×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. Algebra285 (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.
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