arXiv:math/0106254v2 [math.RA] 3 Jul 2001 Self-dual Modules of Semisimple Hopf Algebras Yevgenia Kashina YorckSommerh¨auser Yongchang Zhu Abstract We prove that, over an algebraically closed field of characteristic zero, a semisimple Hopf algebra that has a nontrivial self-dual simple module must have even dimension. This generalizes a classical result of W. Burn- side. As an application, we show under the same assumptions that a semisimple Hopf algebra that has a simple module of even dimension must itself have even dimension. 1 Suppose that H is a finite-dimensional Hopf algebra that is defined over the field K. We denote its comultiplication by Δ, its counit by ε, and its antipode by S. For the comultiplication, we use the sigma notation of R. G. Heyneman and M. E. Sweedler in the following variant: Δ(h)= h (1) ⊗ h (2) We view the dual space H ∗ as a Hopf algebra whose unit is the counit of H , whose counit is the evaluation at 1, whose antipode is the transpose of the antipode of H , and whose multiplication and comultiplication are determined by the formulas (ϕϕ ′ )(h)= ϕ(h (1) )ϕ ′ (h (2) ) ϕ (1) (h)ϕ (2) (h ′ )= ϕ(hh ′ ) for h, h ′ ∈ H and ϕ, ϕ ′ ∈ H ∗ . With H , we can associate its Drinfel’d double D(H ) (cf. [18], § 10.3, p. 187). This is a Hopf algebra whose underlying vector space is D(H )= H ∗ ⊗ H . As a coalgebra, it is the tensor product of H ∗ cop and H , i.e., we have Δ(ϕ ⊗ h)=(ϕ (2) ⊗ h (1) ) ⊗ (ϕ (1) ⊗ h (2) ) as well as ε(ϕ ⊗ h)= ϕ(1)ε(h). Its multiplication is given by the formula (ϕ ⊗ h)(ϕ ′ ⊗ h ′ )= ϕ ′ (1) (S −1 (h (3) ))ϕ ′ (3) (h (1) )ϕϕ ′ (2) ⊗ h (2) h ′ The unit element is ε ⊗ 1 and the antipode is S(ϕ ⊗ h)=(ε ⊗ S(h))(S ∗−1 (ϕ) ⊗ 1). 1