ON THE CLASS EQUATION FOR HOPF ALGEBRAS

MARTIN LORENZ

Abstract. We give a simple proof of the Kac-Zhu class equation for semisimple Hopf algebras over an algebraically closed field of characteristic 0.

1. Introduction

The class equation expresses an arithmetic property of semisimple Hopf algebras over an algebraically closed field of characteristic 0 that is of fundamental importance. It was originally proved 25 years ago by G. I. Kac ([K], Theorem 2) in the context of ring groups, that is, finite dimensional complex Hopf algebras with a C^{∗}-algebra structure which is compatible with the Hopf structure. Ring groups are in particular semisimple. Recently, as a crucial step in the proof of his celebrated dimension p - Theorem, Zhu ([Z], Theorem 1) extended Kac’s class equation to arbitrary semisimple Hopf algebras over an algebraically closed field of characteristic 0. Subsequently, Zhu’s version of the class equation 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). To the best of my knowledge, the class equation was christened thus by H.-J. Schneider in his lecture notes [S] which also contain a proof of the equation. All proofs known heretofore 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 characteristic p ≥ 0, with augmentation ε and antipode S. All H-modules are assumed to be finite dimensional left modules.

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.

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 .

Putting [V]^{∗} = [V^{∗}], where V^{∗} = Hom_{k}(V, k), one obtains an anti-automorphism ∗ of the ring G_{0}(H). Similarly, the antipode S^{∗} of H^{∗} stabilizes R(H) and thus yields an anti-automorphism of R(H). The character map respects these anti-automorphisms:

χ_{V^{∗}} = S^{∗}(χ_{V}) .

We will fix a complete set of simple H-modules V_{1}, …, V_{t}, with V_{1} = k_{ε} the “trivial” H-module, and we put χ_{i} = χ_{V_{i}}, the irreducible characters of H. Then {[V_{1}], …, [V_{t}]} is a Z-basis of G_{0}(H) and {χ_{1}, …, χ_{t}} is a k-basis of R(H), with [V_{1}] = [k_{ε}] and χ_{1} = ε the respective identity elements of G_{0}(H) and R(H).

Examples. (1) H = kG, the group algebra of a finite group G over a splitting field k. The dual Hopf algebra H^{∗} is the algebra of k-valued functions on G, H^{∗} = k^{G}, with pointwise algebra operations, and R(H) is the subalgebra of all k-valued functions f on G that are constant on the conjugacy classes of G and satisfy f(g) = f(g_{p′}) for all g ∈ G. Here, g_{p′} denotes the p-regular part of g, that is, the component of g whose order is not divisible by p = char k. So

R(H) ≅ k^{T (G_{p′})} ,

where T (G_{p′}) denotes the set of G-conjugacy classes of p-regular elements of G. Of course, if p does not divide the order of G then T (G_{p′}) is the set of all conjugacy classes of G. Furthermore, via Brauer characters, G_{0}(H) ⊆ G_{0}(H) ⊗_{Z} C ≅ C^{T (G_{p′})}.

(2) H = (kG)^{∗}, the dual Hopf algebra of kG. Here, R(H) = H^{∗} = kG and G_{0}(H) = ZG.

2.2. The semisimple case. Now assume H split semisimple. Then G_{0}(H) is a semiprime ring and ∗ is an involution (cf. [L], Proposition 3.2 and Remark 2.5(2)). The form ⟨ . , . ⟩ : G_{0}(H) × G_{0}(H) → Z that is defined by

⟨[V], [W]⟩ = dim_{k} Hom_{H}(V, W^{∗})

is bilinear, associative, symmetric, and ∗-invariant ([L], §3.1). A pair of dual Zbases of G_{0}(H) with respect to ⟨ . , . ⟩ is given by {[V_{1}], …, [V_{t}]} and {[V_{1}]^{∗}, …, [V_{t}]^{∗}}: ⟨[V_{i}], [V_{j}]^{∗}⟩ = δ_{i,j}.

By k-linear extension to R(H) ≅ G_{0}(H) ⊗_{Z} k, one obtains a form β : R(H) × R(H) → k with analogous properties. In particular, R(H) is a symmetric algebra, and {χ_{1}, …, χ_{t}}, {S^{∗}(χ_{1}), …, S^{∗}(χ_{t})} is a pair of dual bases for β. If p = char k = 0 then R(H) is semisimple, as a consequence of semiprimeness of G_{0}(H).

