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This is a survey on the state-of-the-art of the classification of finite-dimensional complex Hopf algebras. This general question is addressed through the consideration of different classes of such Hopf algebras. Pointed Hopf algebras constitute the class best understood; the classification of those with abelian group is expected to be completed soon and there is substantial progress in the non-abelian case., Comment: 25 pages. To be presented at the algebra session of ICM 2014. Submitted version

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

We develop a strategy to compute all liftings of a Nichols algebra over a finite dimensional cosemisimple Hopf algebra. We produce them as cocycle deformations of the bosonization of these two. In parallel, we study the shape of any such lifting., Comment: 32 pages

We present examples of color Hopf algebras, i.e. Hopf algebras in color categories (braided tensor categories with braiding induced by a bicharacter on an abelian group), related with quantum doubles of pointed Hopf algebras. We also discuss semisimple color Hopf algebras., Comment: 26 pages

A Hopf algebra is co-Frobenius when it has a nonzero integral. It is proved that the composition length of the indecomposable injective comodules over a co-Frobenius Hopf algebra is bounded. As a consequence, the coradical filtration of a co-Frobenius Hopf algebra is finite; this confirms a conjecture by Sorin D\u{a}sc\u{a}lescu and the first author. The proof is of categorical nature and the same result is obtained for Frobenius tensor categories of subexponential growth. A family of co-Frobenius Hopf algebras that are not of finite type over their Hopf socles is constructed, answering so in the negative another question by the same authors., Comment: Minor changes. Final version, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5); 33 pages

This is a survey on spherical Hopf algebras. We give criteria to decide when a Hopf algebra is spherical and collect examples. We discuss tilting modules as a mean to obtain a fusion subcategory of the non-degenerate quotient of the category of representations of a suitable Hopf algebra., Comment: 27 pages

We introduce a new filtration on Hopf algebras, the standard filtration, generalizing the coradical filtration. Its zeroth term, called the Hopf coradical, is the subalgebra generated by the coradical. We give a structure theorem: any Hopf algebra with injective antipode is a deformation of the bosonization of the Hopf coradical by its diagram, a connected graded Hopf algebra in the category of Yetter-Drinfeld modules over the latter. We discuss the steps needed to classify Hopf algebras in suitable classes accordingly. For the class of co-Frobenius Hopf algebras, we prove that a Hopf algebra is co-Frobenius if and only if its Hopf coradical is so and the diagram is finite dimensional. We also prove that the standard filtration of such Hopf algebras is finite. Finally, we show that extensions of co-Frobenius (resp. cosemisimple) Hopf algebras are co-Frobenius (resp. cosemisimple)., Comment: Subsection 2.2 removed in view of the positive result in new reference [ACE]. Final version, to appear in JNCG; 20 pages

We discuss the relationship between Hopf superalgebras and Hopf algebras. We list the braided vector spaces of diagonal type with generalized root system of super type and give the defining relations of the corresponding Nichols algebras., Comment: 18 pages

We investigate the representation theory of a large class of pointed Hopf algebras, extending results of Lusztig and others. We classify all simple modules in a suitable category and determine the weight multiplicities; we establish a complete reducibility theorem in this category., Comment: 44 pages

The notion of inner linear Hopf algebra is a generalization of the notion of discrete linear group. In this paper, we prove two general results that enable us to enlarge the class of Hopf algebras that are known to be inner linear: the first one is a characterization by using the Hopf dual, while the second one is a stability result under extensions. We also discuss the related notion of inner unitary Hopf *-algebra., Comment: 9 pages. Minor corrections, final version, to appear in Revista de la Union Matematica Argentina.

