1 On finite-dimensional Hopf algebras 2 Dedicado a Biblioco 34 3 4 Nicolás Andruskiewitsch 5 Abstract. This is a survey on the state-of-the-art of the classification of finite-dimensional complex 6 Hopf algebras. This general question is addressed through the consideration of different classes of such 7 Hopf algebras. Pointed Hopf algebras constitute the class best understood; the classification of those 8 with abelian group is expected to be completed soon and there is substantial progress in the non-abelian 9 case. 10 Mathematics Subject Classification (2010). 16T05, 16T20, 17B37, 16T25, 20G42. 11 Keywords. Hopf algebras, quantum groups, Nichols algebras. 12 1. Introduction Hopf algebras were introduced in the 1950’s from three different perspectives: algebraic groups in positive characteristic, cohomology rings of Lie groups, and group objects in the category of von Neumann algebras. The study of non-commutative non-cocommutative Hopf algebras started in the 1960’s. The fundamental breakthrough is Drinfeld’s report [25]. Among many contributions and ideas, a systematic construction of solutions of the quantum Yang-Baxter equation (qYBE) was presented. Let V be a vector space. The qYBE is equivalent to the braid equation: (c ⊗ id)(id ⊗c)(c ⊗ id) = (id ⊗c)(c ⊗ id)(id ⊗c); c 2 GL(V ⊗ V ): (1.1) 13 If c satisfies (1.1), then (V; c) is called a braided vector space; this is a down-to-the-earth 14 version of a braided tensor category [54]. Drinfeld introduced the notion of quasi-triangular 15 Hopf algebra, meaning a pair (H; R) where H is a Hopf algebra and R 2 H ⊗H is invertible 16 and satisfies the approppriate conditions, so that every H-module V becomes a braided vec- 17 tor space, with c given by the action of R composed with the usual flip. Furthermore, every 18 finite-dimensional Hopf algebra H gives rise to a quasi-triangular Hopf algebra, namely the ∗ 19 Drinfeld double D(H) = H ⊗ H as vector space. If H is not finite-dimensional, some pre- 20 cautions have to be taken to construct D(H), or else one considers Yetter-Drinfeld modules, 21 see §2.2. In conclusion, every Hopf algebra is a source of solutions of the braid equation. 22 Essential examples of quasi-triangular Hopf algebras are the quantum groups Uq(g) [25, 53] 23 and the finite-dimensional variations uq(g) [59, 60]. 24 In the approach to the classification of Hopf algebras exposed in this report, braided 25 vector spaces and braided tensor categories play a decisive role; and the finite quantum 26 groups are the main actors in one of the classes that splits off. Proceedings of International Congress of Mathematicians, 2014, Seoul 2 Nicolás Andruskiewitsch 27 By space limitations, there is a selection of the topics and references included. Par- 28 ticularly, we deal with finite-dimensional Hopf algebras over an algebraically closed field 29 of characteristic zero with special emphasis on description of examples and classifications. 30 Interesting results on Hopf algebras either infinite-dimensional, or over other fields, un- 31 fortunately can not be reported. There is no account of the many deep results on tensor 32 categories, see [30]. Various basic fundamental results are not explicitly cited, we refer to 33 [1, 62, 66, 75, 79, 83] for them; classifications of Hopf algebras of fixed dimensions are not 34 evoked, see [21, 71, 86]. 35 2. Preliminaries 36 Let θ 2 N and I = Iθ = f1; 2; : : : ; θg. The base field is C. If X is a set, then jXj is its 37 cardinal and CX is the vector space with basis (xi)i2X . Let G be a group: we denote by 38 Irr G the set of isomorphism classes of irreducible representations of G and by Gb the subset x G 39 of those of dimension 1; by G the centralizer of x 2 G; and by Ox its conjugacy class. 40 More generally we denote by Irr C the set of isomorphism classes of simple objects in an S 41 abelian category C. The group of n-th roots of 1 in C is denoted Gn; also G1 = n≥1 Gn. 42 The group presented by (xi)i2I with relations (rj)j2J is denoted h(xi)i2I j(rj)j2J i. The 43 notation for Hopf algebras is standard: ∆, ", S, denote respectively the comultiplication, 44 the counit, the antipode (always assumed bijective, what happens in the finite-dimensional 45 case). We use Sweedler’s notation: ∆(x) = x(1) ⊗ x(2). Similarly, if C is a coalgebra and 46 V is a left comodule with structure map δ : V ! C ⊗ V , then δ(v) = v(−1) ⊗ v(0). If D; E 0 47 are subspaces of C, then D ^ E = fc 2 C : ∆(c) 2 D ⊗ C + C ⊗ Eg; also ^ D = D and n+1 n 48 ^ D = (^ D) ^ D for n > 0. 