ON THE INTEGRAL FORM OF RANK 1 KAC–MOODY ALGEBRAS

In this paper we shall prove that the ℤ-subalgebra generated by the divided powers of the Drinfeld generators xr±\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {x}_r^{\pm } $$\end{document} (r ∈ ℤ) of the Kac–Moody algebra of type A22\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\textrm{A}}_2^{(2)} $$\end{document} is an integral form (strictly smaller than Mitzman’s; see [Mi]) of the enveloping algebra, we shall exhibit a basis generalizing the one provided in [G] for the untwisted affine Kac–Moody algebras and we shall determine explicitly the commutation relations. Moreover, we prove that both in the untwisted and in the twisted case the positive (respectively negative) imaginary part of the integral form is an algebra of polynomials over ℤ.


Contents
Recall that the twisted affine Kac-Moody algebra of type A (2) 2 isŝ l 3 χ , the χ-invariant subalgebra ofŝ l 3 where χ is the non trivial Dynkin diagram automorphism of A 2 (see [K]) and denote byŨ its enveloping algebra U(ŝ l 3 χ ).
The aim of this paper is to give a basis over Z of the Z-subalgebra ofŨ generated by the divided powers of the Drinfeld generators x ± r 's (r ∈ Z) (see definitions 5.1 and 5.11), thus proving that this Z-subalgebra is an integral form ofŨ.
The integral forms for finite dimensional semisimple Lie algebras were first introduced by Chevalley in [Ch] for the study of the Chevalley groups and of their representation theory. The construction of the "divided power"-Z-form for the simple finite dimensional Lie algebras is due to Kostant (see [Ko]) ; it has been generalized to the untwisted affine Kac-Moody algebras by Garland in [G] as we shall quickly recall. Given a simple Lie algebra g 0 and the corresponding untwisted affine Kac-Moody algebra g = g 0 ⊗ C[t, t −1 ] ⊕ Cc provided with an (ordered) Chevalley basis, the Zsubalgebra U Z of U = U(g) generated by the divided powers of the real root vectors is an integral form of U; a Z-basis of this integral form (hence its Z-module structure) can be described by decomposing U Z as tensor product of its Z-subalgebras relative respectively to the real root vectors (U re,+ Z and U re,− Z ), to the imaginary root vectors (U im,+ Z and U im,− Z ) and to the Cartan subalgebra (U h Z ): U re,+ Z has a basis B re,+ consisting of the (finite) ordered products of divided powers of the distinct positive real root vectors and (U re,− Z , B re,− ) can be described in the same way: ±βN |N ≥ 0, β 1 > ... > β N > 0 real roots, k βj > 0 ∀j}.
Here a real root β of g is said to be positive if there exists a positive root α of g 0 such that β = α or β − α is imaginary; x β is the Chevalley generator corresponding to the real root β.
A basis B h of U h Z , which is commutative, consists of the products of the "binomials" of the (Chevalley) generators h i (i ∈ I) of the Cartan subalgebra of g: it is worth remarking that U h Z is not an algebra of polynomials. U im,+ Z (and its symmetric U im,− Z ) is commutative, too; as Z-module it is isomorphic to the tensor product of the U im,+ i,Z 's (each factor corresponding to the i th copy of U(ŝ l 2 ) inside U), so that it is enough to describe it in the rank 1 case: the basis B im,+ of U im,+ Z (ŝ l 2 ) provided by Garland can be described as a set of finite products of the elements Λ k (ξ(m)) (r ∈ N, m > 0), where the Λ k (ξ(m))'s are the elements of U im,+ = C[h r (= h ⊗ t r )|r > 0] defined recursively (for all m = 0) by Λ −1 (ξ(m)) = 1, kΛ k−1 (ξ(m)) = r≥0,s>0 r+s=k Λ r−1 (ξ(m))h ms : It is not clear from this description that U im,+ Z and U im,− Z are algebras of polynomials. Thanks to the isomorphism of Z-modules a Z-basis B of U Z is produced as multiplication of Z-bases of these subalgebras: The same result has been proved for all the twisted affine Kac-Moody algebras by Mitzman in [Mi], where the author provides a deeper comprehension and a compact description of the commutation formulas by means of a drastic simplification of both the relations and their proofs. This goal is achieved remarking that the generating series of the elements involved in the basis can be expressed as suitable exponentials, observation that allows to apply very general tools of calculus, such as the well known properties Here, too, it is not yet clear that U im,± Z are algebras of polynomials. However this property, namely U im,+ Z = Z[Λ k−1 = Λ k−1 (ξ(1)) = p k,1 |k > 0], is stated in Fisher-Vasta's PhD thesis ( [F]), where the author describes the results of Garland for the untwisted case and of Mitzman for A (2) 2 aiming at a better understanding of the commutation formulas. Yet the proof is missing: the theorem describing the integral form is based on observations which seem to forget some necessary commutations, those between (x + r ) (k) and (x − s ) (l) when |r + s| > 1; in [F] only the cases r + s = 0 and r + s = ±1 are considered, the former producing the binomials appearing in B h , the latter producing the elements p n,1 (and their corresponding negative elements in U im,− Z ).
Comparing the Kac-Moody presentation of the affine Kac-Moody algebras with its "Drinfeld" presentation as current algebra, one can notice a difference between the untwisted and twisted case, which is at the origin of our work. As in the simple finite dimensional case, also in the affine cases the generators of U Z described above are redundant: the Z-subalgebra of U generated by {e (k) i , f (k) i |i ∈ I, k ∈ N}, obviously contained in U Z , is actually equal to U Z . On the other hand, the situation changes when we move to the Drinfeld presentation and study the Z-subalgebra * U Z of U generated by the divided powers of the Drinfeld generators (x ± i,r ) (k) : indeed, while in the untwisted case it is still true that U Z = * U Z and (also in the twisted case) it is always true that * U Z ⊆ U Z , in general we get two different Z-subalgebras of U; more precisely * U Z U Z in case A 2n , that is when there exists a vertex i whose corresponding rank 1 subalgebra is not a copy of U(ŝ l 2 ) but is a copy of U(ŝ l 3 χ ). Thus in order to complete the description of * U Z we need to study the case of A 2 .
