# Hopf Algebras in Noncommutative Geometry and Physics by Stefaan Caenepeel

By Stefaan Caenepeel

This complete reference summarizes the lawsuits and keynote shows from a contemporary convention held in Brussels, Belgium. supplying 1155 show equations, this quantity comprises unique examine and survey papers in addition to contributions from world-renowned algebraists. It makes a speciality of new ends up in classical Hopf algebras in addition to the class concept of finite dimensional Hopf algebras, specific features of Hopf algebras, and up to date advances within the concept of corings and quasi-Hopf algebras. It presents examples and simple houses of corings and their comodules relating to ring and Hopf algebra conception and analyzes entwining buildings and Morita thought for corings.

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**Example text**

Is Assume that a C is locally projective, bA is flat and A /B is C-Galois. e. cr[A*c] = (^[C*c]i injective, for every M G ; an isomorphism, for every A-generated M G . Proof 1. Since A /B is C-Galois, is an isomorphism, hence C is Agenerated. Consequently cr[A*c] ^ £ ^[-^*c]j he. cf[A*c] = cr[C*c]2 . 22] applies. 3. U M £ MP is A-generated, then is surjective, hence bijective by ( 2). 13] (which is itself a generalization of [14, Theorem 2 . 11]). 2. Assume that A /B is C-Galois. 1. If b A is flat, then satisfies the weak structure theorem.

A, A ^ A(M) and Ti. + {n~'^)u + Dv + Duv \{M ) = A(M). 5. i-order. If F is a maximal D-order then it is also an W-order and thus a maximal W-order. Proof. Since L is an 7i-lattice, A(L) is an W-order and since L is a left ideal for F, we have F C A(L). The other statement is obvious from this. 3, we may obtain a first char acterization of maximal W-orders in terms of maximal orders. If A is a maximal H-order in A, then there is a maximal order F in A such that A = \{ H —^ F) = p{H —^ F). 6. Proof.

J. 30 (1981), 675-692. [19] C. Menini, A. Seidel, B. Torrecillas and R. Wisbauer, A-if-bimodules and equivalences. Comm. Algebra 29 (2001), 4619-4640. [20] C. Menini, B. Torrecillas and R. Wisbauer, Strongly rational comodules and semiperfect Hopf algebras over QF Rings, J. Pure Appl. Algebra 155 (2001), 237-255. [21] C. Menini and M. Zuccoli, Equivalence Theorems and Hopf-Galois extensions, J. Algebra 194 (1997), 245-274. [22] H. J. Schneider, Principal homogeneous spaces for arbitrary Hopf algebras, Israel J.