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In this paper we describe the effect on quantum groups -- namely, both QUEA's and QFSHA's -- of deformations by twist and by 2-cocycles, showing how such deformations affect the semiclassical limit. As a second, more important task, we discuss how these deformation procedures can be "stretched" to a new extent, via a formal variation of the original recipes, using "quasi-twists" and "quasi-2-cocycles". These recipes seemingly should make no sense at all, yet we prove that they actually work, thus providing well-defined, more general deformation procedures. Later on, we explain the underlying reason that motivates such a result in light of the "Quantum Duality Principle", through which every "quasi-twist/2-cocycle" for a given quantum group can be seen as a standard twist/2-cocycle for another quantum group, associated to the original one via the appropriate Drinfeld functor. As a third task, we consider standard constructions involving $R$-(co)matrices in the general theory of Hopf algebras. First we adapt them to quantum groups, then we show that they extend to the case of "quasi-$R$-(co)matrices", and finally we discuss how these constructions interact with the Quantum Duality Principle. As a byproduct, this yields new special symmetries (isomorphisms) for the underlying pair of dual Poisson (formal) groups that one gets by specialization., Comment: 72 pages. In this submission, several misprints have been fixed. This is a long, fully detailed version of the work: the version submitted for publication is shorter (58 pages)

We consider a real Abelian Lie supergroup $G$ acting on its complexification $M$, equipped with a $G$-invariant super K\"ahler form. We extend the scheme of classical geometric quantization to this setting and construct a unitary $G$-representation. We show that the occurrences of its irreducible subrepresentations are governed by the image of the moment map of the super K\"ahler form. As an application, we construct a Gelfand model of $G$, namely a unitary $G$-representation in which every unitary irreducible representation occurs exactly once., Comment: 28 pages - Submitted. W.r.t. the previous version, the main core of the content stays the same, yet the paper has been entirely re-written, with several improvements here and there

We introduce the notion of formal multiparameter quantum universal enveloping algebras - in short FoMpQUEA - as a straightforward generalization of Drinfeld's quantum group. Then we show that the class of FoMpQUEA's is closed under deformations by ("toral") twists and deformations by ("toral") 2-cocycles: as a consequence, all "multiparameter formal QUEA's" considered so far are recovered, as falling within this class. In particular, we prove that any FoMpQUEA is isomorphic to a suitable deformation, by twist or by 2-cocycle, of Drinfeld's standard QUEA. We introduce also multiparameter Lie bialgebras (in short, MpLbA's), and we consider their deformations, by twist and by 2-cocycles. The semiclassical limit of every FoMpQUEA is a suitable MpLbA, and conversely each MpLbA can be quantized to a suitable FoMpQUEA. In the end, we prove that, roughly speaking, the two processes of "specialization" (of FoMpQUEA to a MpLbA) and of "deformation (by toral twist or toral 2-cocycle)" - at the level of FoMpQUEA's or of MpLbA's - do commute with each other., Comment: 87 pages. This is the final version, ***strongly improved*** w.r.t. the original submission. To appear in "Annales de l'Institut Fourier"

For the quantized universal enveloping algebra U_h(g_X) associated with a continuous Kac-Moody algebra g_X as in [A. Appel, F. Sala, "Quantization of continuum Kac-Moody algebras", Pure Appl. Math. Q. 16 (2020), no. 3, 439-493], we prove that a suitable formulation of the Quantum Duality Principle holds true, both in a "formal" version - i.e., applying to the original definition of U_h(g_X) as a formal QUEA over the algebra of formal series in h - and in a "polynomial" one - i.e., for a suitable polynomial form of U_h(g_X) over the algebra of Laurent polynomials in q. In both cases, the QDP states that a suitable subalgebra of the given quantization of the Lie bialgebra g_X is in fact a suitable quantization (in formal or in polynomial sense) of a connected Poisson group G^*_X dual to g_X ., Comment: Final version, published in "Journal of Lie theory"

We investigate the notion of real form of complex Lie superalgebras and supergroups, both in the standard and graded version. Our functorial approach allows most naturally to go from the superalgebra to the supergroup and retrieve the real forms as fixed points, as in the ordinary setting. We also introduce a more general notion of compact real form for Lie superalgebras and supergroups, and we prove some existence results for Lie superalgebras that are simple contragredient and their associated connected simply connected supergroups., Comment: 27 pages - This the final, published version

In this paper we study two deformation procedures for quantum groups: deformations by twists, that we call "comultiplication twisting", as they modify the coalgebra structure, while keeping the algebra one -- and deformations by 2-cocycle, that we call "multiplication twisting", as they deform the algebra structure, but save the coalgebra one. We deal with quantum universal enveloping algebras, in short QUEA's, for which we accordingly consider those arising from twisted deformations (in short TwQUEA's) and those arising from 2-cocycle deformations, usually called multiparameter QUEA's (in short MpQUEA's). Up to technicalities, we show that the two deformation methods are equivalent, in that they eventually provide isomorphic outputs, which are deformations (of either kinds) of the "canonical", well-known one-parameter QUEA by Jimbo and Lusztig. It follows that the two notions of TwQUEA's and of MpQUEA's -- which, in Hopf algebra theoretical terms are naturally dual to each other -- actually coincide; thus, that there exists in fact only one type of "pluriparametric deformation" for QUEA's. In particular, the link between the realization of any such QUEA as a MpQUEA and that as a TwQUEA is just a (very simple, and rather explicit) change of presentation., Comment: 49 pages. Final version published in Communications in Contemporary Mathematics. Comparing to the previous one, this version contains only minor changes suggested by the referee - WARNING: the internal labeling of subsections in the printed version does not entirely match with the present, preprint version

The complex Lie superalgebras $\mathfrak{g}$ of type $D(2,1;a)$ - also denoted by $\mathfrak{osp}(4,2;a) $ - are usually considered for "non-singular" values of the parameter $a$, for which they are simple. In this paper we introduce five suitable integral forms of $\mathfrak{g}$, that are well-defined at singular values too, giving rise to "singular specializations" that are no longer simple: this extends the family of simple objects of type $D(2,1;a)$ in five different ways. The resulting five families coincide for general values of $a$, but are different at "singular" ones: here they provide non-simple Lie superalgebras, whose structure we describe explicitly. We also perform the parallel construction for complex Lie supergroups and describe their singular specializations (or "degenerations") at singular values of $a$. Although one may work with a single complex parameter $a$, in order to stress the overall $\mathfrak{S}_3$-symmetry of the whole situation, we shall work (following Kaplansky) with a two-dimensional parameter $\boldsymbol{\sigma} = (\sigma_1,\sigma_2,\sigma_3)$ ranging in the complex affine plane $\sigma_1 + \sigma_2 + \sigma_3 = 0$.