In the present paper we prove that the Z-subalgebra generated by is an integral form of the enveloping algebra also in the case of A (2) 2 , we exhibit a basis generalizing the one provided in [G] and in [Mi] and determine the commutation relations in a compact yet explicit formulation (see theorem 5.42 and appendix A). We use the same approach as Mitzman's, with a further simplification consisting in the remark that an element of the form G(u, v) = exp(xu) exp(yv) is characterized by two properties: G(0, v) = exp(yv) and dG du = xG. Moreover, studying the rank 1 cases we prove that, both in the untwisted and in the twisted case, U im,+ Z and * U im,+ Z are algebras of polynomials: as stated in [F], the generators of U im,+ Z are the elements Λ k introduced in [G] and [Mi] (see proposition 1.18 and remark 4.12); the generators of * U im,+ Z in the case A (2) 2 are elements defined formally as the Λ k 's after a deformation of the h r 's (see definition 5.11 and remark 5.12): describing * U im,+ Z (ŝ l 3 χ ) (denoted byŨ 0,+ Z ) has been the hard part of this work.
We work over Q and dedicate a preliminary particular care to the description of some integral forms of Q[x i |i ∈ I] and of their properties and relations when they appear in some non commutative situations, properties that will be repeatedly used for the computations in g: fixing the notations helps to understand the construction in the correct setting. With analogous care we discuss the symmetries arising both inŝ l 2 and inŝ l 3 χ . We chose to recall also the case of sl 2 and to give in a few lines the proof of the theorem describing its divided power integral form in order to present in this easy context the tools that will be used in the more complicated affine cases.
The paper is organized as follows. Section 1 is devoted to review the description of some integral forms of the algebra of polynomials (polynomials over Z, divided powers,"binomials" and symmetric functions, see [M]): they are introduced together with their generating series as exponentials of suitable series with null constant term, and their properties are rigorously stated, thus preparing to their use in the Lie algebra setting. We have inserted here, in proposition 1.18, a result about the stability of the symmetric functions with integral coefficients under the homomorphism λ m mapping x i to x m i (m > 0 fixed), which is almost trivial in the symmetric function context; it is a straightforward consequence of this observation that U im,+ Z is an algebra of polynomials and so is * U im,+ Z in the twisted case. We also provide a direct, elementary proof of this proposition (see proposition 1.19).
In section 2 we collect some computations in non commutative situations that we shall systematically refer to in the following sections.
Section 3 deals with the case of sl 2 : the one-page formulation and proof that we present (see theorem 3.2) inspire the way we studyŝl 2 andŝ l 3 χ , and offer an easy introduction to the strategy followed also in the harder affine cases: decomposing our Z-algebra as a tensor product of commutative subalgebras; describing these commutative structures thanks to the examples introduced in section 1; and glueing the pieces together applying the results of section 2. Even if the results of this section imply the commutation rules between (x + r ) (k) and (x + −r ) (l) (r ∈ Z, k, l ∈ N) in the enveloping algebra ofŝ l 2 (see remark 4.13), it is worth remarking that section 4 does not depend on section 3, and can be read independently (see remark 4.23).
In section 4 we discuss the case ofŝ l 2 . The first part of the section is devoted to the choice of the notations inÛ = U(ŝ l 2 ); to the definition of its (commutative) subalgebrasÛ ± (corresponding to the real component ofÛ),Û 0,± (corresponding to the imaginary component),Û 0,0 (corresponding to the Cartan), of their integral formsÛ ± Z ,Û 0,± Z ,Û 0,0 Z , and of the Z-subalgebraÛ Z ofÛ; and to a detailed reminder about the useful symmetries (automorphisms, antiautomorphisms, homomorphisms and triangular decomposition) thanks to which we can get rid of redundant computations. In the second part of the section the apparently tough computations involved in the commutation relations are reduced to four formulas whose proofs are contained in a few lines : proposition 4.14, proposition 4.15, lemma 4.24, and proposition 4.25, (together with proposition 1.18) are all what is needed to show thatÛ Z is an integral form ofÛ, to recognize that the imaginary (positive and negative) componentsÛ 0,± Z ofÛ Z are the algebras of polynomials Z[Λ k (ξ(±1))|k ≥ 0] = Z[ĥ ±k |k > 0], and to exhibit a Z-basis ofÛ Z (see theorem 4.29).
In section 5 we finally present the case of A 2 . As forŝ l 2 we first evidentiate some general structures of U(ŝ l 3 χ ) (that we denote hereŨ in order to distinguish it fromÛ = U(ŝ l 2 )): notations, subalgebras and symmetries. Here we introduce the elementsh k through the announced deformation of the formulas defining the elementsĥ k 's (see definition 5.11 and remark 5.12). We also describe a Q[w]-module structure on a Lie subalgebra L ofŝ l 3 χ (see definitions 5.7 and 5.9), thanks to which we can further simplify the notations. In addition, in remark 5.25 we recall the embeddings ofÛ insideŨ thanks to which a big part of the work can be translated from section 4.
The heart of the problem is thus reduced to the commutation of exp(x + 0 u) with exp(x − 1 v) (which is technically more complicated than for A 1 since it is a product involving a higher number of factors) and to deducing from this formula the description of the imaginary part of the integral form as the algebra of the polynomials in theh k 's. To the solution of this problem, which represents the central contribution of this work, we dedicate subsection 5.1, where we concentrate, perform and explain the necessary computations.
At the end of the paper some appendices are added for the sake of completeness.
In appendix A we collect all the straightening formulas: since not all of them are necessary to our proofs and in the previous sections we only computed those which were essential for our argument, we give here a complete explicit picture of the commutation relations.
Appendix B is devoted to the description of a Z-basis of Z (sym) [h r |r > 0] alternative to that introduced in the example 1.12. Z (sym) [h r |r > 0] is the algebra of polynomials Z[ĥ k |k > 0], and as such it has a Z-basis consisting of the monomials in theĥ k 's, which is the one considered in our paper. But, as mentioned above, this algebra, that we are naturally interested in because it is isomorphic to the imaginary positive part of the integral form of the rank 1 Kac-Moody algebras, was not recognized by Garland and Mitzman as an algebra of polynomials: in this appendix the Z-basis they introduce is studied from the point of view of the symmetric functions and thanks to this interpretation it is easily proved to generate freely the same Z-submodule of Q[h r |r > 0] as the monomials in theĥ k 's.
In appendix C we compare the Mitzman integral form of the enveloping algebra of type A (2) 2 with the one studied here, proving the inclusion stated above. We also show that our commutation relations imply Mitzman theorem, too.
Finally, in order to help the reader to orientate in the notations and to find easily their definitions, we conclude the paper with an index of symbols, collected in appendix D.
