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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)

This paper is a first step toward the full description of a family of Hopf algebras whose coradical is isomorphic to a semisimple Hopf algebra K_{n}, n an odd positive integer, obtained by a cocentral abelian cleft extension. We describe the simple Yetter-Drinfeld modules, compute the fusion rules and determine the finite-dimensional Nichols algebras for some of them. In particular, we give the description of the finite-dimensional Nichols algebras over simple modules over K_{3}. This includes a family of 12-dimensional Nichols algebras $B_\xi$ depending on 3rd roots of unity. Here, $B_{1}$ is isomorphic to the well-known Fomin-Kirillov algebra, and $B_{\xi} \simeq B_{\xi^{2}} $ as graded algebras but $B_1$ is not isomorphic to $B_{\xi} $ as algebra for $\xi\neq 1$. As a byproduct we obtain new Hopf algebras of dimension 216., Comment: 27 pages. In this version we include the description of the finite-dimensional Nichols algebras of the simple modules over K_3, which could be completed thanks to a recent result of Heckenberger, Mehir and Vendramin that appeared after the first version of this manuscript. To appear in Communications in Algebra

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"

Based on previous results on the classification of finite-dimensional Nichols algebras over dihedral groups and the characterization of simple modules of Drinfeld doubles, we compute the irreducible characters of the Drinfeld doubles of bosonizations of finite-dimensional Nichols algebras over the dihedral groups $\mathbb{D}_{4t}$ with $t\geq 3$. To this end, we develop new techniques that can be applied to Nichols algebras over any Hopf algebra. Namely, we explain how to construct recursively irreducible representations when the Nichols algebra is generated by a decomposable module, and show that the highest-weight of minimum degree in a Verma module determines its socle. We also prove that tensoring a simple module by a rigid simple module gives a semisimple module., Comment: The paper is mostly self-contained and includes figures to lighten the reading. v2: The major changes are in Sections 3.2, 3.3, 4.4 and the beginning of 5. It is the final version accepted in Documenta Mathematica

We show that all classes that are neither semisimple nor unipotent in finite simple Chevalley or Steinberg groups different from $PSL_n(q)$ collapse (i.e. are never the support of a finite-dimensional Nichols algebra). As a consequence, we prove that the only finite-dimensional pointed Hopf algebra whose group of group-like elements is $PSp_{2n}(q)$, $P\Omega^+_{4n}(q)$, $P\Omega^-_{4n}(q)$, $^3D_4(q)$, $E_7(q)$, $E_8(q)$, $F_4(q)$, or $G_2(q)$ with q even is the group algebra., Comment: Use of CHEVIE for dealing with E6 is introduced. For explicit calculations see version 2. Further minor corrections

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

We describe how to find quantum determinants and antipode formulas from braided vector spaces using the FRT-construction and finite-dimensional Nichols algebras. It generalizes the construction of quantum function algebras using quantum grassmanian algebras. Given a finite-dimensional Nichols algebra B, our method provides a Hopf algebra H such that B is a braided Hopf algebra in the category of H-comodules. It also serves as source to produce Hopf algebras generated by cosemisimple subcoalgebras, which are of interest for the generalized lifting method. We give several examples, among them quantum function algebras from Fomin-Kirillov algebras associated with the symmetric groups on three letters., Comment: 25 pages. Some remarks added after referee's suggestion, particularly, on the existence of localization (with or without Ore condition) and on the type of examples. Some references added

Let p be an odd prime and k an algebraically closed field of characteristic zero. We classify nonsemisimple Hopf algebras over k of dimension 8p with the Chevalley property, and give partial results on nonsemisimple Hopf algebras of dimension 24., Comment: 15 pages. Accepted version: only minor typos corrected. To appear in Contemp. Math

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"

We show that all unipotent classes in finite simple Chevalley or Steinberg groups, different from PSL_n(q) and PSp_{2n}(q), collapse (i.e. are never the support of a finite-dimensional Nichols algebra), with a possible exception on one class of involutions in PSU_n(2^m)., Comment: 38 pages

We classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero such that its coradical is isomorphic to the algebra of functions over a dihedral group D_m, with m=4a> 11. We obtain this classification by means of the lifting method, where we use cohomology theory to determine all possible deformations. Our result provides an infinite family of new examples of finite-dimensional copointed Hopf algebras over dihedral groups., Comment: This version: Minor changes, mainly on exposition following referee's advice. To appear in J. Pure Applied Algebra

Using the standard filtration associated with a generalized lifting method, we determine all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose coradical generates a Hopf subalgebra isomorphic to the smallest non-pointed non-cosemisimple Hopf algebra ${\mathcal{K}}$ and the corresponding infinitesimal module is an indecomposable object in ${}^{{\mathcal{K}}}_{{\mathcal{K}}}\mathcal{YD}$ (we assume that the diagrams are Nichols algebras). As a byproduct, we obtain new Nichols algebras of dimension 8 and new Hopf algebras of dimension 64., Comment: First version: 34 pages, with an Appendix. Second version: 26 pages with minor changes in exposure; we also remove Theorem 5.1. Third version: 26 pages; accepted version to appear in J. Pure Appl. Algebra

Let $G$ be a connected, simply connected simple complex algebraic group and let $\epsilon$ be a primitive $\ell$th root of unity with $\ell$ odd and coprime with $3$ if $G$ is of type $G_{2}$. We determine all Hopf algebra quotients of the twisted multiparameter quantum function algebra $\mathcal{O}_{\epsilon}^{\varphi}(G)$ introduced by Costantini and Varagnolo. This extends the results of Andruskiewitsch and the first author, where the untwisted case is treated., Comment: 30 pages, Several diagrams. This version: minor typos corrected. To appear in Transactions of the AMS

We show that Nichols algebras of most simple Yetter-Drinfeld modules over the projective special linear group over a finite field, corresponding to semisimple orbits, have infinite dimension. We introduce a new criterium to determine when a conjugacy class collapses and prove that for infinitely many pairs (n,q), any finite-dimensional pointed Hopf algebra H with G(H) = PSL(n,q) or SL(n,q) is isomorphic to a group algebra., Comment: Postprint version: Minor changes in Table 1 of Theorem 1.1 and Corollary 3.5

We show that Nichols algebras of most simple Yetter-Drinfeld modules over the projective symplectic linear group over a finite field, corresponding to unipotent orbits, have infinite dimension. We give a criterium to deal with unipotent classes of general finite simple groups of Lie type and apply it to regular classes

We describe quantum groups given by multiparametric deformations of enveloping algebras of Kac-Moody algebras as a family of pointed Hopf algebras introduced by Andruskiewitsch and Schneider associated to a generalized Cartan matrix. We show that under some assumptions, these Hopf algebras depend only on one parameter on each connected component of the Dynkin diagram, up to a cocycle deformation. In particular, we obtain in this way a known result of Hu, Pei and Rosso., Comment: 17 pages. This version: major revision. Abstract is changed, preliminaries are extended and old Section 2 is now merged into the preliminaries. This version is suppose to be more friendly to non-experts. Comments are welcome!

We show that Nichols algebras of most simple Yetter-Drinfeld modules over the projective special linear group over a finite field, corresponding to non-semisimple orbits, have infinite dimension. We spell out a new criterium to show that a rack collapses., Comment: minor changes, to appear in Journal of Algebra

Classifying all Hopf algebras of a given finite dimension over the complex numbers is a challenging problem which remains open even for many small dimensions, not least because few general approaches to the problem are known. Some useful techniques include counting the dimensions of spaces related to the coradical filtration, studying sub- and quotient Hopf algebras, especially those sub-Hopf algebras generated by a simple subcoalgebra, working with the antipode, and studying Hopf algebras in Yetter-Drinfeld categories to help to classify Radford biproducts. In this paper, we add to the classification tools in our previous work [arXiv:1108.6037v1] and apply our results to Hopf algebras of dimension rpq and 8p where p,q,r are distinct primes. At the end of this paper we summarize in a table the status of the classification for dimensions up to 100 to date., Comment: This version of the paper contains a correction on the published version. The statement and proof of Proposition 2.17 are changed and the proof of the results that follow from it are corrected accordingly. We thank H.-S. Ng for kindly communicating the gap to us and for the careful reading of our paper