We address the study of multiparameter quamtum groups (=MpQG's) at roots of unity, namely quantum universal enveloping algebras $ U_{\boldsymbol{\rm q}}(\mathfrak{g}) $ depending on a matrix of parameters $ \boldsymbol{\rm q} = {\big( q_{ij} \big)}_{i, j \in I} \, $. This is performed via the construction of quantum root vectors and suitable "integral forms" of $ U_{\boldsymbol{\rm q}}(\mathfrak{g}) \, $, a \textsl{restricted one} - generated by quantum divided powers and quantum binomial coefficients - and an \textsl{unrestricted\/} one - where quantum root vectors are suitably renormalized. The specializations at roots of unity of either forms are the "MpQG's at roots of unity" we look for. In particular, we study special subalgebras and quotients of our MpQG's at roots of unity - namely, the multiparameter version of small quantum groups - and suitable associated quantum Frobenius morphisms, that link the MpQG's at roots of 1 with MpQG's at 1, the latter being classical Hopf algebras bearing a well precise Poisson-geometrical content. A key point in the discussion - often at the core of our strategy - is that every MpQG is actually a 2-cocycle deformation of the algebra structure of (a lift of) the "canonical" one-parameter quantum group by Jimbo-Lusztig, so that we can often rely on already established results available for the latter. On the other hand, depending on the chosen multiparameter $ \boldsymbol{\rm q} $ our quantum groups yield (through the choice of integral forms and their specialization) different semiclassical structures, namely different Lie coalgebra structures and Poisson structures on the Lie algebra and algebraic group underlying the canonical one-parameter quantum group., Comment: 71 pages. Final version, shortened and polished after the referee's comments - the mathematical content, though, is unchanged with respect to the previous version. To appear in "Journal of Noncommutative Geometry"

It is known that there exists a natural functor $\Phi$ from Lie supergroups to super Harish-Chandra pairs. A functor going backwards, that associates a Lie supergroup with each super Harish-Chandra pair, yielding an equivalence of categories, was found by Koszul [18]; this result was later extended by other authors, to different levels of generality, but always elaborating on Koszul's original idea. In this paper, I provide two new backwards equivalences, i.e. two different functors $\Psi^\circ$ and $\Psi^e$ that construct a Lie supergroup (thought of as a special group-valued functor) out of a given super Harish-Chandra pair, so that any Lie supergroup is recovered from its naturally associated super Harish-Chandra pair; more precisely, both $\Psi^\circ$ and $\Psi^e$ are quasi-inverse to the functor $\Phi$., Comment: 47 pages. The present version is a far-reaching improvement of the first one: roughly, we redouble the main result(s). Sections 2 and 3 are essentially unchanged (but for minimal misprints), while sections 4, 5 and 6 are modified (enriched) a lot; accordingly, section 7 is (slightly) adapted too. I also added *Keywords*: Lie supergroups - Lie superalgebras - super Harish-Chandra pairs

We explore special features of the pair (U^*, U_*) formed by the right and left dual over a (left) bialgebroid U in case the bialgebroid is, in particular, a left Hopf algebroid. It turns out that there exists a bialgebroid morphism S^* from one dual to another that extends the construction of the antipode on the dual of a Hopf algebra, and which is an isomorphism if U is both a left and right Hopf algebroid. This structure is derived from Phung's categorical equivalence between left and right comodules over U without the need of a (Hopf algebroid) antipode, a result which we review and extend. In the applications, we illustrate the difference between this construction and those involving antipodes and also deal with dualising modules and their quantisations., Comment: 25 pages; modified version in which the relation to Phung Ho Hai's comodule equivalence is established; to appear in Algebr. Represent. Theory

This paper dwells upon two aspects of affine supergroup theory, investigating the links among them. First, I discuss the "splitting" properties of affine supergroups, i.e. special kinds of factorizations they may admit - either globally, or pointwise. Second, I present a new contribution to the study of affine supergroups by means of super Harish-Chandra pairs (a method already introduced by Koszul, and later extended by other authors). Namely, I provide an explicit, functorial construction \Psi which, with each super Harish-Chandra pair, associates an affine supergroup that is always globally strongly split (in short, gs-split) - thus setting a link with the first part of the paper. On the other hand, there exists a natural functor \Phi from affine supergroups to super Harish-Chandra pairs: then I show that the new functor \Psi - which goes the other way round - is indeed a quasi-inverse to \Phi, provided we restrict our attention to the subcategory of affine supergroups that are gs-split. Therefore, (the restrictions of) \Phi and \Psi are equivalences between the categories of gs-split affine supergroups and of super Harish-Chandra pairs. Such a result was known in other contexts, such as the smooth differential or the complex analytic one, or in some special cases, via different approaches: the novelty in the present paper lies in that I construct a different functor \Psi and thus extend the result to a much larger setup, with a totally different, more geometrical method (very concrete indeed, and characteristic free). The case of linear algebraic groups is treated also as an intermediate, inspiring step. Some examples, applications and further generalizations are presented at the end of the paper., Comment: La-TeX file, 48 pages. Final revised version, *after correcting the galley proofs* - to appear in "Transactions of the AMS"