The study of the integral form of the affine Kac-Moody algebras from the point of view of the Drinfeld presentation, which differs from the one defined through the Kac-Moody presentation ( [G] and [Mi]) in the case A (2) 2 as outilined above, is motivated by the interest in the representation theory over Z, since for the affine Kac-Moody algebras the notion of highest weight vector with respect to the e i 's has been usefully replaced with that defined through the action of the x + i,r 's (see the works of Chari and Pressley [C] and [CP2]): in order to study what happens over the integers it is useful to work with an integral form defined in terms of the same x + i,r 's. This work is also intended to be the preliminary classical step in the project of constructing and describing the quantum integral form for the twisted affine quantum algebras (with respect to the Drinfeld presentation). It is a joint project with Vyjayanthi Chari (see also [CP]), who proposed it during a period of three months that she passed as a visiting professor at the Department of Mathematics of the University of Rome "Tor Vergata". The commutation relations involved are extremely complicated and appear to be unworkable by hands without a deeper insight; we hope that a simplified approach can open a viable way to work in the quantum setting. §1 Integral form and commutative examples In this section we give the definition of integral form and summarize, fixing the notations useful to our purpose, some well known commutative examples (deeply studied and systematically exposed in [M]), which will play a central role in the non commutative enveloping algebra of finite and affine Kac-Moody algebras.
In particular an integral form of U is (can be identified to) a Z-subalgebra of U , and a Z-basis of an integral form of U is a Q-basis of U .
This can be said also as follows: Let M be a free Z-module and V = Q ⊗ Z M and consider the functor S = "symmetric algebra" from the category of Z-modules (respectively Q-vector spaces) to the category of commutative unitary Z-algebras (respectively commutative unitary Q-algebras). Then SM is an integral form of SV and SM ∩ V = M . By definition, every integral form of SV containing M contains SM , that is SM is the least integral form of SV containing M .
We are interested in other remarkable integral forms of SV containing M .
Remark that the generating series of the a (k) 's is exp(au), that is Example 1.6. Let {x i } i∈I be a Z-basis of M . Then it is well known and trivial that: The viceversa is obviously also true: (1.8) {sdivmv} {sdivmv} Notation 1.9. Let a be an element of a unitary Q-algebra U . The "binomials" of a are the elements Notice that a k is the image of x k through the evaluation ev a : Q[x] → U mapping x to a. Now consider the series exp(a ln(1 + u)): this is a well defined element of U [[u]] (because a ln(1 + u) ∈ uU [[u]]) whose coefficients are polynomials in a; this means that with the notations above exp(a ln(1 + u)) = ev a (exp(xln(1 + u))).
In particular if we want to prove that for all U and for all a ∈ U the generating series of the a k 's is exp(a ln(1 + u)) it is enough to prove the claim in the case a = x ∈ Q[x], and to this aim it is enough to compare the evaluations on an infinite subset of Q (for istance on N), thus reducing the proof to the trivial observation that ∀n ∈ N k∈N n k u k = (1 + u) n = exp(n ln(1 + u)).
Thus in general the generating series exp(a ln(1 + u)) of the a k 's can and will be denoted as (1 + u) a ; more explicitly It is clear from the definition of (1 + u) a that if a and b are commuting elements of U then It is also clear that the Z-submodule of U generated by the coefficients of (1+u) a+m (a ∈ U , m ∈ Z) depends only on a and not on m; it is actually a Z-subalgebra of More precisely for each m ∈ Z and n ∈ N the Z-submodule of U generated by the a+m k 's for k = 0, ..., n (a ∈ U ) depends only on a and n and not on m. Finally notice that in U [[u]] we have d du (1 + u) a = a(1 + u) a−1 . Example 1.11. Let {x i } i∈I be a Z-basis of M . Then it is well known and trivial that: i k ∈ Z[x 1 , ..., x n ] Sn k ⊆ Q[x 1 , ..., x n ] Sn k . It is also well known that for n 1 ≥ n 2 the natural projection and their inverse limits p r ∈ Z[e 1 , ..., e k , ...] (π n1,n2 (p for all r > 0 and all n 1 ≥ n 2 ) give another set of generators of the Q-algebra of the symmetric functions: the p r 's are algebraically independent and Finally Z[e 1 , ..., e k , ...] is an integral form of Q[p 1 , ..., p r , ...] containing p r for all r > 0 (more precisely a linear combination of the p r 's lies in Z[e 1 , ..., e k , ...] if and only if it has integral coefficients), the relation between the e k 's and the p r 's being given by: In this context we use the notation to stress the dependence of the e k 's on the p r 's we set e =p, that iŝ Remark 1.16. With the notations above, let ϕ : Q[p 1 , ..., p r , ...] → U be an algebra-homomorphism and a = ϕ(p 1 ): i) if ϕ(p r ) = 0 for r > 1 then ϕ(p k ) = a (k) for all k ∈ N; ii) if ϕ(p r ) = a for all r > 0 then ϕ(p k ) = a k for all k ∈ N. Hence Z (sym) is a generalization of both Z (div) and Z (bin) .
(it is easy to check that these integral forms are different for example in degree 3); iii) not all the sign changes of the p r 's produce different Z (sym) -forms of Q[p r |r > 0]: (see remark 1.3,1),ii)).
In general it is not trivial to understand whether an element of Q[p r |r > 0] belongs or not to Z (sym) [p r |r > 0]; proposition 1.18 gives an answer to this question, which is generalized in proposition 1.22 (the examples in remark 1.17, ii) and iii) can be obtained also as applications of proposition 1.22).
Proposition 1.18. Let us fix m > 0 and let λ m : Q[p r |r > 0] → Q[p r |r > 0] be the algebra homomorphism defined by λ m (p r ) = p mr for all r > 0. ..., x n ] be the algebra homomorphism defined by λ m (x i ) = x m i for all i = 1, ..., n. We obviously have that We also propose a second, direct, proof of proposition 1.18, which provides in addition an explicit expression of the λ m (p k )'s in terms of thep l 's.
Proposition 1.19. Let m and λ m be as in proposition 1.18 and ω ∈ C a primitive m th root of 1. Then Proof. The equality in the statement is an immediate consequence of whose exponential is the claim. Then for all k > 0 In order to characterize the functions a : Z + → Q such that we introduce the notation 1.20, where we rename the p r 's into h r since in the affine Kac-Moody case the Z (sym) -construction describes the imaginary component of the integral form. Moreover from now on p i will denote a positive prime number.
Notation 1.20. Given a : 1 1 denotes the function defined by 1 1 r = 1 for all r ∈ Z + ; for all m > 0 1 1 (m) denotes the function defined by 1 1 Recall 1.21. The convolution product * in the ring of the Q-valued arithmetic functions The Möbius function µ : Z + → Q defined by Proof. Remark that a = 1 1 * µ * a, that is and, by proposition 1.18 (see also notation 1.20), ii) ∀n < m 0ĥ Proposition 1.23. Let a : Z + → Z be a function satisfying the condition p r |a mp r − a mp r−1 ∀p, m ∈ Z + with p prime and (m, p) = 1.
