33]. The algebra $\displaystyle{\mathcal{L}}_{0,n}^{A}$ is obtained by
replacing $\displaystyle\mathcal{O}_{q}$ in the construction of
$\displaystyle{\mathcal{L}}_{0,n}$ with the restricted dual
$\displaystyle\mathcal{O}_{A}$ of the integral form $\displaystyle
U_{A}^{res}$ of $\displaystyle U_{q}$ defined by Lusztig [52], or equivalently
with the restricted dual of the integral form $\displaystyle\Gamma$ of
$\displaystyle U_{q}$ defined by De Concini-Lyubashenko [36]. The unrestricted
integral form $\displaystyle{\mathcal{M}}_{0,n}^{A}$ of
$\displaystyle{\mathcal{M}}_{0,n}$ is defined as the subalgebra of invariant
elements,

$\displaystyle{\mathcal{M}}_{0,n}^{A}:=({\mathcal{L}}_{0,n}^{A})^{U_{A}}.$

A cornerstone of the theory of $\displaystyle{\mathcal{M}}_{0,n}^{A}$ is a map
originally due to Alekseev [1], building on works of Drinfeld [31] and
Reshetikhin and Semenov-Tian-Shansky [61]. In [23] we showed that it
eventually provides isomorphisms of module algebras and algebras respectively,

$\displaystyle\Phi_{n}\colon{\mathcal{L}}_{0,n}^{A}\rightarrow(U_{A}^{\otimes
n})^{lf},\Phi_{n}\colon{\mathcal{M}}_{0,n}^{A}\rightarrow(U_{A}^{\otimes
n})^{U_{A}}$

where $\displaystyle U_{A}^{\otimes n}$ is endowed with a right adjoint action
of $\displaystyle U_{A}$, and $\displaystyle(U_{A}^{\otimes n})^{lf}$ is the
subalgebra of locally finite elements with respect to this action. When
$\displaystyle n=1$ the algebra $\displaystyle U_{A}^{lf}$ has been studied in
great detail by Joseph-Letzter [44, 45, 43]; their results we use have been
greatly simplified in [70].

All the material we need about the results discussed above is described in
[23], and recalled in Section 2.1-2.2.

Our first result, proved in Section 3, is:

######  Theorem 1.1.

$\displaystyle{\mathcal{L}}_{0,n}$, $\displaystyle{\mathcal{M}}_{0,n}$ and
their unrestricted integral forms and specializations at $\displaystyle
q\in\mathbb{C}\setminus\\{0,1\\}$ are Noetherian rings, and finitely generated
rings.

In [23] we proved that these algebras have no non-trivial zero divisors. Also,
we deduced Theorem 1.1 in the $\displaystyle sl(2)$ case by using an
isomorphism between $\displaystyle{\mathcal{M}}_{0,n}(sl(2))$ and the skein
algebra of a sphere with $\displaystyle n+1$ punctures, which by a result of
[57] is Noetherian and finitely generated. Our approach here is completely
different. For $\displaystyle{\mathcal{L}}_{0,n}$ we adapt the proof given by
Voigt-Yuncken [70] of a result of Joseph [43], which asserts that
$\displaystyle U_{q}^{lf}$ is a Noetherian ring (Theorem 3.1). For
$\displaystyle{\mathcal{M}}_{0,n}$ we deduce the result from the one for
$\displaystyle{\mathcal{L}}_{0,n}$, 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
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$ of
$\displaystyle{\mathcal{L}}_{0,n}^{A}$ at $\displaystyle q=\epsilon$, a root
of unity of odd order $\displaystyle l$ coprime to $\displaystyle 3$ if
$\displaystyle\mathfrak{g}$ has $\displaystyle G_{2}$ components. In [36], De
Concini-Lyubashenko introduced a central subalgebra
$\displaystyle\mathcal{Z}_{0}({\mathcal{O}}_{\epsilon})$ of
$\displaystyle{\mathcal{O}}_{\epsilon}$ isomorphic to the coordinate ring
$\displaystyle{\mathcal{O}}(G)$, and proved that the
$\displaystyle\mathcal{Z}_{0}({\mathcal{O}}_{\epsilon})$-module
$\displaystyle{\mathcal{O}}_{\epsilon}$ is projective of rank $\displaystyle
l^{dim\mathfrak{g}}$. As observed by Brown-Gordon-Stafford [21], Bass’
Cancellation theorem in $\displaystyle K$-theory and the fact that
$\displaystyle K_{0}({\mathcal{O}}(G))\cong\mathbb{Z}$, proved by Marlin [59],
imply that this module is free.

The section 4 proves the analogous property for
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$ and
$\displaystyle({\mathcal{L}}_{0,n}^{\epsilon})^{U_{\epsilon}}$, the subring of
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$ formed by the invariant elements
of $\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$ with respect to the right
coadjoint action of $\displaystyle U_{\epsilon}$. Note that we trivially have
an inclusion
$\displaystyle{\mathcal{M}}_{0,n}^{A,\epsilon}\subset({\mathcal{L}}_{0,n}^{\epsilon})^{U_{\epsilon}}$;
also the center $\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})$ of
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$ is contained in
$\displaystyle({\mathcal{L}}_{0,n}^{\epsilon})^{U_{\epsilon}}$ (this follows
from [23], Proposition 6.17). We have (see Proposition 4.2 and Theorem 4.7):

######  Theorem 1.2.

$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$ has a central subalgebra
$\displaystyle\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon})$ isomorphic to
$\displaystyle\mathcal{O}(G)^{\otimes n}$, and it is a free
$\displaystyle\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon})$-module of rank
$\displaystyle l^{n.dim\mathfrak{g}}$, isomorphic to the
$\displaystyle\mathcal{O}(G)^{\otimes n}$-module
$\displaystyle{\mathcal{O}}_{\epsilon}^{\otimes n}$.

We give a direct and self-contained proof of the theorem by adapting to
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$ the arguments of Theorem 7.2 of
De Concini-Lyubashenko [36]. In particular we study the coregular action of
the braid group of $\displaystyle\mathfrak{g}$ on
$\displaystyle{\mathcal{L}}_{0,1}^{\epsilon}$; by the way, in the Appendix we
provide different proofs of some technical facts shown in [36].

However, Theorem 1.2 may be deduced from the results of [36, 21] recalled
above once the required properties of
$\displaystyle\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon})$ are settled.
The most natural definition of
$\displaystyle\mathcal{Z}_{0}({\mathcal{L}}_{0,1}^{\epsilon})$ is
$\displaystyle\Phi_{1}^{-1}(U_{\epsilon}^{lf}\cap\mathcal{Z}_{0}(U_{\epsilon}))$,
where $\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$ is the De Concini-Kac-
Procesi central subalgebra of $\displaystyle U_{\epsilon}$, and $\displaystyle
U_{\epsilon}^{lf}$ the specialization at $\displaystyle q=\epsilon$ of the
algebra $\displaystyle U_{A}^{lf}$. Thus it is not directly connected to
$\displaystyle\mathcal{Z}_{0}({\mathcal{O}}_{\epsilon})$, and the algebra
structures of $\displaystyle{\mathcal{L}}_{0,1}^{\epsilon}$ and
$\displaystyle{\mathcal{O}}_{\epsilon}$ are completely different indeed. We
show that nevertheless
$\displaystyle\mathcal{Z}_{0}({\mathcal{L}}_{0,1}^{\epsilon})$ and
$\displaystyle\mathcal{Z}_{0}({\mathcal{O}}_{\epsilon})$ coincide and give
$\displaystyle{\mathcal{L}}_{0,1}^{\epsilon}$ and
$\displaystyle{\mathcal{O}}_{\epsilon}$ the same module structures over these
subalgebras. For arbitrary $\displaystyle n$ we set
$\displaystyle\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon})=\mathcal{Z}_{0}({\mathcal{L}}_{0,1}^{\epsilon})^{\otimes
n}$, which we show is indeed central in
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$. All this relies on results of
De Concini-Kac [33], De Concini-Procesi [34, 35], and De Concini-Lyubashenko
[36], that we recall in Section 2.3-2.4. Therefore
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$ and
$\displaystyle{\mathcal{O}}_{\epsilon}^{\otimes n}$ are the same modules over
$\displaystyle{\mathcal{O}}(G)^{\otimes n}$, which proves Theorem 1.2 by using
the results of [36, 21].

It is worth noticing that basis of
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$ over
$\displaystyle\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon})$ are
complicated. For instance, the only basis we know in the case
$\displaystyle\mathfrak{g}=sl(2)$, which is described in [37], is far from
being obvious (see (51)).

In Section 5 we turn to fraction rings. As mentioned above
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$ has no non-trivial zero
divisors. Therefore $\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})$
is an integral domain. Denote by $\displaystyle
Q(\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon}))$ its fraction field. Consider
the rings

$\displaystyle
Q({\mathcal{L}}_{0,n}^{\epsilon})=Q(\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon}))\otimes_{\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})}{\mathcal{L}}_{0,n}^{\epsilon}$

and

$\displaystyle
Q(({\mathcal{L}}_{0,n}^{\epsilon})^{U_{\epsilon}})=Q(\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon}))\otimes_{\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})}({\mathcal{L}}_{0,n}^{\epsilon})^{U_{\epsilon}}.$

Throughout the paper, unless we mention it explicitly we follow the
conventions of Mc Connell-Robson [60] as regards the terminology of ring
theory; in particular, for the notions of central simple algebras, their
classical orders, maximal classical orders, PI degrees and trace ring see in
[60] the sections 5.3 and 13.3.6-13.6.7.

Denote by $\displaystyle m$ the rank of $\displaystyle\mathfrak{g}$, and by
$\displaystyle N$ the number of its positive roots. We prove:

######  Theorem 1.3.

(1) $\displaystyle Q({\mathcal{L}}_{0,n}^{\epsilon})$ is a central simple
algebra of PI degree $\displaystyle l^{nN}$, and
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$ is a maximal order of
$\displaystyle Q({\mathcal{L}}_{0,n}^{\epsilon})$.

(2) $\displaystyle Q(({\mathcal{L}}_{0,n}^{\epsilon})^{U_{\epsilon}})$,
$\displaystyle n\geq 2$, is a central simple algebra of PI degree
$\displaystyle l^{N(n-1)-m}$.

The first claim means that $\displaystyle Q({\mathcal{L}}_{0,n}^{\epsilon})$
is a complex subalgebra of a full matrix algebra $\displaystyle
Mat_{d}(\mathbb{F})$, where $\displaystyle d=l^{nN}$ and
$\displaystyle\mathbb{F}$ is a finite extension of $\displaystyle
Q(\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon}))$ such that

$\displaystyle\mathbb{F}\otimes_{Q(\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon}))}Q({\mathcal{L}}_{0,n}^{\epsilon})=Mat_{d}(\mathbb{F}).$

We deduce it from Theorem 1.2 and the computation of the degree of
$\displaystyle Q(\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon}))$ as a field
extension of $\displaystyle
Q(\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon}))$. This computation uses
$\displaystyle\Phi_{n}$ and the computation of the degree of $\displaystyle
Q(\mathcal{Z}(U_{\epsilon}))$ over $\displaystyle
Q(\mathcal{Z}_{0}(U_{\epsilon}))$ by De Concini-Kac [33] (see Proposition
5.3).

The second claim implies in particular that
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})$ is an integrally
closed domain, and that it coincides with the trace ring of $\displaystyle
Q({\mathcal{L}}_{0,n}^{\epsilon})$. It is proved in Theorem 5.6. More
precisely we prove that $\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$ is
integrally closed in $\displaystyle Q({\mathcal{L}}_{0,n}^{\epsilon})$, in the
sense of [33, 35]. So, before the theorem we show in Lemma 5.5 that a ring
$\displaystyle A$ with no non-trivial zero divisors, Noetherian center, and
finite dimensional classical fraction algebra $\displaystyle Q$, which is the
case of $\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$ and
$\displaystyle({\mathcal{L}}_{0,n}^{\epsilon})^{U_{\epsilon}}$, is integrally
closed in $\displaystyle Q$ if and only if it is maximal as a (classical)
order. For the sake of clarity we have included a general discussion of all
these notions before Theorem 5.6. The proof uses the facts that
$\displaystyle{\mathcal{O}}_{\epsilon}$ is a maximal order of its classical
fraction algebra, which is Theorem 7.4 of [36], and that the twist which
defines the algebra structure of $\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$
from $\displaystyle{\mathcal{O}}_{\epsilon}^{\otimes n}$ keeps the
$\displaystyle\mathcal{Z}_{0}$-module structure unchanged. It seems rather
difficult to prove that $\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$ is a
maximal order without this twist argument, essentially because it is not clear
how to find two “independent” localizations which are maximal orders; however
we can do it in the sl(2) case when $\displaystyle n=1$.

At the end of the section we deduce Theorem 1.3 (2) from Theorem 1.3 (1), the
centralizer theorem for central simple algebras, and a few results of [23] and
[36].

###  1.1. Basic notations

Given a ring $\displaystyle R$, we denote by $\displaystyle\mathcal{Z}(R)$ its
center, by Spec$\displaystyle(R)$ its spectrum, and by SpecM$\displaystyle(R)$
its maximal spectrum. When $\displaystyle R$ is commutative and has no non-
trivial zero divisors, $\displaystyle Q(R)$ denotes its fraction field.

Given a Hopf algebra $\displaystyle H$ with product $\displaystyle m$ and and
coproduct $\displaystyle\Delta$, we denote by $\displaystyle H^{cop}$ (resp.
$\displaystyle H_{op}$) the Hopf algebra with the same algebra (resp.
coalgebra) structure as $\displaystyle H$ but the opposite coproduct
$\displaystyle\sigma\circ{\Delta}$ (resp. opposite product $\displaystyle
m\circ\sigma$), where $\displaystyle\sigma(x\otimes y)=y\otimes x$, and the
antipode $\displaystyle{S}^{-1}$. We use Sweedler’s coproduct notation,
$\displaystyle\textstyle\Delta(x)=\sum_{(x)}x_{(1)}\otimes x_{(2)}$,
$\displaystyle x\in H$.

We let $\displaystyle{\mathfrak{g}}$ be a finite dimensional complex simple
Lie algebra of rank $\displaystyle m$, with Cartan matrix
$\displaystyle(a_{ij})$. We fix a Cartan subalgebra
$\displaystyle\mathfrak{h}\subset{\mathfrak{g}}$ and a basis of simple roots
$\displaystyle\alpha_{i}\in\mathfrak{h}_{\mathbb{R}}^{*}$; we denote by
$\displaystyle d_{1},\ldots,d_{m}$ the unique coprime positive integers such
that the matrix $\displaystyle(d_{i}a_{ij})$ is symmetric, and
$\displaystyle(\ ,\ )$ the unique inner product on
$\displaystyle\mathfrak{h}_{\mathbb{R}}^{*}$ such that $\displaystyle
d_{i}a_{ij}=(\alpha_{i},\alpha_{j})$. For any root $\displaystyle\alpha$ the
coroot is $\displaystyle\alpha\check{}=2\alpha/(\alpha,\alpha)$; in particular
$\displaystyle\alpha\check{}_{i}=d_{i}^{-1}\alpha_{i}$. The root lattice
$\displaystyle Q$ is the $\displaystyle\mathbb{Z}$-lattice in
$\displaystyle\mathfrak{h}_{\mathbb{R}}^{*}$ defined by
$\displaystyle\textstyle Q=\sum_{i=1}^{m}\mathbb{Z}\alpha_{i}$. The weight
lattice $\displaystyle P$ is the $\displaystyle\mathbb{Z}$-lattice formed by
all $\displaystyle\lambda\in\mathfrak{h}_{\mathbb{R}}^{*}$ such that

$\displaystyle(\lambda,\alpha\check{}_{i})\in\mathbb{Z}$

for every $\displaystyle i=1,\ldots,m$. So $\displaystyle\textstyle
P=\sum_{i=1}^{m}\mathbb{Z}\varpi_{i}$, where $\displaystyle\varpi_{i}$ is the
fundamental weight dual to the simple coroot
$\displaystyle\alpha\check{}_{i}$, ie. satisfying
$\displaystyle(\varpi_{i},\alpha\check{}_{j})=\delta_{i,j}$. We denote by
$\displaystyle\textstyle P_{+}:=\sum_{i=1}^{m}\mathbb{Z}_{\geq 0}\varpi_{i}$
the cone of dominant integral weights, by $\displaystyle N$ the number of
positive roots of $\displaystyle{\mathfrak{g}}$, by $\displaystyle\rho$ half
the sum of the positive roots, and by $\displaystyle D$ the smallest positive
integer such that $\displaystyle D(\lambda,\mu)\in{\mathbb{Z}}$ for every
$\displaystyle\lambda,\mu\in P$. Note that
$\displaystyle(\lambda,\alpha)\in\mathbb{Z}$ for every
$\displaystyle\lambda\in P$, $\displaystyle\alpha\in Q$, and $\displaystyle D$
is the smallest positive integer such that $\displaystyle DP\subset Q$. We
denote by $\displaystyle\mathcal{B}(\mathfrak{g})$ the braid group of
$\displaystyle\mathfrak{g}$; we recall its standard defining relations in the
Appendix (Section 6.1).

We let $\displaystyle G$ be the connected and simply-connected Lie group with
Lie algebra $\displaystyle\mathfrak{g}$. We put $\displaystyle
T_{G}=\exp(\mathfrak{h})$, the maximal torus of $\displaystyle G$ generated by
$\displaystyle\mathfrak{h}$; $\displaystyle N(T_{G})$ is the normalizer of
$\displaystyle T_{G}$, $\displaystyle W=N(T_{G})/T_{G}$ is the Weyl group,
$\displaystyle B_{\pm}$ the unique Borel subgroups such that $\displaystyle
B_{+}\cap B_{-}=T_{G}$, and $\displaystyle U_{\pm}\subset B_{\pm}$ their
unipotent subgroups.

We let $\displaystyle q$ be an indeterminate, set $\displaystyle
A={\mathbb{C}}[q,q^{-1}]$, $\displaystyle q_{i}=q^{d_{i}}$, and given integers
$\displaystyle p,k$ with $\displaystyle 0\leq k\leq p$ we put

$\displaystyle\displaystyle[p]_{q}=\frac{q^{p}-q^{-p}}{q-q^{-1}}\ ,\
[0]_{q}!=1\ ,\ [p]_{q}!=[1]_{q}[2]_{q}\ldots[p]_{q}\ ,\
\left[\begin{array}[]{c}p\\\
k\end{array}\right]_{q}=\frac{[p]_{q}!}{[p-k]_{q}![k]_{q}!}$
$\displaystyle\displaystyle(p)_{q}=\frac{q^{p}-1}{q-1}\ ,\ (0)_{q}!=1\ ,\
(p)_{q}!=(1)_{q}(2)_{q}\ldots(p)_{q}\ ,\ \left(\begin{array}[]{c}p\\\
k\end{array}\right)_{q}=\frac{(p)_{q}!}{(p-k)_{q}!(k)_{q}!}.$

We denote by $\displaystyle\epsilon$ a primitive $\displaystyle l$-th root of
unity such that $\displaystyle\epsilon^{2d_{i}}\neq 1$ is also a primitive
$\displaystyle l$-th root of unity for all $\displaystyle
i\in\\{1,\ldots,m\\}$. This means that $\displaystyle l$ is odd, and coprime
to $\displaystyle 3$ if $\displaystyle\mathfrak{g}$ has $\displaystyle
G_{2}$-components.

In this paper we use the definition of the unrestricted integral form
$\displaystyle U_{A}(\mathfrak{g})$ given in [35], [36]; in [23] we used the
one of [33], [34]. The two are (trivially) isomorphic, and have the same
specialization at $\displaystyle q=\epsilon$. Also, we denote here by
$\displaystyle L_{i}$ the generators of $\displaystyle U_{q}(\mathfrak{g})$ we
denoted by $\displaystyle\ell_{i}$ in [23].

To facilitate the comparison with [36] we note that their generators, that we
will denote by $\displaystyle\tilde{K}_{i},\tilde{E}_{i}$ and
$\displaystyle\tilde{F}_{i}$, can be written respectively as $\displaystyle
K_{i},K_{i}^{-1}E_{i}$ and $\displaystyle F_{i}K_{i}$ in our notations. They
satisfy the same algebra relations.