We present a formal algebraic language to deal with quantum deformations of Lie-Rinehart algebras - or Lie algebroids, in a geometrical setting. In particular, extending the ice-breaking ideas introduced by Xu in [Ping Xu, "Quantum groupoids", Comm. Math. Phys. 216 (2001), 539-581], we provide suitable notions of "quantum groupoids". For these objects, we detail somewhat in depth the formalism of linear duality; this yields several fundamental antiequivalences among (the categories of) the two basic kinds of "quantum groupoids". On the other hand, we develop a suitable version of a "quantum duality principle" for quantum groupoids, which extends the one for quantum groups - dealing with Hopf algebras - originally introduced by Drinfeld (cf. [V. G. Drinfeld, "Quantum groups", Proc. ICM (Berkeley, 1986), 1987, pp. 798-820], sec. 7) and later detailed in [F. Gavarini, "The quantum duality principle", Annales de l'Institut Fourier 53 (2002), 809-834]., Comment: La-TeX file, 47 pages. Final version, after galley proofs correction, published in "Journal of Noncommutative Geometry". Compared with the previously posted version, we streamlined the whole presentation, we fixed a few details and we changed a bit the list of references

For a complex or real algebraic group G, with g:=Lie(G), quantizations of global type are suitable Hopf algebras F_q[G] or U_q(g) over C[q,q^{-1}]. Any such quantization yields a structure of Poisson group on G, and one of Lie bialgebra on g : correspondingly, one has dual Poisson groups G^* and a dual Lie bialgebra g^*. In this context, we introduce suitable notions of quantum subgroup and of quantum homogeneous space, in three versions: weak, proper and strict (also called "flat" in the literature). The last two notions only apply to those subgroups which are coisotropic, and those homogeneous spaces which are Poisson quotients; the first one instead has no restrictions. The global quantum duality principle (GQDP) - cf. [F. Gavarini, "The global quantum duality principle", J. Reine Angew. Math. 612 (2007), 17-33] - associates with any global quantization of G, or of g, a global quantization of g^*, or of G^*. In this paper we present a similar GQDP for quantum subgroups or quantum homogeneous spaces. Roughly speaking, this associates with every quantum subgroup, resp. quantum homogeneous space, of G, a quantum homogeneous space, resp. a quantum subgroup, of G^*. The construction is tailored after four parallel paths - according to the different ways one has to algebraically describe a subgroup or a homogeneous space - and is "functorial", in a natural sense. Remarkably enough, the output of the constructions are always quantizations of proper type. More precisely, the output is related to the input as follows: the former is the coisotropic dual of the coisotropic interior of the latter - a fact that extends the occurrence of Poisson duality in the GQDP for quantum groups. Finally, when the input is a strict quantization then the output is strict too - so the special role of strict quantizations is respected. We end the paper with some examples and application., Comment: 43 pages, La-TeX file. Final version, published in "Documenta Mathematica"

Let R be a 1-dimensional integral domain, let h (non-zero) be a prime element, and let \HA be the category of torsionless Hopf algebras over R. We call H in \HA a "quantized function algebra" (=QFA), resp. "quantized restricted universal enveloping algebras" (=QrUEA), at h if H/hH is the function algebra of a connected Poisson group, resp. the (restricted, if R/hR has positive characteristic) universal enveloping algebra of a (restricted) Lie bialgebra. An "inner" Galois correspondence on \HA is established via the definition of two endofunctors, ( )^\vee and ( )', of \HA such that: (a) the image of ( )^\vee, resp. of ( )', is the full subcategory of all QrUEAs, resp. QFAs, at h; (b) if p := Char(R/hR) = 0, the restrictions of ( )^\vee to QFAs and of ( )' to QrUEAs yield equivalences inverse to each other; (c) if p = 0, then starting from a QFA over a Poisson group G, resp. from a QrUEA over a Lie bialgebra g, the functor ( )^\vee, resp. ( )', gives a QrUEA, resp. a QFA, over the dual Lie bialgebra, resp. a dual Poisson group. In particular, (a) provides a machine to produce quantum groups of both types (either QFAs or QrUEAs), (b) gives a characterization of them among objects of \HA, and (c) gives a "global" version of the so-called "quantum duality principle" (after Drinfeld's, cf. [Dr]). These notes draw a sketch of the construction leading to the "global quantum duality principle". Besides, the principle itself, and in particular the above mentioned application, is illustrated by means of several examples. There are no proofs, but all (of them, and any other detail) can be found instead in arXiv:math/0303019v8 [math.QA]., Comment: This is the author's contribution to the Proceedings of the "Rencontres Math\'ematiques de Glanon - 2002"

I present a construction of connected affine algebraic supergroups G_V associated with simple Lie superalgebras g of Cartan type and with g-modules V. Conversely, I prove that every connected affine algebraic supergroup whose tangent Lie superalgebra is of Cartan type is necessarily isomorphic to one of the supergroups G_V that I introduced. In particular, the supergroup constructed in this way associated with g := W(n) and its standard representation is described somewhat more in detail. In addition, *** an "Erratum" is added here *** after the main text to fix a mistake which was kindly pointed out to the author by prof. Masuoka after the paper was published: this "Erratum" is accepted for publication in "Forum Mathematicum", it appears here in its final form (but prior to proofreading). In it, I also explain more in detail the *Existence Theorem* for algebraic supergroups of Cartan type which comes out of the main result in the original paper., Comment: Main file: La-TeX file, 47 pages, already published (see below). Erratum: La-TeX file, 6 pages, to appear (see below). For the main file, the original publication is available at www.degruyter.com (cf. the journal reference here below)

We show that every connected affine algebraic supergroup defined over a field k, with diagonalizable maximal torus and whose tangent Lie superalgebra is a k-form of a complex simple Lie superalgebra of classical type is a Chevalley supergroup, as it is defined and constructed explicitly in [R. Fioresi, F. Gavarini, "Chevalley Supergroups", Memoirs of the AMS 215 (2012), no. 1014]., Comment: La-TeX file, 17 pages. Final accepted version (after the referee's remarks). To appear in "Journal of Lie Theory"

