Quantum Groups SO_q(N),Sp_q(n) have q-Determinants, too

Abstract: We construct the $q$-deformed analogue of the completely antisymmetric tensors and the corresponding $q$-determinants $det_qT$ for the quantum groups $SO_q(N)$, $Sp_q(n)$. The construction is based on the existence of the volume form in the algebra of exterior forms on the corresponding quantum spaces. We show that $det_qT$ is central in $Fun(SO_q(N))$ (resp. $Fun(Sp_q(n))$) is grouplike under the Hopf algebra comultiplication, and that its square is 1.