For n > 1 remark that Recall that if P is the set of the prime factors of n and p ∈ P then (µ * a)(n) = S⊆P (−1) #S a n q∈S q = = S ′ ⊆P \{p} (−1) #S ′ (a n q∈S ′ q − a n p q∈S ′ q ). (

1.24) {mpr} {mpr}
The claim follows from the remark that p r ||n if and only if p r || n q∈S ′ q .
Remark 1.25. The viceversa of proposition 1.23 is trivially true, too, and is immediately proved applying (1.24) to the minimal n > 0 such that there exists p|n and r > 0 (p r |n, n = mp r ) not satisfying the hypothesis of the statement.
Proposition 1.18 will play an important role in the study of the commutation relations in the enveloping algebra ofŝ l 2 (see remarks 4.11,vi) and 4.22) and ofŝ l 3 χ (see remark 5.15 and proposition 5.17,iv)). Proposition 1.22 is based on and generalizes proposition 1.18; it is a key tool in the study of the integral form in the case of A 2 , see corollary 5.39. A more precise connection between the integral form Z (sym) [h r |r > 0] of Q[h r |r > 0] and the homomorphisms λ m 's, namely another Z-basis of Z (sym) [h r |r > 0] (basis defined in terms of the elements λ m (ĥ k )'s and arising from Garland's and Mitzman's description of the integral form of the affine Kac-Moody algebras) is discussed in appendix B. §2 Some non commutative cases We start this section with a basic remark.
ii) Let U be an associative unitary Q-algebra (not necessarily commutative) and Remark 2.1,ii) suggests that if we have a (linear) decomposition of an algebra U as an ordered tensor product of polynomial algebras U i (i = 1, ..., N ), that is we have a linear isomorphism then one can tackle the problem of finding an integral form of U by studying the commutation relations among the elements of some suitable integral forms of the U i 's. Glueing together in a non commutative way the different integral forms of the algebras of polynomials discussed in section 1 is the aim of this section, which collects the preliminary work of the paper: the main results of the following sections are applications of the formulas found here.
Notation 2.2. Let U be an associative Q-algebra and a ∈ U . We denote by L a and R a respectively the left and right multiplication by a; of Lemma 2.3. Let U be an associative unitary Q-algebra.
iv) if exp(a) converges so do exp(L a ) and exp(R a ), and we have Proof. Statements v) and vi) are immediate consequence respectively of the fact that for all n ∈ N: v) a n b = bc n ; vi) ab (n) = b (n) a + b (n−1) c. vii) follows from v) ad vi). viii) follows from vii): The other points are obvious.
Proposition 2.4. Let us fix m ∈ Z and consider the Q-algebra structure on U = are integral forms of U : their images in U are closed under multiplication, and coincide. Indeed or equivalently, with a notation that will be useful in the following, Proof. The relation between x and h can be written as and for all k > 0. In particular it holds for P (h) = h l , that is . ( The conclusion follows multiplying by u k and summing over k.
Proposition 2.9. Let us fix m ∈ Z and consider the Q-algebra structure on Proof. Since z commutes with x and y we just have to straighten y (r) x (s) . Thus the claim is a straightforward consequence of lemma 2.3,vii): Proposition 2.11. Let us fix m, l ∈ Z and consider the Q-algebra structure on Then, recalling the notation is an integral form of U .
Proof. 2.12 follows from lemma 2.3, vii) remarking that which is a straightforward consequence of 2.12.
Proof. By proposition 2.3,iv) If n = 0, 1 the claim is obvious; if n > 1, f n−1 (x) = r>0 a r T r u r (x) with a r ∈ Z for all r > 0, f commutes with T , and by the inductive hypothesis Proposition 2.14. Let us fix integers m d 's (d > 0) and consider elements {h r , Let T be an algebra automorphism of U such that Then, recalling the notation If moreover the subalgebras of U generated by {h r |r > 0} and {x r |r ∈ Z} are isomorphic respectively to Q[h r |r > 0] and Q[x r |r ∈ Z] and there is a Q-linear is an integral form of U .
Proof. This is an application of lemma 2.13: let h = r>0 (−1) r−1 hr r u r ; then Then and the analogous statement for x r follows applying T −r .
the hypothesis on the commutativity of the subalgebra generated by the x r 's implies that ( r≥0 a r x r u r ) (k) lies in the subalgebra of U generated by the divided powers r |r ∈ Z, k ≥ 0}, which allows to conclude the proof thanks to the last hypotheses on the structure of U .
Remark 2.16. Proposition 2.14, implies proposition 2.4: indeed when m 1 = m, m d = 0 ∀d > 1 we have a projection h r → h, x r → x, which maps exp(x 0 u) to exp(xu),ĥ(u) to (1 + u) h and T to the identity. §3 The integral form of sl 2 (A 1 ) The results about sl 2 and the Z-basis of the integral form U Z (sl 2 ) of its enveloping algebra U(sl 2 ) are well known (see [Ko] and [S]). Here we recall the description of U Z (sl 2 ) in terms of the non-commutative generalizations described in section 2, with the notations of the commutative examples given in section 1. The proof expressed in this language has the advantage to be easily generalized to the affine case.
Definition 3.1. sl 2 (respectively U(sl 2 )) is the Lie algebra (respectively the associative algebra) over Q generated by {e, f, h} with relations is an integral form of U(sl 2 ).
Proof. Thanks to proposition 2.4, we just have to study the commutation between e (k) and f (l) for k, l ∈ N. Let us recall the commutation relation which is a direct application of lemma 2.3,iv) and of the relations To obtain this result we derive remarking lemma 2.3,ix) and then apply formulas 2.7 and 3.4: Remarking that it follows that the right hand side of 3.3 is an integer form of U(sl 2 ) (containing U Z (sl 2 )). Finally remark that inverting the exponentials on the right hand side, the formula (3.5) gives an expression of (1 + uv) h in terms of the divided powers of e and f , so The results aboutŝ l 2 and the integral formÛ Z of its enveloping algebraÛ are due to Garland (see [G]). Here we simplify the description of the imaginary positive component ofÛ Z proving that it is an algebra of polynomials over Z and give a compact and complete proof of the assertion that the set given in theorem 4.29 is actually a Z-basis ofÛ Z . This proof has the advantage, following [Mi], to reduce the long and complicated commutation formulas to compact, simply readable and easily proved ones. It is evident from this approach that the results forŝ l 2 are generalizations of those for sl 2 , so that the commutation formulas arise naturally recalling the homomorphism induced by the evaluation of t at 1 . On the other hand these results and the strategy for their proof will be shown to be in turn generalizable toŝ l 3 χ .
As announced in the introduction, the proof of theorem 4.29 is based on a few results: proposition 4.14, proposition 4.15, lemma 4.24, and proposition 4.25.