I present a construction "a` la Chevalley" of affine supergroups associated with simple Lie superalgebras of (classical) type D(2,1;a), for any possible value of the parameter a - in particular, including non-integral values of a. This extends the similar work performed in [R. Fioresi, F. Gavarini, "Chevalley Supergroups", Memoirs of the AMS 215 (2012), no. 1014 - arXiv:0808.0785v8 [math.RA]], where all other simple Lie superalgebras of classical type were considered. The case of simple Lie superalgebras of Cartan type is dealt with in [F. Gavarini, "Algebraic supergroups of Cartan type", Forum Mathematicum (to appear), 92 pages - arXiv:1109.0626v5 [math.RA], so this work completes the program of constructing connected affine supergroups associated with any simple Lie superalgebra., Comment: La-TeX file, 20 pages. All the text was amended after the referee's remarks. To appear in "Proceedings of the Edinburgh Mathematical Society"

Let G be either of Mat(n), GL(n) or SL(n), let O_q(G) be the quantum function algebra - over Z[q,q^{-1}] - associated to G, and let O_e(G) be the specialisation of O_q(G) at a root of unity, of odd order l. Then O_e(G) is a module over the corresponding classical function algebra O(G) via the quantum Frobenius morphism, which embeds O(G) as a central subbialgebra of O_e(G). In this note we prove a PBW-like theorem for O_q(G) - more or less known in literature, but not in this form (to the best of the author's knowledge) - and we show that it yields explicit bases of O_e(G) over O(G) when G is Mat(n) or GL(n): in particular, O_e(G) is free of rank l^{dim(G)}. Also, we apply the latter result to prove that O_e(G) is a free Frobenius extensions over O(G), and to compute explicitly the corresponding Nakayama automorphism, again for G being Mat(n) or GL(n) . This extends previous results by Brown, Gordon and Stroppel (see [BG], [BGS2])., Comment: AMS-TeX file, 10 pages. This paper is a natural evolution - now in *final version* - of preprint math.QA/0608016, which has been withdrawn

This paper has been withdrawn by the author. Its natural evolution is preprint math.QA/0610691, which is different enough to bear a different title (though they share about 2/3s of common content)., Comment: This paper has been withdrawn by the author. Its natural evolution is preprint math.QA/0610691, which is different enough to bear a different title

The radical of the Brauer algebra B_f^x is known to be non-trivial when the parameter x is an integer subject to certain conditions (with respect to f). In these cases, we display a wide family of elements in the radical, which are explicitly described by means of the diagrams of the usual basis of B_f^x . The proof is by direct approach for x=0, and via classical Invariant Theory in the other cases, exploiting then the well-known representation of Brauer algebras as centralizer algebras of orthogonal or symplectic groups acting on tensor powers of their standard representation. This also gives a great part of the radical of the generic indecomposable B_f^x-modules. We conjecture that this part is indeed the whole radical in the case of modules, and it is the whole part in a suitable step of the standard filtration in the case of the algebra. As an application, we find some more precise results for the module of pointed chord diagrams, and for the Temperley-Lieb algebra - realised inside B_f^1 - acting on it., Comment: AMS-TeX file, 2 figures (in EPS format), 25 pages. This is the final version, to appear in "Mathematische Zeitschrift". Comparing to the previous one, it has been streamlined and shortened - yet the mathematical content stands the same. The list of references has been updated

The quantized universal enveloping algebra U_q(gl(n)) has two integral forms - over Z[q,q^{-1}] - the restricted (by Lusztig) and the unrestricted (by De Concini and Procesi) one. Dually, the quantum function algebra F_q[GL(n)] has two integral forms, namely those of all elements - of F_q[GL(n)] - which take values in Z[q,q^{-1}] when paired respectively with the restricted or the unrestricted form of U_q(gl(n)). The first one is the well-known form generated over Z[q,q^{-1}] by the entries of a q-matrix and the inverse of its quantum determinant. In this paper instead we study the second integral form, say F'_q[GL(n)], i.e. that of all elements which are Z[q,q^{-1}]-valued over the unrestricted form of U_q(gl(n)). In particular we yield a presentation of it by generators and relations, and a PBW-like theorem: in short, it is an algebra of "quantum divided powers" and "quantum binomial coefficients". Moreover, we give a direct proof that F'_q[GL(n)] is a Hopf subalgebra of F_q[GL(n)], and that its specialization at q=1 is the Z-hyperalgebra over gl(n)^*, the Lie bialgebra dual to gl(n). In addition, we describe explicitly the specializations of F'_q[GL(n)] at roots of 1, and the associated quantum Frobenius (epi)morphism. The same analysis is done for F'_q[SL(n)] and (as a key step) F'_q[Mat(n)]: in fact, for the latter the strongest results are obtained. This work extends to general n>2 the results for n=2, already treated in math.QA/0411440., Comment: AMS-TeX file, 34 pages. This is the final version (after the journal's proofs correction)

We develop a quantum duality principle for coisotropic subgroups of a (formal) Poisson group and its dual: namely, starting from a quantum coisotropic subgroup (for a quantization of a given Poisson group) we provide functorial recipes to produce quantizations of the dual coisotropic subgroup (in the dual formal Poisson group). By the natural link between subgroups and homogeneous spaces, we argue a quantum duality principle for Poisson homogeneous spaces which are Poisson quotients, i.e. have at least one zero-dimensional symplectic leaf. Only bare results are presented, while detailed proofs can be found in math.QA/0412465, or in the reference [3]. The last section contains new, unpublished material about examples and applications, which is not included in math.QA/0412465 (nor in its printed version)., Comment: La-TeX file, 20 pages. This is a seminal version of math.QA/0412465, as reported by the second author in the conference "Contemporary Geometry and Related Topics 2005" (Belgrade, June 26-July 2, 2005); the last section contains new, unpublished material about examples and applications, *which is not included in math.QA/0412465*. This is the final version

