Integrals, Quantum Galois extensions and the affineness criterion for Yetter-Drinfel'd modules
by
G. Militaru
Univeristy of Bucharest, Faculty of Mathematics
Coauthors: Claudia Menini

Using [2] as a source of inspiration, we introduce a general concept of integral of a treetuple (H, A, C), where H is a Hopf algebra acting on a coalgebra C and coacting on an algebra A. In particular, quantum integrals associate to Yetter-Drinfel'd modules are defined. A theorem of caracterization is proven. Let now, A an H-bicomodule algebra, ^HYD_A the category of quantum Yetter-Drinfel'd modules and B be the subalgebra of coinvariants of the Verma structure A in ^HYD_A. We prove the following quantum affineness criterion: if there exists \gamma:H --> Hom(H, A) a total quantum integral and the canonical map \beta:A\ot_B A --> H\ot A, \beta(a\ot b) = S^-1 (b _{< 1 >} ) b _{< -1 >} \ot ab _{< 0 >} is surjective, then the induction fuctor -\ot_B A: M_B --> ^HYD_A is an equivalence of category. The affiness criteria proven by Cline, Parshall and Scott (for affine algebraic group shemes), Scheider (in the noncommutative case) are recoverd as special cases.

References:

[1] C. Menini and G. Militaru, Integrals, Quantum Galois extensions and the affineness criterion for Yetter-Drinfel'd modules, preprint 2000.

[2] S. Caenepeel, Bogdan Ion, G. Militaru and Shenglin Zhu, Separable functors for the category of Doi-Hopf modules. Applications. Adv. Math. 145(1999), 239-290.

Date received: July 24, 2000