Definition 4.4.Û is endowed with the following anti/auto/homo/morphisms: σ is the antiautomorphism defined on the generators by: Ω is the antiautomorphism defined on the generators by: T is the automorphism defined on the generators by: for all m ∈ Z, λ m is the homomorphism defined on the generators by: Remark 4.6. σΩ = Ωσ, σT = T σ, σλ m = λ m σ for all m ∈ Z; Definition 4.8. Here we define some Z-subalgebras ofÛ: The notations are those of section 1.
We want to prove thatÛ 0 Z =Û 0,− ZÛ 0,0 ZÛ 0,+ Z , so that it is an integral form ofÛ 0 , and thatÛ Z =Û − ZÛ 0 ZÛ + Z , so thatÛ Z is an integral form ofÛ. As in the case of sl 2 , working inÛ [[u]] (see the notation below) simplifies enormously the proofs and gives a deeper insight to the question.
Notation 4.9. We shall consider the following elements inÛ [[u]]: Remark 4.10. Notice that ev • T = ev and Remark 4.11. Here we list some obvious remarks.
thanks to v), to proposition 1.18 and to remarks 4.6 and 4.10.
Remark 4.12. The elementsĥ k 's with k > 0 generate the same Z-subalgebra of U as the elements Λ k 's (k ≥ 0) defined in [G]. Indeed let n≥0 p n u n = P (u) =ĥ(−u) −1 ; then remarks 1.3,1,ii) and 1.17 On the other hand applying λ m we get [G]).
Remark 4.13. Remark that for all r ∈ Z the subalgebra ofŝl 2 generated by maps isomorphically onto sl 2 through the evaluation homomorphism ev (see formula 4.1). On the other hand for each r ∈ Z there is an injection U(sl 2 ) →Û: In particular theorem 3.2, implies that the elements h0+rc k belong toÛ Z for all r ∈ Z, k ∈ N (thus, remarking that the elements c k 's are central and the example 1.11, we get thatÛ 0,0 Z ⊆Û Z ) and proposition 2.4 implies thatÛ 0,0 Proposition 4.14. The following identity holds inÛ: Proof. Since [h r , h s ] = 2rδ r+s,0 c, the claim is proposition 2.11 with m = 2, l = 0.
Z and applying σ we get the reverse inclusions.
We are now left to prove thatÛ +

ZÛ
To this aim we study the commutation relations between (x + r ) (k) and (x − s ) (l) or equivalently between exp(x + r u) and exp(x − s v).
Remark 4.22. Remark 4.13, implies that exp( In order to prove a similar result for exp( , so that remark 4.11,iv),v),vi) allows us to reduce to the case r = 0, s = 1. This case will turn out to be enough also to prove thatÛ 0 Z ⊆Û Z .
Remark 4.23. In the study of the commutation relations inÛ Z remark that (see remark 4.10). Viceversa once we have such an expression for exp(x + 0 u) exp(x − 1 v) applying T −r λ r+s we can deduce from it the identity (3.5) and the expression for exp(x + r u) exp(x − s v) for all r, s ∈ Z (also in the case r + s = 0). Remark that exp(vx − (−uv))ĥ + (uv)exp(ux + (−uv)) is an element ofÛ [[u, v]] which has the required properties (see remark 4.10) and belongs toÛ − Our aim is to prove that Proof. The claim follows from lemma 2.3,iv) remarking that Remark that, thanks to the derivation rules (lemma 2.3,ix)), to proposition 4.15, and to lemma 4.24, we have: Proof. ThatÛ 0,+ Z ⊆Û Z is a consequence of proposition 4.25 inverting the exponentials (see the proof theorem 3.2), which implies also (applying Ω) thatÛ 0,− Z ⊆Û Z ; the claim then follows thanks to remark 4.13.
Proof. We want to prove thatÛ − , thus we just need to perform the correct induction to deal with the general y ± ∈Û ± Z . Remark that setting induces a Z-gradation onÛ (since the relations definingÛ are homogeneous) and onÛ Z (since its generators are homogeneous), which is preserved by σ, T ±1 and λ m ∀m ∈ Z; in particular it induces N-gradationŝ We want to prove that b) if k 1 , k 2 > 0 are such that k 1 + k 2 = k or l 1 , l 2 > 0 are such that l 1 + l 2 = l, thenÛ (4.28) follows from a) and b).
Theorem 4.29. The Z-subalgebraÛ Z ofÛ generated by and a Z-basis ofÛ Z is given by the product where B ± , B 0,± and B 0,0 are the Z-bases respectively ofÛ ± Z ,Û 0,± Z andÛ 0,0 Z given as follows: 2 ) In this section we describe the integral formŨ Z of the enveloping algebraŨ of the Kac-Moody algebra of type A 2 generated by the divided powers of the Drinfeld generators x ± r ; unlike the untwisted case, this integral form is strictly smaller than the one (studied in [Mi]) generated by the divided powers of the Chevalley generators e 0 , e 1 , f 0 , f 1 (see appendix C). However, the construction of a Z-basis ofŨ Z follows the idea of the analogous construction in the case A (1) 1 , seen in the previous section; this method allows us to overcome the technical difficulties arising in case A (2) 2 -difficulties which seem otherwise overwhelming. The commutation relations needed to our aim can be partially deduced from the case A (1) 1 : indeed, underlining some embeddings ofŝ l 2 intoŝl 3 χ (see remark 5.25), the commutation relations inÛ can be directly translated into a class of commutation relations inŨ (see corollary 5.26, proposition 5.27 and the appendix A for more details). Yet, there are some differences between A 1 and A 2 . First of all, the real (positive and negative) components ofŨ are no more commutative (this is well known: it happens in all the affine cases different from A (1) 1 , as well as in all the finite cases different from A 1 ), hence the study of their integral form requires some -easy -additional observations (see lemma 5.20). The non commutativity of the real components ofŨ makes the general commutation formula between the exponentials of positive and negative Drinfeld generators technically more complicated to compute and express than in the case ofŝ l 2 ; nevertheless, general and explicit compact formulas can be given in this case, too, always thanks to the exponential notation. As already seen, the simplification provided by the exponential approach lies essentially on lemma 2.3,iv), which allows to perform the computations inŨ reducing to much simpler computations inŝ l 3 χ , and as we shall show, we need to somehow "deform" the h r 's (by changhing some of their signs) to get a basis ofŨ 0,+ Z by the (sym)-construction (see definition 5.11, example 1.12 and remark 1.17). Notice that in order to prove thatŨ Z is an integral form ofŨ and that B is a Zbasis ofŨ Z (theorem 5.42) it is not necessary to find explicitly all the commutation formulas between the basis elements. In any case, for completeness, we shall collect them in the appendix A.