##  2\. Background results

###  2.1. On $\displaystyle U_{q}$, $\displaystyle{\mathcal{O}}_{q}$,
$\displaystyle{\mathcal{L}}_{0,n}$, $\displaystyle{\mathcal{M}}_{0,n}$, and
$\displaystyle\Phi_{n}$

Except when stated differently, we refer to [23], Sections 2-4 and 6, and the
references therein for details about the material of this section.

The simply-connected quantum group $\displaystyle U_{q}=U_{q}(\mathfrak{g})$
is the Hopf algebra over $\displaystyle\mathbb{C}(q)$ with generators
$\displaystyle E_{i}$, $\displaystyle F_{i}$, $\displaystyle L_{i}$,
$\displaystyle L_{i}^{-1}$, $\displaystyle 1\leq i\leq m$, and defining
relations

$\displaystyle\displaystyle L_{i}L_{j}=L_{j}L_{i}\ ,\
L_{i}L_{i}^{-1}=L_{i}^{-1}L_{i}=1\ ,\
L_{i}E_{j}L_{i}^{-1}=q_{i}^{\delta_{i,j}}E_{j}\ ,\
L_{i}F_{j}L_{i}^{-1}=q_{i}^{-\delta_{i,j}}F_{j}$ $\displaystyle\displaystyle
E_{i}F_{j}-F_{j}E_{i}=\delta_{i,j}\frac{K_{i}-K_{i}^{-1}}{q_{i}-q_{i}^{-1}}$
(5)
$\displaystyle\displaystyle\sum_{r=0}^{1-a_{ij}}(-1)^{r}\left[\begin{array}[]{c}1-a_{ij}\\\
r\end{array}\right]_{q_{i}}E_{i}^{1-a_{ij}-r}E_{j}E_{i}^{r}=0\quad{\rm if}\
i\neq j$ (8)
$\displaystyle\displaystyle\sum_{r=0}^{1-a_{ij}}(-1)^{r}\left[\begin{array}[]{c}1-a_{ij}\\\
r\end{array}\right]_{q_{i}}F_{i}^{1-a_{ij}-r}F_{j}F_{i}^{r}=0\quad{\rm if}\
i\neq j.$

where for $\displaystyle\textstyle\lambda=\sum_{i=1}^{m}m_{i}\varpi_{i}\in P$
we set $\displaystyle\textstyle K_{\lambda}=\prod_{i=1}^{m}L_{i}^{m_{i}}$, and
$\displaystyle\textstyle K_{i}=K_{\alpha_{i}}=\prod_{j=1}^{m}L_{j}^{a_{ji}}$.
The coproduct $\displaystyle\Delta$, antipode $\displaystyle S$, and counit
$\displaystyle\varepsilon$ of $\displaystyle U_{q}$ are given by

$\begin{array}[]{c}\Delta(L_{i})=L_{i}\otimes L_{i}\ ,\
\Delta(E_{i})=E_{i}\otimes K_{i}+1\otimes E_{i}\ ,\
\Delta(F_{i})=K_{i}^{-1}\otimes F_{i}+F_{i}\otimes 1\\\
S(E_{i})=-E_{i}K_{i}^{-1}\ ,\ S(F_{i})=-K_{i}F_{i}\ ,\ S(L_{i})=L_{i}^{-1}\\\
\varepsilon(E_{i})=\varepsilon(F_{i})=0,\ \varepsilon(L_{i})=1.\end{array}$

We fix a reduced expression $\displaystyle s_{i_{1}}\ldots s_{i_{N}}$ of the
longest element $\displaystyle w_{0}$ of the Weyl group of
$\displaystyle\mathfrak{g}$. It induces a total ordering of the positive
roots,

$\displaystyle\beta_{1}=\alpha_{i_{1}},\beta_{2}=s_{i_{1}}(\alpha_{i_{2}}),\ldots,\beta_{N}=s_{i_{1}}\ldots
s_{i_{N-1}}(\alpha_{i_{N}}).$

The root vectors of $\displaystyle U_{q}$ with respect to such an ordering are
defined by

$\displaystyle E_{\beta_{k}}=T_{i_{1}}\ldots T_{i_{k-1}}(E_{i_{k}})\ ,\
F_{\beta_{k}}=T_{i_{1}}\ldots T_{i_{k-1}}(F_{i_{k}})$

where $\displaystyle T_{i}$ is Lusztig’s algebra automorphism of
$\displaystyle U_{q}$ associated to the simple root $\displaystyle\alpha_{i}$
([53, 52], see also [30], Ch. 8). In the Appendix we recall the relation
between $\displaystyle T_{i}$ and the generator $\displaystyle\hat{w}_{i}$ of
the quantum Weyl group, which we will mostly use. Let us just recall here that
the monomials $\displaystyle F_{\beta_{1}}^{r_{1}}\ldots
F_{\beta_{N}}^{r_{N}}K_{\lambda}E_{\beta_{N}}^{t_{N}}\ldots
E_{\beta_{1}}^{t_{1}}$ ($\displaystyle r_{i},t_{i}\in\mathbb{N}$,
$\displaystyle\lambda\in P$) form a basis of $\displaystyle U_{q}$.

$\displaystyle U_{q}$ is a pivotal Hopf algebra, with pivotal element

$\displaystyle\textstyle\ell:=K_{2\rho}=\prod_{j=1}^{m}L_{j}^{2}.$

So $\displaystyle\ell$ is group-like, and $\displaystyle S^{2}(x)=\ell
x\ell^{-1}$ for every $\displaystyle x\in U_{q}$.

The adjoint quantum group $\displaystyle U_{q}^{ad}=U_{q}^{ad}(\mathfrak{g})$
is the Hopf subalgebra of $\displaystyle U_{q}$ generated by the elements
$\displaystyle E_{i}$, $\displaystyle F_{i}$ ($\displaystyle i=1,\ldots,m$)
and $\displaystyle K_{\alpha}$ with $\displaystyle\alpha\in Q$; so
$\displaystyle\ell\in U_{q}^{ad}$. When $\displaystyle\mathfrak{g}=sl(2)$, we
simply write the above generators $\displaystyle E=E_{1}$, $\displaystyle
F=F_{1}$, $\displaystyle L=L_{1}$, $\displaystyle K=K_{1}$.

We denote by $\displaystyle U_{q}(\mathfrak{n}_{+})$, $\displaystyle
U_{q}(\mathfrak{n}_{-})$ and $\displaystyle U_{q}(\mathfrak{h})$ the
subalgebras of $\displaystyle U_{q}$ generated respectively by the
$\displaystyle E_{i}$, the $\displaystyle F_{i}$, and the $\displaystyle
K_{\lambda}$ ($\displaystyle\lambda\in P$), and by $\displaystyle
U_{q}(\mathfrak{b}_{+})$ and $\displaystyle U_{q}(\mathfrak{b}_{-})$ the
subalgebras generated by the $\displaystyle E_{i}$ and the $\displaystyle
K_{\lambda}$, and by the $\displaystyle F_{i}$ and the $\displaystyle
K_{\lambda}$, respectively (they are the two-sided ideals generated by
$\displaystyle U_{q}(\mathfrak{n}_{\pm})$). We do similarly with
$\displaystyle U_{q}^{ad}$.

$\displaystyle U_{q}^{ad}$ is not a braided Hopf algebra in a strict sense,
but it has braided categorical completions. Namely, denote by
$\displaystyle\mathcal{C}$ the category of type $\displaystyle 1$ finite
dimensional $\displaystyle U_{q}^{ad}$-modules, by $\displaystyle Vect$ the
category of finite dimensional $\displaystyle\mathbb{C}(q)$-vector spaces, and
by $\displaystyle F_{\mathcal{C}}:{\mathcal{C}}\to Vect$ the forgetful
functor. The categorical completion $\displaystyle\mathbb{U}_{q}^{ad}$ of
$\displaystyle U_{q}^{ad}$ is the set of natural transformations
$\displaystyle F_{\mathcal{C}}\rightarrow F_{\mathcal{C}}$.

Let us recall briefly what this means and implies. For details we refer to the
sections 2 and 3 of [23] (see also [70], Section 2.10, where
$\displaystyle\mathbb{U}_{q}$ below is formulated in terms of multiplier Hopf
algebras). An element of $\displaystyle\mathbb{U}_{q}^{ad}$ is a collection
$\displaystyle(a_{V})_{V\in Ob(\mathcal{C})}$, where $\displaystyle a_{V}\in
End_{\mathbb{C}(q)}(V)$ satisfies $\displaystyle F_{\mathcal{C}}(f)\circ
a_{V}=a_{W}\circ F_{\mathcal{C}}(f)$ for any objects $\displaystyle V,W$ of
$\displaystyle\mathcal{C}$ and any arrow $\displaystyle f\in
Hom_{U_{q}^{ad}}(V,W)$. It is not hard to see that
$\displaystyle\mathbb{U}_{q}^{ad}$ inherits from $\displaystyle\mathcal{C}$ a
natural structure of Hopf algebra such that the map

$\displaystyle\begin{array}[]{lcll}\iota:&U_{q}^{ad}&\longrightarrow&\mathbb{U}_{q}^{ad}\\\
&x&\longmapsto&(\pi_{V}(x))_{V\in Ob(\mathcal{C})}\end{array}$

is a morphism of Hopf algebras, where
$\displaystyle\pi_{V}:U_{q}^{ad}\rightarrow{\rm End}(V)$ is the representation
associated to a module $\displaystyle V$ in $\displaystyle\mathcal{C}$. It is
a theorem that this map is injective; $\displaystyle\mathbb{U}_{q}^{ad}$ can
be understood as a weak-$\displaystyle*$ completion of $\displaystyle
U_{q}^{ad}$ by means of the pairing $\displaystyle\langle.,.\rangle$
introduced below. From now on, let us extend the coefficient ring of the
modules and morphisms in $\displaystyle\mathcal{C}$ to
$\displaystyle\mathbb{C}(q^{1/D})$. Put

$\displaystyle\mathbb{U}_{q}=\mathbb{U}_{q}^{ad}\otimes_{\mathbb{C}(q)}\mathbb{C}(q^{1/D})$

One can show that the map $\displaystyle\iota$ above extends to an embedding
of $\displaystyle U_{q}\otimes_{\mathbb{C}(q)}\mathbb{C}(q^{1/D})$ in
$\displaystyle\mathbb{U}_{q}$. The category $\displaystyle\mathcal{C}$, with
coefficients in $\displaystyle\mathbb{C}(q^{1/D})$, is braided and ribbon. We
postpone a discussion of that fact to Section 2.3, where it will be developed.
As a consequence, $\displaystyle\mathbb{U}_{q}$ is a quasitriangular and
ribbon Hopf algebra. The R-matrix of $\displaystyle\mathbb{U}_{q}$ is the
family of morphisms

$\displaystyle R=((R_{h})_{V,W})_{V,W\in Ob(\mathcal{C})}$

where $\displaystyle q=e^{h}$, $\displaystyle R_{h}$ is the universal R-matrix
of the quantized universal enveloping algebra $\displaystyle
U_{h}(\mathfrak{g})$, and $\displaystyle(R_{h})_{V,W}\in End(V\otimes W)$, for
every modules $\displaystyle V,W$ in $\displaystyle\mathcal{C}$, is the
endomorphism defined by the action of $\displaystyle R_{h}$ on $\displaystyle
V\otimes W$ (which is well-defined). The ribbon element $\displaystyle v_{h}$
of $\displaystyle U_{h}(\mathfrak{g})$ defines similarly the ribbon element
$\displaystyle v=((v_{h})_{V})_{V}$ of $\displaystyle\mathbb{U}_{q}$. One
defines the categorical tensor product
$\displaystyle\mathbb{U}_{q}^{\hat{\otimes}2}$ similarly as
$\displaystyle\mathbb{U}_{q}$; it contains all the infinite series of elements
of $\displaystyle\mathbb{U}_{q}^{\otimes 2}$ having only a finite number of
non-zero terms when evaluated on a given module $\displaystyle V\otimes W$ of
$\displaystyle\mathcal{C}$. The expansion of $\displaystyle R_{h}$ as an
infinite series in $\displaystyle U_{h}(\mathfrak{g})^{\hat{\otimes}2}$
induces an expansion of $\displaystyle R$ as an infinite series in
$\displaystyle\mathbb{U}_{q}^{\hat{\otimes}2}$. Adapting Sweedler’s coproduct
notation $\displaystyle\textstyle\Delta(x)=\sum_{(x)}x_{(1)}\otimes x_{(2)}$
we find convenient to write this series as

(9) $R=\sum_{(R)}R_{(1)}\otimes R_{(2)}.$

We put $\displaystyle R^{+}:=R$, $\displaystyle R^{-}:=(\sigma\circ R)^{-1}$
where $\displaystyle\sigma$ is the flip map $\displaystyle x\otimes y\mapsto
y\otimes x$.

The quantum function Hopf algebra
$\displaystyle{\mathcal{O}}_{q}={\mathcal{O}}_{q}(G)$ is the restricted dual
of $\displaystyle U_{q}^{ad}$, ie. the set of
$\displaystyle{\mathbb{C}}(q)$-linear maps $\displaystyle f\colon
U_{q}^{ad}\rightarrow\mathbb{C}(q)$ such that $\displaystyle{\rm Ker}(f)$
contains a cofinite two sided ideal $\displaystyle I$ (ie. such that
$\displaystyle I\oplus M=U_{q}$ for some finite dimensional vector space
$\displaystyle M$), and
$\displaystyle\textstyle\prod_{s=-r}^{r}(K_{i}-q_{i}^{s})\in I$ for some
$\displaystyle r\in\mathbb{N}$ and every $\displaystyle i$. The structure maps
of $\displaystyle{\mathcal{O}}_{q}$ are defined dually to those of
$\displaystyle U_{q}^{ad}$. We denote by $\displaystyle\star$ its product. The
algebras $\displaystyle{\mathcal{O}}_{q}(T_{G})$,
$\displaystyle{\mathcal{O}}_{q}(U_{\pm})$,
$\displaystyle{\mathcal{O}}_{q}(B_{\pm})$ are defined similarly, by replacing
$\displaystyle U_{q}^{ad}$ with $\displaystyle U_{q}^{ad}(\mathfrak{h})$,
$\displaystyle U_{q}^{ad}(\mathfrak{n}_{\pm})$, $\displaystyle
U_{q}^{ad}(\mathfrak{b}_{\pm})$ respectively. $\displaystyle{\mathcal{O}}_{q}$
is generated as an algebra by the functionals $\displaystyle x\mapsto
w(\pi_{V}(x)v)$, $\displaystyle x\in U_{q}^{ad}$, for every object
$\displaystyle V\in Ob(\mathcal{C})$ and vectors $\displaystyle v\in V$,
$\displaystyle w\in V^{*}$. Such functionals are called matrix coefficients.
We can uniquely extend the (non-degenerate) evaluation pairing
$\displaystyle\langle.,.\rangle\colon{\mathcal{O}}_{q}\otimes
U_{q}^{ad}\rightarrow\mathbb{C}(q)$ to a bilinear pairing
$\displaystyle\langle.,.\rangle\colon{\mathcal{O}}_{q}\otimes\mathbb{U}_{q}\rightarrow\mathbb{C}(q^{1/D})$
such that the following diagram is commutative:

$\displaystyle\textstyle{{\mathcal{O}}_{q}\otimes
U_{q}^{ad}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{\;\langle.,.\rangle}$$\displaystyle\scriptstyle{id\otimes\iota}$$\displaystyle\textstyle{{\mathbb{C}}(q)}$$\displaystyle\textstyle{{\mathcal{O}}_{q}\otimes\mathbb{U}_{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{\langle.,.\rangle}$

This pairing is defined by

$\displaystyle\langle{}_{Y}\phi{}^{w}_{v},(a_{X})_{X}\rangle=w(a_{Y}v)$

for every $\displaystyle(a_{X})_{X}\in\mathbb{U}_{q}$,
$\displaystyle{}_{Y}\phi{}^{w}_{v}\in{\mathcal{O}}_{q}$. It is a perfect
pairing, and reflects the properties of the R-matrix $\displaystyle
R\in\mathbb{U}_{q}^{\hat{\otimes}2}$ in a subtle way. In particular, these
properties imply that the maps

(10)
$\begin{array}[]{lcll}\Phi^{\pm}:&{\mathcal{O}}_{q}&\longrightarrow&U_{q}^{cop}\\\
&\alpha&\longmapsto&(\alpha\otimes
id)(R^{\pm})=\sum_{(R^{\pm})}\langle\alpha,R_{(1)}^{\pm}\rangle
R_{(2)}^{\pm}\end{array}$

are well-defined morphisms of Hopf algebras. Here we stress that it is the
simply-connected quantum group $\displaystyle U_{q}^{cop}$ that is the range
of $\displaystyle\Phi^{\pm}$. This will be explained in more details in
Section 2.3.

The quantum loop algebra
$\displaystyle{\mathcal{L}}_{0,1}={\mathcal{L}}_{0,1}(\mathfrak{g})$ is
defined by twisting the product $\displaystyle\star$ of
$\displaystyle{\mathcal{O}}_{q}$, keeping the same underlying linear space.
The new product is equivariant with respect to the right coadjoint action
$\displaystyle coad^{r}$ of $\displaystyle U_{q}^{ad}$; noting that
$\displaystyle coad^{r}$ extends to an action of the simply-connected quantum
group $\displaystyle U_{q}$, the new product thus gives
$\displaystyle{\mathcal{L}}_{0,1}$ a structure of $\displaystyle U_{q}$-module
algebra. Recall that

$\displaystyle\ coad^{r}(x)(\alpha)=\sum_{(x)}{S}(x_{(2)})\rhd\alpha\lhd
x_{(1)}$

for all $\displaystyle x\in U_{q}$ and
$\displaystyle\alpha\in{\mathcal{O}}_{q}$, where $\displaystyle\rhd$,
$\displaystyle\lhd$ are the left and right coregular actions of $\displaystyle
U_{q}$ on $\displaystyle{\mathcal{O}}_{q}$, defined by

$\displaystyle
x\rhd\alpha:=\sum_{(\alpha)}\alpha_{(1)}\langle\alpha_{(2)},x\rangle,\
\alpha\lhd x:=\sum_{(\alpha)}\langle\alpha_{(1)},x\rangle\alpha_{(2)}.$

Using the fact that $\displaystyle U_{q}\otimes\mathbb{C}(q^{1/D})$ can be
regarded as a subspace of $\displaystyle\mathbb{U}_{q}$, these actions extend
naturally to actions of $\displaystyle\mathbb{U}_{q}$. The product of
$\displaystyle{\mathcal{L}}_{0,1}$ is expressed in terms of
$\displaystyle\star$ by the formula ([23], Proposition 4.1):

(11)
$\alpha\beta=\sum_{(R),(R)}(R_{(2^{\prime})}{S}(R_{(2)})\rhd\alpha)\star(R_{(1^{\prime})}\rhd\beta\lhd
R_{(1)}),$

where $\displaystyle\textstyle\sum_{(R)}R_{(1)}\otimes R_{(2)}$ and
$\displaystyle\textstyle\sum_{(R)}R_{(1^{\prime})}\otimes R_{(2^{\prime})}$
are expansions of two copies of $\displaystyle
R\in\mathbb{U}_{q}^{\hat{\otimes}2}$. Note that the sum in (11) has only a
finite number of non zero terms. This product gives
$\displaystyle{\mathcal{L}}_{0,1}$ (like $\displaystyle{\mathcal{O}}_{q}$) a
structure of module algebra for the actions $\displaystyle\rhd$,
$\displaystyle\lhd$, and also for $\displaystyle coad^{r}(x)$. Spelling this
out for $\displaystyle coad^{r}$, this means

$\displaystyle
coad^{r}(x)(\alpha\beta)=\sum_{(x)}coad^{r}(x_{(1)})(\alpha)coad^{r}(x_{(2)})(\beta).$

The relations between $\displaystyle{\mathcal{O}}_{q}$,
$\displaystyle{\mathcal{L}}_{0,1}$ and $\displaystyle U_{q}$ (the simply-
connected quantum group) are encoded by the map

(12)
$\begin{array}[]{lcll}\Phi_{1}:&{\mathcal{O}}_{q}&\longrightarrow&\mathbb{U}_{q}\\\
&\alpha&\longmapsto&(\alpha\otimes id)(RR^{\prime})\end{array}$

where $\displaystyle R^{\prime}=\sigma\circ R$, and as usual
$\displaystyle\sigma\colon x\otimes y\mapsto y\otimes x$. Note that

$\displaystyle\Phi_{1}=m\circ(\Phi^{+}\otimes(S^{-1}\circ\Phi^{-}{}))\circ\Delta.$

We call $\displaystyle\Phi_{1}$ the RSD map, for Drinfeld, Reshetikhin and
Semenov-Tian-Shansky introduced it first (see [31, 61],[58]). Recall that
$\displaystyle U_{q}$ embeds in $\displaystyle\mathbb{U}_{q}$. It is a
fundamental result of the theory ([28, 43, 9]) that $\displaystyle\Phi_{1}$
affords an isomorphism of $\displaystyle U_{q}$-modules

$\displaystyle\Phi_{1}\colon{\mathcal{O}}_{q}\rightarrow U_{q}^{lf}.$

For full details on that result we refer to Section 2.12 of [70] (where
different conventions are used). Here, $\displaystyle U_{q}^{lf}$ is the set
of _locally finite_ elements of $\displaystyle U_{q}$, endowed with the right
adjoint action $\displaystyle ad^{r}$ of $\displaystyle U_{q}$. It is defined
by

$\displaystyle U_{q}^{lf}:=\\{x\in U_{q}\ |\
rk_{\mathbb{C}(q)}(ad^{r}(U_{q})(x))<\infty\\}$

and

$\displaystyle ad^{r}(y)(x)=\sum_{(y)}{S}(y_{(1)})xy_{(2)}$

for every $\displaystyle x,y\in U_{q}$. The action $\displaystyle ad^{r}$
gives in fact $\displaystyle U_{q}^{lf}$ a structure of right $\displaystyle
U_{q}$-module algebra. Moreover, $\displaystyle\Phi_{1}$ affords an
isomorphism of $\displaystyle U_{q}$-module algebras

(13) $\Phi_{1}\colon{\mathcal{L}}_{0,1}\rightarrow U_{q}^{lf}.$

The centers $\displaystyle\mathcal{Z}({\mathcal{L}}_{0,1})$ of
$\displaystyle{\mathcal{L}}_{0,1}$, and $\displaystyle\mathcal{Z}(U_{q})$ of
$\displaystyle U_{q}$, coincide respectively with
$\displaystyle{\mathcal{L}}_{0,1}^{U_{q}}$ and $\displaystyle U_{q}^{U_{q}}$,
the subsets of $\displaystyle U_{q}$-invariants elements of
$\displaystyle{\mathcal{L}}_{0,1}$ and $\displaystyle U_{q}$. As a
consequence, $\displaystyle\Phi_{1}$ affords an isomorphism between
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,1})$ and
$\displaystyle\mathcal{Z}(U_{q})$.

