## The homological determinant of quantum groups of type $A$

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## Abstract:

Let $R$ be a Hecke symmetry depending algebraically on a parameter $q\in \mathbb {C}$. We show that the homology of the Koszul complex associated with $R$ is one-dimensional when $q$ is not a root of unity. A generator of this homology group then induces the homological determinant of the quantum group associated with $R$.## References

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## Additional Information

**Phùng Hồ Hai**- Affiliation: Institute of Mathematics, P.O. Box 631, 10000 Boho, Hanoi, Vietnam
- Address at time of publication: FB6 Mathematik, Universität Duisburg–Essen, 45117 Essen, Germany
- Email: phung@math.ac.vn, ho-hai.phung@uni-essen.de
- Received by editor(s): September 19, 2002
- Received by editor(s) in revised form: February 22, 2004
- Published electronically: February 15, 2005
- Additional Notes: This work was supported by the National Program of Basic Sciences Research of Vietnam
- Communicated by: Martin Lorenz
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**133**(2005), 1897-1905 - MSC (2000): Primary 16W30, 17B37; Secondary 17A45, 17A70
- DOI: https://doi.org/10.1090/S0002-9939-05-07739-7
- MathSciNet review: 2137853