Definition 5.3.ŝ l 3 χ andŨ are endowed with the following anti/auto/homo/morphisms: σ is the antiautomorphism defined on the generators by: ; Ω is the antiautomorphism defined on the generators by: T is the automorphism defined on the generators by: for all odd integer m ∈ Z, λ m is the homomorphism defined on the generators by: Remark that if m is even λ m is not defined onŨ, but it is still defined onŨ 0,+ = Q[h r |r > 0].
Definition 5.9. L is endowed with the Q[w]-module structure defined by Proof. The assertions are just a translation of the defining relations ofŨ: For iv), remark that 2(2 + (−1) r−1 )w r = 4w r − 2(−w) r .
Definition 5.11. Here we define some Z-subalgebras ofŨ: Z is the Z-subalgebra ofŨ generated byŨ 0,− Z ,Ũ 0,0 Z andŨ 0,+ Z . The notations are those of section 1. In particular remark the definition ofŨ 0,± Z (where the ε r 's represent the necessary "deformation" announced in the introduction of this section, and discussed in details in proposition 1.22) and introduce the notation Remark 5.12. It is worth underlining thath More precisely the Z-subalgebras generated respectively by {ĥ k |k > 0} and {h k |k > 0} are different and not included in each other: Before entering the study of the integral forms just introduced, we still dwell on the comparison betweenh + (u) andĥ + (u), proving lemma 5.15, that will be useful later.
(1 + r>0 a r u r ) m = 1 + m 2 u implies 1 + m 2 u = 1 + m r>0 a r u r + k>1 m k r>0 a r u r k .
Let us prove by induction on s that a s ∈ mZ: if s = 1 we have that ma 1 = m 2 ; if s > 1 the coefficient c s of u s in k>1 m k r>0 a r u r k is a combination with integral coefficients of products of the a t 's with t < s, which are all multiple of m. Then, since k ≥ 2, m 2 |c s . But ma s + c s = 0, thus m|a s .
Proposition 5.17. The following stability properties under the action of σ, Ω, T ±1 and λ m (m ∈ Z odd) hold: Proof. The only non-trivial assertion is the claim thatŨ 0,+ Z is λ m -stable when m > 0, which was proved in lemma 5.15,i). The assertion about λ m (Ũ 0,± Z ) in the general case follows using that Remark that Remark 5.18. The stability properties described in proposition 5.17 imply that: is T ±2 -stable and λ mstable (m ∈ Z odd); in particular: Proposition 5.19. The following identities hold inŨ: Proof. Since [h r , h s ] = [ε r h r , ε s h s ] = δ r+s,0 2r(2 + (−1) r−1 )c, the claim is proposition 2.11 with m = 4, l = −2.
Lemma 5.20. The following identity holds inŨ for all r, s ∈ Z: Proof. The claim follows by immediate application of 2.5.
Applying σ we get the reverse inclusion and applying Ω we obtain the claim for U − Z .
Now that we have describedŨ 0 Z ,Ũ ± Z and the Z-subalgebras generated byŨ 0 Z and Before attaching this problem in its generality it is worth evidentiating the existence of some copies ofŝl 2 insideŝ l 3 χ , hence of embeddingsÛ ֒→Ũ, that induce some useful commutation relations inŨ.
Proposition 5.27.Ũ +,0 The assertion forŨ ±,1 Z follows applying T , see proposition 5.17,i),ii) and iv). §5.1 exp(x + 0 u) exp(x − 1 v) andŨ 0,+ Z : here comes the hard work We shall deal with the commutation betweenŨ +,0 Z andŨ −,1 Z following the strategy already proposed forÛ Z and recalling remark 5.18,iv): finding an explicit expression involving suitable exponentials for u, v]] and proving that all its coefficients lie iñ Since here there are more factors involved, the computation is more complicated than in the case ofŝ l 2 and the simplification provided by this approach is even more evident. On the other hand it is not immediately clear from the commutation formula that our element belongs toŨ − ZŨ 0 ZŨ + Z , or better: the factors relative to the (negative, resp. positive) real root vectors will be evidently elements ofŨ − Z , resp. U + Z , while proving that the null part lies indeed inŨ 0 Z is not evident at all and will require a deeper inspection (see remark 5.37, lemma 5.38 and corollary 5.39). As we shall see, in order to complete the proof thatŨ 0,+ Z ⊆Ũ Z (see proposition 5.41), it is useful to compute also exp(x + 0 u) exp(X − 1 v). The two computations (exp(x + 0 u) exp(yv) with y = x − 1 or y = X − 1 ) are essentially the same and will be performed together (see the considerations from remark 5.28 to lemma 5.32, of which the propositions 5.33 and 5.34 are straightforward applications); even though exp(x + 0 u) exp(x − 1 v) presents more symmetries than exp(x + 0 u) exp(X − 1 v) (see remark 5.30,iii)), its interpretation will require more work, since it is not evident the connection withŨ 0,+ Z , as just mentioned. Remark 5.28. Let G = G(u, v) ∈Ũ[[u, v]] and y ∈ L − (see definition 5.7); then G(u, v) = exp(x + 0 u) exp(yv) if and only if the following two conditions hold: Notation 5.29. In the following G − , G 0 , G + will denote elements ofŨ [[u, v]] of the form u, v]].x + 1 , η ∈ wQ[w] [[u, v]].h 0 . G(u, v) will denote the element G(u, v u, v]] be as in notation 5.29. Then: ii) If moreover G = exp(x + 0 u) exp(yv) with y ∈ L − , the property b) of remark 5.28 translates into Observe that T λ −1 Ω(X + 2r+1 ) = −X − 2r+3 ∀r ∈ Z.