The quantum graph algebra
$\displaystyle{\mathcal{L}}_{0,n}={\mathcal{L}}_{0,n}(\mathfrak{g})$ is the
braided tensor product of $\displaystyle n$ copies of
$\displaystyle{\mathcal{L}}_{0,1}$ (considered as a
$\displaystyle\mathbb{U}_{q}$-module algebra). Thus it coincides with
$\displaystyle{\mathcal{L}}_{0,1}^{\otimes n}$ as a linear space, and it is a
right $\displaystyle U_{q}$-module algebra, the action of $\displaystyle
U_{q}$ (extending $\displaystyle coad^{r}$ on
$\displaystyle{\mathcal{L}}_{0,1}$) being given by

$\displaystyle\displaystyle
coad_{n}^{r}(y)(\alpha^{(1)}\otimes\ldots\otimes\alpha^{(n)})=$
$\displaystyle\displaystyle\sum_{(y)}coad^{r}(y_{(1)})(\alpha^{(1)})\otimes\ldots\otimes
coad^{r}(y_{(n)})(\alpha^{(n)})$

for all $\displaystyle y\in U_{q}$ and
$\displaystyle\alpha^{(1)}\otimes\ldots\otimes\alpha^{(n)}\in{\mathcal{L}}_{0,n}$.
The algebra structure can be explicited as follows. For every $\displaystyle
1\leq a\leq n$ define
$\displaystyle\mathfrak{i}_{a}\colon{\mathcal{L}}_{0,1}\rightarrow{\mathcal{L}}_{0,n}$
by $\displaystyle\mathfrak{i}_{a}(x)=1^{\otimes(a-1)}\otimes x\otimes
1^{\otimes(n-a)}$; $\displaystyle\mathfrak{i}_{a}$ is an embedding of
$\displaystyle U_{q}$-module algebras. We will use the notations

$\displaystyle{\mathcal{L}}_{0,n}^{(a)}:={\rm Im}(\mathfrak{i}_{a})\ ,\
(\alpha)^{(a)}:=\mathfrak{i}_{a}(\alpha).$

Take
$\displaystyle(\alpha)^{(a)},(\alpha^{\prime})^{(a)}\in{\mathcal{L}}_{0,n}^{(a)}$
and
$\displaystyle(\beta)^{(b)},(\beta^{\prime})^{(b)}\in{\mathcal{L}}_{0,n}^{(b)}$
with $\displaystyle a<b$. Then the product of
$\displaystyle{\mathcal{L}}_{0,n}$ is given by the following formula (see in
[23] the proposition 6.2-6.3 and the formulas (13)-(41)-(42)):

(14)
$\begin{array}[]{ll}\left((\alpha)^{(a)}\otimes(\beta)^{(b)}\right)&\\!\\!\left((\alpha^{\prime})^{(a)}\otimes(\beta^{\prime})^{(b)}\right)\\\
&\hskip
28.45274pt=\sum_{(R^{1}),\ldots,(R^{4})}\left(\alpha\left(S(R^{3}_{(1)}R^{4}_{(1)})\rhd\alpha^{\prime}\lhd
R^{1}_{(1)}R^{2}_{(1)}\right)\right)^{(a)}\\\ &\hskip
128.0374pt\otimes\left(\left(S(R^{1}_{(2)}R^{3}_{(2)})\rhd\beta\lhd
R^{2}_{(2)}R^{4}_{(2)}\right)\beta^{\prime}\right)^{(b)}\end{array}$

where $\displaystyle R^{i}=\textstyle\sum_{(R^{i})}R_{(1)}^{i}\otimes
R_{(2)}^{i}$, $\displaystyle i\in\\{1,2,3,4\\}$, are expansions of four copies
of $\displaystyle R\in\mathbb{U}_{q}^{\hat{\otimes}2}$, and on the right-hand
side the product is componentwise that of $\displaystyle{\mathcal{L}}_{0,1}$.
Later we will use the fact that the product of
$\displaystyle{\mathcal{L}}_{0,n}$ is obtained from the standard
(componentwise) product of $\displaystyle{\mathcal{L}}_{0,1}^{\otimes n}$ by a
process that may be inverted. Indeed, (14) can be rewritten as

(15)
$\left((\alpha)^{(a)}\otimes(\beta)^{(b)}\right)\\!\\!\left((\alpha^{\prime})^{(a)}\otimes(\beta^{\prime})^{(b)}\right)=\sum_{(F)}\
(\alpha)^{(a)}\\!\\!\left((\alpha^{\prime})^{(a)}\cdot
F_{(2)}\right)\otimes\left((\beta)^{(b)}\cdot
F_{(1)}\right)\\!\\!(\beta^{\prime})^{(b)}$

where $\displaystyle\textstyle F=\sum_{(F)}F_{(1)}\otimes
F_{(2)}:=(\Delta\otimes\Delta)(R^{\prime})$, and the symbol
“$\displaystyle\cdot$” stands for the right action of
$\displaystyle\mathbb{U}_{q}^{\otimes 2}$ on
$\displaystyle{\mathcal{L}}_{0,1}$ that may be read from (14). The tensor
$\displaystyle F$ is known as a twist. Then, by replacing $\displaystyle F$
with its inverse
$\displaystyle\bar{F}=(\Delta\otimes\Delta)(R^{\prime}{}^{-1})$, one can
express the product of $\displaystyle{\mathcal{L}}_{0,1}^{\otimes n}$ in terms
of the product of $\displaystyle{\mathcal{L}}_{0,n}$ by

(16)
$(\alpha)^{(a)}(\alpha^{\prime})^{(a)}\otimes(\beta)^{(b)}(\beta^{\prime})^{(b)}=\sum_{(\bar{F})}\
\left((\alpha)^{(a)}\otimes\left((\beta)^{(b)}\cdot\bar{F}_{(1)}\right)\right)\\!\\!\left(\left((\alpha^{\prime})^{(a)}\cdot\bar{F}_{(2)}\right)\otimes(\beta^{\prime})^{(b)}\right).$

We call quantum moduli algebra and denote by

$\displaystyle{\mathcal{M}}_{0,n}={\mathcal{M}}_{0,n}(\mathfrak{g})$

the subalgebra $\displaystyle{\mathcal{L}}_{0,n}^{U_{q}}$ of
$\displaystyle{\mathcal{L}}_{0,n}$ formed by the $\displaystyle
U_{q}$-invariant elements.

Consider the following action of $\displaystyle U_{q}$ on the tensor product
algebra $\displaystyle U_{q}^{{\otimes}n}$, which extends $\displaystyle
ad^{r}$ on $\displaystyle U_{q}$:

(17)
$ad_{n}^{r}(y)(x)=\sum_{(y)}\Delta^{(n)}({S}(y_{(1)}))x\Delta^{(n)}(y_{(2)})$

for all $\displaystyle y\in U_{q}$, $\displaystyle x\in U_{q}^{{\otimes}n}$.
This action gives $\displaystyle U_{q}^{{\otimes}n}$ a structure of right
$\displaystyle U_{q}$-module algebra. In [1] Alekseev introduced a morphism of
$\displaystyle U_{q}$-module algebras
$\displaystyle\Phi_{n}\colon{\mathcal{L}}_{0,n}\rightarrow U_{q}^{\otimes n}$
which extends $\displaystyle\Phi_{1}$. In Proposition 6.5 and Lemma 6.8 of
[23] we showed that $\displaystyle\Phi_{n}$ affords isomorphisms

(18) $\Phi_{n}\colon{\mathcal{L}}_{0,n}\rightarrow(U_{q}^{\otimes n})^{lf}\ ,\
\Phi_{n}:{\mathcal{M}}_{0,n}\rightarrow(U_{q}^{{\otimes}n})^{U_{q}}$

where $\displaystyle(U_{q}^{\otimes n})^{lf}$ is the set of $\displaystyle
ad_{n}^{r}$-locally finite elements of $\displaystyle U_{q}^{{\otimes}n}$. We
call $\displaystyle\Phi_{n}$ the Alekseev map; we will not use the definition
of $\displaystyle\Phi_{n}$ in this paper.

It is a key argument of the proof of (18), to be used later, that the set of
locally finite elements of $\displaystyle U_{q}^{{\otimes}n}$ for
$\displaystyle(ad^{r})^{\otimes n}\circ\Delta^{(n-1)}$ coincides with
$\displaystyle(U_{q}^{lf})^{\otimes n}$; this follows from the main result of
[49]. Using that the map

(19) $\psi_{n}=\Phi_{n}\circ(\Phi_{1}^{-1})^{\otimes n}$

extends to a linear automorphism of $\displaystyle U_{q}^{\otimes n}$ which
intertwines the actions $\displaystyle(ad^{r})^{\otimes n}\circ\Delta^{(n-1)}$
and $\displaystyle ad_{n}^{r}$ of $\displaystyle U_{q}$, we deduced that
$\displaystyle\psi_{n}((U_{q}^{lf})^{\otimes n})=(U_{q}^{\otimes n})^{lf}$,
whence $\displaystyle{\rm Im}(\Phi_{n})=(U_{q}^{\otimes n})^{lf}$.

######  Remark 2.1.

We have $\displaystyle(U_{q}^{lf})^{\otimes n}\neq(U_{q}^{\otimes n})^{lf}$,
and in fact there is not even an inclusion. Indeed let
$\displaystyle\Omega=(q-q^{-1})^{2}FE+qK+q^{-1}K^{-1}$ be the standard Casimir
element of $\displaystyle U_{q}(sl(2))$. We trivially have
$\displaystyle\Delta(\Omega)\in(U_{q}^{\otimes 2})^{lf}$ but

$\displaystyle\Delta(\Omega)=(q-q^{-1})^{2}(K^{-1}E\otimes FK+F\otimes E)+\\\
\Omega\otimes K+K^{-1}\otimes\Omega-(q+q^{-1})K^{-1}\otimes K$

and therefore $\displaystyle\Delta(\Omega)\notin(U_{q}^{lf})^{\otimes 2}$,
since $\displaystyle K\notin U_{q}^{lf}$ (see eg. Theorem 2.2 (2)).

Let us point out here two important consequences of (18). First,
$\displaystyle\Phi_{n}$ yields isomorphisms between centers,
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n})\cong\mathcal{Z}(U_{q})^{\otimes
n}$ and
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{U_{q}})\cong\mathcal{Z}((U_{q}^{\otimes
n})^{U_{q}})$, where one can show that ([23], Lemma 6.25)

$\displaystyle\mathcal{Z}((U_{q}^{\otimes
n})^{U_{q}})\cong\Delta^{(n-1)}(\mathcal{Z}(U_{q}))\otimes_{{\mathbb{C}}(q)}\mathcal{Z}(U_{q})^{\otimes
n}.$

Second, we see that $\displaystyle{\mathcal{L}}_{0,n}$ (and therefore
$\displaystyle{\mathcal{M}}_{0,n}$) has no non-trivial zero divisors, by using
the isomorphisms
$\displaystyle\Phi_{n}\colon{\mathcal{L}}_{0,n}\rightarrow(U_{q}^{\otimes
n})^{lf}\subset U_{q}^{\otimes n}$ and $\displaystyle U_{q}^{\otimes n}\cong
U_{q}(\mathfrak{g}^{\oplus n})$, and the fact that $\displaystyle
U_{q}(\mathfrak{g}^{\oplus n})$ has no non-trivial zero divisors (proved eg.
in [33]).

###  2.2. Integral forms and specializations

An integral form of a (Hopf) $\displaystyle\mathbb{C}(q)$-algebra is a (Hopf)
$\displaystyle A$-subalgebra, where $\displaystyle A=\mathbb{C}[q,q^{-1}]$,
that becomes isomorphic to the algebra after tensoring it with
$\displaystyle\mathbb{C}(q)$. We consider three integral forms related by the
pairing $\displaystyle\langle\ ,\ \rangle$, one of $\displaystyle U_{q}$, one
of $\displaystyle U_{q}^{ad}$, and one of $\displaystyle{\mathcal{O}}_{q}$.

The unrestricted integral form of $\displaystyle U_{q}$ is the $\displaystyle
A$-subalgebra $\displaystyle U_{A}=U_{A}(\mathfrak{g})$ introduced by De
Concini–Kac–Procesi in [35], Section 12 (and in a differently normalized form
in [33] and [34]). It is generated by the elements ($\displaystyle
i=1,\ldots,m$)

$\displaystyle\bar{E}_{i}=(q_{i}-q_{i}^{-1})E_{i}\ ,\
\bar{F}_{i}=(q_{i}-q_{i}^{-1})F_{i}\ ,L_{i}\ ,\ L_{i}^{-1}.$

Clearly, the subalgebra of locally finite elements of $\displaystyle U_{A}$ is
$\displaystyle U_{A}^{lf}=U_{A}\cap U_{q}^{lf}$. Similarly, we define the
unrestricted integral form of $\displaystyle U_{q}^{ad}$ as the $\displaystyle
A$-subalgebra $\displaystyle U_{A}^{ad}\subset U_{A}$ generated by the
elements $\displaystyle\bar{E}_{i}$, $\displaystyle\bar{F}_{i}$ and
$\displaystyle K_{i}^{\pm 1}$, for $\displaystyle i=1,\ldots,m$.

The restricted integral form of $\displaystyle U_{q}^{ad}$ is the
$\displaystyle A$-subalgebra $\displaystyle\Gamma=\Gamma(\mathfrak{g})$
introduced by De Concini-Lyubashenko in [36], Sections 2-3. It is generated by
the elements ($\displaystyle i=1,\ldots,m$)

$\displaystyle E_{i}^{(k)}=\frac{E_{i}^{k}}{[k]_{q_{i}}!}\ ,\
F_{i}^{(k)}=\frac{F_{i}^{k}}{[k]_{q_{i}}!}\ ,\
(K_{i};t)_{q_{i}}=\prod_{s=1}^{t}\frac{K_{i}q_{i}^{-s+1}-1}{q_{i}^{s}-1}\ ,\
K_{i}^{-1}$

where $\displaystyle k\in\mathbb{N}$, $\displaystyle t\in\mathbb{N}$ (setting
$\displaystyle(K_{i};0)_{q_{i}}=1$ by convention).

Note that $\displaystyle\Gamma$ contains the elements $\displaystyle K_{i}$,
and the unrestricted integral form $\displaystyle U_{A}^{ad}$. It plays a
fundamental rôle in relation with the integral pairings
$\displaystyle\pi_{A}^{\pm}$ considered in Section 2.3; it is by this rôle
that $\displaystyle\Gamma$ is more suited to our purposes than the more
standard restricted integral form $\displaystyle U_{A}^{res}$ defined by
Lusztig, and discussed below.

The integral forms $\displaystyle U_{A}(\mathfrak{h})$, $\displaystyle
U_{A}(\mathfrak{b}_{\pm})$ and $\displaystyle\Gamma(\mathfrak{h})$,
$\displaystyle\Gamma(\mathfrak{b}_{\pm})$ associated to the subalgebras
$\displaystyle\mathfrak{h}$,
$\displaystyle\mathfrak{b}_{\pm}\subset\mathfrak{g}$ are the subalgebras of
$\displaystyle U_{A}$ and $\displaystyle\Gamma$ defined in the obvious way.
For instance the “Cartan” subalgebra $\displaystyle\Gamma(\mathfrak{h})$ is
generated by the elements $\displaystyle(K_{i};t)_{q_{i}}$ and $\displaystyle
K_{i}^{-1}$.

Denote by $\displaystyle\mathcal{C}_{A}$ the category of
$\displaystyle\Gamma$-modules which are free $\displaystyle A$-modules of
finite rank, and semisimple as $\displaystyle\Gamma(\mathfrak{h})$-modules; so
they have a basis where $\displaystyle K_{i}$ and
$\displaystyle(K_{i};t)_{q_{i}}$ act diagonally with respective eigenvalues of
the form

$\displaystyle q_{i}^{k}\ ,\ \left(\begin{array}[]{c}k\\\
t\end{array}\right)_{q_{i}}\quad k\in\mathbb{Z},t\in\mathbb{N}^{*}.$

The integral quantum function Hopf algebra
$\displaystyle{\mathcal{O}}_{A}={\mathcal{O}}_{A}(G)$ is the restricted dual
of $\displaystyle\Gamma$, ie. the set of $\displaystyle A$-linear maps
$\displaystyle f\colon\Gamma\rightarrow A$ such that $\displaystyle{\rm
Ker}(f)$ contains a cofinite two sided ideal $\displaystyle I$, and
$\displaystyle\textstyle\prod_{s=-r}^{r}(K_{i}-q_{i}^{s})\in I$ for some
$\displaystyle r\in\mathbb{N}$ and every $\displaystyle i$.
$\displaystyle{\mathcal{O}}_{A}$ is an integral form of
$\displaystyle{\mathcal{O}}_{q}$. The algebras
$\displaystyle{\mathcal{O}}_{A}(T_{G})$,
$\displaystyle{\mathcal{O}}_{A}(U_{\pm})$,
$\displaystyle{\mathcal{O}}_{A}(B_{\pm})$ are defined similarly, by replacing
$\displaystyle\Gamma$ with $\displaystyle\Gamma(\mathfrak{h})$,
$\displaystyle\Gamma(\mathfrak{n}_{\pm})$,
$\displaystyle\Gamma(\mathfrak{b}_{\pm})$ respectively.
$\displaystyle{\mathcal{O}}_{A}$ is generated as an algebra by the matrix
coefficients $\displaystyle x\mapsto v^{i}(\pi_{V}(x)v_{i})$, $\displaystyle
x\in\Gamma$, for every module $\displaystyle V$ in
$\displaystyle\mathcal{C}_{A}$ where $\displaystyle(v_{i})$ is an
$\displaystyle A$-basis of $\displaystyle V$ and $\displaystyle(v^{i})$ the
dual $\displaystyle A$-basis of the dual module $\displaystyle V^{*}$.

