SYMMETRIC YETTER-DRINFELD CATEGORIES ARE TRIVIAL

BODO PAREIGIS

November 27, 1995

We show the following theorem answering a question of S. Montgomery and D. Fischman.

Theorem. Let H be a k-Hopf algebra such that the canonical braiding of the category YD_{H}^{H} of Yetter-Drinfeld modules is a symmetry. Then H = k.

Proof. Let M := H with the coregular cooperation of H as the H-comodule structure (δ(h) = ∑ h_{1} ⊗ h_{2}) and with the adjoint operation of H on itself as module structure (x · h = ∑ S(h_{1})xh_{2}). Then M is a Yetter-Drinfeld module or a crossed module, i.e. a right H-module and a right H-comodule such that ∑(x · h)_{0} ⊗ (x · h)_{1} = ∑(x_{0} · h_{2}) ⊗ S(h_{1})x_{1}h_{3}.

Let N := H with the regular operation of H on itself as module structure (x · h = xh) and the coadjoint cooperation of H as the H-comodule structure (δ(h) = ∑ h_{2} ⊗ S(h_{1})h_{3}). Then N is a Yetter-Drinfeld module.

Apply the square of the braiding τ(x⊗y) = ∑ y_{0}⊗x·y_{1} to the element x⊗1 ∈ M⊗N and use the given structures to get x ⊗ 1 = τ^{2}(x ⊗ 1) = τ(1 ⊗ x) = ∑ x_{1} ⊗ x_{2} for all x ∈ H hence H = k.

Corollary. For a finite-dimensional Hopf algebra H ≠ k the Drinfeld double D(H) is quasitriangular, but not triangular.

References

[M] Susan Montgomery: Hopf Algebras and Their Actions on Rings. CBMS 82, AMS-NSF, 1993.

Mathematisches Institut der Universität, Theresienstr.39, 80333 München, Germany

E-mail address: pareigis@rz.mathematik.uni-muenchen.de

1991 Mathematics Subject Classification. Primary 16W30.