2.3. The Class Equation (Kac-Zhu). Assume H semisimple and k algebraically closed of characteristic p = 0. Then, for every primitive idempotent e = e^{2} ∈ R(H), dim_{k} eH^{∗} divides dim_{k} H.

Examples. For H = kG, the class equation reduces to the fact that the size of each conjugacy class of G divides |G|. For H = (kG)^{∗}, it says that the dimension of each simple kG-module divides |G| (Frobenius’ Theorem).

3. The proof

Letting A denote the ring of algebraic integers in k, it suffices to show that

d := \frac{dim_{k} H}{dim_{k} eH^{∗}} ∈ A .

To this end, we first express d in terms of the central primitive idempotent ê ∈ Z(R(H)), the center of R(H), with e = eê. In other words, êR(H) is the simple component of the split semisimple k-algebra R(H) containing e. Thus, if m denotes the dimension of the irreducible R(H)-module eR(H), then êR(H) ≅ (eR(H))^{m}. This implies êH^{∗} ≅ (eH^{∗})^{m}, and hence

d = \frac{m dim_{k} H}{dim_{k} êH^{∗}} .

Central primitive idempotents can be expressed in terms of dual bases (cf. [CR], Proposition (9.17)). In our situation, in view of §2.2, this expression takes the following form:

ê = ω(ẽ)^{−1}ẽ with ẽ = ∑_{i} μ(S^{∗}(χ_{i}))χ_{i} ,

where ω : Z(R(H)) → k denotes the central character of the irreducible R(H)-module eR(H) and μ its (ordinary) character; so m = μ(ε)and ω(z) = m^{−1}μ(z). We claim that

d = ω(ẽ) .

Indeed, defining L_{f} ∈ End_{k}(H^{∗}) by L_{f}(g) = fg for f, g ∈ H^{∗}, the element x = (f ↦ trace(L_{f})) ∈ H^{∗∗} = H is an integral of H ([LS], Proposition 9; recall that H is involutory, by [LR]). Since ê is idempotent, we have dim_{k} êH^{∗} = trace(Lê), and so dim_{k} êH^{∗} = ê(x) = ω(ẽ)^{−1}ẽ(x). Finally, χ_{1}(x) = x(ε) = dim_{k} H^{∗} = dim_{k} H and χ_{i}(x) = 0 for i > 1, since xV_{i} consists of H-invariants in V_{i} and hence must vanish. Thus, ẽ(x) = μ(ε) dim_{k} H = m dim_{k} H and our claim follows.

To complete the proof, we remark that if ϕ is any character of R(H) then, identifying G_{0}(H) with its image in R(H) under the character map, we have ϕ(G_{0}(H)) ⊆ A. For, any α ∈ G_{0}(H) satisfies a monic polynomial over Z, and hence so does the image Φ(α) under the representation Φ belonging to ϕ, and all eigenvalues of Φ(α) as well. Thus, ϕ(α) = trace(Φ(α)) ∈ A. In particular, ẽ = ∑_{i} μ(S^{∗}(χ_{i}))χ_{i} belongs to G_{0}(H) ⊗_{Z} A. Hence ẽ is integral over Z and, therefore, so is d = ω(ẽ).

References

[CR] C. W. Curtis and I. Reiner, Methods of Representation Theory, Vol. 1, Wiley-Interscience, New York, 1981.

[K] G. I. Kac, Certain arithmetic properties of ring groups, Functional Anal. Appl. 6 (1972), 158– 160.

[LR] R. G. Larson and D. E. Radford, Semisimple cosemisimple Hopf algebras, Amer. J. Math. 109 (1987), 187–195.

[LS] R. G. Larson and M. E. Sweedler, An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 (1969), 75–93.

[L] M. Lorenz, Representations of finite dimensional Hopf algebras, J. Algebra (to appear).

[M] A. Masuoka, Some further classification results of semisimple Hopf algebras, Commun. Algebra 24 (1996), 307–329.

[S] H.-J. Schneider, Lectures on Hopf Algebras, Lecture Notes, Universidad Nacional de Cordoba, 1995.

[Z] Y. Zhu, Hopf algebras of prime dimension, Internat. Math. Res. Notices 1 (1994), 53–59.

Department of Mathematics, Temple University, Philadelphia, PA 19122-6094

   E-mail address: lorenz@math.temple.edu