It is immediate that the $\displaystyle U_{q}$-module structure of
$\displaystyle{\mathcal{O}}_{q}$ restricts to an $\displaystyle U_{A}$-module
structure on $\displaystyle{\mathcal{O}}_{A}$.

We note that $\displaystyle{\mathcal{O}}_{A}$ is also the restricted dual of
$\displaystyle U_{A}^{res}$, the Lusztig integral form of $\displaystyle
U_{q}^{ad}$ [52, 53], defined as $\displaystyle\Gamma$ except that the
$\displaystyle(K_{i};t)_{q_{i}}$ ($\displaystyle i=1,\ldots,m$), are replaced
by the elements

$\displaystyle[K_{i};t]_{q_{i}}=\prod_{s=1}^{t}\frac{K_{i}q_{i}^{-s+1}-K_{i}^{-1}q_{i}^{s-1}}{q_{i}^{s}-q_{i}^{-s}}.$

Indeed, $\displaystyle\Gamma(\mathfrak{h})$ contains $\displaystyle
U_{A}^{res}(\mathfrak{h})$ strictly, but the restriction functor
$\displaystyle\mathcal{C}_{A}\rightarrow\mathcal{C}_{A}^{res}$ is an
equivalence of categories, where $\displaystyle\mathcal{C}_{A}^{res}$ is the
category of $\displaystyle U_{A}^{res}$-modules defined as
$\displaystyle\mathcal{C}_{A}$ above, but replacing the condition on
$\displaystyle(K_{i};t)_{q_{i}}$ by its analog for
$\displaystyle[K_{i};t]_{q_{i}}$, ie. that it acts diagonally with eigenvalues

$\displaystyle\left[\begin{array}[]{c}k\\\ t\end{array}\right]_{q_{i}}\quad
k\in\mathbb{Z},t\in\mathbb{N}^{*}.$

The integral form $\displaystyle{\mathcal{L}}_{0,1}^{A}$ of
$\displaystyle{\mathcal{L}}_{0,1}$ is defined as the $\displaystyle
U_{A}$-module $\displaystyle{\mathcal{O}}_{A}$ endowed with the product of
$\displaystyle{\mathcal{L}}_{0,1}$, and the integral form
$\displaystyle{\mathcal{L}}_{0,n}^{A}$ of $\displaystyle{\mathcal{L}}_{0,n}$
is the braided tensor product of $\displaystyle n$ copies of
$\displaystyle{\mathcal{L}}_{0,1}^{A}$. That these two products are well-
defined over $\displaystyle A$ is elementary (see Definition 4.10 and 6.7 of
[23] for the details). The integral quantum moduli algebra is

$\displaystyle{\mathcal{M}}_{0,n}^{A}=({\mathcal{L}}_{0,n}^{A})^{U_{A}}.$

Finally, given $\displaystyle q={\epsilon^{\prime}}\in\mathbb{C}^{\times}$ we
define $\displaystyle U_{\epsilon^{\prime}}$,
$\displaystyle\Gamma_{\epsilon^{\prime}}$,
$\displaystyle{\mathcal{O}}_{\epsilon^{\prime}}$,
$\displaystyle{\mathcal{L}}_{0,n}^{{\epsilon^{\prime}}}$ and
$\displaystyle{\mathcal{M}}_{0,n}^{A,{\epsilon^{\prime}}}$ as the
$\displaystyle\mathbb{C}$-algebras obtained by tensoring $\displaystyle
U_{A}$, $\displaystyle\Gamma$, $\displaystyle{\mathcal{O}}_{A}$,
$\displaystyle{\mathcal{L}}_{0,n}^{A}$ and
$\displaystyle{\mathcal{M}}_{0,n}^{A}$ respectively with
$\displaystyle\mathbb{C}_{\epsilon^{\prime}}$, the $\displaystyle A$-module
$\displaystyle\mathbb{C}$ where $\displaystyle q$ acts by multiplication by
$\displaystyle{\epsilon^{\prime}}$. They are the specializations of the latter
algebras at $\displaystyle q={\epsilon^{\prime}}$; they can also be defined as
the quotients by the ideal generated by $\displaystyle q-{\epsilon^{\prime}}$.
We find convenient to use the notations

(20) $(U_{A}^{\otimes n})^{U_{A}}_{\epsilon^{\prime}}:=(U_{A}^{\otimes
n})^{U_{A}}\otimes_{A}\mathbb{C}_{\epsilon^{\prime}}\ ,\ (U^{\otimes
n})^{lf}_{\epsilon^{\prime}}:=(U_{A}^{\otimes
n})^{lf}\otimes_{A}\mathbb{C}_{\epsilon^{\prime}}.$

Let us stress here that when $\displaystyle{\epsilon^{\prime}}$ is a root of
unity, taking the locally finite part and taking the specialization at
$\displaystyle{\epsilon^{\prime}}$ are non commuting operations. Indeed, when
$\displaystyle{\epsilon^{\prime}}$ has odd order, it follows from Theorem 2.14
below that $\displaystyle U_{\epsilon^{\prime}}$ is finite over
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon^{\prime}})$ and therefore has all
its elements locally finite for $\displaystyle ad^{r}$; on another hand
$\displaystyle U_{A}^{lf}\otimes_{A}\mathbb{C}_{\epsilon^{\prime}}$, ie.
$\displaystyle U_{\epsilon^{\prime}}^{lf}$ in the notations above, does not
contain the elements $\displaystyle L_{i}$.

In a similar manner, taking invariants and taking the specialization at
$\displaystyle{\epsilon^{\prime}}$ are non commuting operations when
$\displaystyle{\epsilon^{\prime}}$ is a root of unity: indeed, it is easily
checked that in this case $\displaystyle(U_{A}^{\otimes
n})^{U_{A}}_{\epsilon^{\prime}}$ and
$\displaystyle(U_{\epsilon^{\prime}}^{\otimes n})^{U_{\epsilon^{\prime}}}$, or
$\displaystyle{\mathcal{M}}_{0,n}^{A,{\epsilon^{\prime}}}={\mathcal{M}}_{0,n}^{A}\otimes_{A}\mathbb{C}_{\epsilon^{\prime}}$
and
$\displaystyle({\mathcal{L}}_{0,n}^{\epsilon^{\prime}})^{U_{\epsilon^{\prime}}}$,
are distinct spaces. As explained in the introduction, when
$\displaystyle{\epsilon^{\prime}}$ is a root of unity, we will not consider
the algebras $\displaystyle{\mathcal{M}}_{0,n}^{A,{\epsilon^{\prime}}}$ in
this paper.

The morphism $\displaystyle\Phi_{n}$ has also an integral form. In order to
define it, we first consider the relations between $\displaystyle U_{A}$ and
$\displaystyle U_{A}^{lf}$. Denote by $\displaystyle T\subset U_{A}$ the
multiplicative Abelian group formed by the elements $\displaystyle
K_{\lambda}$, $\displaystyle\lambda\in P$, and by $\displaystyle T_{2}\subset
T$ the subgroup formed by the $\displaystyle K_{\lambda}$,
$\displaystyle\lambda\in 2P$. Consider the subset $\displaystyle T_{2-}\subset
T_{2}$ formed by the elements $\displaystyle K_{-\lambda}$,
$\displaystyle\lambda\in 2P_{+}$. It is easily seen to be an Ore subset of
$\displaystyle U_{A}$. Clearly $\displaystyle T_{2}=T_{2-}^{-1}T_{2-}$ and
$\displaystyle{\rm Card}(T/T_{2})=2^{m}$.

######  Theorem 2.2.

(1) $\displaystyle U_{A}^{lf}=\oplus_{\lambda\in
2P_{+}}ad^{r}(U_{A})(K_{-\lambda})$.

(2) $\displaystyle U_{A}=T_{2-}^{-1}U_{A}^{lf}[T/T_{2}]$, so $\displaystyle
U_{A}$ is free of rank $\displaystyle 2^{m}$ over $\displaystyle
T_{2-}^{-1}U_{A}^{lf}$.

(3) The ring $\displaystyle U_{A}^{lf}$ is (left and right) Noetherian.

Proof. These results are immediate adaptations to $\displaystyle U_{A}^{lf}$
of those for $\displaystyle U_{q}^{lf}$, proved in Theorem 4.10 of [45],
Theorem 6.4 of [44], and Theorem 7.4.8 of [43], respectively (see also the
sections 7.1.6, 7.1.13 and 7.1.25 in [43]). For (1) and (3) we refer to
Theorem 2.113 and 2.137 in [70], which provides simpler proofs.
$\displaystyle\Box$

######  Remark 2.3.

The summands in (1) are finite-dimensional $\displaystyle U_{A}$-modules (by
eg. (22) below), so the action $\displaystyle ad^{r}$ is completely reducible
on $\displaystyle U_{A}^{lf}$. In fact, $\displaystyle U_{A}^{lf}$ is the
socle of $\displaystyle ad^{r}$ on $\displaystyle U_{A}$, and by the theorem
of separation of variables ([45, 43, 9], see also [70]), $\displaystyle
U_{A}^{lf}$ has an $\displaystyle U_{A}$-invariant subspace
$\displaystyle\mathbb{H}$ such that the multiplication in $\displaystyle
U_{A}$ affords an isomorphism of $\displaystyle U_{A}$-modules from
$\displaystyle\mathbb{H}\otimes_{\mathbb{C}(q)}\mathcal{Z}(U_{A})$ onto
$\displaystyle U_{A}^{lf}$. In particular, $\displaystyle U_{A}^{lf}$ is free
over $\displaystyle\mathcal{Z}(U_{A})$. Moreover, any simple finite
dimensional $\displaystyle U_{A}$-module has in $\displaystyle\mathbb{H}$ a
multiplicity equal to the dimension of its zero-weight subspace.

Recall the RSD map $\displaystyle\Phi_{1}\colon{\mathcal{O}}_{q}\rightarrow
U_{q}^{lf}$. By construction $\displaystyle\langle.,.\rangle$ induces a
perfect pairing
$\displaystyle\langle.,.\rangle\colon{\mathcal{O}}_{A}\otimes\mathbb{U}_{\Gamma}\rightarrow
A$. Let $\displaystyle V_{-\lambda}$ be the lowest weight
$\displaystyle\Gamma$-module of lowest weight $\displaystyle-\lambda\in-P_{+}$
(ie. the highest weight $\displaystyle\Gamma$-module $\displaystyle
V_{-w_{0}(\lambda)}$ of highest weight $\displaystyle-w_{0}(\lambda)$, where
$\displaystyle w_{0}$ is the longest element of the Weyl group; note that
$\displaystyle-w_{0}$ permutes the simple roots). Let $\displaystyle v\in
V_{-\lambda}$ be a lowest weight vector, and $\displaystyle v^{*}\in
V_{-\lambda}^{*}$ be such that $\displaystyle v^{*}(v)=1$ and $\displaystyle
v^{*}$ vanishes on a $\displaystyle\Gamma(\mathfrak{h})$-invariant complement
of $\displaystyle v$. Define
$\displaystyle\psi_{-\lambda}\in{\mathcal{O}}_{A}$ by
$\displaystyle\langle\psi_{-\lambda},x\rangle=v^{*}(xv)$, $\displaystyle
x\in\Gamma$. From the definition (12) it is quite easy to see that

(21) $\Phi_{1}(\psi_{-\lambda})=K_{-2\lambda}.$

######  Corollary 2.4.

$\displaystyle\Phi_{1}$ restricts on $\displaystyle{\mathcal{O}}_{A}$ to an
isomorphism of $\displaystyle U_{A}$-modules
$\displaystyle\Phi_{1}\colon{\mathcal{O}}_{A}\rightarrow U_{A}^{lf}$ and an
isomorphism of $\displaystyle U_{A}$-module algebras
$\displaystyle\Phi_{1}\colon{\mathcal{L}}_{0,1}^{A}\rightarrow U_{A}^{lf}$.

Proof. An elementary computational proof of this result in the $\displaystyle
sl(2)$ case is given in Section 5 of [23]. A proof of the general case can be
found in Lemma 4.11 of [23]. It uses Theorem 2.2 (1). We point out an
alternative proof in Remark 2.13 (1). $\displaystyle\Box$

######  Corollary 2.5.

Let us denote $\displaystyle d=\psi_{-\rho}\in{\mathcal{L}}_{0,1}^{A}$. We
have:

(1) The set $\displaystyle\\{d^{n}\\}_{n\in{\mathbb{N}}}$ is a left and right
multiplicative Ore set in $\displaystyle{\mathcal{L}}_{0,1}^{A}.$ We can
therefore define the localization
$\displaystyle{\mathcal{L}}_{0,1}^{A}[d^{-1}].$

(2) $\displaystyle\Phi_{1}$ extends to an isomorphism of $\displaystyle
U_{A}$-module algebras
$\displaystyle\Phi_{1}\colon{\mathcal{L}}_{0,1}^{A}[d^{-1}]\rightarrow
T_{2-}^{-1}U_{A}^{lf}$.

Proof. (1) Because $\displaystyle{\mathcal{L}}_{0,1}^{A}$ has no non-trivial
zero divisors, $\displaystyle d$ is a regular element. We have to show that
for all $\displaystyle x\in{\mathcal{L}}_{0,1}^{A}$ there exists elements
$\displaystyle y,y^{\prime}\in{\mathcal{L}}_{0,1}^{A}$ such that
$\displaystyle xd=dy$ and $\displaystyle dx=y^{\prime}d$. But
$\displaystyle\Phi_{1}(x)\Phi_{1}(d)=\Phi_{1}(x)K_{-2\rho}=K_{-2\rho}ad^{r}(K_{2\rho})(\Phi_{1}(x))$,
and $\displaystyle
ad^{r}(K_{2\rho})(\Phi_{1}(x))=\Phi_{1}(coad^{r}(K_{2\rho})(x))$. Therefore
the Ore condition is satisfied with $\displaystyle y=coad^{r}(K_{2\rho})(x)$.

(2) It follows from the fact that
$\displaystyle\textstyle\Phi_{1}(d)=K_{-2\rho}=\prod_{j=1}^{m}L_{j}^{-2}$, so
if we localize in $\displaystyle d$ we obtain $\displaystyle\textstyle
L_{j}^{2}=\prod_{k\not=j}L_{k}^{-2}\Phi_{1}(d^{-1})=\Phi_{1}(\prod_{k\not=j}\psi_{-\varpi_{k}}d^{-1}))\in\Phi_{1}({\mathcal{L}}_{0,1}^{A}[d^{-1}])$.
Therefore $\displaystyle
T_{2-}^{-1}\subset\Phi_{1}({\mathcal{L}}_{0,1}^{A}[d^{-1}])$, which implies
the assertion (2). $\displaystyle\Box$

######  Remark 2.6.

When $\displaystyle\mathfrak{g}=sl(2)$ the element $\displaystyle d$ is the
generator of $\displaystyle{\mathcal{L}}_{0,1}(sl(2))$ appearing in (52)
below. In this case we had already shown in [23] that
$\displaystyle\Phi_{1}\colon{\mathcal{L}}_{0,1}^{A}[d^{-1}]\rightarrow
U_{A}^{ad}=T_{2-}^{-1}U_{A}^{lf}$ is an isomorphism of algebras.

Denote by $\displaystyle C(\mu)$, $\displaystyle\mu\in P^{+}$, the linear
subspace of $\displaystyle{\mathcal{L}}_{0,1}$ generated by the matrix
coefficients of $\displaystyle V_{\mu}$, the $\displaystyle U_{q}$-module of
type $\displaystyle 1$ and highest weight $\displaystyle\mu$. The formula (21)
can be used to prove (see Section 7.1.22 in [43], or page 112 of [70]) that
$\displaystyle\Phi_{1}$ yields the following linear isomorphism, which
illuminates the claim (1) of Theorem 2.2:

(22) $\Phi_{1}\colon C(\mu)\rightarrow ad^{r}(U_{q})(K_{-2w_{0}(\mu)}).$

Working over the ground ring $\displaystyle A$ one has to consider for
$\displaystyle V_{\mu}$ the highest weight $\displaystyle\Gamma$-module of
highest weight $\displaystyle\mu$. In that situation $\displaystyle\Phi_{1}$
affords an isomorphism from $\displaystyle C(\mu)_{A}=End_{A}(V_{\mu})^{*}$ to
$\displaystyle ad^{r}(U_{A})(K_{-2w_{0}(\mu)})$.

By (21) we have $\displaystyle\Phi_{1}(\psi_{-\rho})=\ell^{-1}$, where as
usual $\displaystyle\ell$ is the pivotal element of $\displaystyle U_{A}$.
Because the latter has the elementary factorization
$\displaystyle\textstyle\ell=\prod_{j=1}^{m}L_{j}^{2}$, this naturally raises
the question of the factorization of $\displaystyle\psi_{-\rho}$. This
question is considered in [46], where
$\displaystyle{\mathcal{L}}_{0,1}({\mathfrak{g}})$ for
$\displaystyle{\mathfrak{g}}=gl(r+1)$ is analysed and quantum minors are
extensively studied. Let us review here some of their results in relation with
$\displaystyle\psi_{-\rho}$.

First note that for for $\displaystyle{\mathfrak{g}}=sl(r+1)$ the irreducible
representation $\displaystyle V_{-\rho}$ of lowest weight $\displaystyle-\rho$
is isomorphic to the representation of highest weight $\displaystyle V_{\rho}$
because $\displaystyle-w_{0}(\rho)=\rho$. By the Weyl formula the dimension of
this representation is
$\displaystyle\textstyle\prod_{\alpha>0}\frac{(2\rho,\alpha)}{(\rho,\alpha)}=2^{N}$.
In [50] a presentation of $\displaystyle U_{q}(gl(r+1))$ is given, which
differs from our presentation of $\displaystyle U_{q}(sl(r+1))$ only by its
subalgebra $\displaystyle U_{q}({\mathfrak{h}})$, generated by $\displaystyle
r+1$ elements $\displaystyle{\mathbb{K}}_{1},...,{\mathbb{K}}_{r+1}$. The
inclusion $\displaystyle U_{q}(sl(r+1))\subset U_{q}(gl(r+1))$ is such that
$\displaystyle K_{i}={\mathbb{K}}_{i}^{2}{\mathbb{K}}_{i+1}^{-2},i=1,...,r$.
The quantum minors, properly defined in [46], of the matrix of matrix elements
of the natural representation of $\displaystyle U_{q}(gl(r+1))$ are denoted
$\displaystyle det_{q}(A_{\geq k})$ for $\displaystyle k=1,...,r+1$. We have
$\displaystyle det_{q}(A_{\geq 1})=1$ in the case of $\displaystyle sl(r+1).$
Then [46] proves that $\displaystyle det_{q}(A_{\geq
k})=({\mathbb{K}}_{k}...{\mathbb{K}}_{r+1})^{2}$, and there exists an element
$\displaystyle\mathbb{K}\in U_{q}(gl(r+1))$ such that

$\displaystyle{\mathbb{K}}^{-2\rho}=det_{q}(A_{\geq 1})^{-r}det_{q}(A_{\geq
2})...det_{q}(A_{\geq r+1}).$

This has to be interpreted in the $\displaystyle sl(r+1)$ case as
$\displaystyle K_{-2\rho}=\Phi_{1}(det_{q}(A_{\geq 2})...det_{q}(A_{\geq
r+1}))$. As a result this gives the equality

$\displaystyle\psi_{-\rho}=det_{q}(A_{\geq 2})...det_{q}(A_{\geq r+1}).$

Corollary 2.4 can be extended as follows:

######  Theorem 2.7.

$\displaystyle\Phi_{n}$ restricts to an isomorphism of $\displaystyle
U_{A}$-module algebras
$\displaystyle\Phi_{n}\colon{\mathcal{L}}_{0,n}^{A}\rightarrow(U_{A}^{\otimes
n})^{lf}$, and it restricts to an isomorphism of algebras
$\displaystyle\Phi_{n}\colon{\mathcal{M}}_{0,n}^{A}\rightarrow(U_{A}^{\otimes
n})^{U_{A}}$.

The proof relies on (18) and the expression of $\displaystyle\Phi_{n}$ in
terms of $\displaystyle\Phi_{1}$ and $\displaystyle R$-matrices (see [23],
Proposition 6.5 and Lemma 6.8).

