THE AFFINENESS CRITERION FOR WEAK YETTER-DRINFEL’D MODULES

Abstract

In this paper, we introduce the concept of total quantum integrals in the case of weak Hopf algebras and study the affineness criterion for weak Yetter-Drinfel’d modules, which is a generalization of the results studied by Menini and Militaru (J. Algebra, 247 (2002), 467-508).

Keywords

• Weak Hopf algebra, total quantum integral, comodule algebra

References

1. G. B¨ohm, Doi-Hopf modules over weak Hopf algebras, Comm. Algebra, 28 (2000), 4687–4698.
2. G. B¨ohm, F. Nill and K. Szlach´anyi, Weak Hopf algebras I: Integral theory and C*-structure, J. Algebra, 221 (1999) 385–438.
3. S. Caenepeel and E. De Groot, Modules over weak entwining structures, Con- temp. Math., 267 (2000), 31–54.
4. S. Caenepeel, G. Militaru, Bogdan Ion and S. L. Zhu, Separable functors for the category of Doi-Hopf modules, applications, Adv. Math., 145 (1999), 239–290.
5. S. Caenepeel, G. Militaru and S. L. Zhu, Crossed modules and Doi-Hopf mod- ules, Israel J. Math., 100 (1997), 221–247.
6. S. Caenepeel, D. G. Wang and Y. M. Yin, Yetter-Drinfel’d modules over weak bialgebras, Ann. Univ. Ferrara-Sez. VII-Sc. Mat., 51 (2005), 69–98.
7. Y. Doi and M. Takeuchi, Cleft comodule algebras for a bialgebra, Comm. Al- gebra, 13 (1985), 2137–2159.
8. R. G. Larson and M. E. Sweedler, An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math., 91 (1969), 75–94.

9. C. Menini and G. Militaru, Integrals, quantum Galois extensions and the affine- ness criterion for quantum Yetter-Drinfel’d modules, J. Algebra, 247 (2002), –508
10. M. E. Sweedler, Hopf Algebras, Mathematics Lecture Note Series, W.A. Ben- jamin, Inc. New York, 1969.
11. L. Y. Zhang, The structure theorem of weak comodule algebras, Comm. Alge- bra, 38 (2010), 254–260. Shuangjian Guo
12. School of Mathematics and Statistics Guizhou University of Finance and Economics Guiyang, Guizhou 550025, China e-mail: [e-mail]