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^2dimensional bicovariant first order differential calculi over A and let q be transcendental. For Woronowicz' external algebra we determine the dimension of the space of leftinvariant and of biinvariant kforms. Biinvariant forms are closed and represent different de Rham cohomology classes. The algebra of biinvariant forms is graded anticommutative. For N>2 the three differential Hopf algebras coincide. However, in case of the 4D_\pmcalculi on SL_q(2) the universal differential Hopf algebra is strictly larger than Woronowicz' external algebra. The biinvariant 1form is not closed.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 1998
 DOI:
 10.48550/arXiv.math/9805139
 arXiv:
 arXiv:math/9805139
 Bibcode:
 1998math......5139S
 Keywords:

 Quantum Algebra;
 Mathematical Physics;
 58B30;
 81R50;
 17B37
 EPrint:
 30 pages