In the case of $\displaystyle\mathfrak{g}=sl(2)$ we proved in [25] the
existence of elements $\displaystyle\xi^{(i)}\in{\mathcal{L}}_{0,n}^{A}$
($\displaystyle i=1,...,n$), and we defined an algebra
$\displaystyle{}_{loc}{\mathcal{L}}_{0,n}^{A}$ generalizing
$\displaystyle{\mathcal{L}}_{0,1}^{A}[d^{-1}]$ above, containing
$\displaystyle{\mathcal{L}}_{0,n}^{A}$ as a subalgebra and the inverses of the
elements $\displaystyle\xi^{(i)}$. We showed that $\displaystyle\Phi_{n}$
extends to $\displaystyle{}_{loc}{\mathcal{L}}_{0,n}^{A}$, and that
$\displaystyle\Phi_{n}({}_{loc}{\mathcal{L}}_{0,n}^{A})=U_{A}^{ad}(sl(2))^{\otimes
n}$. The key property of $\displaystyle\xi^{(i)}$ is

(23) $\Phi_{n}(\xi^{(i)})=(K^{-1})^{(i)}\cdots(K^{-1})^{(n)}.$

For general $\displaystyle\mathfrak{g}$ we now describe a partial
generalization of this result. Define elements
$\displaystyle\xi^{(i)}_{j}\in{\mathcal{L}}_{0,n}^{A}$, for $\displaystyle
i=1,...,n$ and $\displaystyle j=1,...,m$, by

(24) $\xi_{j}^{(i)}=v^{*}(M_{j}^{(i)}\cdots M_{j}^{(n)})(v)$

where $\displaystyle M_{j}^{(i)}\in
End(V_{-\varpi_{j}})\otimes{\mathcal{L}}_{0,n}^{A}$ is the matrix of matrix
coefficients $\displaystyle
1^{\otimes(i-1)}\otimes{}_{V_{-\varpi_{j}}}\phi_{e_{k}}^{e_{l}}\otimes
1^{\otimes(n-i)}$, where $\displaystyle\\{e_{k}\\}$ is the canonical basis of
weight vectors of $\displaystyle V_{-\varpi_{j}}$, $\displaystyle v$ is a
lowest non-zero weight vector of $\displaystyle V_{-\varpi_{j}}$, and
$\displaystyle v^{*}$ the associated linear form, vanishing on a
$\displaystyle\Gamma(\mathfrak{h})$-invariant complement of $\displaystyle v$.
Similarly to (23) the elements $\displaystyle\xi_{j}^{(i)}$ satisfy

(25) $\Phi_{n}(\xi_{j}^{(i)})=(L_{j}^{-2})^{(i)}\cdots(L_{j}^{-2})^{(n)}.$

These elements commute and the multiplicative sets
$\displaystyle\\{\xi_{j}^{(1)}{}^{k}\\}_{k\in{\mathbb{N}}}$ are Ore set of
$\displaystyle{\mathcal{L}}_{0,n}^{A}$, but for $\displaystyle i\geq 2$ the
multiplicative sets
$\displaystyle\\{\xi_{j}^{(i)}{}^{k}\\}_{k\in{\mathbb{N}}}$ are not Ore sets
of $\displaystyle{\mathcal{L}}_{0,n}^{A}$. In fact, as in the proof of Theorem
2.5 we see that $\displaystyle\\{\xi_{j}^{(i)}{}^{k}\\}_{k\in{\mathbb{N}}}$ is
only an Ore set of the subalgebra $\displaystyle{\mathcal{L}}_{0,n}^{(i\leq)}$
of $\displaystyle{\mathcal{L}}_{0,n}^{A}$ generated by the subalgebras
$\displaystyle{\mathcal{L}}_{0,n}^{(a)}$, $\displaystyle a\geq i$. We
therefore cannot localize $\displaystyle{\mathcal{L}}_{0,n}^{A}$ with respect
to the elements $\displaystyle\xi_{j}^{(i)}$ as easily as in the case where
$\displaystyle n=1$.

In order to proceed let us explain the case $\displaystyle n=2$. Since the
elements $\displaystyle\xi_{j}^{(1)}$, $\displaystyle j\in\\{1,\ldots,m\\}$,
are commuting regular Ore elements of $\displaystyle{\mathcal{L}}_{0,2}^{A}$
we can define the localisation of $\displaystyle{\mathcal{L}}_{0,2}^{A}$ with
respect to the multiplicative sets
$\displaystyle\\{\xi_{1}^{(1)}{}^{k}\\}_{k\in{\mathbb{N}}},\ldots,\\{\xi_{m}^{(1)}{}^{k}\\}_{k\in{\mathbb{N}}}$.
Denote it
$\displaystyle{\mathcal{L}}_{0,2}^{A}[\\{\xi_{j_{1}}^{(1)}{}^{-1}\\}]$. Let us
add new elements $\displaystyle\nu_{j_{1}}^{(1)}$ such that
$\displaystyle(\nu_{j_{1}}^{(1)})^{2}=\xi_{j_{1}}^{(1)}$ and
$\displaystyle\Phi_{2}(\nu_{j_{1}}^{(1)})=(L_{j_{1}}^{-1})^{(1)}(L_{j_{1}}^{-1})^{(2)}$.
They are Ore elements, and we can define similarly the localisation
$\displaystyle{\mathcal{L}}_{0,2}^{A}[\\{\nu_{j_{1}}^{(1)}{}^{-1}\\}]$ (see
the following remark for an explanation of this additional construction). We
want to define the inverses of the elements $\displaystyle\xi_{j}^{(2)}$,
$\displaystyle j\in\\{1,\ldots,m\\}$, and a new algebra
$\displaystyle{\mathcal{L}}_{0,2}^{A}[\\{\xi_{j_{1}}^{(1)}{}^{-1}\\}][\\{\xi_{j_{2}}^{(2)}{}^{-1}\\}]$
such that
$\displaystyle{\mathcal{L}}_{0,2}^{A}[\\{\xi_{j_{1}}^{(1)}{}^{-1}\\}]\subset{\mathcal{L}}_{0,2}^{A}[\\{\xi_{j_{1}}^{(1)}{}^{-1}\\}][\\{\xi_{j_{2}}^{(2)}{}^{-1}\\}]$
and $\displaystyle\Phi_{2}$ extends naturally to an algebra homomorphism
$\displaystyle\Phi_{2}:{\mathcal{L}}_{0,2}^{A}[\\{\xi_{j_{1}}^{(1)}{}^{-1}\\}][\\{\xi_{j_{2}}^{(2)}{}^{-1}\\}]\rightarrow
U_{A}^{\otimes 2}$ such that
$\displaystyle\Phi_{n}(\xi_{j_{2}}^{(2)})=(L_{j_{2}}^{-2})^{(2)}$ for all
$\displaystyle j_{2}\in\\{1,\ldots,m\\}$. As in the $\displaystyle sl(2)$ case
described in [23], this can be done by writing explicitly, for every
$\displaystyle j_{2}\in\\{1,\ldots,m\\}$, the exchange relations between the
matrices $\displaystyle M_{j_{1}}^{(1)}$ and $\displaystyle M_{j_{2}}^{(2)}$
involving $\displaystyle\xi_{j_{2}}^{(2)}$, for every $\displaystyle
j_{1}\in\\{1,\ldots,m\\}$ (these matrices are defined in (24)). Such exchange
relations have the form (39) in the graded algebra $\displaystyle
Gr_{\mathcal{F}_{2}}({\mathcal{L}}_{0,n}^{A})$ defined in (41) below.
Similarly, by replacing the elements $\displaystyle\xi_{j_{1}}^{(1)}$,
$\displaystyle\xi_{j_{2}}^{(2)}$ with square roots
$\displaystyle\nu_{j_{1}}^{1)}$, $\displaystyle\nu_{j_{2}}^{(2)}$ we get a
localization
$\displaystyle{\mathcal{L}}_{0,2}^{A}[\\{\nu_{j_{1}}^{(1)}{}^{-1}\\}][\\{\nu_{j_{2}}^{(2)}{}^{-1}\\}]$
such that
$\displaystyle{\mathcal{L}}_{0,2}^{A}[\\{\nu_{j_{1}}^{(1)}{}^{-1}\\}]\subset{\mathcal{L}}_{0,2}^{A}[\\{\nu_{j_{1}}^{(1)}{}^{-1}\\}][\\{\nu_{j_{2}}^{(2)}{}^{-1}]\\}$
and $\displaystyle\Phi_{2}$ extends to an algebra homomorphism
$\displaystyle\Phi_{2}:{\mathcal{L}}_{0,2}^{A}[\\{\nu_{j_{1}}^{(1)}{}^{-1}\\}][\\{\nu_{j_{2}}^{(2)}{}^{-1}\\}]\rightarrow
U_{A}^{\otimes 2}$ such that
$\displaystyle\Phi_{n}(\nu_{j_{2}}^{(2)})=(L_{j_{2}}^{-1})^{(2)}$ for all
$\displaystyle j_{2}\in\\{1,\ldots,m\\}$. This morphism of algebras will be
shown to be an isomorphism.

For any $\displaystyle n\geq 2$ we can proceed in the same way:

######  Definition 2.8.

By iterating the above construction we define:

$\displaystyle{}_{loc}{\mathcal{L}}_{0,n}^{A}={\mathcal{L}}_{0,n}^{A}[\\{\xi_{j_{n}}^{(n)}{}^{-1}\\}][\\{\xi_{j_{n-1}}^{(n-1)}{}^{-1}\\}]\cdots[\\{\xi_{j_{1}}^{(1)}{}^{-1}\\}],$
$\displaystyle{}_{loc^{\prime}}{\mathcal{L}}_{0,n}^{A}={\mathcal{L}}_{0,n}^{A}[\\{\nu_{j_{n}}^{(n)}{}^{-1}\\}][\\{\nu_{j_{n-1}}^{(n-1)}{}^{-1}\\}]\cdots[\\{\nu_{j_{1}}^{(1)}{}^{-1}\\}].$

In the sequel it will be convenient to define invertible elements
$\displaystyle\sqrt{\delta_{j}}^{(i)}\in{}_{loc^{\prime}}{\mathcal{L}}_{0,n}^{A}$,
for $\displaystyle i=1,...,n$ and $\displaystyle j=1,...,m$, satisfying
$\displaystyle\nu_{j}^{(i)}=\sqrt{\delta}_{j}^{(i)}\cdots\sqrt{\delta}_{j}^{(n)},$
i.e $\displaystyle\sqrt{\delta}_{j}^{(i)}=\nu_{j}^{(i)}/\nu_{j}^{(i+1)}$.

The elements $\displaystyle\sqrt{\delta}_{j}^{(i)}$ are invertible, commute
and satisfy

$\displaystyle\Phi_{n}(\sqrt{\delta}_{j}^{(i)})=(L_{j}^{-1})^{(i)}.$

######  Theorem 2.9.

$\displaystyle\Phi_{n}$ restricts to an isomorphism of $\displaystyle
U_{A}$-module algebras
$\displaystyle\Phi_{n}:{}_{loc^{\prime}}{\mathcal{L}}_{0,n}^{A}\rightarrow
U_{A}^{\otimes n}.$

Proof. We know from Corollary 2.4 that
$\displaystyle\Phi_{1}:{\mathcal{L}}_{0,n}^{A}\rightarrow U_{A}^{lf}$ is an
isomorphism of algebra. Using $\displaystyle
U_{A}=T_{2-}^{-1}U_{A}^{lf}[T/T_{2}]$ and the fact that the image by
$\displaystyle\Phi_{1}$ of the elements
$\displaystyle(\sqrt{\delta}_{j}^{(1)})^{\pm 1}$ generates the group
$\displaystyle T$ we get the result for $\displaystyle n=1$. The result for
$\displaystyle\Phi_{n}$ is obtained by induction. We have

$\displaystyle\displaystyle(id\otimes\Phi_{n})(\stackrel{{\scriptstyle
V}}{{M}}{}^{\\!\\!(n)})$ $\displaystyle\displaystyle=R_{0n}R_{0n}^{\prime}$
$\displaystyle\displaystyle(id\otimes\Phi_{n})(\stackrel{{\scriptstyle
V}}{{M}}{}^{\\!\\!(a)})$ $\displaystyle\displaystyle=\left(R_{0n}\ldots
R_{0a+1}\right)R_{0a}R_{0a}^{\prime}\left(R_{0n}\ldots
R_{0a+1}\right)^{-1},1\leq a<n.$

Because the matrix elements of
$\displaystyle(id\otimes\Phi_{n})(\stackrel{{\scriptstyle
V}}{{M}}{}^{\\!\\!(n)})$ generate $\displaystyle 1^{\otimes n-1}\otimes
U_{A}^{lf}$ when $\displaystyle V$ varies, the image of
$\displaystyle({\mathcal{L}}_{0,n}^{A})^{(n)}[\\{\nu_{j_{n}}^{(n)}{}^{-1}\\}]$
by $\displaystyle\Phi_{n}$ is $\displaystyle 1^{\otimes(n-1)}\otimes U_{A}$.
Since the matrix elements of $\displaystyle R_{0n}$ and $\displaystyle
R_{0n}^{-1}$ are in $\displaystyle 1^{\otimes(n-1)}\otimes U_{A},$ they belong
to
$\displaystyle\Phi_{n}({}{\mathcal{L}}_{0,n}^{A}[\\{\nu_{j_{n}}^{(n)}{}^{-1}\\}])$
by the preceding remark. It follows that
$\displaystyle\Phi_{n}({}_{loc^{\prime}}{\mathcal{L}}_{0,n}^{A})$ contains the
matrix elements of $\displaystyle
R_{0n}^{-1}(id\otimes\Phi_{n})(\stackrel{{\scriptstyle
V}}{{M}}{}^{\\!\\!(n-1)})R_{0n}$, whence the matrix elements of $\displaystyle
R_{0n-1}R_{0n-1}^{\prime}$, and therefore the space $\displaystyle
1^{\otimes(n-2)}\otimes U_{A}^{lf}\otimes 1.$ It contains also the elements
$\displaystyle\Phi_{n}(\sqrt{\delta}_{j}^{(n-1)})=(L_{j}^{-1})^{(n-1)}$, so
$\displaystyle\Phi_{n}({}_{loc^{\prime}}{\mathcal{L}}_{0,n}^{A})$ contains
$\displaystyle 1^{\otimes(n-2)}\otimes U_{A}\otimes 1.$ By a trivial induction
we finally obtain that
$\displaystyle\Phi_{n}({}_{loc^{\prime}}{\mathcal{L}}_{0,n}^{A})=U_{A}^{\otimes
n}.$ $\displaystyle\Box$

######  Remark 2.10.

It is a natural problem to determine the image by $\displaystyle\Phi_{n}$ of
$\displaystyle{}_{loc}{\mathcal{L}}_{0,n}^{A}$, and it is natural to expect
that it would be $\displaystyle(T_{2-}^{-1}U_{A}^{lf})^{\otimes n}$, because
this is true for $\displaystyle n=1$, as well as for any $\displaystyle n$ in
the $\displaystyle sl(2)$ case, as shown in [23]. Unfortunately this is not
so. This comes from the fact, eg. for $\displaystyle n=2$, that the matrix
elements of $\displaystyle R_{02}R_{01}R_{01}^{\prime}R_{02}^{-1}$ do not
belong to $\displaystyle(T_{2-}^{-1}U_{A}^{lf})^{\otimes 2}$ as can be shown
by an explicit computation in the $\displaystyle sl(3)$ case. This explains
the reason why we had to introduce the square roots
$\displaystyle\nu_{j}^{(i)}$ in the previous theorem.

Arguments similar to those mentioned at the end of Section 2.1 imply that the
algebras $\displaystyle{\mathcal{L}}_{0,n}^{A}$,
$\displaystyle{\mathcal{M}}_{0,n}^{A}$ and
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon^{\prime}}$,
$\displaystyle{\mathcal{M}}_{0,n}^{A,{\epsilon^{\prime}}}$,
$\displaystyle{\epsilon^{\prime}}\in\mathbb{C}^{\times}$, have no non-trivial
zero divisors (see [23], Proposition 7.1). By Theorem 2.7 the Alekseev map
yields isomorphisms of $\displaystyle U_{\epsilon^{\prime}}$-module algebras,
and of algebras for the latter,

(26)
$\Phi_{n}\colon{\mathcal{L}}_{0,n}^{\epsilon^{\prime}}\rightarrow(U^{\otimes
n})^{lf}_{\epsilon^{\prime}}\ ,\
\Phi_{n}\colon{}_{loc^{\prime}}{\mathcal{L}}_{0,n}^{\epsilon^{\prime}}\rightarrow
U^{\otimes
n}_{\epsilon^{\prime}}\;,\;\Phi_{n}\colon{\mathcal{M}}_{0,n}^{A,{\epsilon^{\prime}}}\rightarrow(U_{A}^{\otimes
n})^{U_{A}}_{\epsilon^{\prime}}\subset(U_{\epsilon^{\prime}}^{\otimes
n})^{U_{\epsilon^{\prime}}}$

where we use the notations (20).

###  2.3. Perfect pairings

We will need restrictions on the integral forms
$\displaystyle{\mathcal{O}}_{A}(B_{+})$,
$\displaystyle{\mathcal{O}}_{A}(B_{-})$ of the morphisms
$\displaystyle\Phi^{+}$, $\displaystyle\Phi^{-}$ in (10). We collect their
properties in Theorem 2.11 and the discussion thereafter. In order to state
it, we recall first a few facts about $\displaystyle R$-matrices and related
pairings.

In [52, 53] Lusztig proved that the category of $\displaystyle
U_{A}^{res}$-modules
$\displaystyle\mathcal{C}_{A}^{res}\otimes_{A}\mathbb{C}[q^{\pm 1/D}]$ (ie.
with coefficients extended to $\displaystyle\mathbb{C}[q^{\pm 1/D}]$) is
braided and ribbon, with braiding given by the collection of endomorphisms

$\displaystyle R_{A}=((R_{h})_{V,W})_{V,W\in Ob(\mathcal{C}_{A}^{res})}.$

Actually, $\displaystyle(R_{h})_{V,W}$ is represented by a matrix with
coefficients in $\displaystyle q^{\pm 1/D}\mathbb{Z}[q^{\pm 1}]$ on the basis
of $\displaystyle V\otimes W$ formed by the tensor products of the canonical
(Kashiwara-Lusztig) basis vectors of $\displaystyle V$ and $\displaystyle W$.
The restriction functor
$\displaystyle\mathcal{C}_{A}\rightarrow\mathcal{C}_{A}^{res}$ is an
equivalence of categories, so
$\displaystyle\mathcal{C}_{A}\otimes_{A}\mathbb{C}[q^{\pm 1/D}]$ has the same
braided and ribbon structure. This can be rephrased as follows in Hopf algebra
terms. Denote by $\displaystyle\mathbb{U}_{\Gamma}$ the categorical completion
of $\displaystyle\Gamma$, ie. the Hopf algebra of natural transformations
$\displaystyle F_{\mathcal{C}_{A}}\rightarrow F_{\mathcal{C}_{A}}$. Then
$\displaystyle\mathbb{U}_{\Gamma}\otimes_{A}\mathbb{C}[q^{\pm 1/D}]$ is quasi-
triangular and ribbon with R-matrix

$\displaystyle
R_{A}\in\mathbb{U}_{\Gamma}^{\hat{\otimes}2}\otimes_{A}\mathbb{C}[q^{\pm
1/D}].$

As in (9), we can write

$\displaystyle R^{\pm}_{A}=\sum_{(R)}R^{\pm}_{(1)}\otimes R^{\pm}_{(2)}.$

There are pairings of Hopf algebras naturally related to the R-matrix
$\displaystyle R\in\mathbb{U}_{q}^{\hat{\otimes}2}$. What follows is standard
(see eg. [47, 48, 51]), for details we refer to the results 2.73, 2.75, 2.92,
2.106 and 2.107 in [70]:

  * •

There is a unique pairing of Hopf algebras $\displaystyle\rho\colon
U_{q}(\mathfrak{b}_{-})^{cop}\otimes
U_{q}(\mathfrak{b}_{+})\rightarrow\mathbb{C}(q^{1/D})$ such that, for every
$\displaystyle\alpha,\lambda\in P$ and $\displaystyle l,k\in
U_{q}(\mathfrak{h})$,

$\rho(K_{\lambda},K_{\alpha})=q^{(\lambda,\alpha)}\ ,\
\rho(F_{i},E_{j})=\delta_{i,j}(q_{i}-q_{i}^{-1})^{-1}\
,\rho(l,E_{j})=\rho(F_{i},k)=0.$

  * •

The Drinfeld pairing $\displaystyle\tau\colon
U_{q}(\mathfrak{b}_{+})^{cop}\otimes
U_{q}(\mathfrak{b}_{-})\rightarrow\mathbb{C}(q^{1/D})$ is the bilinear map
defined by $\displaystyle\tau(X,Y)=\rho(S(Y),X)$; it satisfies

$\tau(K_{\lambda},K_{\alpha})=q^{-(\lambda,\alpha)}\ ,\
\tau(E_{j},F_{i})=-\delta_{i,j}(q_{i}-q_{i}^{-1})^{-1}\ ,\
\tau(l,F_{i})=\tau(E_{j},k)=0.$

  * •

$\displaystyle\rho$ and $\displaystyle\tau$ are perfect pairings; this means
that they yield isomorphisms of Hopf algebras $\displaystyle i_{\pm}\colon
U_{q}(\mathfrak{b}_{\pm})\rightarrow{\mathcal{O}}_{q}(B_{\mp})_{op}$ (with
coefficients a priori extended to $\displaystyle\mathbb{C}(q^{1/D})$, but see
below) defined by, for every $\displaystyle X\in U_{q}(\mathfrak{b}_{+})$,
$\displaystyle Y\in U_{q}(\mathfrak{b}_{-})$,

$\displaystyle\langle i_{+}(X),Y\rangle=\tau(S(X),Y)\ ,\ \langle
i_{-}(Y),X\rangle=\tau(X,Y).$

Since $\displaystyle{\mathcal{O}}_{q}(B_{\mp})_{op}$ is equipped with the
inverse of the antipode $\displaystyle S_{{\mathcal{O}}_{q}}$ of
$\displaystyle{\mathcal{O}}_{q}(B_{\mp})$, it follows that $\displaystyle
i_{\pm}\circ S=S_{{\mathcal{O}}_{q}}^{-1}\circ i_{\pm}$.