We consider issues related to the origins, sources and initial motivations of the theory of Hopf algebras. We consider the two main sources of primeval development: algebraic topology and algebraic group theory. Hopf algebras are named from the work of Heinz Hopf in the 1940's. In this note we trace the infancy of the subject back to papers from the 40's, 50's and 60's in the two areas mentioned above. Many times we just describe -- and/or transcribe parts of -- some of the relevant original papers on the subject., Comment: 19 pages, to appear in Acta Applicandae Mathematicae

We prove that every slim double Lie groupoid with proper core action is completely determined by a factorization of a certain canonically defined "diagonal" Lie groupoid., Comment: 15 pages

It is an important open problem whether the dimension of the Nichols algebra B(O,\rho) is finite when O is the class of the transpositions and \rho is the sign representation, with m>= 6. In the present paper, we discard most of the other conjugacy classes showing that very few pairs (O,\rho) might give rise to finite-dimensional Nichols algebras., Comment: 14 pages; typos added in abstract

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 extend the main result of [N. Andruskiewitsch and H.-J. Schneider, A characterization of quantum groups], see math/0201095, to braided vector spaces of generic diagonal type using results of Heckenberger., Comment: 17 pages. The second version incorporates some changes suggested by the referee. To appear in "Modules and comodules", Proceedings of a conference dedicated to R. Wisbauer

We classify finite-dimensional complex pointed Hopf algebra with group of group-like elements isomorphic to A_5. We show that any pointed Hopf algebra with infinitesimal braiding associated with the conjugacy class of $\pi$ \in $A_n$ is infinite-dimensional if the order of $\pi$ is odd except for $\pi=(1 2 3)$ in $A_4$. We also study pointed Hopf algebras over the dihedral groups., Comment: v2: minor corrections, we remove Table 1 and change Remark 3.4; v3: minor corrections, we modify reference [FGV]; final version to appear in Rev. Uni\'on Mat. Argent., 17 pages

We show that indecomposable exact module categories over the category Rep H of representations of a finite-dimensional Hopf algebra H are classified by left comodule algebras, H-simple from the right and with trivial coinvariants, up to equivariant Morita equivalence. Specifically, any indecomposable exact module categories is equivalent to the category of finite-dimensional modules over a left comodule algebra. This is an alternative approach to the results of Etingof and Ostrik. For this, we study the stabilizer introduced by Yan and Zhu and show that it coincides with the internal Hom. We also describe the correspondence of module categories between Rep H and Rep (H^*)., Comment: 33 pages, minor misprints corrected, to appear in J. Algebra

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 study Poisson symmetric spaces of group type with Cartan subalgebra "adapted" to the Lie cobracket., Comment: Several corrections of notation. Samo proof have been corrected. Some minor corrections

We give a general description of the structure of a discrete double groupoid (with an extra, quite natural, filling condition) in terms of groupoid factorizations and groupoid 2-cocycles with coefficients in abelian group bundles. Our description goes as follows: To any double groupoid, we associate an abelian group bundle and a second double groupoid, its frame. The frame satisfies that every box is determined by its edges, and thus is called a `slim' double groupoid. In a first step, we prove that every double groupoid is obtained as an extension of its associated abelian group bundle by its frame. In a second, independent, step we prove that every slim double groupoid with filling condition is completely determined by a factorization of a certain canonically defined `diagonal' groupoid., Comment: amslatex, 28 pages, revised version to appear in J. Pure Appl. Algebra

The construction of a quantum groupoid out of a double groupoid satisfying a filling condition and a perturbation datum is given. This extends previous work that appeared in math.QA/0308228. Several important classes of examples of tensor categories are shown to fit into this construction. Certain invariants such as a pivotal group-like element and quantum and Frobenius-Perron dimensions of simple objects are computed., Comment: Revised version to appear in Adv. Math., 45 pages. Added examples where the square of the antipode is not the identity, necessary and sufficient conditions for the representation category to be fusion, and the computation of quantum and Frobenius-Perron dimensions of simple representations

We construct explicit examples of weak Hopf algebras (actually face algebras in the sense of Hayashi) via vacant double groupoids as explained in \http://arxiv.org/abs/math.QA/0308228. To this end, we first study the Kac exact sequence for matched pairs of groupoids and show that it can be computed via group cohomology. Then we describe explicit examples of finite vacant double groupoids., Comment: 19 pages

