DIFFERENTIAL ALGEBRAS FOR QUANTUM GROUPS OF TYPE B, C, AND D*)

Abstract

We study higher order bicovariant differential calculi on the quantum groups Oq(N) and Spq(N). We show that the second antisymmetrizer exterior algebra uΓ∧ is the quotient of the universal exterior algebra uΓ∧ by the principal ideal generated by Θ ∧ Θ. Here θ denotes the unique up to scalars biinvariant 1-form. Moreover Θ ∧ Θ is central in uΓ∧ and uΓ∧ is an inner differential calculus. We show that the quadratic dual to the left-invariant algebra sΓL∧ is isomorphic to the reflection equation algebra. Let Γ be an arbitrary left-covariant first order differential calculus. We show that the dimension of the space of left-invariant 2-forms in the universal exterior algebra equals the number of linearly independent quadratic-linear relations in the quantum tangent space.