  * •

Denote by $\displaystyle
p_{\pm}\colon{\mathcal{O}}_{q}(G)\rightarrow{\mathcal{O}}_{q}(B_{\pm})$ the
canonical projection map, ie. the Hopf algebra homomorphism dual to the
inclusion map $\displaystyle U_{q}(\mathfrak{b}_{\pm})\hookrightarrow
U_{q}(\mathfrak{g})$. For every
$\displaystyle\alpha,\beta\in{\mathcal{O}}_{q}(G)$ we have

(27)
$\langle\alpha\otimes\beta,R\rangle=\tau(i_{+}^{-1}(p_{-}(\beta)),i_{-}^{-1}(p_{+}(\alpha)).$

Note that it is the use of weights $\displaystyle\alpha,\lambda\in P$ that
forces the pairings $\displaystyle\rho$, $\displaystyle\tau$ to be defined
over $\displaystyle\mathbb{C}(q^{1/D})$, instead of
$\displaystyle\mathbb{C}(q)$. Then, let us consider the restrictions
$\displaystyle\pi_{q}^{+}$ of $\displaystyle\rho$, and
$\displaystyle\pi_{q}^{-}$ of $\displaystyle\tau$, obtained by taking
$\displaystyle\alpha\in Q$ and $\displaystyle l\in U_{q}(\mathfrak{h})$,
$\displaystyle k\in U_{q}^{ad}(\mathfrak{h})$. They take values in
$\displaystyle\mathbb{C}(q)$, and define pairings

$\displaystyle\pi^{+}_{q}\colon U_{q}(\mathfrak{b}_{-})^{cop}\otimes
U_{q}^{ad}(\mathfrak{b}_{+})\rightarrow\mathbb{C}(q)\ ,\ \pi^{-}_{q}\colon
U_{q}(\mathfrak{b}_{+})^{cop}\otimes
U_{q}^{ad}(\mathfrak{b}_{-})\rightarrow\mathbb{C}(q).$

By the same arguments as for $\displaystyle\rho$ and $\displaystyle\tau$ (eg.
in [70], Proposition 2.92), it follows that $\displaystyle\pi^{\pm}_{q}$ are
perfect pairings. Note also that
$\displaystyle\pi^{-}_{q}=\kappa\circ\pi^{+}_{q}\circ(\kappa\otimes\kappa)$,
where $\displaystyle\kappa$ is the conjugate-linear automorphism of
$\displaystyle U_{q}$, viewed as a Hopf algebra over
$\displaystyle\mathbb{C}(q)$ with conjugation given by
$\displaystyle\kappa(q)=q^{-1}$, defined by

(28) $\kappa(E_{i})=F_{i}\ ,\ \kappa(F_{i})=E_{i}\ ,\
\kappa(K_{\lambda})=K_{-\lambda}\ ,\ \kappa(q)=q^{-1}.$

In [36], De Concini-Lyubashenko described integral forms of
$\displaystyle\pi_{q}^{\pm}$ as follows. Denote by $\displaystyle
m^{*}\colon{\mathcal{O}}_{A}\rightarrow{\mathcal{O}}_{A}(B_{+})\otimes{\mathcal{O}}_{A}(B_{-})$
the map dual to the multiplication map
$\displaystyle\Gamma(\mathfrak{b}_{+})\otimes\Gamma(\mathfrak{b}_{-})\rightarrow\Gamma$,
so $\displaystyle m^{*}=(p_{+}\otimes p_{-})\circ\Delta_{{\mathcal{O}}_{A}}$.
Let $\displaystyle U_{A}(H)$ be the sub-Hopf algebra of $\displaystyle
U_{A}(\mathfrak{b}_{-})^{cop}\otimes U_{A}(\mathfrak{b}_{+})^{cop}$ generated
by the elements ($\displaystyle i\in\\{1,\ldots,m\\}$)

$\displaystyle 1\otimes K_{i}^{-1}\bar{E}_{i}\ ,\ \bar{F}_{i}K_{i}\otimes 1\
,\ L_{i}^{\pm 1}\otimes L_{i}^{\mp 1}.$

Note that $\displaystyle U_{A}(H)$ is free over $\displaystyle A$, and that a
basis is given by the elements

$\displaystyle\bar{F}_{\beta_{1}}^{n_{1}}\ldots\bar{F}_{\beta_{N}}^{n_{N}}K_{n_{1}\beta_{1}+\ldots+n_{N}\beta_{N}}K_{\lambda}\otimes
K_{-\lambda}K_{-p_{1}\beta_{1}\ldots-
p_{N}\beta_{N}}\bar{E}_{\beta_{1}}^{p_{1}}\ldots\bar{E}_{\beta_{N}}^{p_{N}}$

where $\displaystyle\lambda\in P$ and $\displaystyle
n_{1},...,n_{N},p_{1},...,p_{N}\in{\mathbb{N}}$.

Recall the lowest weight $\displaystyle\Gamma$-module $\displaystyle
V_{-\lambda}$, $\displaystyle\lambda\in P_{+}$, the lowest weight vector
$\displaystyle v\in V_{-\lambda}$, the dual vector $\displaystyle v^{*}\in
V_{-\lambda}^{*}$, and $\displaystyle\psi_{-\lambda}\in{\mathcal{O}}_{A}$ (see
before Corollary 2.4). For every positive root $\displaystyle\alpha$ define
elements
$\displaystyle\psi_{-\lambda}^{\alpha},\psi_{-\lambda}^{-\alpha}\in{\mathcal{O}}_{A}$
by the formulas (where $\displaystyle x\in\Gamma$, and we note that the root
vectors $\displaystyle E_{\alpha}$, $\displaystyle F_{\alpha}\in\Gamma$):

$\displaystyle\langle\psi^{\alpha}_{-\lambda},x\rangle=v^{*}(xE_{\alpha}v)\ ,\
\langle\psi_{-\lambda}^{-\alpha},x\rangle=v^{*}(F_{\alpha}xv).$

Consider the maps $\displaystyle
j_{q}^{\pm}\colon{\mathcal{O}}_{q}(B_{\pm})\rightarrow
U_{q}(\mathfrak{b}_{\mp})^{cop}$ defined by

$\displaystyle\langle\alpha_{+},X\rangle=\pi_{q}^{+}(j_{q}^{+}(\alpha_{+}),X)\
,\ \langle\alpha_{-},Y\rangle=\pi_{q}^{-}(j_{q}^{-}(\alpha_{-}),Y)$

where $\displaystyle\alpha_{\pm}\in{\mathcal{O}}_{q}(B_{\pm})$, $\displaystyle
X\in U_{q}^{ad}(\mathfrak{b}_{+})$, $\displaystyle Y\in
U_{q}^{ad}(\mathfrak{b}_{-})$.

The following theorem summarizes results proved in the sections 3 and 4 of
[36]. For the sake of clarity, let us spell out the correspondence between
statements. First, $\displaystyle\pi^{+}_{q}$, $\displaystyle\pi^{-}_{q}$,
$\displaystyle U_{q}(\mathfrak{b}_{\mp})^{cop}$, $\displaystyle
U_{A}(\mathfrak{b}_{\mp})^{cop}$, $\displaystyle{\mathcal{O}}_{A}(B_{\pm})$,
$\displaystyle U_{A}(H)$ and $\displaystyle J$ are denoted in [36]
respectively by $\displaystyle\pi^{\prime\prime}$,
$\displaystyle\bar{\pi}^{\prime\prime}$, $\displaystyle
U_{q}(\mathfrak{b}_{\mp})_{op}$, $\displaystyle
R_{q}[B_{\pm}]^{\prime\prime}$, $\displaystyle R_{q}[B_{\pm}]$, $\displaystyle
A^{\prime\prime}$ and $\displaystyle\mu^{\prime\prime}$. Also, the definition
of $\displaystyle j_{A}^{\pm}$ is implicit in the section 4.2 of [36], and the
formulas in Theorem 2.11 (3) are related to those in Lemma 4.5 of [36] by
observing that their generators $\displaystyle\tilde{E}_{i}$ and
$\displaystyle\tilde{F}_{i}$ are respectively $\displaystyle K_{i}^{-1}E_{i}$
and $\displaystyle F_{i}K_{i}$ in our notations; this also explains the
appearance of $\displaystyle q_{i},q_{i}^{-1}$ in the formulas in (3).
Finally, $\displaystyle\kappa$ in (28) maps $\displaystyle\bar{E}_{i}$,
$\displaystyle\bar{F}_{i}$ to $\displaystyle-\bar{F}_{i}$,
$\displaystyle-\bar{E}_{i}$, whence the sign for the expression of
$\displaystyle J(\psi^{\alpha_{i}}_{-\varpi_{j}})$.

######  Theorem 2.11.

(1) $\displaystyle\pi^{\pm}_{q}$ restricts to a perfect Hopf pairing between
the unrestricted and restricted integral forms,
$\displaystyle\pi^{\pm}_{A}\colon
U_{A}(\mathfrak{b}_{\mp})^{cop}\otimes\Gamma(\mathfrak{b}_{\pm})\rightarrow
A$.

(2) $\displaystyle j_{q}^{\pm}$ yields an isomorphism of Hopf algebras
$\displaystyle j_{A}^{\pm}\colon{\mathcal{O}}_{A}(B_{\pm})\rightarrow
U_{A}(\mathfrak{b}_{\mp})^{cop}$, satisfying
$\displaystyle\langle\alpha_{\pm},x_{\pm}\rangle=\pi^{\pm}_{A}(j_{A}^{\pm}(\alpha_{\pm}),x_{\pm})$
for every $\displaystyle\alpha_{\pm}\in{\mathcal{O}}_{A}(B_{\pm})$,
$\displaystyle x_{\pm}\in\Gamma(\mathfrak{b}_{\pm})$.

(3) The map $\displaystyle J=(j_{A}^{+}\otimes j_{A}^{-})\circ
m^{*}\colon{\mathcal{O}}_{A}\rightarrow U_{A}(H)\subset
U_{A}(\mathfrak{b}_{-})^{cop}\otimes U_{A}(\mathfrak{b}_{+})^{cop}$ is an
embedding of Hopf algebras, and it extends to an isomorphism $\displaystyle
J\colon{\mathcal{O}}_{A}[\psi_{-\rho}^{-1}]\rightarrow U_{A}(H)$. In
particular it satisfies (where $\displaystyle\lambda\in P_{+}$):

$\displaystyle J(\psi_{-\lambda})=K_{-\lambda}\otimes K_{\lambda}\ ,\
J(\psi^{\alpha_{i}}_{-\varpi_{j}})=-\delta_{i,j}q_{i}L_{i}^{-1}\otimes
L_{i}K_{i}^{-1}\bar{E}_{i}\ ,\
J(\psi^{-\alpha_{i}}_{-\varpi_{j}})=\delta_{i,j}q_{i}^{-1}\bar{F}_{i}K_{i}L_{i}^{-1}\otimes
L_{i}.$

For our purposes it is necessary to reformulate this result. Consider the
morphisms of Hopf algebras
$\displaystyle\Phi^{\pm}\colon{\mathcal{O}}_{A}(B_{\pm})\rightarrow
U_{A}(\mathfrak{b}_{\mp})^{cop}$, $\displaystyle\alpha\mapsto(\alpha\otimes
id)(R^{\pm}_{A})$.

######  Lemma 2.12.

We have $\displaystyle\Phi^{\pm}=j_{A}^{\pm}$.

Thus, the theorem above tells us that $\displaystyle\Phi^{\pm}$ is an
isomorphism of Hopf algebras, such that
$\displaystyle\langle\alpha_{\pm},x_{\pm}\rangle=\pi^{\pm}_{A}(\Phi^{\pm}(\alpha_{\pm}),x_{\pm})$
for every $\displaystyle\alpha_{\pm}\in{\mathcal{O}}_{A}(B_{\pm})$,
$\displaystyle x_{\pm}\in\Gamma(\mathfrak{b}_{\pm})$. Moreover, changing the
notation $\displaystyle J$ for $\displaystyle\Phi$,

(29) $\Phi:=(\Phi^{+}\otimes\Phi^{-})\circ
m^{*}\colon{\mathcal{O}}_{A}\rightarrow U_{A}(H)\subset
U_{A}(\mathfrak{b}_{-})^{cop}\otimes U_{A}(\mathfrak{b}_{+})^{cop}$

is an embedding of Hopf algebras, and it extends to an isomorphism
$\displaystyle\Phi\colon{\mathcal{O}}_{A}[\psi_{-\rho}^{-1}]\rightarrow
U_{A}(H)$ which in particular satisfies:

(30) $\Phi_{1}(\psi_{-\lambda})=K_{-2\lambda}\ ,\
\Phi_{1}(\psi^{\alpha_{i}}_{-\varpi_{j}})=\delta_{i,j}L_{i}^{-2}\bar{E}_{i}.\
,\
\Phi_{1}(\psi^{-\alpha_{i}}_{-\varpi_{j}})=\delta_{i,j}q_{i}^{-1}\bar{F}_{i}K_{i}L_{i}^{-2}.$

Proof of Lemma 2.12. By definitions, for every $\displaystyle X\in
U_{q}(\mathfrak{b}_{+})^{cop}$, $\displaystyle Y\in
U_{q}^{ad}(\mathfrak{b}_{-})$ we have $\displaystyle\langle
i_{+}(S^{-1}(X)),Y\rangle=\pi_{q}^{-}(X,Y)$, and similarly for every
$\displaystyle X\in U_{q}^{ad}(\mathfrak{b}_{+})$, $\displaystyle Y\in
U_{q}(\mathfrak{b}_{-})^{cop}$ we have $\displaystyle\langle
i_{-}(S^{-1}(Y)),X\rangle=\pi_{q}^{+}(Y,X)$. By keeping these respective
notations for $\displaystyle X$ and $\displaystyle Y$, we deduce
$\displaystyle j_{q}^{-}(i_{+}(S^{-1}(X)))=X$ and $\displaystyle
j_{q}^{+}(i_{-}(S^{-1}(Y)))=Y$, ie.

(31) $j_{q}^{\pm}=S\circ i_{\mp}^{-1}.$

Because $\displaystyle S_{{\mathcal{O}}_{q}}^{-1}\circ i_{\pm}=i_{\pm}\circ
S$, it follows that

(32) $j_{q}^{\pm}\circ S_{{\mathcal{O}}_{q}}=S^{-1}\circ j_{q}^{\pm}.$

Also, for every $\displaystyle\alpha_{-}\in{\mathcal{O}}_{q}(B_{-})$ we have

$\displaystyle\langle\alpha_{-},\Phi^{+}(i_{-}(Y))\rangle=\langle
i_{-}(Y)\otimes\alpha_{-},R\rangle=\tau(i_{+}^{-1}(\alpha_{-}),Y)=\pi^{-}_{q}(j_{q}^{-}(S_{{\mathcal{O}}_{q}}(\alpha_{-})),Y)=\langle\alpha_{-},S(Y)\rangle$

where the first equality is by definition of $\displaystyle\Phi^{+}$ (see
(10)), the second is (27), the third follows from (32), and the last from the
definition of $\displaystyle j_{q}^{-}$. Similarly, for every
$\displaystyle\alpha_{+}\in{\mathcal{O}}_{q}(B_{+})$ we have

$\displaystyle\displaystyle\langle\alpha_{+},\Phi^{-}(i_{+}(X))\rangle$
$\displaystyle\displaystyle=\langle i_{+}(X)\otimes\alpha_{+},R^{-}\rangle$
$\displaystyle\displaystyle=\langle\alpha_{+}\otimes
S_{{\mathcal{O}}_{q}}^{-1}\circ i_{+}(X),R\rangle$
$\displaystyle\displaystyle=\langle\alpha_{+}\otimes i_{+}(S(X)),R\rangle$
$\displaystyle\displaystyle=\tau(S(X),i_{-}^{-1}(\alpha_{+}))$
$\displaystyle\displaystyle=\pi^{+}_{q}(S(i_{-}^{-1}(\alpha_{+})),S(X))=\pi^{+}_{q}(j_{q}^{+}(\alpha_{+}),S(X))=\langle\alpha_{+},S(X)\rangle.$

These computations imply $\displaystyle\Phi^{\pm}=S\circ
i_{\mp}^{-1}=j_{q}^{\pm}$, and the result follows by taking integral forms.
$\displaystyle\Box$

######  Remark 2.13.

(1) Since $\displaystyle\Phi_{1}=m\circ(id\otimes S^{-1})\circ\Phi$ and
$\displaystyle{\rm Im}(\Phi)\subset U_{A}(\mathfrak{b}_{-})^{cop}\otimes
U_{A}(\mathfrak{b}_{+})^{cop}$,
$\displaystyle\Phi_{1}({\mathcal{O}}_{A})\subset U_{A}$. Because
$\displaystyle\Phi_{1}({\mathcal{O}}_{q})=U_{q}^{lf}$, we have also
$\displaystyle\Phi_{1}({\mathcal{O}}_{A})\subset U_{A}^{lf}.$ The converse
inclusion $\displaystyle\Phi_{1}({\mathcal{O}}_{A})\supset U_{A}^{lf}$ holds
true as well, since $\displaystyle\Phi_{1}({\mathcal{O}}_{q})=U_{q}^{lf}$ and
$\displaystyle{\mathcal{O}}_{A}$ is an $\displaystyle A$-lattice of
$\displaystyle{\mathcal{O}}_{q}$.

(2) The components of $\displaystyle R^{\pm}_{A}$ may be described explicitly:
if $\displaystyle\\{\xi_{i}\\}_{i}$ is a basis of
$\displaystyle\Gamma(\mathfrak{b}_{+})$ (say, as obtained in section 3 of
[36]), one can determine the dual basis $\displaystyle\\{\xi^{*}_{i}\\}_{i}$
of $\displaystyle U_{A}(\mathfrak{b}_{-})$ by using the perfect pairing
$\displaystyle\pi_{A}^{+}$; then $\displaystyle\textstyle
R^{+}_{A}=\sum_{i}\xi_{i}\otimes\xi_{i}^{*}$. Note that, like $\displaystyle
U_{A}^{ad}$ is contained in $\displaystyle\Gamma$, $\displaystyle U_{A}$ is
contained in the restricted integral form of $\displaystyle U_{q}$, whose
categorical completion is
$\displaystyle\mathbb{U}_{\Gamma}\otimes\mathbb{C}[q^{\pm 1/D}]$. Therefore
the components $\displaystyle\xi_{i}^{*}$ of $\displaystyle R_{A}^{+}$ can be
viewed as elements of
$\displaystyle\mathbb{U}_{\Gamma}\otimes\mathbb{C}[q^{\pm 1/D}]$. This is
compatible with the fact that $\displaystyle\textstyle R^{+}_{A}$ is an
element of
$\displaystyle\mathbb{U}_{\Gamma}^{\hat{\otimes}2}\otimes\mathbb{C}[q^{\pm
1/D}]$.