The quantum duality principle (QDP) for homogeneous spaces gives four recipes to obtain, from a quantum homogeneous space, a dual one, in the sense of Poisson duality. One of these recipes fails (for lack of the initial ingredient) when the homogeneous space we start from is not a quasi-affine variety. In this work we solve this problem for the quantum Grassmannian, a key example of quantum projective homogeneous space, providing a suitable analogue of the QDP recipe., Comment: La-TeX file, 20 pages. Final version after the referee's corrections/remarks. To appear in: Proceedings of the Workshop "Quantum Groups and Noncommutative Geometry" (Max Planck Institut fur Mathematik, Bonn, Germany, August 6-8, 2007)

We develop a quantum duality principle for coisotropic subgroups of a (formal) Poisson group and its dual. Namely, starting from a quantum coisotropic subgroup (for a quantization of a given Poisson group) we provide functorial recipes to produce quantizations of the dual coisotropic subgroup (in the dual formal Poisson group). By the natural link between subgroups and homogeneous spaces, we argue a quantum duality principle for Poisson homogeneous spaces which are Poisson quotients, i.e. have at least one zero-dimensional symplectic leaf. As an application, we provide an explicit quantization of the homogeneous SL(n)^*-space of Stokes matrices, with the Poisson structure given by Dubrovin and Ugaglia., Comment: 27 pages, AMS-TeX file. Final version, with some typoes (also occurring in the printed version) corrected

Let U_q(sl_2) be the standard Drinfeld-Jimbo quantized universal enveloping algebra over sl_2, let F_q[SL_2] be the corresponding quantum function algebra, and let R be the ring of Laurent polynomials in q with coefficients in the ring of integers. Let \Cal{U}_q(sl_2) be the unrestricted R-integer form of U_q(sl_2) introduced by De Concini, Kac and Procesi. Within the quantum function algebra F_q[SL_2], we study the subset \Cal{F}_q[SL_2] of all elements which give values in the ring R when paired with \Cal{U}_q(sl_2). In this paper we describe \Cal{F}_q[SL_2]. In particular we provide a presentation of it by generators and relations, and a nice R-spanning set (of PBW type). Moreover, we give a direct proof that \Cal{F}_q[SL_2] is a Hopf subalgebra of F_q[SL_2], and that the specialization of \Cal{F}_q[SL_2] at q=1 is the hyperalgebra U_Z(sl_2^*) associated to the Lie bialgebra sl_2^* dual to sl_2: in other words, \Cal{F}_q[SL_2] is a "quantum hyperalgebra". In fact, our description of \Cal{F}_q[SL_2] is much like the presentation of Lusztig's restricted R-integer form of U_q(sl_2). We describe explicitly also the specializations of \Cal{F}_q[SL_2] at roots of 1, and the associated quantum Frobenius (epi)morphism; these results again closely resemble Lusztig's ones. All this improve results proved in previous work by the first named author basing upon the results of De Concini, Kac and Procesi. The same analysis is done for the analogue algebra \Cal{F}_q[GL_2], with similar results, and also (as a key, intermediate step) for \Cal{F}_q[Mat_2], for which even stronger results hold, in particular a PBW-like theorem., Comment: 21 pages, AMS-TeX file. This version is much shorter and streamlined (the discussion of the three cases is made parallelwise); several details have been corrected here and there - even the title has slightly changed! Final version, to appear in "Communications in Algebra"

We prove that the logarithm of a group-like element in a free algebra coincides with its image by a certain linear map. We use this result and the formula of Le and Murakami for the Knizhnik-Zamolodchikov (KZ) associator $\Phi$ to derive a formula for $\log(\Phi)$ in terms of MZV's (multiple zeta values)., Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

We provide an alternative approach to the Faddeev-Reshetikhin-Takhtajan presentation of the quantum group U_q(g), with L-operators as generators and relations ruled by an R-matrix. We look at U_q(g) as being generated by the quantum Borel subalgebras U_q(b_+) and U_q(b_-), and we use the standard presentation of the latters as quantum function algebras. When g = gl(n) these Borel quantum function algebras are generated by the entries of a triangular q-matrix, thus eventually U_q(gl(n)) is generated by the entries of an upper triangular and a lower triangular q-matrix, which share the same diagonal. The same elements generate over the ring of Laurent polynomials the unrestricted integer form of U_q(gl(n)) of De Concini and Procesi, which we present explicitly, together with a neat description of the associated quantum Frobenius morphisms at roots of 1. All this holds, mutatis mutandis, for g = sl(n) too., Comment: AMS-TeX, 17 pages. Final version appeared in "Proceedings of the Edinburgh Mathematical Society"

We develop a quantum duality principle for subgroups of a Poisson group and its dual, in two formulations. Namely, in the first one we provide functorial recipes to produce quantum coisotropic subgroups in the dual Poisson group out of any quantum subgroup (in a tautological sense) of the initial Poisson group, while in the second one similar recipes are given only starting from coisotropic subgroups. In both cases this yields a Galois-type correspondence, where a quantum coisotropic subgroup is mapped to its complementary dual; moreover, in the first formulation quantum coisotropic subgroups are characterized as being the fixed points in this Galois' reciprocity. By the natural link between quantum subgroups and quantum homogeneous spaces then we argue a quantum duality principle for homogeneous spaces too, where quantum coisotropic spaces are the fixed elements in a suitable Galois' reciprocity. As an application, we provide an explicit quantization of the space of Stokes matrices with the Poisson structure given by Dubrovin and Ugaglia. The *paper is actually under revision* to fix some minor, technical issues. The geometric objects considered here (Poisson groups, subgroups and homogeneous spaces) are "global" - as opposed to "formal" - and quantizations are considered as standard - i.e.non topological - Hopf algebras over the ring of Laurent polynomials (or other non-topological rings). Instead, a "local/formal" version of this work is developed in math.QA/0412465 - which is in final form - due to appear in "Advances in Mathematics". The example of Stokes matrices mentioned above is considered in math.QA/0412465 as well., Comment: This paper has been withdrawn, as it is logically replaced by the new preprint arXiv:1210.1597 [math.QA]

