PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 10, October 1998, Pages 2841–2844 S 0002-9939(98)04392-5 ON THE CLASS EQUATION FOR HOPF ALGEBRAS MARTIN LORENZ (Communicated by Ken Goodearl) Abstract. We give a simple proof of the Kac-Zhu class equation for semisim- ple Hopf algebras over an algebraically closed field of characteristic 0. 1. Introduction The class equation ([K], Theorem 2, [Z], Theorem 1) expresses a fundamental arithmetic property of semisimple Hopf algebras over an algebraically closed field of characteristic 0. It has featured prominently in many classification results for semisimple Hopf algebras of small or otherwise restricted dimension, notably in the work of Masuoka (cf. [M] and the references therein). All proofs heretofore ([K], [S]) ultimately depend on the finiteness of the class number of algebraic number fields. The purpose of this short note is to give a simpler, more representation theoretic proof whose main ingredient is the known explicit form of the central primitive idempotents in split semisimple algebras. In §2, we gather the requisite background material and give the statement of the class equation. The reader is referred to [L] or [S] for complete details. §3 then presents our proof of the class equation. Throughout, H denotes a finite dimensional Hopf algebra over a field k of char- acteristic p ≥ 0, with augmentation ε and antipode S . All H -modules are assumed to be finite dimensional left modules. 2. The class equation 2.1. Character algebra and Grothendieck ring. The character algebra R(H ) of H is the k-subalgebra of the dual Hopf algebra H * that is generated by the characters χ V of H -modules V .A Z-form of R(H ) is provided by the Grothendieck ring G 0 (H ), the abelian group generated by the isomorphism classes [V ] of H - modules V modulo the relations [V ]=[U ]+[W ] for each short exact sequence 0 → U → V → W → 0. Multiplication in G 0 (H ) is given by [V ] · [W ]=[V ⊗ k W ]. The character map χ : G 0 (H ) → R(H ), [V ] → χ V , induces an isomorphism of k-algebras R(H ) ∼ = G 0 (H ) ⊗ Z k. Received by the editors December 16, 1996 and, in revised form, March 13, 1997. 1991 Mathematics Subject Classification. Primary 16W30; Secondary 16G10. Key words and phrases. Hopf algebra, Grothendieck ring, character algebra, idempotent. Research supported in part by NSF Grant DMS-9400643. c 1998 American Mathematical Society 2841 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use