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Lectures on Hopf Algebras LECTURES ON HOPF ALGEBRAS HANS-JURGENÄ SCHNEIDER Notes by Sonia Natale Preface These notes contain the material presented in a series of ¯ve lectures at the University of C¶ordobain September 1994. The intent of this brief course was to give a quick introduction to Hopf algebras and to prove as directly as possible (to me) some recent results on ¯nite- dimensional Hopf algebras conjectured by Kaplansky in 1975. In particular, in the last part of the course I gave a complete proof from scratch of Zhou's theorem (1994): Any ¯nite- dimensional Hopf algebra over the complex numbers of prime dimension p is isomorphic to the group algebra of the group of order p. I would like to thank Nicol¶asAndruskiewitsch and the members of the Mathematics Department of the FaMAF (University of C¶ordoba)for the kind invitation to visit C¶ordoba and for their warm hospitality. I also wish to thank the students in my class for their attention and for most enjoyable hours after the lectures. My special thanks are due to Sonia Natale who has written up the notes of a condensed course with great care and insight. H.-J. Schneider Supported by a grant of DAAD-Antorchas, a Subsidio of the SECyT (U.N. C¶ordoba)and the CIEM. Typeset by AMS-TEX 1 2 HANS-JURGENÄ SCHNEIDER x1 Definitions and examples In what follows, we will consider a commutative ring k (later on k will be a ¯eld); the symbols Hom and ­ will mean Homk and ­k respectively. By an algebra R over k (or simply an algebra) we understand a unitary, associative k-algebra R, with identity 1 = 1R. The category of k-algebras will be denoted by Algk. If R is an algebra, then Rop denotes the opposite algebra (i.e. the k-module R with multiplication a:opb = ba). For an algebra R, RM (respectively MR) will denote the category of left (respectively right) R-modules. Recall that a k-module M is a left (respectively right) R-module, if and only if, there exists an algebra map: R ¡! End(M) (respectively Rop ¡! End(M)). Remarks on representation theory. 1. Let G be a group, H := kG its group algebra; V , W in H M. Then k, V ­ W , and V ¤ = Hom(V; k) can be made into left H-modules by setting: g:1 = 1; g:(v ­ w) = g:v ­ g:w; (g:Á)(v) = Á(g¡1:v); for all g 2 G; v 2 V; w 2 W; Á 2 V ¤. We note that the algebra maps which de¯ne the module structures in each case are given by: ² : kG ¡! k; kG ¡!¢ kG ­ kG ¡! End(V ) ­ End(W ) ¡! End(V ­ W ); transpose kG ¡!S kGop ¡¡¡¡¡¡! End(V ¤); where ²(g) = 1, ¢(g) = g ­ g, S(g) = g¡1, 8g 2 G. 2. Let us now consider a Lie algebra g, H = U(g) its universal enveloping algebra. Then the Lie algebra maps: g ¡! k; x 7! 0; g ¡! g £ g; x 7! (x; x); g ¡! gop; x 7! ¡x; together with the universal properties de¯ning H, give rise to algebra maps: ² : H ¡! k; ¢ : H ¡! H ­ H w U(g £ g); S : H ¡! Hop w U(gop): LECTURES ON HOPF ALGEBRAS 3 Explicitly ²(x) = 0; ¢(x) = 1 ­ x + x ­ 1; S(x) = ¡x; x 2 g. If V and W are left H-modules, with corresponding actions H ¡! End(V ) and H ¡! End(W ), then the composition H ¡!¢ H ­ H ¡! End(V ) ­ End(W ) ¡! End(V ­ W ) provides V ­ W a left H-module structure. This is uniquely determined by the condition x:(v ­ w) = x:v ­ w + v ­ x:w; x 2 g, v 2 V , w 2 W . In an analogous way, but now using the antipode, we may let H act over V ¤, via: transpose H ¡!S Hop ¡¡¡¡¡¡! End(V ¤) This dual action is determined by (x:Á)(v) = Á(¡x:v), x 2 g , v 2 V , Á 2 V ¤. Finally, we shall consider k as H-module via ² : H ¡! k, this is uniquely determined by x:1 = 0, for all x 2 g. We want to consider algebras such that tensor products and duals of modules are again modules, as in the examples above. First we need the de¯nition of coalgebra. Observe that an associative, unitary k-algebra is a pair (A; m), where A is a k-module and m : A ­ A ¡! A is a k-linear map, called the multiplication, such that: 1. The following diagram is commutative: A ­ A ­ A ¡¡¡¡!m­id A ­ A ? ? ? ? id ­my ym A ­ A ¡¡¡¡! A m 2. There exists a k-linear map u : k ¡! A such that the following diagrams commute: k ­ A ¡¡¡¡!u­id A ­ A á¡¡¡id ­u A ­ k ? ? ? ? ? ? y ym y A A A; where the maps k ­ A ¡! A and A ­ k ¡! A are the canonical ones. Such a u is necessarily unique. The ¯rst of these diagrams says that the algebra A is associative and the second gives the existence of a unit u(1) = 1A in A. By reversing arrows, we get the dual notion. 