(3) The dualities of Theorem 2.11 (2) afford a refinement defined over
$\displaystyle A$ of the quantum Killing form $\displaystyle\kappa\colon
U_{q}\otimes_{\mathbb{C}(q)}U_{q}\rightarrow\mathbb{C}(q^{1/D})$ (studied eg.
in [70], Section 2.8). This form is the duality realizing the isomorphism
$\displaystyle ad^{r}(U_{A})(K_{-2w_{0}(\mu)})\cong
End_{A}({}_{A}V_{\mu})^{*}$ stated after (22).

###  2.4. Structure theorems for $\displaystyle U_{\epsilon}$ and
$\displaystyle{\mathcal{O}}_{\epsilon}$

As usual we denote by $\displaystyle\epsilon$ a primitive $\displaystyle l$-th
root of unity, where $\displaystyle l$ is odd, and coprime to $\displaystyle
3$ if $\displaystyle\mathfrak{g}$ has $\displaystyle G_{2}$-components.

Let $\displaystyle G^{0}=B_{+}B_{-}$ (the big cell of $\displaystyle G$), and
define the group

$\displaystyle H=\\{(u_{+}t,u_{-}t^{-1}),t\in T_{G},u_{\pm}\in U_{\pm}\\}.$

Consider the map

$\displaystyle\begin{array}[]{lcll}\sigma:&B_{+}\times
B_{-}&\longrightarrow&G^{0}\\\
&(b_{+},b_{-})&\longmapsto&b_{+}b_{-}^{-1}.\end{array}$

The restriction of $\displaystyle\sigma$ to $\displaystyle H$ is an unramified
covering of degree $\displaystyle 2^{m}$. It can be seen as the classical
analog of the map $\displaystyle m\circ(id\otimes
S^{-1})\colon{\mathcal{O}}_{\epsilon}(B_{+})\otimes{\mathcal{O}}_{\epsilon}(B_{-})\rightarrow{\mathcal{O}}_{\epsilon}(G)$.

Denote by $\displaystyle\mathcal{Z}_{1}(U_{\epsilon})$ the image of
$\displaystyle\mathcal{Z}(U_{q})$ in $\displaystyle\mathcal{Z}(U_{\epsilon})$
under the specialization map $\displaystyle U_{q}\rightarrow U_{\epsilon}$,
and by $\displaystyle\mathcal{Z}_{0}(U_{\epsilon})\subset U_{\epsilon}$ the
subalgebra generated by $\displaystyle E_{\beta_{k}}^{l}$, $\displaystyle
F_{\beta_{k}}^{l}$, $\displaystyle L_{i}^{\pm l}$, for $\displaystyle
k\in\\{1,\ldots,N\\}$ and $\displaystyle i\in\\{1,\ldots m\\}$. In [33],
Section 1.8-3.3-3.8, and [35], Theorem 14.1-21.5, the following results are
proved:

######  Theorem 2.14.

(1) $\displaystyle U_{\epsilon}$ has no non-trivial zero divisors,
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$ is a central Hopf subalgebra of
$\displaystyle U_{\epsilon}$, and $\displaystyle U_{\epsilon}$ is a free
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$-module of rank $\displaystyle
l^{dim\mathfrak{g}}$. Moreover $\displaystyle U_{\epsilon}$ is a maximal order
of its classical fraction algebra $\displaystyle
Q(U_{\epsilon})=Q(\mathcal{Z}(U_{\epsilon}))\otimes_{\mathcal{Z}(U_{\epsilon})}U_{\epsilon}$,
and $\displaystyle Q(U_{\epsilon})$ is a central simple algebra of PI degree
$\displaystyle l^{N}$.

(2) SpecM$\displaystyle(\mathcal{Z}_{0}(U_{\epsilon}))$ is a group isomorphic
to $\displaystyle H$ above, and the multiplication map yields an isomorphism
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})\otimes_{\mathcal{Z}_{0}\cap\mathcal{Z}_{1}}\mathcal{Z}_{1}(U_{\epsilon})\rightarrow\mathcal{Z}(U_{\epsilon})$.

It follows from (1) and $\displaystyle dim\mathfrak{g}=m+2N$ that the field
$\displaystyle Q(\mathcal{Z}(U_{\epsilon}))$ is an extension of $\displaystyle
Q(\mathcal{Z}_{0}(U_{\epsilon}))$ of degree $\displaystyle l^{m}$. Conversely,
this degree and the rank of $\displaystyle U_{\epsilon}$ over
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$ imply that $\displaystyle
Q(U_{\epsilon})$ has PI degree $\displaystyle l^{N}$.

As for (2), note that $\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$ being an
affine and commutative algebra, the set
SpecM($\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$), viewed as the set of
characters of $\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$, acquires by
duality a structure of affine algebraic group. Thus, the first claim means
precisely the identification of this group with $\displaystyle H$.

In addition to (2), SpecM($\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$) and
$\displaystyle H$ have natural Poisson structures, that the isomorphism
identifies. Moreover we have the following identifications (see [35], Section
21.2). Consider the $\displaystyle l^{m}$-fold covering
$\displaystyle\tilde{T}_{G}\rightarrow T_{G}$. Recall that $\displaystyle T$
is the group formed by the elements $\displaystyle K_{\lambda}\in U_{A}$,
$\displaystyle\lambda\in P$. We can identify $\displaystyle T$ with the
additive group $\displaystyle P$, $\displaystyle
U_{A}(\mathfrak{h})=\mathbb{C}[T]=\mathbb{C}[P]$ with
$\displaystyle{\mathcal{O}}(\tilde{T}_{G})$, and therefore
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})\cap
U_{\epsilon}(\mathfrak{h})=\mathbb{C}[lP]$ with
$\displaystyle{\mathcal{O}}(T_{G})$. The quantum Harish-Chandra isomorphism
then identifies $\displaystyle\mathcal{Z}_{1}(U_{\epsilon})$ with
$\displaystyle\mathbb{C}[2P]^{W}\cong{\mathcal{O}}(\tilde{T}_{G}/(2))^{W}$,
where we denote by $\displaystyle(2)$ the subgroup of $\displaystyle
2$-torsion elements in $\displaystyle\tilde{T}_{G}$. Composing
$\displaystyle\sigma\colon H\rightarrow G^{0}$ with the quotient map under
conjugation, $\displaystyle G^{0}\hookrightarrow G\rightarrow G/\\!/G$, we get
dually an embedding of
$\displaystyle{\mathcal{O}}(G/\\!/G)=\mathcal{O}(G)^{G}$ in
$\displaystyle\mathcal{O}(H)$. The isomorphism of Theorem 2.14 (2) then
affords identifications

$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})\cap\mathcal{Z}_{1}(U_{\epsilon})\cong\mathcal{O}(G)^{G}$

as a subalgebra of
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})\cong{\mathcal{O}}(H)$, and

$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})\cap\mathcal{Z}_{1}(U_{\epsilon})=\mathbb{C}[2lP]^{W}\cong{\mathcal{O}}(\tilde{T}_{G}/(2l))^{W}\cong{\mathcal{O}}(T_{G}/(2))^{W}$

as a subalgebra of
$\displaystyle\mathcal{Z}_{1}(U_{\epsilon})\cong{\mathcal{O}}(\tilde{T}_{G}/(2))^{W}$.

A result similar to Theorem 2.14 holds true for
$\displaystyle{\mathcal{O}}_{\epsilon}$. Namely, take the specializations at
$\displaystyle q=\epsilon$ in Theorem 2.11. Denote by
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon}(H))$ the subalgebra of
$\displaystyle U_{\epsilon}(H)$ generated by the elements ($\displaystyle
k\in\\{1,\ldots,N\\},i\in\\{1,\ldots m\\}$)

$\displaystyle 1\otimes K_{-l\beta_{k}}E_{\beta_{k}}^{l}\ ,\
F_{\beta_{k}}^{l}K_{l\beta_{k}}\otimes 1\ ,\ L_{i}^{\pm l}\otimes L_{i}^{\mp
l}.$

It is a central Hopf subalgebra. Recall that $\displaystyle{\mathcal{O}}(G)$
can be realized as a Hopf subalgebra of $\displaystyle
U(\mathfrak{g})^{\circ}$, the restricted dual of the envelopping algebra
$\displaystyle U(\mathfrak{g})$ over $\displaystyle\mathbb{C}$. In [36] De
Concini-Lyubashenko introduced an epimorphism of Hopf algebras
$\displaystyle\eta:\Gamma_{\epsilon}\rightarrow U(\mathfrak{g})$ (essentially
a version of Lusztig’s “Frobenius” epimorphism in [52]). Let us put

(33) $\mathcal{Z}_{0}({\mathcal{O}}_{\epsilon}):=\eta^{*}({\mathcal{O}}(G))$

where $\displaystyle\eta^{*}\colon
U(\mathfrak{g})^{\circ}\rightarrow\Gamma_{\epsilon}^{\circ}$ is the
monomorphism dual to $\displaystyle\eta$.

######  Theorem 2.15.

(1) $\displaystyle\mathcal{Z}_{0}({\mathcal{O}}_{\epsilon})$ is a central Hopf
subalgebra of
$\displaystyle{\mathcal{O}}_{\epsilon}\subset\Gamma_{\epsilon}^{\circ}$, and
$\displaystyle Q(\mathcal{Z}({\mathcal{O}}_{\epsilon}))$ is an extension of
$\displaystyle Q(\mathcal{Z}_{0}({\mathcal{O}}_{\epsilon}))$ of degree
$\displaystyle l^{m}$ if $\displaystyle l$ is coprime to the coefficients of
the Cartan matrix of $\displaystyle\mathfrak{g}$.

(2) $\displaystyle\psi_{-l\rho}\in\mathcal{Z}_{0}({\mathcal{O}}_{\epsilon})$,
and $\displaystyle\mathcal{Z}_{0}({\mathcal{O}}_{\epsilon})$ is generated by
the matrix coefficients of the irreducible $\displaystyle\Gamma$-modules of
highest weight $\displaystyle l\lambda$, $\displaystyle\lambda\in P_{+}$.
Moreover, the map $\displaystyle\Phi$ in (29) affords an algebra embedding
$\displaystyle\mathcal{Z}_{0}({\mathcal{O}}_{\epsilon})\rightarrow\mathcal{Z}_{0}(U_{\epsilon}(H))$
and algebra isomorphisms
$\displaystyle\mathcal{Z}_{0}({\mathcal{O}}_{\epsilon})[\psi_{-l\rho}^{-1}]\rightarrow\mathcal{Z}_{0}(U_{\epsilon}(H))$,
$\displaystyle{\mathcal{O}}_{\epsilon}[\psi_{-l\rho}^{-1}]\rightarrow
U_{\epsilon}(H)$.

(3) $\displaystyle{\mathcal{O}}_{\epsilon}$ has no non-trivial zero divisors,
and it is a free
$\displaystyle\mathcal{Z}_{0}({\mathcal{O}}_{\epsilon})$-module of rank
$\displaystyle l^{dim\mathfrak{g}}$. Moreover
$\displaystyle{\mathcal{O}}_{\epsilon}$ is a maximal order of its classical
fraction algebra $\displaystyle
Q({\mathcal{O}}_{\epsilon})=Q(\mathcal{Z}({\mathcal{O}}_{\epsilon}))\otimes_{\mathcal{Z}({\mathcal{O}}_{\epsilon})}{\mathcal{O}}_{\epsilon}$,
and $\displaystyle Q({\mathcal{O}}_{\epsilon})$ is a central simple algebra of
PI degree $\displaystyle l^{N}$.

For the proof, see in [36]: the proposition 6.4 for the first claim of (1)
(where $\displaystyle\mathcal{Z}_{0}({\mathcal{O}}_{\epsilon})$ and
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon}(H))$ are denoted $\displaystyle
F_{0}$ and $\displaystyle A_{0}$ respectively), the appendix of Enriquez and
[38] for the second claim of (1), the propositions 6.4-6.5 for (2), and for
(3) the theorems 7.2-7.4 (where $\displaystyle{\mathcal{O}}_{\epsilon}$ is
shown to be projective over
$\displaystyle\mathcal{Z}_{0}({\mathcal{O}}_{\epsilon})$) and [21] (which
provides the additional K-theoretic arguments to deduce that
$\displaystyle{\mathcal{O}}_{\epsilon}$ is free).

As above for $\displaystyle U_{\epsilon}$, it follows from (3) that
$\displaystyle Q(\mathcal{Z}({\mathcal{O}}_{\epsilon}))$ has degree
$\displaystyle l^{m}$ over $\displaystyle
Q(\mathcal{Z}_{0}({\mathcal{O}}_{\epsilon}))$. Generators of
$\displaystyle\mathcal{Z}({\mathcal{O}}_{\epsilon})$ are described in
Enriquez’ Appendix in [36] under the assumptions on $\displaystyle l$ stated
in (1). We do not know a presentation by generators and relations for general
$\displaystyle G$, nor a basis of $\displaystyle{\mathcal{O}}_{\epsilon}$ over
$\displaystyle\mathcal{Z}_{0}({\mathcal{O}}_{\epsilon})$, but see [37] for the
case of $\displaystyle SL_{2}$.

We will recall the known results in this case of $\displaystyle SL_{2}$ before
Lemma 4.3.

There is a natural action of the braid group
$\displaystyle\mathcal{B}(\mathfrak{g})$ on
$\displaystyle{\mathcal{O}}_{\epsilon}$, that we will use. Namely, let
$\displaystyle n_{i}\in N(T_{G})$ be a representative of the reflection
$\displaystyle s_{i}\in W=N(T_{G})/T_{G}$ associated to the simple root
$\displaystyle\alpha_{i}$. In [67, 66] Soibelman-Vaksman introduced
functionals $\displaystyle t_{i}:\mathcal{O}_{A}\rightarrow A$ which quantize
the elements $\displaystyle n_{i}$. They correspond dually to generators of
the quantum Weyl group of $\displaystyle\mathfrak{g}$; in the Appendix we
recall their main properties (see also [30], Section 8.2, and [47, 67, 51, 48,
36]). Denote by $\displaystyle\lhd$ the natural right action of functionals on
$\displaystyle{\mathcal{O}}_{A}$, namely (using Sweedler’s notation)

$\displaystyle\alpha\lhd h=\sum_{(\alpha)}h(\alpha_{(1)})\alpha_{(2)}$

for every $\displaystyle\alpha\in{\mathcal{O}}_{A}$ and $\displaystyle
h\in{\mathcal{O}}_{A}\rightarrow A$. Let us identify
$\displaystyle\mathcal{Z}_{0}({\mathcal{O}}_{\epsilon})$ with
$\displaystyle{\mathcal{O}}(G)$ by means of (33). We have ([36], Proposition
7.1):

######  Proposition 2.16.

The maps $\displaystyle\lhd t_{i}$ on $\displaystyle{\mathcal{O}}_{\epsilon}$
preserve $\displaystyle\mathcal{Z}_{0}({\mathcal{O}}_{\epsilon})$, and satisfy
$\displaystyle(f\lhd t_{i})(a)=f(n_{i}a)$ and $\displaystyle(f\star\alpha)\lhd
t_{i}=(f\lhd t_{i})(\alpha\lhd t_{i})$ for every $\displaystyle
f\in\mathcal{Z}_{0}({\mathcal{O}}_{\epsilon})$, $\displaystyle a\in G$,
$\displaystyle\alpha\in\mathcal{O}_{\epsilon}$.

We provide an alternative, non computational, proof of this result in the
Appendix (Section 6.2).

##  3\. 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 $\displaystyle{\mathcal{L}}_{0,n}$,
$\displaystyle{\mathcal{L}}_{0,n}^{A}$ and
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon^{\prime}}$,
$\displaystyle{\epsilon^{\prime}}\in\mathbb{C}^{\times}$, are Noetherian.

Let us note that the algebras in this theorem are generated by a finite number
of elements over their respective ground rings $\displaystyle{\mathbb{C}}(q)$,
$\displaystyle A$ and $\displaystyle\mathbb{C}$. Indeed, by the formula (14)
it is enough to verify this for $\displaystyle{\mathcal{L}}_{0,1}^{A}$, but
$\displaystyle{\mathcal{L}}_{0,1}^{A}={\mathcal{O}}_{A}$ as a vector space,
and $\displaystyle{\mathcal{O}}_{A}$ with its product $\displaystyle\star$ is
well-known to be finitely generated by the matrix coefficients of the
fundamental $\displaystyle\Gamma$-modules $\displaystyle{}_{A}V_{\varpi_{k}}$,
$\displaystyle k\in\\{1,\ldots,m\\}$. Then the claim follows from the formula
inverse to (11), expressing the product $\displaystyle\star$ in terms of the
product of $\displaystyle{\mathcal{L}}_{0,1}$ (see (18) in [23]).

Proof of Theorem 3.1. The result for $\displaystyle{\mathcal{L}}_{0,1}$ and
$\displaystyle{\mathcal{L}}_{0,1}^{A}$ follows immediately from Theorem 2.2
(3) by identifying $\displaystyle{\mathcal{L}}_{0,1}^{A}$ with $\displaystyle
U_{A}^{lf}$ via $\displaystyle\Phi_{1}$. Assume now that $\displaystyle n>1$.
We are going to develop the proof for $\displaystyle{\mathcal{L}}_{0,n}$; the
arguments can be repeated verbatim for $\displaystyle{\mathcal{L}}_{0,n}^{A}$,
and the result for $\displaystyle{\mathcal{L}}_{0,n}^{\epsilon^{\prime}}$ will
then follow immediately by lifting ideals by the quotient map
$\displaystyle{\mathcal{L}}_{0,n}^{A}\rightarrow{\mathcal{L}}_{0,n}^{\epsilon^{\prime}}={\mathcal{L}}_{0,n}^{A}/(q-{\epsilon^{\prime}}){\mathcal{L}}_{0,n}^{A}$.

Recall the isomorphism of $\displaystyle U_{q}$-modules (see (19)):

(34)
${\mathcal{L}}_{0,n}\stackrel{{\scriptstyle\Phi_{n}}}{{\longrightarrow}}(U_{q}({\mathfrak{g}})^{\otimes
n})^{lf}\stackrel{{\scriptstyle\psi_{n}^{-1}}}{{\longrightarrow}}U_{q}^{lf}({\mathfrak{g}})^{\otimes
n}=U_{q}^{lf}({\mathfrak{g}}^{\oplus n})$

where $\displaystyle lf$ means respectively locally finite for the action
$\displaystyle ad_{n}^{r}$ of $\displaystyle U_{q}({\mathfrak{g}})$ on
$\displaystyle U_{q}({\mathfrak{g}})^{\otimes n}$, locally finite for the
action $\displaystyle ad^{r}$ of $\displaystyle U_{q}({\mathfrak{g}})$ on
$\displaystyle U_{q}({\mathfrak{g}})$, and locally finite for the action
$\displaystyle ad^{r}$ of $\displaystyle U_{q}^{lf}({\mathfrak{g}}^{\oplus
n})$ on itself. It is a fact that Theorem 2.2 (3) holds true by replacing
$\displaystyle U_{q}^{lf}({\mathfrak{g}})$ with $\displaystyle
U_{q}^{lf}({\mathfrak{g}}^{\oplus n})$, but one cannot use this to deduce the
result because $\displaystyle\psi_{n}$ 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 [70]. Let us begin by recalling these arguments.