Quivers over a fixed base set form a monoidal category with tensor product given by pullback. The quantum Yang-Baxter equation, or more properly the braid equation, is investigated in this setting. A solution of the braid equation in this category is called a "solution" for short. Results of Etingof-Schedler-Soloviev, Lu-Yan-Zhu and Takeuchi on the set-theoretical quantum Yang-Baxter equation are generalized to the context of quivers, with groupoids playing the r\^ole of groups. The notion of "braided groupoid" is introduced. Braided groupoids are solutions and are characterized in terms of bijective 1-cocycles. The structure groupoid of a non-degenerate solution is defined; it is shown that it is braided groupoid. The reduced structure groupoid of a non-degenerate solution is also defined. Non-degenerate solutions are classified in terms of representations of matched pairs of groupoids. By linearization we construct star-triangular face models and realize them as modules over quasitriangular quantum groupoids introduced in recent papers by M. Aguiar, S. Natale and the author., Comment: 40 pages. An Appendix by M. Takeuchi is included

We introduce the category of set-theoretic representations of a matched pair of groupoids. This is a monoidal category endowed with a monoidal functor to the category of quivers over the common base of the groupoids in the matched pair (the forgetful functor). We study monoidal functors between two such categories of representations which preserve the forgetful functor. We show that the centralizer of such a monoidal functor is the category of representations of a new matched pair, which we construct explicitly. We introduce the notions of {\em double} of a matched pair of groupoids and {\em generalized double} of a morphism of matched pairs. We show that the centralizer of the forgetful functor is the category of representations of the dual matched pair, and the centralizer of the identity functor (the center) is the category of representations of the double. We use these constructions to classify the braidings in the category of representations of a matched pair. Such braidings are parametrized by certain groupoid-theoretic structures which we call {\em matched pairs of rotations}. Finally, we express our results in terms of the weak Hopf algebra associated to a matched pair of groupoids. A matched pairs of rotations gives rise to a quasitriangular structure for the associated weak Hopf algebra. The Drinfeld double of the weak Hopf algebra of a matched pair is the weak Hopf algebra associated to the double matched pair.

We give the construction of a class of weak Hopf algebras (or quantum groupoids) associated to a matched pair of groupoids and certain cocycle data. This generalizes a now well-known construction for Hopf algebras, first studied by G. I. Kac in the sixties. Our approach is based on the notion of double groupoids, as introduced by Ehresmann., Comment: 34 pages, amslatex. The statement in Proposition 3.10 has been corrected

Let g be a simple complex finite dimensional Lie algebra and let U_q^+(g) be the positive part of the quantum enveloping algebra of g. If g is of type A_2, the group of algebra automorphisms of U_q^+(g) is a semidirect product of a 2-dimensional torus and the group of automorphisms of the Dynkin diagram; any algebra automorphism is an automorphism of braided Hopf algebra, and preserves the standard grading [AD1]. This intriguing smallness of the group of algebra automorphisms raises questions about the extent of these phenomena. We discuss some of them in the present paper. We introduce the notion of ``algebra with few automorphisms" and establish some consequences. We prove some exploratory results concerning the group of algebra automorphisms for the type B_2. We study Hopf algebra automorphisms of Nichols algebras and their bosonizations and compute in particular the group of Hopf algebra automorphisms of U_q^+(g)., Comment: Completely revised version, now in English

Two algebras can be attached to a braided vector space $(V, c)$ in an intrinsic way; the FRT-bialgebra and the Nichols algebra $\toba(V, c)$. The FRT-bialgebra plays the r\^ole of the algebra of quantum matrices, whereas the r\^ole of the Nichols algebra is less understood. Some authors call $\toba(V)$ a quantum symmetric algebra. The purpose of this paper is to discuss some properties of certain Nichols algebras, in an attempt to establish classes of Nichols algebras which are worth of further study.