4 HANS-JURGENÄ SCHNEIDER De¯nition. A coalgebra over k is a pair (C; ¢), where C is a k-module and ¢ : C ¡! C­C is a k-linear map called the comultiplication, such that: 1. The following diagram commutes: C ­ C ­ C á¡¡¡¢­id C ­ C x x ? ? id ­¢? ?¢ C ­ C á¡¡¡ C: ¢ 2. There exists a k-linear map ² : C ¡! k, such that the following diagrams commute: k ­ C á¡¡¡²­id C ­ C ¡¡¡¡!id ­² C ­ k x x x ? ? ? ? ?¢ ? C C C: The map ² is called the counit and is uniquely determined by the pair (C; ¢). The kernel of ² will be denoted by C+. If (C; ¢C ), (D; ¢D) are coalgebras, a k-linear map: f : C ¡! D is said a coalgebra map, if the following diagrams commute: f C ¡¡¡¡! D ? ? ? ? ¢C y y¢D C ­ C ¡¡¡¡! D ­ D; f­f f C ¡¡¡¡! D ? ? ? ? ²C y y²D k k: Remark. More generally, one can de¯ne algebras and coalgebras in monoidal categories, that is k-linear categories C provided with a "tensor" functor ­ : C £ C ¡!C, plus an associativity constraint (see below). The opposite category Cop of a category C has the same objects but the arrows are reversed: HomCop (A; B) = HomC(B; A). In this way, a coalgebra in C is the same as an algebra in Cop. Examples. 1. If S is any set and C = kS is the free k-module with basis S, then C becomes a coalgebra if we set: ¢(s) = s ­ s, ²(s) = 1, s 2 S. 2. The universal enveloping algebra of a Lie algebra g is a coalgebra with the coproduct ¢ and counit ² just considered. Now we dualize the de¯nition of a module over a k-algebra. LECTURES ON HOPF ALGEBRAS 5 De¯nition. Let C be a coalgebra over k.A right comodule over C is a pair (M; ¢M ), where M is a k-module and ¢M : M ¡! M ­C is a k-linear map (the comodule structure), such that the following diagrams commute: M ¡¡¡¡!¢M M ­ C ? ? ? ? ¢M y y¢M ­id M ­ C ¡¡¡¡! M ­ C ­ C; id­¢ M ¡¡¡¡!¢M M ­ C ? ? ? ? y yid­² M ­ k M ­ k: A k-linear map Á : M ¡! N between right C-comodules M, N, is said a comodule map if the following diagram commutes: Á M ¡¡¡¡! N ? ? ? ? ¢M y y¢N M ­ C ¡¡¡¡! N ­ C Á­id The left C-comodules are de¯ned in a similar fashion. We will denote MC and C M, respectively, the categories of right and left C-comodules. Consider a k-module A; it could happen that A has both an algebra and coalgebra structure. In case these structures "paste" well, we give A a special name: De¯nition. We say that a triple (A; m; ¢) is a bialgebra, if (A; m) is an algebra with unit u,(A; ¢) is a coalgebra with counit ² and ¢ : A ¡! A ­ A, ² : A ¡! k are algebra maps. A k-linear map Á : A ¡! B, where A and B are bialgebras is said a bialgebra map if it is both an algebra and a coalgebra map. Remarks. 1. In the de¯nition A ­ A is considered with the natural algebra structure. In general the tensor product of two algebras A and B has a natural algebra structure determined by (a ­ b)(c ­ d) = ac ­ bd; 8a; c 2 A; b; d 2 B: Equivalently, the multiplication mA­B is the composition A ­ B ­ A ­ B ¡¡¡¡¡!id ­¿­id A ­ A ­ B ­ B ¡¡¡¡¡!mA­mB A ­ B: Here ¿ denotes the "twist" map: ¿ : a ­ b 7! b ­ a. 6 HANS-JURGENÄ SCHNEIDER Now, if C and D are coalgebras, then the tensor product C ­ D can be made into a coalgebra in a natural way, with the comultiplication C ­ D ¡¡¡¡¡!¢C ­¢D C ­ C ­ D ­ D ¡¡¡¡¡!id ­¿­id C ­ D ­ C ­ D: The counit is given by C ­ D ¡¡¡¡!²C ­²D k ­ k w k: One can then check that in the de¯nition of bialgebra the condition of ¢ and ² being algebra maps may be replaced by the (equivalent) condition of m and u being coalgebra maps. 2. The kernel of the counit ² in a bialgebra A is a two sided ideal of codimension 1, called the augmentation ideal. Examples of bialgebras are kG, the group algebra of a group G, with the algebra and coalgebra structures considered at the beginning (notice that we do not make use here of the existence of inverses for elements of G), and the universal enveloping algebra of a Lie algebra g, where ¢ and ² are as treated earlier.
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