Let G be the group of all formal power series starting with x with coefficients in a field k of zero characteristic (with the composition product), and let F[G] be its function algebra. C. Brouder and A. Frabetti introduced a non-commutative, non-cocommutative graded Hopf algebra H, via a direct process of ``disabelianisation'' of F[G], i.e. taking the like presentation of the latter as an algebra but dropping the commutativity constraint. In this paper we apply a general method to provide four one-parameters deformations of H, which are quantum groups whose semiclassical limits are Poisson geometrical symmetries such as Poisson groups or Lie bialgebras, namely two quantum function algebras and two quantum universal enveloping algebras. In particular the two Poisson groups are extensions of G, isomorphic as proalgebraic Poisson varieties but not as proalgebraic groups. This analysis easily extends to a hudge family of Hopf algebras of similar nature, thus yielding a method to associate to such "generalized symmetries" some classical geometrical symmetries (such as Poisson groups and Lie bialgebras) in a natural way: the present case then stands as a simplest, toy model for the general situation., Comment: AMS-TeX file, 34 pages. To appear in Communications in Mathematical Physics. Minor corrections have been fixed here and there

We give functorial recipes to get, out of any Hopf algebra over a field, two pairs of Hopf algebras which have some geometrical content. When the ground field has characteristic zero, the first pair is made by a function algebra over a connected Poisson group and a universal enveloping algebra over a Lie bialgebra. In addition, the Poisson group as a variety is an affine space, and the Lie bialgebra as a Lie algebra is graded. Forgetting these last details, the second pair is of the same type. When the Hopf algebra H we start from is already of geometric type the result involves Poisson duality: the first Lie bialgebra associated to H = F[G] is g^* (with g := Lie(G)), and the first Poisson group H = U(g) is of type G^*, i.e. it has g as cotangent Lie bialgebra. If the ground field has positive characteristic, then the same recipes give similar results, but for the fact that the Poisson groups obtained have dimension 0 and height 1, and restricted universal enveloping algebras are obtained. We show how these "geometrical" Hopf algebras are linked to the original one via 1-parameter deformations, and explain how these results follow from quantum group theory. The cases of hyperalgebras and group algebras are examined in some detail (thus recovering some well-known, classical construction), along with some relevant examples., Comment: 37 pages, AMS-TeX file. Minor typoes have been corrected in pages 23, 24, 26, 32 and 36. Final version, to appear in Journal of Algebra

Let R be an integral domain, h non-zero in R such that R/hR is a field, and HA the category of torsionless (or flat) Hopf algebras over R. We call any H in HA "quantized function algebra" (=QFA), resp. "quantized (restricted) universal enveloping algebra" (=QrUEA), at h if H/hH is the function algebra of a connected Poisson group, resp. the (restricted, if R/hR has positive characteristic) universal enveloping algebra of a (restricted) Lie bialgebra. We establish an "inner" Galois' correspondence on HA, via the definition of two endofunctors, ()^\vee and ()', of HA such that: (a) the image of ()^\vee, resp. of ()', is the full subcategory of all QrUEAs, resp. all QFAs, at h; (b) if R/hR has zero characteristic, the restriction of ()^\vee to QFAs and of ()' to QrUEAs yield equivalences inverse to each other; (c) if R/hR has zero characteristic, starting from a QFA over a Poisson group, resp. from a QrUEA over a (restricted) Lie bialgebra, the functor ()^\vee, resp. ()', gives a QrUEA, resp. a QFA, over the dual Lie bialgebra, resp. the dual Poisson group. In particular, (a) yields a recipe to produce quantum groups of both types (QFAs or QrUEAs), (b) gives a characterization of them within HA, and (c) gives a "global" version of the "quantum duality principle" after Drinfeld. We then apply our result to Hopf algebras defined over a field k and extended to the polynomial ring k[h]: this yields quantum groups, hence "classical" geometrical symmetries of Poisson type (via specialization) associated to the "generalized symmetry" encoded by the original Hopf algebra over k. Both the main result and the above mentioned application are illustrated via several examples of many different kinds, which are studied in full detail., Comment: 120 pages, AMS-TeX file. This paper is NOT for publication: the results herein will be published in several, separated articles (ask the author for reference(s))

If g is a quasitriangular Lie bialgebra, one can asks what is the geometrical meaning of its r-matrix. A first answer was given in a paper by Weinstein and Xu, using purely geometrical means: roughly, one has that the formal Poisson group F[[g^*]] is endowed with a "braiding", i.e. a distinguished operator on its tensor square which satisfy quasitriangularity conditions (in particular, it is a solution of the QYBE). Independently, the authors also found, by means of quantum groups, that F[[g^*]] has a braiding. In this paper we compare these two approaches and their outcomes. First, we show that the two braidings obtained in the two processes do share several similar properties (in particular, the construction is functorial). Second, in the simplest case (G = SL_2) we prove that the two braidings do coincide. The question then rises of whether they are always the same: this problem is addressed and solved in math.QA/0207235, in a much broader context, in which unicity of braidings is proved., Comment: AMS-TeX file, 28 pages. To appear in "Journal of Geometry and Physics", 856 (2002)

The "quantum duality principle" states that a quantisation of a Lie bialgebra provides also a quantisation of the dual formal Poisson group and, conversely, a quantisation of a formal Poisson group yields a quantisation of the dual Lie bialgebra as well. We extend this to a much more general result: namely, given any principal ideal domain R, for each prime h in R we establish sort of an "inner" Galois' correspondence on the category HA of torsionless Hopf algebras over R, via the definition of two functors (from HA to itself) such that the image of the first, resp. of the second, is the full subcategory of those Hopf algebras which are commutative, resp. cocommutative, modulo h (i.e. they are "quantum function algebras" (=QFA), resp. "quantum universal enveloping algebras" (=QUEA), at h). In particular we provide a machine to get two quantum groups - a QFA and a QUEA - out of any Hopf algebra H over a field k: just plug in a parameter x and apply the functors to H[x] for h = x. Several relevant examples are studied in full detail: the trivial quantisations, the semisimple groups, the Euclidean group, the Heisenberg group, and the Kostant-Kirillov structure on any Lie algebra; furthermore, an interesting application to renormalisation theory in quantum electro-dynamics is studied, as a sample of application of the principle to a quite large class of problems. This work is a far-reaching "evolution" of the same author's preprint math.QA/9912186: the present paper is entirely self-contained, is more general from the mathematical point of view, and contains additional examples. WARNING: This preprint has been overtaken by a new, deeply enhanced and improved version, available as {\tt math.QA/0303019}; the interested reader is kindly asked to refer to that new preprint., Comment: This paper has been withdrawn by the author because it is replaced by a far-reaching "evolution", namely preprint arXiv:math/0303019 (2003)