49 2.1. Basic constructions and results. The first examples of finite-dimensional Hopf alge- G 50 bras are the group algebra CG of a finite group G and its dual, the algebra of functions C . 51 Indeed, the dual of a finite-dimensional Hopf algebra is again a Hopf algebra by transpos- 52 ing operations. By analogy with groups, several authors explored the notion of extension of 53 Hopf algebras at various levels of generality; in the finite-dimensional context, every exten- 54 sion C ! A ! C ! B ! C can be described as C with underlying vector space A ⊗ B, 55 via a heavy machinery of actions, coactions and non-abelian cocycles, but actual examples 56 are rarely found in this way (extensions from a different perspective are in [9]). Relevant 57 exceptions are the so-called abelian extensions [56] (rediscovered by Takeuchi and Majid): 58 here the input is a matched pair of groups (F; G) with mutual actions ., / (or equivalently, an G 59 exact factorization of a finite group). The actions give rise to a Hopf algebra C #CF . The 60 multiplication and comultiplication can be further modified by compatible cocycles (σ; κ), G Gκ 61 producing to the abelian extension C ! C ! C #σCF ! CF ! C. Here (σ; κ) turns 62 out to be a 2-cocycle in the total complex associated to a double complex built from the 2 63 matched pair; the relevant H is computed via the so-called Kac exact sequence. 64 It is natural to approach Hopf algebras by considering algebra or coalgebra invariants. 65 There is no preference in the finite-dimensional setting but coalgebras and comodules are 66 locally finite, so we privilege the coalgebra ones to lay down general methods. The basic 67 coalgebra invariants of a Hopf algebra H are: 68 ◦ The group G(H) = fg 2 H − 0 : ∆(g) = x ⊗ gg of group-like elements of H. On finite-dimensional Hopf algebras 3 69 ◦ The space of skew-primitive elements Pg;h(H), g; h 2 G(H); P(H) := P1;1(H). 70 ◦ The coradical H0, that is the sum of all simple subcoalgebras. n S 71 ◦ The coradical filtration H0 ⊂ H1 ⊂ ::: , where Hn = ^ H0; then H = n≥0 Hn. 72 2.2. Modules. The category H M of left modules over a Hopf algebra H is monoidal with H 73 tensor product defined by the comultiplication; ditto for the category M of left comod- 74 ules, with tensor product defined by the multiplication. Here are two ways to deform Hopf 75 algebras without altering one of these categories. 76 • Let F 2 H ⊗H be invertible such that (1⊗F )(id ⊗∆)(F ) = (F ⊗1)(∆⊗id)(F ) and F 77 (id ⊗")(F ) = (" ⊗ id)(F ) = 1. Then H (the same algebra with comultiplication F −1 78 ∆ := F ∆F ) is again a Hopf algebra, named the twisting of H by F [26]. The 79 monoidal categories H M and HF M are equivalent. If H and K are finite-dimensional 80 Hopf algebras with H M and K M equivalent as monoidal categories, then there exists F 81 F with K ' H as Hopf algebras (Schauenburg, Etingof-Gelaki). Examples of 82 twistings not mentioned elsewhere in this report are in [31, 65]. 83 • Given a linear map σ : H ⊗H ! C with analogous conditions, there is a Hopf algebra 84 Hσ (same coalgebra, multiplication twisted by σ) such that the monoidal categories H Hσ 85 M and M are equivalent [24]. 86 A Yetter-Drinfeld module M over H is left H-module and left H-comodule with the 87 compatibility δ(h:m) = h(1)m(−1)S(h(3)) ⊗ h(2) · m(0), for all m 2 M and h 2 H. The H 88 category H YD of Yetter-Drinfeld modules is braided monoidal. That is, for every M; N 2 H 89 H YD, there is a natural isomorphism c : M ⊗ N ! N ⊗ M given by c(m ⊗ n) = H 90 m(−1) · n ⊗ m(0), m 2 M, n 2 N. When H is finite-dimensional, the category H YD is 91 equivalent, as a braided monoidal category, to D(H)M. 92 The definition of Hopf algebra makes sense in any braided monoidal category. Hopf H 93 algebras in H YD are interesting because of the following facts–discovered by Radford and 94 interpreted categorically by Majid, see [62, 75]: H 95 If R is a Hopf algebra in H YD, then R#H := R ⊗ H with semidirect product and 96 coproduct is a Hopf algebra, named the bosonization of R by H. ι { coπ 97 Let π; ι be Hopf algebra maps as in K π / / H with πι = idH .