The following lemma is based on lemma 2.3, iv) and on the defining relations ofŨ (definition 5.1).
where, with the notations of lemma 5.32, In particular: Proof. We use the notation fixed in 5.29. It is obvious that G(0, v) = exp(X − 1 v), so that the condition a) of remark 5.28 is fulfilled, and we need to verify condition b), following lemmas 5.31,vi) and 5.32. Remark that

Now let us recall that for all
hence, fixing a = 4 2 w 2 v 2 , we get The relations to prove are then equivalent to the following: which are easily verified. Then, since α ± , β ± , γ ± have integral coefficients, i) follows from example 1.6, remark 5.13 and lemma 5.15,iii). ii) follows at once from the above considerations, inverting the exponentials.
where, with the notations of lemma 5.32, Proof. We use the notations fixed in 5.29. It is obvious that G(0, v) = exp(x − 1 v), so that the condition a) of remark 5.28 is fulfilled, and we need to verify condition b), following lemma 5.32. First of all remark that and that thus, replacing t by wuv, we get Hence the relations of lemma 5.32 involving η are easily proved: and exp(−4η(w) + 2η(−w)) = 1 − 2wuv − w 2 u 2 v 2 (1 + 2wuv − w 2 u 2 v 2 ) 2 while, on the other hand, In order to prove the remaining relations remark that for all n, m ∈ N d dt which helps to compute the derivative of α ± (w 2 ), β ± (−w 2 ), γ − (w 2 ), fixing t = wuv and recalling that d du = wv d dt : The relations to prove are then equivalent to the following: which are easily verified. u, v]] (see notation 5.29). Then, in order to prove that we just need to show that exp(η) ∈Ũ 0 Z [[u, v]]. This will imply thatŨ − ZŨ 0 ZŨ + Z is closed under multiplication, hence it is an integral form ofŨ, obviously containing U Z . In order to prove thatŨ Z =Ũ − ZŨ 0 ZŨ + Z we need to show in addition thatŨ 0 Z ⊆Ũ Z . The last part of this paper is devoted to prove that (see corollary 5.39) and thatŨ 0 Z ⊆Ũ Z (see proposition 5.41). Notation 5.36. In the following d : Z + → Q denotes the function defined by n>0 (−1) n−1 d n n u n = 1 2 ln(1 + 2u − u 2 ) andd = εd (that isd n = ε n d n for all n > 0, where ε n has been defined in definition 5.11). Remark that with this notation we have exp(η) =ĥ Remark 5.37. From 1 + 2u − u 2 = (1 + (1 + √ 2)u)(1 + (1 − √ 2)u), we get that: i) for all n ∈ Z + d n = 1 2 ((1 + √ 2) n + (1 − √ 2) n ); equivalently ∃δ n ∈ Z such that ii) d n is odd for all n ∈ Z + ; δ n is odd if and only if n is odd.
, which is not a multiple of 4, see propositions 1.22 and 1.23).
Lemma 5.38. Let p, m, r ∈ Z + be such that p is prime and (m, p) = 1. Then Proof. The claim is obvious for p r = 2 since the d n 's are all odd. In general if n is any positive integer it follows from remark 5.37 that If p = 2 this means that d 2n = d 2 n + 2δ 2 n , δ 2n = 2d n δ n , hence 2 r ||δ 2 r m (recall that δ m is odd since m is odd) d 2 r m ≡ d 2 2 r−1 m (mod 2 2r−1 ), from which it follows that d 2m ≡ −1 (mod 4), d 2 r m ≡ 1 (mod 2 r+1 ) if r > 1 : indeed, since d m and δ m are odd, while if r ≥ 2 then 2r − 1 ≥ r + 1 and by induction on r we get d 2 r m ≡ d 2 2 r−1 m = (±1 + 2 r k) 2 ≡ 1 (mod 2 r+1 ).
These last relations immediately imply the claim for p = 2. Now let p = 2. Then Suppose that d n = d + p r−1 k, δ n = δ + p r−1 k ′ with k = k ′ = 0 if r = 1. Then The above relations allow us to prove by induction on r > 0 that if ζ p is defined by the properties ζ p ∈ {±1}, ζ p ≡ (p) 2 p−1 2 then d p r m ≡ d p r−1 m (mod p r ) and δ p r m ≡ ζ p δ p r−1 m (mod p r ) : Proof. The claim follows from propositions 1.22 and 1.23, remark 5.37 and lemma 5.38, remarking that if m is odd then while if (m, p) = 1 and p r = 4 then Thus for all n > 0ĥ Proof. The proof is identical to that of proposition 4.27 replacingÛ withŨ, having care to remark that in this case, too, if r + s is even this follows at once comparing proposition 5.27 with the properties of the gradation, while if r + s is odd it is true by proposition 5.34 and remark 5.18,iv).
But (d j , 2 2j−1 ) = 1 because d j is odd, henceh j ∈ Z. We can now collect all the results obtained till now in the main theorem of this work.
Theorem 5.42. The Z-subalgebraŨ Z ofŨ generated by is an integral form ofŨ. More preciselỹ and a Z-basis ofŨ Z is given by the product where B ±,0 , B ±,1 , B ±,c , B 0,± and B 0,0 are the Z-bases respectively ofŨ ±,0 Z ,Ũ ±,1 Z , U ±,c Z ,Ũ 0,± Z andŨ 0,0 Z given as follows: For the sake of completeness we collect here the commutation formulas of A 2 , inserting also the formulas that we didn't need for the proof of theorem 5.42. Notation A.1 and remark A.2 will help writing some of the following straightening relations and to understand the origin of some apparently misterious terms.
Remark that the maps p(t) → p + (t) and We shall now list a complete set of straightening formulas inŨ Z .
I) Zero commutations regardingŨ 0,0 Z : II) Relations inŨ 0,+ Z (from which those inŨ 0,− Z follow as well): where ω is a primitive m th root of 1, that is where the k m 's are integers defined by the identity The corresponding relations inŨ 0,− Z are obtained applying Ω, that is just replacing IV) Commuting elements and straightening relations inŨ + Z (and inŨ − Z ): uv) exp(x + r u). All the relations inŨ − Z are obtained from those inŨ + Z applying the antiautomorphism Ω; in particular if r + s is odd which can be written in a more compact way observing that that is more symmetric but less explicit in terms of the given basis ofŨ Z . Applying the homomorphism Applying Ω one analogously gets the expression for exp(X + 2r+1 u) exp(x − s v). VII,d) The remaining relations: The general straightening formula for exp(x + r u) exp(x − s v) when r + s is odd is obtained from the case r = 0, s = 1 applying T −r λ r+s , remarking that w L ± → T ∓(r+s) . §B Garland's description of U im,+ Z In this appendix we focus on the imaginary positive part U im,+ Z of U Z = U Z (g) (see section 0) when g is an affine Kac-Moody algebra of rank 1 (that is g =ŝ l 2 or g =ŝ l 3 χ : these cases are enough to understand also the cases of higher rank): we aim at a better understanding of Garland's (and Mitzman's) basis of U im,+ Z and of its connection with the basis consisting of the monomials in theĥ k 's, basis which arises naturally from the description of U im,+ Z as Z (sym) [h r |r > 0] = Z[ĥ k |k > 0]. First of all let us fix some notations and recall Garland's description of U im,+ Z . Definition B.1. With the notations of example 1.12 and proposition 1.18 let us define the following elements and subsets in Q[h r |r > 0]: Then, with our notation, Garland's description of U im,+ Z can be stated as follows: Remark B.4. Once proved that U im,+ Z is the Z-subalgebra of U generated by {λ m (ĥ k )|m > 0, k ≥ 0} (hence by B λ or equivalently by Z λ [h r |r > 0]), proceeding in two different directions leads to the two descriptions of U im,+ Z that we want to compare: Hence ⋆) and ⋆⋆) imply that U im,+ . ⋆) has been proved in [G] by induction on a suitably defined degree. The first step of the induction is the second assertion of [G]-lemma 5.11(b), proved in [G]-section 9: for all k, l ∈ Nĥ kĥl − k+l k ĥ k+l is a linear combination with integral coefficients of elements of B λ of degree lower than the degree ofĥ k+l . In the proof the author uses that B λ is a Q-basis of Q[h r |r > 0] and concentrates on the integrality of the coefficients: he studies the action of h onŝ l 3 ⊗N where h is the commutative Lie-algebra with basis {h r |r > 0} and N ∈ N is large enough (N is the maximum among the degrees of the elements of B λ appearing inĥ kĥl with non-integral coefficient, assuming that such an element exists): h is a subalgebra of sl 2 and there is an embedding ofŝ l 2 inŝ l 3 for every vertex of the Dynkin diagram of sl 3 , so that fixing a vertex of the Dynkin diagram of sl 3 induces an embedding h ⊆ŝ l 2 ֒→ŝl 3 , hence an action of h onŝl 3 . But the integral form ofŝ l 3 defined as the Z-span of a Chevalley basis is U Z (ŝ l 3 )-stable; since the stability under U Z (ŝ l 3 ) is preserved by tensor products ([G]-section 6), the author can finally deduce the desired integrality property ofĥ kĥl from the study of the h-action onŝ l 3 ⊗N .