Let R be an integral domain, let a non-zero h in R be such that k := R/hR is a field, and let HA be the category of torsionless (or flat) Hopf algebras over R. We call H in HA a "quantized function algebra" (=QFA), resp. "quantized restricted universal enveloping algebras" (=QrUEA), at h if - roughly speaking - H/hH is the function algebra of a connected Poisson group, resp. the (restricted, if R/hR has positive characteristic) universal enveloping algebra of a (restricted) Lie bialgebra. Extending a result of Drinfeld, we establish an "inner" Galois' correspondence on HA, via two endofunctors, ( )^\vee and ( )' , of HA such that H^\vee is a QrUEA and H' is a QFA (for all H in HA). In addition: (a) the image of ( )^\vee , resp. of ( )' , is the full subcategory of all QrUEAs, resp. of all QFAs; (b) if p := Char(R/hR) = 0 , the restrictions of ( )^\vee to the QFAs and of ( )' to the QrUEAs yield equivalences inverse to each other; (c) if p = 0 , starting from a QFA over a Poisson group G, resp. from a QrUEA over a Lie bialgebra g, the functor ( )^\vee , resp. ( )' , gives a QrUEA, resp. a QFA, over the dual Lie bialgebra, resp. the dual Poisson group. Several, far-reaching applications are developed in detail in [Ga2-4]., Comment: AMS-TeX file, 13 pages. This is the author's file of the final version (after the refereeing process), as sent for publication

The "quantum duality principle" states that the quantization of a Lie bialgebra - via a quantum universal enveloping algebra (QUEA) - provides also a quantization of the dual Lie bialgebra (through its associated formal Poisson group) - via a quantum formal series Hopf algebra (QFSHA) - and, conversely, a QFSHA associated to a Lie bialgebra (via its associated formal Poisson group) yields a QUEA for the dual Lie bialgebra as well; more precisely, there exist functors QUEA --> QFSHA and QFSHA --> QUEA, inverse of each other, such that in either case the Lie bialgebra associated to the target object is the dual of that of the source object. Such a result was claimed true by Drinfeld, but seems to be unproved in literature: we give here a complete detailed proof of it., Comment: 19 pages, AMS-TeX file. The paper has been entirely re-written: in particular, we add a discussion of the possible generalisation of the main result to the infinite dimensional case. This is the author's file of the final version (after the refereeing process), as sent for publication

Let g be a quasitriangular Lie bialgebra over a field k of characteristic zero, and let g^* be its dual Lie bialgebra. We prove that the formal Poisson group F[[g^*]] is a braided Hopf algebra. More generally, we prove that if (U_h,R) is any quasitriangular QUEA, then (U_h', Ad(R)|_{U_h' \otimes U_h'}) --- where U_h' is defined by Drinfeld --- is a braided QFSHA. The first result is then just a consequence of the existence of a quasitriangular quantization (U_h,R) of U(g) and of the fact that U_h' is a quantization of F[[g^*]]. ----- Soit g une big\`ebre de Lie quasitriangulaire sur un corps k de characteristique zero, et soit g^* sa big\`ebre de Lie duale. Nous prouvons que le groupe de Poisson formel F[[g^*]] est une algebre de Hopf tress\'ee. Plus en g\'en\'eral, nous prouvons que, si (U_h,R) est une QUEA quasitriangulaire, alors (U_h', Ad(R)|_{U_h' \otimes U_h'}) --- o\`u U_h' est definie par Drinfeld --- est une QFSHA tress\'ee. Le premier r\'esultat est alors une consequence de l'existence d'une quantification quasitriangulaire (U_h,R) de U(g) et du fait que U_h' est une quantification de F[[g^*]]., Comment: 11 pages, AMS-TeX file; English and French version available. The whole paper (even the title) has been entirely rewritten. Some new references, and a new section on the relation with the "global R-matrix" of Xu and Weinstein, have been added

Let $\hat{\frak g}$ be an untwisted affine Kac-Moody algebra. The quantum group $U_h(\hat{\frak g})$ (over $\mathbb{C}[[h]]$) is known to be a quasitriangular Hopf algebra: in particular, it has a universal $ R $--matrix, which yields an $ R $--matrix for each pair of representations of $U_h(\hat{\frak g})$. On the other hand, the quantum group $U_q(\hat{\frak g})$ (over $\mathbb{C}(q) $) also has an $ R $--matrix for each pair of representations, but it has not a universal $ R $--matrix so that one cannot say that it is quasitriangular. Following Reshetikin, one introduces the (weaker) notion of braided Hopf algebra: then $ U_q(\hat{\frak g})$ is a braided Hopf algebra. In this work we prove that also the unrestricted specializations of $U_q(\hat{\frak g})$ at roots of 1 are braided: in particular, specializing $q$ at 1 we have that the function algebra $F \big[ \hat{H} \big]$ of the Poisson proalgebraic group $\hat{H}$ dual of $\hat{G}$ (a Kac-Moody group with Lie algebra $\hat{\frak g} \,$) is braided. This is useful because, despite these specialized quantum groups are not quasitriangular, the braiding is enough for applications, mainly for producing knot invariants. As an example, the action of the $ R $--matrix on (tensor products of) Verma modules can be specialized at odd roots of 1., Comment: 12 pages, AMS-TeX C, Version 2.1c - this is the author's file of the final version (after the refereeing process), as sent for publication

