Differential algebras for quantum groups of type B, C, and D

We study higher order bicovariant differential calculi on the quantum group s O-q(N) and Sp(q)(N). We show that the second antisymmetrizer exterior alg ebra (s)Gamma(boolean AND) is the quotient of the universal exterior algebr a (u)Gamma(boolean AND) by the principal ideal generated by theta boolean A ND theta. Here theta denotes the unique up to scalars biinvariant 1-form. M oreover theta boolean AND theta is central in (u)Gamma(boolean AND) and (u) Gamma(boolean AND) is an inner differential calculus. We show that the quad ratic dual to the left-invariant algebra (s)Gamma(L)(boolean AND) is isomor phic to the reflection equation algebra. Let Gamma be an arbitrary left-cov ariant first order differential calculus. We show that the dimension of the space of left-invariant 2-forms in the universal exterior algebra equals t he number of Linearly independent quadratic-linear relations in the quantum tangent space.