The explicit list of all almost factorizable Lie bialgebra structures on real absolutely simple Lie algebras is given., Comment: Final version, to appear in IRMN. A discussion on Manin triples is added

Let k be a field. Let also (F, G) be a matched pair of groups. We give necessary and sufficient conditions on a pair (\sigma, \tau) of 2-cocycles in order that the crossed product algebra and the crossed coproduct coalgebra k^G{}^{\tau}#_{\sigma} kF combine into a braided Hopf algebra. We also discuss diagonal realizations of such braided Hopf algebras in the category of Yetter-Drinfeld modules over a finite group., Comment: 25 pages, to appear in J. of Pure and Applied Algebra, appendix added at the end of the paper by suggestion of the referee

A fundamental step in the classification of finite-dimensional complex pointed Hopf algebras is the determination of all finite-dimensional Nichols algebras of braided vector spaces arising from groups. The most important class of braided vector spaces arising from groups is the class of braided vector spaces (CX, c^q), where C is the field of complex numbers, X is a rack and q is a 2-cocycle on X with values in C^*. Racks and cohomology of racks appeared also in the work of topologists. This leads us to the study of the structure of racks, their cohomology groups and the corresponding Nichols algebras. We will show advances in these three directions. We classify simple racks in group-theoretical terms; we describe projections of racks in terms of general cocycles; we introduce a general cohomology theory of racks contaninig properly the existing ones. We introduce a "Fourier transform" on racks of certain type; finally, we compute some new examples of finite-dimensional Nichols algebras., Comment: 54 pages. Several minor corrections. Some references added. Same version as will appear in Adv. Math

We prove that a Hopf algebra with a finite coradical filtration is co-Frobenius, i. e. there is a non-zero integral on it. As a consequence, we show that algebras of functions on quantum groups at roots of one are co-Frobenius. We also characterize co-Frobenius Hopf algebras with coradical a Hopf subalgebra. This characterization is in the framework of the lifting method due to H.-J. Schneider and the first-named author. Here is our main result. Let H be a Hopf algebra whose coradical is a Hopf subalgebra. Let gr H be the associated graded Hopf algebra and let R be the diagram of H. Then the following are equivalent: (1) H is co-Frobenius, (2) gr H is co-Frobenius, (3) R is finite dimensional, (4) the coradical filtration of H is finite. This Theorem allows to construct many new examples of co-Frobenius Hopf algebras and opens the way to the classification of ample classes of such Hopf algebras., Comment: 13 pages

Since the discovery of quantum groups (Drinfeld, Jimbo) and finite dimensional variations thereof (Lusztig, Manin), these objects were studied from different points of view and had many applications. The present paper is part of a series where we intend to show that important classes of Hopf algebras are quantum groups and therefore belong to Lie theory. One of our main results is the explicit construction of a general family of pointed Hopf algebras from Dynkin diagrams. All the Frobenius-Lusztig kernels and their parabolic subalgebras belong to this family, but in addition we get many new examples. We show that any finite dimensional pointed Hopf algebra with group of prime exponent (greater than 17) is indeed in this family. An important step in the proof follows from a another result, where we show that a wide family of finite dimensional pointed Hopf algebras is generated by group-like and skew-primitive elements, giving additional support to a conjecture in our previous paper "Finite quantum groups and Cartan matrices", Adv. Math. 154 (2000), 1--45., Comment: 26 pages

We say that a Hopf algebra has the Chevalley property if the tensor product of any two simple modules over this Hopf algebra is semisimple. In this paper we classify finite dimensional triangular Hopf algebras with the Chevalley property, over the field of complex numbers. Namely, we show that all of them are twists of triangular Hopf algebras with R-matrix having rank <=2, and explain that the latter ones are obtained from group algebras of finite supergroups by a simple modification procedure. We note that all examples of finite dimensional triangular Hopf algebras which are known to the authors, do have the Chevalley property, so one might expect that our classification potentially covers all finite dimensional triangular Hopf algebras., Comment: 26 pages, latex. The first proof of Theorem 5.2.1, which is given in Section 5, was completed. The original proof missed a step, and we are grateful to A. Davydov for pointing it out

We define new quantizations of the Heisenberg group by introducing new quantizations in the universal enveloping algebra of its Lie algebra. Matrix coefficients of the Stone--von Neumann representation are preserved by these new multiplications on the algebra of functions on the Heisenberg group. Some of the new quantizations provide also a new multiplication in the algebra of theta functions; we obtain in this way Sklyanin algebras., Comment: 13 pages