Classification of bicovariant differential calculi on the quantum groups SLq(n+1) and Spq(2n)

Abstract

For transcendental values of q all bicovariant first order differential calculi on the coordinate Hopf algebras of the quantum groups SLq(n + 1) and Spq(2n) are classified. It is shown that the irreducible bicovariant first order calculi are determined by an irreducible corepresentation of the quantum group and a complex number ζ such that ζ n+1 = 1 for SLq(n + 1) and ζ 2 = 1 for Spq(2n). Any bicovariant calculus is inner and its quantum Lie algebra is generated by a central element. The main technical ingredient is a result of the Hopf algebra R(Gq) • for arbitrary simple Lie algebras.

Quantum Symmetric Pairs and the Reflection Equation

“…The following proposition is a generalization of a result proved for sl(r + 1) and sp(2r) in [10,Lemma 4.7]. The proof given here is significantly simpler.…”

Commutator representations of differential calculi on the quantum group SUq(2)

“…Remark 1. As noted in [4], the 4D ± -calculus Γ ± is an irreducible bicovariant FODC, because it is derived from the fundamental corepresentation of SL q (2) which is irreducible. By definition this means that Γ ± has no non-trivial bicovariant quotient FODC.…”

On the mock Peter-Weyl theorem and the Drinfeld double of a double

“…Baumann and Schmitt [1] and Heckenberger and Schmudgen [3] have classified such calculi on A = Rq [G]under the assumption that the space ΓL of left covariants is finite dimensional. Here we give the classification without this assumption.…”