QUANTUM GROUPS AND DIFFERENTIAL FORMS

ABSTRACT

We construct a quantum semigroup and an algebra of forms appropriate for the generalised homological algebra of N-complexes (Kapranov, arXiv:q-alg/9611005). This is an analogue to the picture for usual homological algebra, where one has the quantum general linear group (Kassel, 1995) and the differential forms constructed by Wess and Zumino (1990). The two pictures are different and we explain why the dichotomy arises.

Key Words:

• FRT construction
• Generalised homology
• Hecke algebra
• Yang Baxter equation

2000 Mathematics Subject Classification:

• Primary 12H05
• 16W30
• Secondary 18G99

ACKNOWLEDGMENTS

I would like to thank K. Brown for many discussions on this subject and M. Dubois-Violette, M. Kapranov, R. Kerner, Yu. Manin and F. Ngakeu for useful correspondence. I would also like to thank the referee for useful comments regarding the exposition and for pointing out the references of Mayer and Spanier. This research was supported by the project G.0278.01 “Construction and applications of non-commutative geometry: from algebra to physics′′ from FWO Vlaanderen.

Notes

^#Communicated by M. Cohen