×r-bialgebras associated with iterative q-difference rings

Abstract

Realizing the possibility suggested by Hardouin [Iterative q-difference Galois theory, J. Reine Angew. Math. 644 (2010) 101-144], we show that her own Picard-Vessiot (PV) theory for iterative q-difference rings is covered by the (consequently, more general) framework, settled by Amano and Masuoka [Picard-Vessiot extensions of artinian simple module algebras, J. Algebra 285 (2005) 743-767], of artinian simple module algebras over a cocommutative pointed Hopf algebra. An essential point is to represent iterative q-difference modules over an iterative q-difference ring R, by modules over a certain cocommutative ×R-bialgebra. Recall that the notion of ×R- bialgebras was defined by Sweedler [Groups of simple algebras, Publ. Math. Inst. Hautes Études Sci. 44 (1974) 79-189], as a generalization of bialgebras. © 2013 World Scientific Publishing Company.