Garland's argument has been sometimes misunderstood: it is the case for instance of [JM] where the authors affirm (in lemma 1.5) that [G]-lemma 5.11(b) implies that U im,+ Z = Z[ĥ k |k > 0], while, as discussed above, it just implies the inclusion On the other hand Garland's argument strongly involves many results of the (integral) representation theory of the Kac-Moody algebras, while ⋆) is a property of the algebra Q[h r |r > 0] and of its integral forms that can be stated in a way completely independent of the Kac-Moody algebra setting: The above considerations motivate the present appendix, whose aim is to propose a self-contained proof of ⋆), independent of the Kac-Moody algebra context: on one hand we think that a direct proof can help evidentiating the essential structure of the integral form of Q[h r |r > 0] arising from our study; on the other hand the idea of isolating the single pieces and glueing them together after studying them separately is much in the spirit of this work, so that it is natural for us to explain also Garland's basis of U im,+ Z through this approach; and finally we hope that presenting a different proof can also help to clarify the steps which appear more difficult in Garland's proof.
In the following we go back to the description of Z[ĥ k |k > 0] as the algebra of the symmetric functions and we show that B λ is a basis of Z[ĥ k |k > 0] by comparing it with a well known Z-basis of this algebra.
Notation B.6. As in remark B.5, for all k : Z + → N finitely supported let us denote by (σx) k the limit of the elements a 1 ,...,an #{i|a i =m}=km ∀m>0 n i=1 x ai i (n ∈ N).
By abuse of notation, when n ≥ m>0 k m we shall write which is justified because, under the hypothesis that n ≥ m>0 k m , k is determined by the set {(a 1 , ..., a n )|#{i = 1, ..., n|a i = m} = k m ∀m > 0}.
is the Z-module generated by B (n) x .
Remark B.8. By the very definition of B x , see remark B.5, ii); ii) h ∈ Z (n) x means that for all N ≥ n each monomial in the x i 's appearing in π N (h) with nonzero coefficient involves no more that n indeterminates Lemma B.9. Let n, n ′ , n ′′ ∈ N and k ′ , k ′′ : Z + → N be such that n ′ + n ′′ = n, x follows from remark B.8,ii), so we just need to: i) prove that if n i=1 x ai i with a i = 0 ∀i = 1, ..., n is the product of two monomials M ′ and M ′′ appearing with nonzero coefficient respectively in (σx) k ′ and in (σx) k ′′ then #{i|a i = m} = k ′ m + k ′′ m for all m > 0; ii) compute the coefficient of (σx) k ′ +k ′′ in the expression of (σx) k ′ · (σx) k ′′ as a linear combination of the (σx) k 's when ∀m > 0 k ′ m and k ′′ m are not simultaneously non zero, and find that it is 1. i) is obvious because the condition a i = 0 ∀i = 1, ..., n implies that the indeterminates involved in M ′ and those involved in M ′′ are disjoint sets. For ii) it is enough to show that, under the further condition on k ′ m and k ′′ m , the monomial n i=1 x ai i chosen in i) uniquely determines M ′ and M ′′ such that Lemma B.10. Let k : Z + → N, n ∈ N be such that m>0 k m = n. Then: ; but m>0 k [m] = k and the claim follows.
Proof. We prove by induction on n that B ∀n ∈ N, the case n = 0 being obvious. Let n > 0: by the inductive hypothesis B (n−1) λ and B is a Z-basis of Z (n) x ; but B In the present appendix we compare the integral formŨ Z = * U Z (ŝ l 3 χ ) ofŨ described in section 5 with the integral form U Z (ŝ l 3 χ ) of the same algebraŨ introduced and studied by Mitzman in [Mi], that we denote here byŨ Z,M and that is easily defined as the Z-subalgebra ofŨ generated by the divided powers of the Kac-Moody generators e i , f i (i = 0, 1): see also remark C.11. More precisely: Definiton C.1.Ũ is the enveloping algebra of the Kac-Moody algebra whose generalized Cartan matrix is A (2) 2 = (a i,j ) i,j∈{0,1} = 2 −1 −4 2 (see [K] i |i = 0, 1, k ∈ N}. Remark C.3. The Kac-Moody presentation ofŨ (definition C.1) and its presentation given in definition 5.1 are identified through the following isomorphism: Notation C.4. In order to avoid in the following any confusion and heavy notations, we set: where the X ± 2r+1 's, the h r 's and c are those introduced in definition 5.1 (thus e 0 = y − 1 , f 0 = y + −1 , while the Kac-Moody h 0 and h 1 appearing in definition C.1 are respectivelyc − h 0 and 2h 0 ; moreoverŨ Z,M is the Z-subalgebra ofŨ generated by {(x ± 0 ) (k) , (y ± ∓1 ) (k) |k ∈ N}).
Mitzman completely described the integral form generated by the divided powers of the Kac-Moody generators in all the twisted cases; in case A (2) 2 his result can be stated as follows, using our notations (see examples 1.6, 1.11 and 1.12, definition B.1 and notation C.4): The isomorphisms are all induced by the product inŨ.