Differential Hopf Algebras on Quantum Groups of Type A

Abstract

Let A be a Hopf algebra and Γ be a bicovariant first order differential calculus over A. It is known that there are three possibilities to construct a differential Hopf algebra Γ∧=Γ⊗/Jthat contains Γ as its first order part. Corresponding to the three choices of the idealJ, we distinguish the “universal” exterior algebra, the “second antisymmetrizer” exterior algebra, and Woronowicz' external algebra, respectively. Let Γ be one of theN2-dimensional bicovariant first order differential calculi on the quantum groupGLq(N) orSLq(N), and letqbe a transcendental complex number. For Woronowicz' external algebra we determine the dimension of the space of left-invariant and of bi-invariantk-forms, respectively. Bi-invariant forms are closed and represent different de Rham cohomology classes. The algebra of bi-invariant forms is graded anti-commutative. ForN≥3 the three differential Hopf algebras coincide. However, in case of the 4D±-calculi onSLq(2) the universal differential Hopf algebra is strictly larger than Woronowicz' external algebra. The bi-invariant 1-form is not closed.