As usual let $\displaystyle C(\mu)$ be the vector space generated by the
matrix coefficients of $\displaystyle V_{\mu}$, the simple $\displaystyle
U_{q}$-module of type $\displaystyle 1$ and highest weight
$\displaystyle\mu\in P_{+}$. Denote by $\displaystyle C(\mu)_{\lambda}\subset
C(\mu)$ the subspace of weight $\displaystyle\lambda$ for the left coregular
action of $\displaystyle U_{q}({\mathfrak{h}})$; so $\displaystyle\alpha\in
C(\mu)_{\lambda}$ if

$\displaystyle K_{\nu}\rhd\alpha=q^{(\nu,\lambda)}\alpha\ ,\nu\in P.$

Consider the ordered semigroup

$\displaystyle\Lambda=\\{(\mu,\lambda)\in P_{+}\times P,\lambda\;\text{is a
weight of}\;V_{\mu}\\}$

with the partial order
$\displaystyle(\mu,\lambda)\leq(\mu^{\prime},\lambda^{\prime})$ if and only if
$\displaystyle\mu^{\prime}-\mu\in P_{+},\lambda^{\prime}-\lambda\in P_{+}$.
Since $\displaystyle{\mathcal{L}}_{0,1}$ and $\displaystyle{\mathcal{O}}_{q}$
are isomorphic vector spaces we have
$\displaystyle\textstyle{\mathcal{L}}_{0,1}=\bigoplus_{\mu\in
P_{+}}C(\mu)=\bigoplus_{(\mu,\lambda)\in\Lambda}C(\mu)_{\lambda}$. Consider
the filtration $\displaystyle\mathcal{F}_{2}$ of the vector space
$\displaystyle{\mathcal{L}}_{0,1}$ given by the family of subspaces

$\displaystyle{\mathcal{F}}_{2}^{\mu,\lambda}=\bigoplus_{(\mu^{\prime},\lambda^{\prime})\leq(\mu,\lambda)}C(\mu^{\prime})_{\lambda^{\prime}}\
,(\mu,\lambda)\in\Lambda.$

Denote by $\displaystyle Gr_{\mathcal{F}_{2}}({\mathcal{L}}_{0,1})$ the
associated graded vector space. The standard vector space isomorphism
$\displaystyle{\mathcal{L}}_{0,1}\rightarrow
Gr_{{\mathcal{F}}_{2}}({\mathcal{L}}_{0,1})$, assigning to $\displaystyle x\in
C(\mu)_{\lambda}$ its coset
$\displaystyle\bar{x}\in{\mathcal{F}}_{2}^{\mu,\lambda}/\left(\oplus_{(\mu^{\prime},\lambda^{\prime})<(\mu,\lambda)}C(\mu^{\prime})_{\lambda^{\prime}}\right)$,
implies

$\displaystyle\textstyle
Gr_{{\mathcal{F}}_{2}}({\mathcal{L}}_{0,1})=\bigoplus_{(\mu,\lambda)\in\Lambda}C(\mu)_{\lambda}.$

Now, one has the following facts:

(i) First, taking the product in $\displaystyle{\mathcal{L}}_{0,1}$ we have

(35)
$\alpha\beta\in{\mathcal{F}}_{2}^{\mu_{1}+\mu_{2},\lambda_{1}+\lambda_{2}}\quad\mathrm{for}\
\alpha\in C(\mu_{1})_{\lambda_{1}},\beta\in C(\mu_{2})_{\lambda_{2}}.$

Therefore $\displaystyle\mathcal{F}_{2}$ is an algebra filtration of
$\displaystyle{\mathcal{L}}_{0,1}$, and $\displaystyle
Gr_{\mathcal{F}_{2}}({\mathcal{L}}_{0,1})$ a graded algebra. Denote by
$\displaystyle\alpha\circ\beta$ the product in $\displaystyle\textstyle
Gr_{{\mathcal{F}}_{2}}({\mathcal{L}}_{0,1})$ of
$\displaystyle\alpha,\beta\in{\mathcal{L}}_{0,1}$; by definition, if
$\displaystyle\alpha\in C(\mu_{1})_{\lambda_{1}}$, $\displaystyle\beta\in
C(\mu_{2})_{\lambda_{2}}$ then $\displaystyle\alpha\circ\beta$ is the
projection of $\displaystyle\alpha\beta$ onto $\displaystyle
C(\mu_{1}+\mu_{2})_{\lambda_{1}+\lambda_{2}}$.

(ii) Second, denote by $\displaystyle\bar{\star}$ the product
$\displaystyle\star$ of $\displaystyle{\mathcal{O}}_{q}$ followed by the
projection onto the component $\displaystyle C(\mu+\nu)$. Then we have

(36) $C(\mu)\circ C(\nu)=C(\mu)\ \bar{\star}\ C(\nu)=C(\mu+\nu).$

(iii) Finally, for every $\displaystyle\mu\in P_{+}$ fix a basis of weight
vectors $\displaystyle e_{1}^{\mu},\ldots,e_{m}^{\mu}$ of $\displaystyle
V_{\mu}$. Denote by $\displaystyle e^{1}_{\mu},\ldots,e^{m}_{\mu}\in
V_{\mu}^{*}$ the dual basis, and by $\displaystyle w(e_{i}^{\mu})$ the weight
of $\displaystyle e_{i}^{\mu}$. One can assume that the ordering of
$\displaystyle e_{1}^{\mu},\ldots,e_{m}^{\mu}$ is such that $\displaystyle
w(e_{i}^{\mu})>w(e_{j}^{\mu})$ implies $\displaystyle i<j$; indeed,
$\displaystyle e_{1}^{\mu}$ generates the subspace of weight
$\displaystyle\mu$, then come (in any order) the $\displaystyle e_{i}^{\mu}$
such that $\displaystyle w(e_{i}^{\mu})=\mu-\alpha_{s}$ for some
$\displaystyle s$, then those such that $\displaystyle
w(e_{i}^{\mu})=\mu-\alpha_{s}-\alpha_{t}$ for some $\displaystyle s$ and
$\displaystyle t$, etc. Consider the matrix coefficients
$\displaystyle{}_{\mu}\phi_{i}^{j}(x):=e^{i}_{\mu}(\pi_{V}(x)(e_{j}^{\mu}))$,
$\displaystyle x\in U_{q}$. By (11), using the explicit form of the
$\displaystyle R$-matrix it can be shown that

(37)
$\displaystyle\displaystyle{}_{{\nu}}\phi_{k}^{l}\circ{}_{{\mu}}\phi_{i}^{j}-q_{ijkl}\
{}_{{\mu}}\phi_{i}^{j}\circ{}_{{\nu}}\phi_{k}^{l}=\sum_{r=i}^{m}\sum_{s=1}^{k}\sum_{u=1}^{l-1}$
$\displaystyle\displaystyle\sum_{v=j+1}^{m}\delta^{ijkl}_{rsuv}\
{}_{\mu}\phi_{r}^{v}\circ{}_{{\nu}}\phi_{s}^{u}$
$\displaystyle\displaystyle-\sum_{r=i+1}^{m}\sum_{s=1}^{k-1}q_{ijkl}\gamma_{rs}^{ijkl}\
{}_{\mu}\phi_{r}^{j}\circ{}_{{\nu}}\phi_{s}^{l}$

where $\displaystyle
q_{ijkl}=q^{(w(e_{j}^{\mu})+w(e_{i}^{\mu}),w(e_{k}^{\nu})-w(e_{l}^{\nu}))}$,
and
$\displaystyle\gamma_{rs}^{ijkl},\delta^{ijkl}_{rsuv}\in\mathbb{C}(q^{1/D})$
are such that $\displaystyle\gamma_{rs}^{ijkl}=0$ unless $\displaystyle
w(e_{r}^{\mu})<w(e_{i}^{\mu})$ and $\displaystyle
w(e_{s}^{\nu})>w(e_{k}^{\nu})$, and $\displaystyle\delta^{ijkl}_{rsuv}=0$
unless $\displaystyle w(e_{u}^{\nu})>w(e_{l}^{\nu})$, $\displaystyle
w(e_{v}^{\mu})<w(e_{j}^{\mu})$, $\displaystyle w(e_{r}^{\mu})\leq
w(e_{i}^{\mu})$ and $\displaystyle w(e_{s}^{\nu})\geq w(e_{k}^{\nu})$.

By (36) (or more simply by using (11), as observed before the proof),
$\displaystyle\textstyle Gr_{{\mathcal{F}}_{2}}({\mathcal{L}}_{0,1})$ is
generated by the matrix coefficients
$\displaystyle{}_{{\varpi_{k}}}\\!\phi_{i}^{j}$ of the fundamental
representations $\displaystyle V_{\varpi_{k}}$. One can list these matrix
coefficients, say $\displaystyle M$ in number, in an ordered sequence
$\displaystyle u_{1},\ldots,u_{M}$ such that the following condition holds: if
$\displaystyle w(e_{k}^{\varpi_{s}})<w(e_{i}^{\varpi_{r}})$, or $\displaystyle
w(e_{k}^{\varpi_{s}})=w(e_{i}^{\varpi_{r}})$ and $\displaystyle
w(e_{l}^{\varpi_{s}})<w(e_{j}^{\varpi_{r}})$, then $\displaystyle
u_{a}:={}_{{\varpi_{r}}}\\!\phi_{i}^{j}$ and $\displaystyle
u_{b}:={}_{{\varpi_{s}}}\\!\phi_{k}^{l}$ satisfy $\displaystyle b<a$. Then
denoting $\displaystyle{}_{{\mu}}\phi_{i}^{j}$,
$\displaystyle{}_{{\nu}}\phi_{k}^{l}$ in (37) by $\displaystyle u_{j}$,
$\displaystyle u_{i}$ respectively, and assuming $\displaystyle u_{j}<u_{i}$,
one finds that all terms $\displaystyle u_{s}:={}_{\mu}\phi_{r}^{v}$,
$\displaystyle{}_{\mu}\phi_{r}^{j}$ in the sums are $\displaystyle<u_{j}$.
Therefore, for all $\displaystyle 1\leq j<i\leq M$ it takes the form:

(38) $u_{i}\circ u_{j}-q_{ij}u_{j}\circ
u_{i}=\sum_{s=1}^{j-1}\sum_{t=1}^{M}\alpha_{ij}^{st}u_{s}\circ u_{t}$

for some $\displaystyle
q_{ij}\in\mathbb{C}(q^{1/D})^{\times},\alpha_{ij}^{st}\in\mathbb{C}(q^{1/D})$.
By Proposition I.8.17 of [20] (see also Proposition 2.133 of [70]) an algebra
$\displaystyle A$ over a field $\displaystyle\mathbb{K}$ generated by elements
$\displaystyle u_{1},\ldots,u_{M}$ such that

(39) $u_{i}\circ u_{j}-q_{ij}u_{j}\circ
u_{i}=\sum_{s=1}^{j-1}\sum_{t=1}^{M}\alpha_{ij}^{st}u_{s}\circ
u_{t}+\beta_{ij}^{st}u_{t}\circ u_{s}$

for all $\displaystyle 1\leq j<i\leq M$ and some $\displaystyle
q_{ij}\in\mathbb{K}^{\times}$ and
$\displaystyle\alpha_{ij}^{st},\beta_{ij}^{st}\in\mathbb{K}$, is Noetherian.
In fact $\displaystyle A$ has an algebra filtration, say
$\displaystyle{\mathcal{F}}_{3}$, such that $\displaystyle
Gr_{{\mathcal{F}}_{3}}(A)$ 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. [60], 1.6.9-1.6.11).
Applying this to $\displaystyle A=Gr_{{\mathcal{F}}_{2}}({\mathcal{L}}_{0,1})$
and going up the filtration $\displaystyle\mathcal{F}_{2}$ it follows that
$\displaystyle{\mathcal{L}}_{0,1}$ is Noetherian too.

We are going to extend all these facts to $\displaystyle{\mathcal{L}}_{0,n}$.
The main point is to generalize the filtration
$\displaystyle{\mathcal{F}}_{2}$, which we do first. Consider the semigroup

$\displaystyle[\Lambda]=\left\\{([\mu],[\lambda])\in P_{+}^{n}\times P^{n}\
\mid\ (\mu_{i},\lambda_{i})\in\Lambda\ \mathrm{where}\
[\mu]=(\mu_{i})_{i=1}^{n},[\lambda]=(\lambda_{i})_{i=1}^{n}\right\\}.$

Put the lexicographic partial order on $\displaystyle\textstyle[\Lambda]$,
starting from the tail: so
$\displaystyle([\mu^{\prime}],[\lambda^{\prime}])\leq([\mu],[\lambda])$ if
$\displaystyle\mu_{n}-\mu_{n}^{\prime}\in P_{+}\setminus\\{0\\}$, or
$\displaystyle\mu_{n}=\mu_{n}^{\prime}$ and
$\displaystyle\lambda_{n}-\lambda_{n}^{\prime}\in P_{+}\setminus\\{0\\}$, or
there is $\displaystyle k\in\\{n,\ldots,2\\}$ such that
$\displaystyle\mu_{i}=\mu_{i}^{\prime},\lambda_{i}=\lambda_{i}^{\prime}$ for
$\displaystyle i\in\\{n,\ldots,k\\}$ and
$\displaystyle\mu_{k-1}-\mu_{k-1}^{\prime}\in P_{+}\setminus\\{0\\}$, or
$\displaystyle\mu_{k-1}=\mu_{k-1}^{\prime}$ and
$\displaystyle\lambda_{k-1}-\lambda_{k-1}^{\prime}\in P_{+}\setminus\\{0\\}$,
replacing this last condition by
$\displaystyle\lambda_{1}-\lambda_{1}^{\prime}\in P_{+}$ when $\displaystyle
k=2$. Now recall that
$\displaystyle{\mathcal{L}}_{0,n}={\mathcal{L}}_{0,1}^{\otimes
n}={\mathcal{O}}_{q}^{\otimes n}$ as vector spaces. For every
$\displaystyle([\mu],[\lambda])\in[\Lambda]$ consider the subspaces
$\displaystyle C([\mu])_{[\lambda]}\subset C([\mu])\subset{\mathcal{L}}_{0,n}$
defined by

$\displaystyle\displaystyle C([\mu])$
$\displaystyle\displaystyle=C(\mu_{1})\otimes\ldots\otimes C(\mu_{n})$
$\displaystyle\displaystyle C([\mu])_{[\lambda]}$
$\displaystyle\displaystyle=C(\mu_{1})_{\lambda_{1}}\otimes\ldots\otimes
C(\mu_{n})_{\lambda_{n}}.$

Then $\displaystyle\textstyle{\mathcal{L}}_{0,n}=\bigoplus_{[\mu]\in
P_{+}^{n}}C({[\mu]})$ and $\displaystyle\textstyle
C({[\mu]})=\bigoplus_{([\mu],[\lambda])\in[\Lambda]}C([\mu])_{[\lambda]}$. For
every $\displaystyle([\mu],[\lambda])\in[\Lambda]$ define

(40)
${\mathcal{F}}_{2}^{[\mu],[\lambda]}=\bigoplus_{([\mu^{\prime}],[\lambda^{\prime}])\leq([\mu],[\lambda])}\bigotimes_{j=1}^{n}C(\mu^{\prime}_{j})_{\lambda^{\prime}_{j}}.$

Clearly
$\displaystyle{\mathcal{F}}_{2}^{[\mu^{\prime}],[\lambda^{\prime}]}\subset{\mathcal{F}}_{2}^{[\mu],[\lambda]}$
for $\displaystyle([\mu^{\prime}],[\lambda^{\prime}])\leq([\mu],[\lambda])$,
and the vector space $\displaystyle{\mathcal{L}}_{0,n}$ is the union of the
subspaces $\displaystyle{\mathcal{F}}_{2}^{[\mu],[\lambda]}$ over all
$\displaystyle([\mu],[\lambda])\in[\Lambda]$, so these form a filtration of
$\displaystyle{\mathcal{L}}_{0,n}$. Let us denote it
$\displaystyle{\mathcal{F}}_{2}$, as when $\displaystyle n=1$. As usual, write
$\displaystyle([\mu^{\prime}],[\lambda^{\prime}])<([\mu],[\lambda])$ for
$\displaystyle([\mu^{\prime}],[\lambda^{\prime}])\leq([\mu],[\lambda])$ and
$\displaystyle([\mu^{\prime}],[\lambda^{\prime}])\neq([\mu],[\lambda])$, and
put

$\displaystyle{\mathcal{F}}_{2}^{<[\mu],[\lambda]}=\sum_{([\mu^{\prime}],[\lambda^{\prime}])<([\mu],[\lambda])}{\mathcal{F}}_{2}^{[\mu^{\prime}],[\lambda^{\prime}]}.$

Then define

$\displaystyle
Gr_{{\mathcal{F}}_{2}}({\mathcal{L}}_{0,n})_{[\mu],[\lambda]}={\mathcal{F}}_{2}^{[\mu],[\lambda]}/{\mathcal{F}}_{2}^{<[\mu],[\lambda]}.$

This space is canonically identified with $\displaystyle
C({[\mu]})_{[\lambda]}$, so the graded vector space associated to
$\displaystyle\mathcal{F}_{2}$ is

(41)
$Gr_{{\mathcal{F}}_{2}}({\mathcal{L}}_{0,n})=\bigoplus_{([\mu],[\lambda])\in[\Lambda]}Gr_{{\mathcal{F}}_{2}}({\mathcal{L}}_{0,n})_{[\mu],[\lambda]}=\bigoplus_{([\mu],[\lambda])\in[\Lambda]}C({[\mu]})_{[\lambda]}.$

We claim that $\displaystyle{\mathcal{F}}_{2}$ is an algebra filtration with
respect to the product of $\displaystyle{\mathcal{L}}_{0,n}$, and therefore
$\displaystyle Gr_{{\mathcal{F}}_{2}}({\mathcal{L}}_{0,n})$ is a graded
algebra.

For notational simplicity let us prove it for $\displaystyle n=2$, the general
case being strictly similar. Recall that the product of
$\displaystyle{\mathcal{L}}_{0,n}$ is given by the formula (14). Take
$\displaystyle([\mu],[\lambda]),([\mu^{\prime}],[\lambda^{\prime}])\in[\Lambda]$,
and elements $\displaystyle\alpha\otimes\beta\in
C(\mu_{1})_{\lambda_{1}}\otimes C(\mu_{2})_{\lambda_{2}}$ and
$\displaystyle\alpha^{\prime}\otimes\beta^{\prime}\in
C(\mu_{1}^{\prime})_{\lambda_{1}^{\prime}}\otimes
C(\mu_{2}^{\prime})_{\lambda_{2}^{\prime}}$. The $\displaystyle R$-matrix
expands as $\displaystyle R=\Theta\hat{R}$, where
$\displaystyle\textstyle\Theta=q^{\sum_{i,j=1}^{m}(B^{-1})_{ij}H_{i}\otimes
H_{j}}\in\mathbb{U}_{q}^{\otimes 2}$, with $\displaystyle B\in
M_{m}(\mathbb{Q})$ the matrix with entries $\displaystyle
B_{ij}:=d_{j}^{-1}a_{ij}$, and
$\displaystyle\textstyle\hat{R}=\sum_{(\hat{R})}\hat{R}_{(1)}\otimes\hat{R}_{(2)}\in\mathbb{U}_{q}(\mathfrak{n}_{+})\otimes\mathbb{U}_{q}(\mathfrak{n}_{-})$
(see eg. [30], Theorem 8.3.9, or [70], Theorem 2.108). If $\displaystyle x$,
$\displaystyle y$ are weight vectors of weights $\displaystyle\mu$,
$\displaystyle\nu$ respectively, then $\displaystyle\Theta(x\otimes
y)=q^{(\mu,\nu)}x\otimes y$. Moreover, $\displaystyle\hat{R}$ has weight
$\displaystyle 0$ for the adjoint action of $\displaystyle
U_{q}(\mathfrak{h})$; that is, complementary components
$\displaystyle\hat{R}_{(1)}$ and $\displaystyle\hat{R}_{(2)}$ have opposite
weights. Note also that the coregular actions $\displaystyle\rhd$,
$\displaystyle\lhd$ fix globally each component $\displaystyle C(\mu)$,
$\displaystyle\mu\in P_{+}$. Then, for every $\displaystyle\nu\in P$ and any
of the components $\displaystyle R^{1}_{(2)},\ldots,R^{4}_{(2)}$ we have

$\displaystyle\displaystyle
K_{\nu}\rhd\left(S(R^{1}_{(2)}R^{3}_{(2)})\rhd\beta\lhd
R^{2}_{(2)}R^{4}_{(2)}\right)$
$\displaystyle\displaystyle=\sum_{(\beta),(\beta)}\beta_{(1)}(R^{2}_{(2)}R^{4}_{(2)})\left(K_{\nu}S(R^{1}_{(2)}R^{3}_{(2)})\rhd\beta_{(2)}\right)$