We introduce a method to construct explicitly multiplicative 2-cocycles for bosonizations of Nichols algebras B(V) over Hopf algebras H. These cocycles arise as liftings of H-invariant linear functionals on V tensor V and give a close formula to deform braided commutator-type relations. Using this construction, we show that all known finite dimensional pointed Hopf algebras over the dihedral groups D_m with m=4t > 11, over the symmetric group S_3 and some families over S_4 are cocycle deformations of bosonizations of Nichols algebras., Comment: 20 pages. This version: extended version following the referee's suggestions. Intended for non-experts

The classification of all Hopf algebras of a given finite dimension over an algebraically closed field of characteristic 0 is a difficult problem. If the dimension is a prime, then the Hopf algebra is a group algebra. If the dimension is the square of a prime then the Hopf algebra is a group algebra or a Taft Hopf algebra. The classification is also complete for dimension 2p or 2p^2, p a prime. Partial results for some other cases are available. For example, for dimension p^3 the classification of the semisimple Hopf algebras was done by Masuoka, and the pointed Hopf algebras were classified by Andruskiewitsch and Schneider, Caenepeel and Dascalescu, and Stefan and van Oystaeyen independently. Many classification results for the nonsemisimple, nonpointed, non-copointed case have been proved by the second author but the classification in general for dimension p^3 is still incomplete, up to now even for dimension 27. In this paper we outline some results and techniques which have been useful in approaching this problem and add a few new ones. We give some further results on Hopf algebras of dimension p^3 and finish the classification for dimension 27., Comment: 19 pages

Let k be an algebraically closed field of characteristic 0. We conclude the classification of finite dimensional pointed Hopf algebras whose group of group-likes is S_4. We also describe all pointed Hopf algebras over S_5 whose infinitesimal braiding is associated to the rack of transpositions., Comment: 22 pages. Some results extended

Let a, b be non-zero complex numbers and l an odd natural number bigger that 2. We determine all Hopf algebra quotients of the quantized coordinate algebra O_{a,b}(GL_{n}) when a^{-1}b is a primitive l-th root of unity and a, b satisfy certain mild conditions, and we caracterize all finite-dimensional quotients when a^{-1}b is not a root of unity. As a byproduct we give a new family of non-semisimple and non-pointed Hopf algebras with non-pointed duals which are quotients of O_{a, b}(GL_{n})., Comment: 42 pages

We complete the classification of Hopf algebras of dimension 16 over an algebraically closed field of characteristic zero. We show that a non-semisimple Hopf algebra of dimension 16, has either the Chevalley property or its dual is pointed., Comment: 23 pages

Let G be a connected, simply connected, simple complex algebraic group and let e be a primitive l-th root of 1, with l odd and 3 does not divide l if G is of type G_{2}. We determine all Hopf algebra quotients of the quantized coordinate algebra of G at e., Comment: 29 pages, accepted in Compositio Mathematica

We give a necessary and sufficient condition for two Hopf algebras presented as central extensions to be isomorphic, in a suitable setting. We then study the question of isomorphism between the Hopf algebras constructed in 0707.0070v1 as quantum subgroups of quantum groups at roots of 1. Finally, we apply the first general result to show the existence of infinitely many non-isomorphic Hopf algebras of the same dimension, presented as extensions of finite quantum groups by finite groups., Comment: final version, to appear in Transformation Groups

We discuss some general results on finite-dimensional Hopf algebras over an algebraically closed field k of characteristic zero and then apply them to Hopf algebras H of dimension p^{3} over k. There are 10 cases according to the group-like elements of H and H^{*}. We show that in 8 of the 10 cases, it is possible to determine the structure of the Hopf algebra. We give also a partial classification of the quasitriangular Hopf algebras of dimension p^{3} over k, after studying extensions of a group algebra of order p by a Taft algebra of dimension p^{2}. In particular, we prove that every ribbon Hopf algebra of dimension p^{3} over k is either a group algebra or a Frobenius-Lusztig kernel. Finally, using some previous results on bounds for the dimension of the first term H_{1} in the coradical filtration of H, we give the complete classification of the quasitriangular Hopf algebras of dimension 27., Comment: 27 pages, minor changes. Accepted for publication in the Tsukuba Journal of Mathematics