Let $ \mathfrak{g} $ be a quasitriangular Lie bialgebra over a field $ K $ of characteristic zero, and let $ \mathfrak{g}^* $ be its dual Lie bialgebra. We prove that the formal Poisson group $ K\big[\big[\mathfrak{g}^*\big]\big] $ is a braided Hopf algebra, thus generalizing a result due to Reshetikhin (in the case $ \, \mathfrak{g} = \mathfrak{sl}(2,K) \, $). The proof is via quantum groups, using the existence of a quasitriangular quantization of $ \mathfrak{g}^* $, as well as the fact that this one provides also a quantization of $ K\big[\big[\mathfrak{g}^*\big]\big] \, $., Comment: AMS-TeX file, 13 pages, in French; this is the authors' file of the final version (after the refereeing process), as sent for publication. There exists also an English version, posted on arXiv as preprint math/9909065

Let $\hat{\mathfrak{g}}$ be an untwisted affine Kac-Moody algebra, with its Sklyanin-Drinfel'd structure of Lie bialgebra, and let $\hat{\mathfrak{h}}$ be the dual Lie bialgebra. By dualizing the quantum double construction - via formal Hopf algebras - we construct a new quantum group $U_q(\hat{\mathfrak{h}})$, dual of $U_q(\hat{\mathfrak{g}})$. Studying its restricted and unrestricted integer forms and their specializations at roots of 1 (in particular, their classical limits), we prove that $U_q(\hat{\mathfrak{h}})$ yields quantizations of $\hat{\mathfrak{h}}$ and $\hat{G}^\infty$ (the formal group attached to $\hat{\mathfrak{g}}$), and we construct new quantum Frobenius morphisms. The whole picture extends to the untwisted affine case the results known for quantum groups of finite type., Comment: 36 pages, AMS-TeX file. This the author's final version, corresponding to the pronted journal version. arXiv admin note: text overlap with arXiv:q-alg/9511022

Let $ \mathfrak{g} $ be an untwisted affine Kac-Moody algebra over the field $ K \, $, and let $ U_q(\mathfrak{g}) $ be the associated quantum enveloping algebra; let $ \mathfrak{U}_q(g) $ be the Lusztig's integer form of $ U_q(\mathfrak{g}) \, $, generated by $ q $-divided powers of Chevalley generators over a suitable subring $ R $ of $ K(q) \, $. We prove a Poincar\'e-Birkhoff-Witt like theorem for $ \mathfrak{U}_q(\mathfrak{g}) \, $, yielding a basis over $ R $ made of ordered products of $ q $-divided powers of suitable quantum root vectors., Comment: 22 pages, AMS-TeX C, Version 2.1c. This is the author's final version, corresponding to the printed journal version

Inspired by a result in [Ga], we locate two $ k[q,q^{-1}] $-integer forms of $ F_q[SL(n+1)] $, along with a presentation by generators and relations, and prove that for $ q=1 $ they specialize to $ U({\mathfrak{h}}) $, where $ {\mathfrak{h}} $ is the Lie bialgebra of the Poisson Lie group $ H $ dual of $ SL(n+1) $; moreover, we explain the relation with [loc. cit.]. In sight of this, we prove two PBW-like theorems for $ F_q[SL(n+1)] $, both related to the classical PBW theorem for $ U({\mathfrak{h}}) $., Comment: 27 pages, AMS-TeX C, Version 3.0 - Author's file of the final version, as it appears in the journal printed version, BUT for a formula in Subsec. 3.5 and one in Subsec. 5.2 - six lines after (5.1) - that in this very pre(post)print have been corrected

The present work splits in two parts: first, we perform a straightforward generalization of results from [Re], proving autoquasitriangularity of quantum groups $ U_q(\frak{g}) $ and their unrestricted specializations at roots of 1, in particular the function algebra $ F[H] $ of the Poisson group $ H $ dual of $ G $; second, as a main contribution, we prove the convergence of the (specialized) $ R $--matrix action to a birational automorphism of a 2$\ell$--fold ramified covering of the specialization of $ U_q(\frak{g}) $ at a primitive $ \ell $--th root of 1, and of a 2-fold ramified covering of $ H $, thus giving a geometric content to the notion of triangularity (or autoquasitriangularity) for quantum groups., Comment: 23 pages, AMS-TeX C, Version 3.0; final author's version, as appeared in the printed paper

Let $ G^\tau $ be a connected simply connected semisimple algebraic group, endowed with generalized Sklyanin-Drinfeld structure of Poisson group; let $ H^\tau $ be its dual Poisson group. By means of Drinfeld's double construction and dualization via formal Hopf algebras, we construct new quantum groups $ U_{q,\phi}^M ({\frak h}) $ --- dual of $ U_{q,\phi}^{M'} ({\frak g}) $ --- which yield infinitesimal quantization of $ H^\tau $ and $ G^\tau $; we study their specializations at roots of 1 (in particular, their classical limits), thus discovering new quantum Frobenius morphisms. The whole description dualize for $ H^\tau $ what was known for $ G^\tau $, completing the quantization of the pair $ (G^\tau,H^\tau) $., Comment: This preprint was withdrawn because its content is included in the (now updated) preprint arXiv:q-alg/9511022

Let $ G^\tau $ be a connected simply connected semisimple algebraic group, endowed with generalized Sklyanin-Drinfel'd structure of Poisson group, let $ H^\tau $ be its dual Poisson group. By means of quantum double construction and dualization via formal Hopf algebras, we construct new quantum groups $ U_{q,\varphi}^M(\frak{h}) $ --- dual of $ U_{q,\varphi}^{M'}(\frak{g}) $ --- which yield infinitesimal quantization of $ H^\tau $ and $ G^\tau \, $, we study their specializations at roots of 1 (in particular, their classical limits), thus discovering new quantum Frobenius morphisms. The whole description dualize for $ H^\tau $ what was known for $ G^\tau $, completing the quantization of the pair $ (G^\tau,H^\tau) $., Comment: AMS-TeX file, 45 pages, final author's version of the printed paper