Differential Hopf Algebras on Quantum Groups of Type A

Abstract

Let A be a Hopf algebra and $Gamma$ be a bicovariant first order differential calculus over A. It is known that there are three possibilities to construct a differential Hopf algebra $Gamma^wedge$ that contains $Gamma$ as its first order part; namely the universal exterior algebra, the second antisymmetrizer exterior algebra, and Woronowicz' external algebra. Now let A be one of the quantum groups GL_q(N) or SL_q(N). Let $Gamma$ be one of the N^2-dimensional bicovariant first order differential calculi over A and let q be transcendental. For Woronowicz' external algebra we determine the dimension of the space of left-invariant and of bi-invariant k-forms. Bi-invariant forms are closed and represent different de Rham cohomology classes. The algebra of bi-invariant forms is graded anti-commutative. For N>2 the three differential Hopf algebras coincide. However, in case of the 4D_\pm-calculi on SL_q(2) the universal differential Hopf algebra is strictly larger than Woronowicz' external algebra. The bi-invariant 1-form is not closed.